PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA CENTRO TECNOLÓGICO PRÓ-REITORIA DE PESQUISA E PÓS-GRADUAÇÃO UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO RUAN SCHULTZ RIGUETTI DIMENSIONLESS GENERAL TRANSIENT MODELING FOR SMOLDERING COMBUSTION REACTORS Vitória, ES 2025 PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA CENTRO TECNOLÓGICO PRÓ-REITORIA DE PESQUISA E PÓS-GRADUAÇÃO UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO RUAN SCHULTZ RIGUETTI DIMENSIONLESS GENERAL TRANSIENT MODELING FOR SMOLDERING COMBUSTION REACTORS Dissertação apresentada ao Programa de Pós-Graduação em Engenharia Mecânica da Universidade Federal do Espírito Santo, como requisito parcial para obtenção do Grau de Mestre em Engenharia Mecânica. Orientador: Prof. Dr. Márcio Ferreira Martins. Coorientador: Prof. Dr. Flávio Lopes Francisco Bittencourt. Vitória, ES 2025 Ficha catalográfica disponibilizada pelo Sistema Integrado de Bibliotecas - SIBI/UFES e elaborada pelo autor S387d Schultz Riguetti, Ruan, 1999- SchDimensionless general transient modeling for smoldering combustion reactors / Ruan Schultz Riguetti. - 2025. Sch136 p. SchOrientador: Márcio Ferreira Martins. SchCoorientador: Flávio Lopes Francisco Bittencourt. SchTese (Mestrado em Engenharia Mecânica) - Universidade Federal do Espírito Santo, Centro Tecnológico. Sch1. Combustão. 2. Simulação (Computadores). 3. Calor - Transmissão. 4. Termoquímica. 5. Materiais porosos. I. Ferreira Martins, Márcio. II. Lopes Francisco Bittencourt, Flávio. III. Universidade Federal do Espírito Santo. Centro Tecnológico. IV. Título. CDU: 621 PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA CENTRO TECNOLÓGICO UNIVERSIDADE FEDERAL DO ESPÍRITO SANTO MODELAGEM TRANSIENTE GERAL ADIMENSIONAL PARA REATORES DE COMBUSTÃO SMOLDERING RUAN SCHULTZ RIGUETTI COMISSÃO EXAMINADORA ________________________________________ Prof. Dr. Márcio Ferreira Martins (Orientador – PPGEM/UFES) ________________________________________ Prof. Dr. Flávio Lopes Francisco Bittencourt (Coorientador – IFES) ________________________________________ Prof. Dr. Marcelo Risso Errera (Examinador Externo – UFPR) ________________________________________ Dr. Marco Aurelio Bazelatto Zanoni (Examinador Externo – Canadian Nuclear Laboratories) ________________________________________ Dra. Miriam Suely Klippel (Examinadora Interna – PPGEM/UFES) Dissertação apresentada ao Programa de Pós-Graduação em Engenharia Mecânica da Universidade Federal do Espírito Santo como parte dos requisitos necessários à obtenção do título de Mestre em Engenharia Mecânica. 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Universidade Federal do Espírito Santo 03/12/2025 às 16:41:57 Folha aprovação - RUAN SCHULTZ RIGUETTI Data e Hora de Criação: 03/12/2025 às 16:24:06 Documentos que originaram esse envelope: - Folha aprovação - RUAN SCHULTZ RIGUETTI.pdf (Arquivo PDF) - 1 página(s) Hashs únicas referente à esse envelope de documentos [SHA256]: 21b173eeca10c21faf2916606b1ae4469b1076bd1e196c6ed165c4d6a36c9bef [SHA512]: 925fade62072d6a0b40bf556ceecd5e02959c65f8a290d63b6a95e562a13d2da62112b30556a3dcf1cb3a09d3e022539e8ef8ed220442e68bb09e56ca64d8786 Lista de assinaturas solicitadas e associadas à esse envelope ASSINADO - Flávio Lopes Francisco Bittencourt (flavio.lopes@ifes.edu.br) Data/Hora: 03/12/2025 - 16:30:27, IP: 200.137.83.3, Geolocalização: [-20.8375, -40.7271] [SHA256]: 770a279c80a9b6ff440a107bf5286cc8115002d22f153b74a67f9b00f679a768 Assinatura Eletrônica Avançada (Conforme Lei nº 14.063/20, art. 4º, II) ASSINADO - Márcio Ferreira Martins (marcio.martins@ufes.br) Data/Hora: 03/12/2025 - 16:33:52, IP: 200.137.65.104 [SHA256]: e29e6cc1068b66d329acac1da0a7fa1976988963c1cedbe5c8a5f31ff20159d0 Assinatura Eletrônica Avançada (Conforme Lei nº 14.063/20, art. 4º, II) ASSINADO - Marco Aurelio Bazelatto Zanoni (marco.bazelattozanoni@cnl.ca) Data/Hora: 03/12/2025 - 16:38:18, IP: 206.47.148.202 [SHA256]: 7bb7b1abd70dfc3944352774a07a8f01d37fca27d9e1c5408c4148598fe0e63c Assinatura Eletrônica Avançada (Conforme Lei nº 14.063/20, art. 4º, II) ASSINADO - Marcelo Risso Errera (merrera@gmail.com) Data/Hora: 03/12/2025 - 16:26:45, IP: 177.220.182.178, Geolocalização: [-25.465972, -49.225563] [SHA256]: 9eac6e3cb434a7408698c656e9b5f9fd9bc371dfa13e9a02c2c379577d09829e Assinatura Eletrônica Avançada (Conforme Lei nº 14.063/20, art. 4º, II) ASSINADO - Miriam Suely Klippel (miriam.klippel@edu.ufes.br) Data/Hora: 03/12/2025 - 16:41:57, IP: 200.137.67.20 [SHA256]: 0db810ea4537754e60a57a03d24a48a42ac9d7e07b95c8533b1526bb73450533 Assinatura Eletrônica Avançada (Conforme Lei nº 14.063/20, art. 4º, II) Histórico de eventos registrados neste envelope 03/12/2025 16:41:57 - Envelope finalizado por miriam.klippel@edu.ufes.br, IP 200.137.67.20 03/12/2025 16:41:57 - Assinatura realizada por miriam.klippel@edu.ufes.br, IP 200.137.67.20 03/12/2025 16:38:18 - Assinatura realizada por marco.bazelattozanoni@cnl.ca, IP 206.47.148.202 03/12/2025 16:33:52 - Assinatura realizada por marcio.martins@ufes.br, IP 200.137.65.104 03/12/2025 16:30:27 - Assinatura realizada por flavio.lopes@ifes.edu.br, IP 200.137.83.3 03/12/2025 16:30:22 - Envelope visualizado por flavio.lopes@ifes.edu.br, IP 200.137.83.3 03/12/2025 16:26:45 - Assinatura realizada por merrera@gmail.com, IP 177.220.182.178 03/12/2025 16:26:42 - Envelope visualizado por merrera@gmail.com, IP 177.220.182.178 03/12/2025 16:25:30 - Envelope registrado na Blockchain por andreia.eyng@ufes.br, IP 200.137.65.106 03/12/2025 16:25:30 - Envelope encaminhado para assinaturas por andreia.eyng@ufes.br, IP 200.137.65.106 03/12/2025 16:24:06 - Envelope criado por andreia.eyng@ufes.br, IP 200.137.65.106 Abstract Smoldering combustion is a slow, flameless process that occurs at relatively low temperatures and reaction rates, typically under limited oxygen condi- tions. Beyond its scientific interest, this process offers environmental, tech- nological, and social benefits, which make it relevant for both industrial applications and sustainable development. In this context, the present re- search develops and applies a general dimensionless numerical model for smoldering combustion reactors, aiming to simulate the phenomenon on a small scale. The approach relies on a 2D axisymmetric model implemented in COMSOL Multiphysics (v5.4) using the Local Thermal Non-Equilibrium (LTNE) consideration, which allows separate treatment of the solid and fluid phases. Conservation equations for mass, momentum, energy, and species transport were implemented in a dimensionless form and enable a compre- hensive and generalized analysis of the physical and chemical processes involved. Novel dimensional and dimensionless groups emerged during the non-dimensionalization process, associated with the effects of particle-bed burning and the interstitial chemical kinetic dynamics. Classical numbers such as Prandtl, Grashof, Darcy, Schmidt, and Peclet numbers also appeared. The model proposed in the methodology was validated through three case studies. The first involved combustion at the fluid–porous interface, highlight- ing the influence of natural convection. In this case, the model reproduced the same recirculation patterns reported in the reference study and also allowed vi investigation of how the velocity profile was distorted by these recircula- tions. The second case addressed the cooling of a porous bed and was used to calibrate convective heat transfer under transient conditions. The results showed that the model is capable of simulating studies without a reactive porous bed, although a maximum discrepancy of 25% was observed in the temperature profiles when comparing the simulations with the experimental data. The third case consisted of a full simulation of smoldering combustion, which included the ignition process through a heat source, propagation of the combustion front, and coupled interactions between heat and mass. This case allows analysis of solid fuel consumption over time and comparison of tem- perature profiles with experimental data obtained at different axial positions of the reactor. In general, the results demonstrate that the model created is capable of capturing the main behaviors with good agreement compared to the experimental data and the results from the literature. Therefore, the pro- posed methodology provides a reliable model that allows one to understand smoldering dynamics. Keywords: combustion; dimensionless; 2D axisymmetric model; temperature profiles; smoldering dynamics. vii Author’s Declaration The contributions of others are clearly stated in my dissertation. Márcio Ferreira Martins: initiated the research topic, provided guidance on methodology development and numerical simulations, assisted in the devel- opment of dimensionless analysis with the mathematical equations governing the physics present in the model, contributed with the implementation in COMSOL Multiphysics, assisted in data interpretation. Flávio Lopes Francisco Bittencourt: Assisted in the development of the di- mensionless analysis with the mathematical equations governing the physics present in the model, and provided data from previous numerical simulations that contributed to comparison between the results obtained. Gabriel Gusmão Almeida: provided guidance on the numerical simulation structure and contributed to the implementation in COMSOL Multiphysics. Tarek L. Rashwan: provided guidance on the numerical simulation struc- ture and assisted in the development of the dimensionless analysis with the mathematical equations governing the physics present in the model. During the Master’s degree period (2023–2025), the author participated in various academic, technical, and scientific activities. The main activities included attending mandatory and elective courses in the pos graduated program, participating in scientific events and submitting and publishing articles. Among the main technical and scientific contributions during the course, the following stand out: viii • Participation and presentation in a workshop on smoldering combustion called NanoSymp, held on April 3, 2024. The event was organized by MMLABS at UFES and featured renowned authors in the smoldering combustion field, such as Tarek Rashwan and Marco Zanoni. • Publication of the article "A Two-Dimensional Modeling of a Permeable Fluid–Porous Interface in COMSOL Multiphysics" for the 20th Brazilian Congress of Thermal Sciences and Engineering (ENCIT), in November 2024, as described in Appendix A1. • Participation in a technical-scientific visit to The Open University, in Milton Keynes, England, from 30 April to 9 May, 2025, to meet and collaborate with the team of students led by Professor Dr. Tarek Rash- wan, discussing the parameters that govern smoldering combustion, the optimization the computational simulation model developed by MM LABS (UFES), as well as to foster future partnerships and support the production of scientific articles, as described in Appendix A3. • Publication of the work "Calibration of a Numerical Model for Cool- ing Simulation in a Porous Bed" to the 28th International Congress of Mechanical Engineering (COBEM), in November 2025, as described in Appendix A2. • Development of the computational model described in this dissertation, focusing on the transient and dimensionless simulation of smoldering combustion in porous beds. These activities contributed significantly to the consolidation of the au- thor’s academic and scientific training, as reflected in the results presented in this work. ix Acknowledgments It was an impressive period of about two years of development and learning. On 8 February, 2023, my best friend and a great researcher joined me on a car trip to the Federal University of Espírito Santo in Goiabeiras, Vitória–ES, to take the entrance test for the Master’s degree in Mechanical Engineering. Since we lived in São Mateus–ES, we faced an exhausting 450-kilometer round trip on the same day. A few days later, the results were released, and it was confirmed that both, my friend and I, had been accepted into the program. After that, we took this long trip countless times, always happy and confident that we were doing what we loved and that our future was taking shape in academic life. That day, I met my advisor, Prof. Márcio Ferreira Martins, who administered the exam to all the candidates. Months later, we developed friendship and mutual trust. I am grateful for his guidance and for sharing his knowledge. I would also like to commend his humanity in understanding the different contexts in which his students are immersed, as well as his energy to collaborate with them regardless of the day of the week or the time of day. That is striking! I would like to thank God for giving me the strength to face the challenges of the course, guidance in the trips from São Mateus to Vitória, and for opening my mind during the development of my dissertation. I am grateful to my parents, Elaine and Advaldo, who have been my constant support every week. To my fiancée, Daniela, I thank you for your wise advice and for all your patience during my less joyful moments, always striving to make each x day happier. To my best friend, Gabriel, thank you for all your support and assistance during the mathematical modeling of the equations used in my numerical simulation. Your shared knowledge of fluid mechanics was very important to my development. I thank the Federal University of Espírito Santo (UFES). I’m very proud to have coursed my Master’s degree at such a respected institution. I am also grateful to the Federal Institute of Espírito Santo (IFES), where I studied for almost nine years (2014-2022), including four years in the Mining Technician program and almost five years in the undergraduate program in Mechanical Engineering. This institution was my second home throughout that time, where I made many friends and met outstanding professors who remain my friends to this day. I thank my co-advisor, Prof. Dr Flávio Lopes Francisco Bittencourt, whom I had the privilege of knowing and being mentored by during this time. I am grateful for sharing his knowledge in the area of smoldering combustion. I thank the team MMLABS, especially André, Míriam, Bruno Lourenço, Cleyton, Bruno Gobi, João Victor, Marciellyo, Julio, Eron, Mateus, Eduardo, Alexandre, João Freitas and Prof Ramon, for the companionship during the routine in the laboratory. I thank SENAI (National Industrial Learning Service) and Multivix College, where I worked during my Master’s degree. These institutions allowed me to develop my teaching skills and grow into a better professor. xi Table of Contents Table of Contents xii List of Tables xiv List of Figures xvi Nomenclature xix 1 Introduction 1 1.1 Smoldering Combustion . . . . . . . . . . . . . . . . . . . . . . 1 1.2 General objective . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Specific objectives . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Dissertation structure . . . . . . . . . . . . . . . . . . . . . . . . 6 2 State of Art 10 2.1 Numerical simulations challenges . . . . . . . . . . . . . . . . 10 2.2 Classical dimensionless scales . . . . . . . . . . . . . . . . . . . 17 3 Methodology 29 xii 3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Dimensionless group . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Dimensionless equations . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.1 Combustion at a fluid-porous interface . . . . . . . . . 41 3.4.2 Cooling of a porous bed . . . . . . . . . . . . . . . . . . 48 3.4.3 Combustion in porous bed . . . . . . . . . . . . . . . . 55 4 Results and Discursion 64 4.1 Resulting scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Results - Combustion in a fluid-porous interface . . . . . . . . 68 4.3 Results - Cooling of a porous bed . . . . . . . . . . . . . . . . . 73 4.4 Results - Combustion in porous bed . . . . . . . . . . . . . . . 76 5 Conclusions and further work 86 A Appendix 89 A.1 Presented paper in the ENCIT 2024 . . . . . . . . . . . . . . . . 89 A.2 Presented paper in the COBEM 2025 . . . . . . . . . . . . . . . 97 A.3 Technical visit at The Open University . . . . . . . . . . . . . . 108 xiii List of Tables 3.1 Value of dimensionless numbers. Results obtained to Bitten- court (2023) in a real experimental measure. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Initial condition in numerical simulation - case 01. . . . . . . . 48 3.3 Value of boundary conditions to numerical analyses. Results obtained to Bittencourt (2023) in a real experimental measure. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . 48 3.4 Numerical model input parameters - case 02 . . . . . . . . . . 52 3.5 Initial condition in numerical simulation - case 02. . . . . . . . 52 3.6 Boundary condition in numerical simulation - case 02. . . . . . 53 3.7 Flux/source boundary conditions - case 02. . . . . . . . . . . . 53 3.8 Numerical model input parameters - case 03. . . . . . . . . . . 57 3.9 Initial condition in numerical simulation - case 03. . . . . . . . 59 3.10 Boundary condition in numerical simulation - case 03. . . . . . 59 3.11 Flux/source boundary conditions - case 03. . . . . . . . . . . . 61 xiv 4.1 Significance of dimensionless terms obtained in the dimen- sional analysis. Developed by the author. . . . . . . . . . . . . 65 4.2 Term’s magnitudes of the different portions of the equations - case 03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 xv List of Figures 1.1 Difference between flaming and smoldering combustion and their residues. Developed by Santoso (2019). . . . . . . . . . . 1 2.1 General physical behavior of porous interface in smoldering combustion. Developed by the authors. . . . . . . . . . . . . . 13 2.2 Classical dimensionless numbers in smoldering combustion, divided into five important subgroups. Developed by the authors. 18 3.1 Boundary conditions and crucial parameters involved in the general modeling. Developed by the authors. . . . . . . . . . . 31 3.2 (a) Physical behavior of a vertical cylindrical - case 01. (b) Interstitial behavior of heat and flux - case 01. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Representation of boundary conditions and crucial parameters involved in case 01. Developed by the authors. . . . . . . . . . 47 3.4 (a) Physical behavior of a vertical cylindrical - case 02. (b) Interstitial behavior of heat and flux - case 02. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xvi 3.5 Representation of boundary conditions and crucial parameters involved in case 02. Developed by the authors. . . . . . . . . . 54 3.6 (a) Physical behavior of a vertical cylindrical - case 03. (b) Interstitial behavior of heat and flux - case 03. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 Representation of boundary conditions and crucial parameters involved in case 03. Developed by the authors. . . . . . . . . . 60 4.1 Comparison with Bittencourt (2023): Streamlines at various stages of pore bed contraction. Subfigures (a), (b), (c), and (d) are derived from the work of Bittencourt, while subfigures (e), (f), (g), and (h) correspond to the same stages of pore bed contraction for this work, respectively. Developed by the authors. 70 4.2 Deformation of the longitudinal velocity profile, fluid temper- ature variation and compression of the boundary layer that feeding thermochemical phenomena, considering (h∗ = 0.6). Blue line: Velocity profile. Red line: Temperature profile. Green line: Compression of fluid boundary layer that feeds the ther- mochemical phenomena. Developed by the authors. . . . . . . 72 4.3 Comparison between a experimental test and simulation pro- pose in the Section 3.4.2. Developed by the authors. . . . . . . 75 xvii 4.4 Comparison between the experimental test and the simulation proposed in Section 3.4.3. The continuous line corresponds to the temperature measured by thermocouples, while the dashed line represents the result obtained through the numerical simu- lation. Developed by the authors. . . . . . . . . . . . . . . . . . 76 4.5 Comparison between each experimental and simulation ther- mocouple. Developed by the authors. . . . . . . . . . . . . . . 78 4.6 Front combustion evolution downward across the porous bed. Developed by the authors. . . . . . . . . . . . . . . . . . . . . . 80 4.7 Pressure, solid mass fraction and fluid specific mass behavior across the reactor for 2500 seconds. Developed by the authors. 83 A.1 Main entrance of The Open University. . . . . . . . . . . . . . . 108 A.2 First contact and casual discussion. . . . . . . . . . . . . . . . . 109 A.3 Presentation of my dissertation theme. . . . . . . . . . . . . . . 110 A.4 Focused technical meetings on smoldering combustion. . . . . 111 A.5 Particular technical meetings focused on smoldering combus- tion with Tarek Rashwan. . . . . . . . . . . . . . . . . . . . . . 112 A.6 Engineering Department – location of meetings and study ac- tivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 xviii Nomenclature Latin letter Ṙoxi f Fluid generation reaction rate Ṙs General reaction rate Ṙoxi s Solid consumption reaction rate As Pre-exponential factor Arr Arrhenius number b Stoichiometric coefficient Be Bejan number Bi Biot number cp Specific heat D Mass diffusivity d Diameter Da f Darcy number xix Das Damköhler number E f Activation energy for fluid phase Es Activation energy for solid phase g Gravity GF Dimensionless geometric factor Gr Grashof number H Enthalpy h Height K Intrinsic permeability L Reactor’s length m Mass m∗ part Particle mass fraction Nu Nusselt number P Power p Pressure Pe Peclet number Pr Prandtl number Q Heat Storage xx q Radiation heat source R Ideal gas constant r Radial coordinates Sc Schmidt number T Temperature t Time u, U Velocity u∗ z Dimensionless longitudinal velocity of fluid Y Mass fraction z Longitudinal coordinates v̇ f Variation rate specific volume of the fluid phase Greek symbols α Thermal diffusivity β Thermal expansion coefficient Γ Convection heat transfer coefficient λ Thermal conductivity µ Dynamic viscosity ν Kinematic viscosity xxi ρ Specific mass σ Stefan-Boltzmann constant τ Time scale ε Surface emissivity φ Porosity Subscripts/superscripts 0 Initial ∞ Referrer to the ambient ∗ Dimensionless variable abs Absolute ext Refer an external parameter f Fluid f r Front propagation h hot i Index variable for solid phase ig Ignition in Inlet int Refer an interstitial parameter xxii k Index variable for fluid phase m Porous medium o2 Referrer to the oxygen out Outlet oxi Oxidation p Porous bed part Particle r Radial direction rad Radiation s Solid sg Between solid and fluid phase stg Storage w Wall w1 Wall in the fluid domain w2 Wall in the solid domain z Longitudinal direction xxiii Chapter 1 Introduction 1.1 Smoldering Combustion Smoldering combustion is a process characterized by low temperatures and low reaction rates that occur without visible flames. Its difference from flaming combustion can be observed in Figure 1.1 [1]. Figure 1.1: Difference between flaming and smoldering combustion and their residues. Developed by Santoso (2019). 1 Introduction The main difference between flaming and smoldering combustion is that the flaming occurs in the gas phase, with high temperatures and rapid propa- gation, while smoldering is a heterogeneous combustion in the solid phase, slow and with low heat release. In wildfires, flaming typically occurs in the initial phase and smoldering in the final phase, when less volatile fuels remain. In the environment scenario, smoldering combustion is the predominant com- bustion phenomenon in megafires that occur in natural deposits of peat and coal [2]. In addition, smoldering combustion is one of the primary causes of residential fires and is associated with significant safety concerns in industrial facilities. Although it may be undesirable in certain scenarios, such as forest fires, this mechanism can be used effectively in heating systems. This process offers high efficiency in converting the energy stored in the fuel into thermal energy while producing relatively low emissions [3], compared to other forms of combustion. There are many examples of applied smoldering in the desired way. In some industrial processes, smoldering combustion can be used to gener- ate heat, an example is in situ combustion, this process works with a local controlled burn of petroleum inside the reservatory to reduce viscosity and consequently to increase API gravity, facilities the recovery of crude oil [4]. Another example is the recovery of oil sludge-contaminated soil (OSS) [5, 6], after all, smoldering combustion is a new approach to organic waste treat- ment in which organic waste is efficiently destroyed with minimal energy input. For this case application, the influence of four key parameters was observed: the moisture content of the oil sludge, the filling ratio, the grain 2 Introduction size, and the airflow rate on the smoldering [7]. In addition to contributions to environmental causes, the smoldering combustion process can be used to achieve a relevant social impact. Recently, it was shown that this process can be applied as an energy efficient destruction technique for human feces [3] and a continuous device was created to perform pyrolysis in this type of material using smoldering combustion. This proposal can be utilized to address the sanitation challenges in some countries. An important concept related to the phenomena of smoldering combustion is pyrolysis. When a coal particle or any type of material is placed inside a hot environment, without oxygen, it passes through a process called pyrolysis. This process leads to the release of combustion gases and a solid carbon-rich material, called ’char’ [8]. This endothermic process converts materials into valuable products such as biochar, oil, and gas. This phenomenon helps to create new sources of energy, making it an important technology for sustain- able waste management. To work effectively, pyrolysis needs a steady and controlled heat supply, which can be achieved through smoldering combus- tion. Smoldering combustion operates with low chemical reactions and does not need a continuous fuel supply, making it a controllable and efficient way to execute pyrolysis. Considering the physics involved in smoldering combustion or other pro- cesses that occur during burn materials, it is important to mention that the characteristics of these combustion are governed by a complex interaction of chemical phenomena that occur close at the level of the particle scale. Within this type of context, there exist devolatilization, phase changes, and 3 Introduction heterogeneous chemical reactions, which are responsible for controlling cru- cial functionalities, from ignition behavior up to complete combustion of the material [9]. Devolatilization is the thermal decomposition of a fuel particle by heating, which releases volatile gases, and represents the initial phase of ther- mal conversion and is especially influenced by factors such as the heating rate, final temperature, and particle size [9]. After the devolatilization process, the char undergoes heterogeneous chemical reactions that appear in this type of system as slower processes, occurring throughout the entire combustion time of the particle. To understand the transition between the different combustion regimes, it is necessary to model the burn rate and the total fuel conversion time with precision [8, 9]. Following these concepts, the processes of phase change and mass trans- port are also present in smoldering combustion systems, especially in combus- tion with moisture. For example, the drying process is frequently controlled by heat transport through an evaporation front, where liquid water is converted to steam. This steam diffuses through the porous medium to colder areas and can condense, releasing latent heat and influencing the internal temperature profile of the particle [8]. The diffusion of mass is not restricted only to water vapor, for example, the gases generated in pyrolysis or devolatization and the oxidation products of char also move through the porous structure by diffu- sion and convection. The interaction between volatile gases and the diffusion of oxidants at the particle surface determines whether ignition occurs in the gaseous phase, called homogeneous, or on the char surface, in this case named heterogeneous [10]. Therefore, it is evident that the smoldering combustion 4 Introduction process has the highest complexity, the number of physical and chemical pro- cesses in smoldering is larger and demanded a thorough analysis of scales and parameters. The application of smoldering simulations should understand in detail the process to optimize operational conditions, for example, since fuel can be trapped inside the pores of the material, directly affecting the spread of combustion [2]. Although many studies investigate small scale smoldering combustion using experiments and numerical models, most of them are limited to spe- cific fixed geometries and simplify assumptions. In addition, the transient characteristics of smoldering, where heat and mass transfer interact with heterogeneous reactions and porous contraction, are frequently simplified, re- ducing the accuracy of the simulations. For this reason, it is necessary to have a generalized and dimensionless transient modeling that can represent the fundamentals of smoldering combustion and be applied to different reactor configurations and operational conditions. Developing new studies in this area and proposing a new standard of modeling is crucial to simulate with more accuracy smoldering combustion phenomena. 1.2 General objective Develop a general dimensionless transient model for axisymmetric smoldering combustion reactors. 5 Introduction 1.2.1 Specific objectives • Create a new dimensionless analysis of smoldering combustion in an LTNE domain, based on the important physical existence in this phe- nomenon; • Test and validate three case studies using real experimental parameters from the literature; • Show that the model is capable of simulating both inert and reactive porous beds; • Demonstrate that the model can simulate different axisymmetric geome- tries; • Visualize that the model can operate with different air-supply configura- tions for smoldering combustion. 1.3 Dissertation structure The dissertation structure is divided into chapters responsible for address- ing the following subjects: the introduction presents the theme, providing context for the subjects and the objectives of the research; the state of the art section reviews what the main authors in the literature state about the smoldering combustion area and the most relevant improvements observed in recent years; the methodology describes the main dimensionless equations considered in the model and details each simulated case study; the results section discusses the main parameters and dimensionless terms obtained 6 Introduction from the nondimensionalization process, as well as the behaviors and results obtained for each numerical simulation; finally, the Appendix includes two papers developed and published in important international congresses, along with the description of a technical visit at the England conducted during the master’s degree. 7 References 1. Santoso, M. A., Christensen, E. G., Yang, J. & Rein, G. Review of the transition from smouldering to flaming combustion in wildfires. Frontiers in Mechanical Engineering 5, 49 (2019). 2. Torero, J. L. et al. Processes defining smouldering combustion: Integrated review and synthesis. Progress in Energy and Combustion Science 81, 100869 (2020). 3. Bittencourt, F. L. F. TOWARD A SAFE AND CIRCULAR THERMOCHEM- ICAL PROCESS TO SANITIZE HUMAN FECES IN RESOURCE-POOR ENVIRONMENTS PhD thesis (Universidade Federal do Espírito Santo, 2023). 4. Castanier, L. & Brigham, W. Upgrading of crude oil via in situ com- bustion. Journal of Petroleum Science and Engineering 39, 125–136. ISSN: 0920-4105. https://www.sciencedirect.com/science/article/pii/ S0920410503000445 (2003). 5. Gan, Z. et al. Method of smoldering combustion for the treatment of oil sludge-contaminated soil. Waste Management 175, 73–82. ISSN: 0956-053X. https://www.sciencedirect.com/science/article/pii/S0956053X23007845 (2024). 6. Zhao, C., Li, Y., Gan, Z. & Nie, M. Method of smoldering combustion for refinery oil sludge treatment. Journal of Hazardous Materials 409, 124995. ISSN: 0304-3894. https://www.sciencedirect.com/science/article/ pii/S0304389420329861 (2021). 7. Gan, Z. et al. Experimental investigation on smoldering combustion for oil sludge treatment: Influence of key parameters and product analysis. Fuel 316, 123354. ISSN: 0016-2361. https://www.sciencedirect.com/ science/article/pii/S001623612200223X (2022). 8 https://www.sciencedirect.com/science/article/pii/S0920410503000445 https://www.sciencedirect.com/science/article/pii/S0920410503000445 https://www.sciencedirect.com/science/article/pii/S0956053X23007845 https://www.sciencedirect.com/science/article/pii/S0304389420329861 https://www.sciencedirect.com/science/article/pii/S0304389420329861 https://www.sciencedirect.com/science/article/pii/S001623612200223X https://www.sciencedirect.com/science/article/pii/S001623612200223X REFERENCES 8. He, F. & Behrendt, F. A new method for simulating the combustion of a large biomass particle—A combination of a volume reaction model and front reaction approximation. Combustion and flame 158, 2500–2511 (2011). 9. Annamalai, K. & Ryan, W. Interactive processes in gasification and com- bustion—II. Isolated carbon, coal and porous char particles. Progress in energy and combustion science 19, 383–446 (1993). 10. Anca-Couce, A., Zobel, N., Berger, A. & Behrendt, F. Smouldering of pine wood: Kinetics and reaction heats. Combustion and flame 159, 1708–1719 (2012). 9 Chapter 2 State of Art 2.1 Numerical simulations challenges Elaborating numerical simulations allows for a better interpretation of phe- nomena and enables comparisons with experimental results, as well as the exploration of different conditions and scenarios. Some studies attempt to reproduce smoldering combustion experimentally on a small scale through devices named smoldering reactors. In general, smoldering combustion reac- tors can vary in geometry, ignition source, burning direction, and propagation modes [1]. Consequently, even a slight change in the way air is supplied to the system can have a significant impact on the entire combustion process [2]. Within this context of small-scale experiments in the scientific field of smol- dering simulations, several studies focus on forward smoldering, where air is injected in the same direction as the propagating reaction, while other authors investigate feeding the reaction with airflow in the opposite direction to the propagation. In numerical studies that address these aspects is visualized also porous contraction, which occurs when the process is simulated by accounting 10 State of Art for a permeable fluid–porous interface that decreases in size over time. As expected in this situation, several distinct behaviors are observed: the solid phase undergoes mass and volume loss, chemical reactions release heat and mass into the fluid flow, resulting in gas transport, heat transfer through the porous domain, and the generation of new chemical species in the gas phase [3]. In a smoldering combustion process, it is acknowledged that modeling results depend on assumptions about heat transfer processes and chemical reactions. When used in simulations, the physics of these two subareas re- quire careful interpretation [4]. For example, the work proposed by Huang et al (2015) presented a computational 1D model to investigate smoldering combustion of natural fuels with emphasis on the roles of moisture and inert contents, and simulated the driving phenomenon of wildfires in peatlands. This approach made it possible to understand the influences of kinetic param- eters, physical properties, and ignition protocol on this type of phenomenon [5]. To achieve this, several mathematical equations were applied, including mass, energy, and momentum conservation, as well as chemical balance equa- tions related to char oxidation, heterogeneous chemical processes, and the Arrhenius equation. This demonstrates the complexity involved in simulating this type of phenomenon. Yet in the mathematical context of the smoldering phenomena, several authors have contributed to the development of computational approaches that advance the understanding of smoldering processes. For example, Bit- tencourt and Martins (2022) conceptualized a cylindrical smoldering reactor 11 State of Art filled with a reactive porous material. In this configuration, airflow is induced to enter the reactor due to a pressure gradient as the combustion front prop- agates downward [6]. The temperature difference between the combustion front and the stream also initiates buoyancy forces that oppose downward airflow. These physical characteristics were modeled in a 2D numerical anal- ysis with steady-state equations in a dimensionless structure, highlighting the complexity of capturing such coupled mechanisms. To better illustrate this information, Fig. 2.1 presents a vacuum-induced reactor proposed by the author, which typically uses smoldering to perform pyrolysis on specific materials with a forward burning direction [7]. In this device, a pump located at the bottom draws air to sustain combustion. The reactor was also open at the top, which increases natural convection and causes recirculation. While the effects of recirculation are minor with small contractions, as the combus- tion front begins to widen and the porous bed contraction increases, natural convection also increases and starts to compress the airflow boundary layer against the combustion chamber walls, this last behavior also proves in the work proposed by Riguetti et al (2024) [8]. This process reduces the progress of the combustion front and gradually extinguishes it. When this condition occurs, the front of the propagation and contraction of the porous bed stop. This type of behavior was also identified by [9]. 12 State of Art Figure 2.1: General physical behavior of porous interface in smoldering combustion. Developed by the authors. To simulate this type of vacuum-induced smoldering reactor, several math- ematical equations must be modeled. Beyond the equations of mass, energy, and momentum conservation, it is also necessary to capture the phenomenon of recirculation, which can be achieved through adaptations in the momen- tum conservation equations. Due to the high temperatures, the boundary conditions of heat loss in the upper, lateral, and lower regions should also be considered. Furthermore, the inclusion of transient conditions can provide more detailed insights, since the simulation proposed by this author revealed the recirculation phenomenon at a specific time and at a certain level of porous 13 State of Art bed contraction. Capturing this type of behavior in a time-dependent simula- tion makes it possible to visualize new processes that depend on the physical interactions represented in the model, promoting greater realism in the results. Following this line of reasoning, the author Zanoni et al (2020) conducted smoldering column experiments under self-sustaining conditions, with the aim of gradually extinguishing combustion by reducing air flux and fuel concentration [10]. To simulate the experiments, a previously developed 1D numerical model validated for robust smoldering was used. The simula- tions showed that the model was adequate for robust conditions but failed in scenarios with weak self-sustaining combustion, revealing the limitations of simplified approaches in marginal cases. This numerical model incorporated a general set of physical parameters, which can provide greater flexibility to test specific conditions, but also can introduce challenges. Because smoldering combustion depends on multiple parameters such as kinetics, heat, flow, and mass, this big quantitative of parameters can lead to convergence problems and make it difficult to identify which factor is responsible for errors. There- fore, it is necessary to be careful in this type of system modeling, because the use of many dependent parameters requires attention to ensure correct detail in the implementation inside of numerical software, and the numerical simulations need to present correct results so that the modeling is reliable. Another relevant aspect in numerical simulations of smoldering to ensure correct results is the consideration of heat transfer mechanisms in porous domains, which can be approached through LTE (Local Thermal Equilibrium) or LTNE (Local Thermal Non-Equilibrium) formulations [11]. While the 14 State of Art first assumes a unique temperature for the solid and gas phases, the latter considers them at different temperatures. When LTNE is implemented, the model becomes more reliable and better represents reality. In this sense, [12] presented a comparison between the LTE and LTNE approaches, showing that in some time steps the difference between the two could reach 100°C. This result clearly demonstrates the importance of the assumptions chosen in numerical simulations and how they can significantly affect the results. This highlights that accurately capturing the complex behavior of smolder- ing requires careful consideration of both physical phenomena and numerical assumptions. Continuing comments about considerations in the numerical smoldering area, for instance, the author Zanoni et al (2022) observed that temperatures close to the reactor center followed the advancement of the com- bustion front, a key physical characteristic in smoldering reactors. The study also showed that oxygen consumption was higher in the center, corresponding to more intense chemical reactions [13]. The authors proposed a permeability heterogeneity analysis, concluding that regions with higher permeability ex- hibited faster combustion fronts and higher peak temperatures. Based on this numerical model, a new correlation for the interstitial heat transfer coefficient was implemented [12, 14], and, to better simulate reality, a heating process until ignition conditions was included. As a result, it was concluded that increasing airflow strengthens self-sustained combustion by enhancing the oxidation rate. All of these implementations proposed by the authors can help new researchers as a starting point for mathematical equations and additional processes necessary to simulate this type of combustion. 15 State of Art More recently, multidimensional simulations have been explored. Yuan (2023) presented a 2D case study to simulate the lateral and vertical smol- dering spread [15]. The model predicted the effects of peat conditions on smoldering propagation and shows how the mechanisms controlling these two propagation modes are validated using a shallow-reactor experiment previously reported in the literature. Similarly, Hansan (2015) detailed the development and validation of a computational model that simulates the spread of a smoldering front through a porous medium. Two-dimensional smoldering propagation experiments were conducted in a steel box, and after calibration to a baseline experiment, the model accurately predicted the out- comes of subsequent tests [16]. These works show that, although such models introduce more complexity and require greater computational power, the pos- sibility of simulating multiple physical directions facilitates the visualization of behaviors occurring inside the porous bed, which could only be captured experimentally with extensive setups, raising the cost of such experimental processes. Finally, Cheung (2023) [17], who investigated the transport and hazards of CO from smoldering fires in the context of building design, emphasized the importance of accurate input data and model calibration. This work re- produced two full-scale smoldering fire experiments from the late 1970s to calibrate the model settings, shows that the reliability of numerical simula- tions depends not only on the equations implemented but also on the quality of experimental understanding and precision data collection. This emphasizes 16 State of Art that the choice of initial conditions, boundary conditions, and values of physi- cal parameters can be crucial for the numerical convergence and reliability of the results obtained. Perceptual errors and discrepancies observed between numerical simulations and experimental results in reality may often be caused by wrongly chosen parameters. Taken together, the works cited previously demonstrate how numerical simulations of smoldering combustion have progressively evolved from sim- plified one-dimensional approaches to multidimensional, time-dependent, and more physically detailed models. They also reveal how modeling choices such as LTE versus LTNE assumptions, permeability distribution, heat transfer correlations and boundary conditions play a decisive role in capturing the dynamics of smoldering. 2.2 Classical dimensionless scales As previously mentioned in Section 1, several distinct physical behaviors are observed in smoldering combustion. To optimize numerical analyzes, dimen- sionless equations can be applied to these problems, enabling the reduction of complex interactions into representative dimensionless groups. Several studies have explored the role of classical scales in interpreting the governing mechanisms of smoldering. The dimensionless numbers most significant for smoldering combustion are presented in Figure 2.2. 17 State of Art Fluid Flow in Porous Media Fluid and Heat Transfer Mass and Heat Transfer Chemical Reactions Dynamics Geometric scales Lewis Grashof Rayleigh Damköhler Frank-Kamenetski Arrhenius Biot Peclet Nusselt Prandtl Reynolds Darcy Bejan Ratio of diameters Aspect ratio Figure 2.2: Classical dimensionless numbers in smoldering combustion, divided into five important subgroups. Developed by the authors. The first group of relevant numbers is associated with fluid flow in porous media. The most common is the Reynolds number, which accounts for both 18 State of Art fluid and porous bed material characteristics. Parameters such as the average flow velocity through the cross-sectional area and porosity directly influence its value [18]. Flow resistance within the porous matrix is quantified by the Darcy number, which expresses the relation between viscous and inertial effects. This number is directly linked to air supply, one of the main drivers of smoldering propagation. Jang (19992) and Tong (1986) showed that the Nusselt number is sensitive to permeability and reaches a minimum value at specific porous layer thicknesses according to the varying Darcy number [19] [20]. More recently, Shruti (2023) assessed the combined impact of Darcy and Rayleigh numbers on natural convection [21], showing that with the increase of cylinder reactor size, both Darcy, Rayleigh numbers and heat transfer rates increase, where doubling the diameter of the cylinder enhances heat transport by 41.5 percent. In forced flow conditions, specific scales may be more suitable, such as the Reynolds number when the flux velocity is imposed in the experiment [22], or the Bejan number when pressure differences are the driving force of fluid flow [23–25]. When flow and heat transfer are considered together, the Nusselt number becomes very important. This dimensionless number reveals how thermal energy is transferred from a surface to the surrounding fluid, bridging flow dynamics and your thermal properties [2, 26]. It is related to the convective heat transfer coefficient and depends on correlations involving Reynolds and Prandtl numbers. For smoldering systems, Wakao (1982) proposed a valid correlation for Reynolds between 15 and 8000 [27], while Pallares (2010) [28] modified the correlation of Fujio (2001) [29] to better fit the experimental 19 State of Art results. This shows the evolution of research in convective heat transfer applied to smoldering systems. In these correlations, the Prandtl number also plays a key role, as fluids with large values behave very differently from those with small values [30]. Other scales help describe the balance of conduction and convection. The Biot number indicates whether internal conduction or surface convection dominates, influencing the temperature profiles in solids and fluids [31, 32]. The Peclet number evaluates the relative importance of advection and diffusion and can be combined with other dimensionless groups to describe complex transport behaviors in multiphase porous systems, as mentioned by [33]. Dimensionless geometric scale relations are also widely used. One of them is the ratio of diameters, which influences permeability and flow resistance, where larger pores relative to particles favor higher permeability and lower resistance [2, 26]. This ratio also affects regime transitions, where higher val- ues often result in non-Darcian flows described by the Forchheimer equation [18]. Another example is the aspect ratio of the reactor, given by the relation between height and diameter. In smoldering reactors, this parameter deter- mines important dynamic characteristics, since low aspect ratios modify the distribution of experimental duration time and high aspect ratios enhance vertical mixing, favoring internal convective movements that better distribute mass and energy between different regions of the reactor. Time itself can also be expressed as dimensionless, and studies have applied Dimensionless Time to scale behaviors under different experimental conditions, supporting predictions and making them independent of bed size or absolute duration of 20 State of Art the experiment [12, 34]. Mass and heat transfer coupling is often expressed through the Lewis and Schmidt numbers. The Lewis number measures the balance between heat and mass transfer, while the Schmidt number expresses the ratio between diffusion and viscosity. Since Lewis can be written as Schmidt over Prandtl [35], their combined interpretation becomes important and is frequently ap- plied in dimensionless analyzes of smoldering systems. Experimental studies confirm the importance of these numbers in porous combustion [36, 37]. Buoyancy-driven processes are usually represented by the Grashof number, which quantifies the ratio of buoyancy to viscous forces. Together with the Prandtl number, it can be applied to estimate heat transfer coefficients even un- der vacuum conditions [38]. The Rayleigh number, defined as the product of Grashof and Prandtl, has long been used in smoldering analyzes. Poulikakos (1986) studied its role in natural convection in coupled fluid-porous systems for Rayleigh values ranging from 102 to 106 [39], which are values usually above the critical value needed for the onset of convection. The study shows how the domain affects the behavior of recirculations through simple alter- ations of the porous–fluid interface model to a unic fluid. Finally, the dynamics of chemical reactions can also be described by di- mensionless groups. The Damköhler number compares characteristic times of chemical kinetics with those of mass transport, showing under which con- ditions reactions dominate or are limited by diffusion [2, 31]. Leach (1998) demonstrated its effect on extinguishing reactions in porous media [40]. The Frank-Kamenetskii number is associated with ignition and thermal stability, 21 State of Art as it indicates when heat release exceeds conduction, potentially leading to instabilities in numerical simulations [2, 31]. Applications of this number include the burning of coal dust layers [40] and is often combined with Darcy and Arrhenius numbers to provide a better interpretation [41]. The Arrhe- nius number itself captures the exponential dependence of reaction rates on temperature [2, 42] and can be extended to free convection problems [43]. Understanding the classical behaviors expected in smoldering combustion systems is crucial to the development of more reliable numerical simulations that capture the main quantitative phenomena and physical behaviors ob- served in this type of combustion. Due to the complexity, new comparative frameworks that connect different reactor scales, operating conditions, and physical assumptions are necessary to better understanding the smoldering. 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ISSN: 1540-7489. https://www.sciencedirect.com/science/article/pii/ S1540748918305261 (2019). 26 https://aiche.onlinelibrary.wiley.com/doi/abs/10.1002/aic.690010211 https://aiche.onlinelibrary.wiley.com/doi/abs/10.1002/aic.690010211 https://www.sciencedirect.com/science/article/pii/S0735193310001788 https://www.sciencedirect.com/science/article/pii/S0735193310001788 https://www.sciencedirect.com/science/article/pii/S0017931000001666 https://www.sciencedirect.com/science/article/pii/S0017931000001666 https://www.sciencedirect.com/science/article/pii/S0735193321000233 https://www.sciencedirect.com/science/article/pii/S0735193321000233 https://www.sciencedirect.com/science/article/pii/0009250984801402 https://www.sciencedirect.com/science/article/pii/0009250984801402 https://www.sciencedirect.com/science/article/pii/S0017931020334748 https://www.sciencedirect.com/science/article/pii/S0017931020334748 https://www.sciencedirect.com/science/article/pii/S1540748918305261 https://www.sciencedirect.com/science/article/pii/S1540748918305261 REFERENCES 35. 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The Canadian Journal of Chemical Engineering 95, 1721–1729. https://onlinelibrary.wiley.com/doi/10.1002/ 9781119975465.ch8 (2017). 43. Huang, C.-J. Arrhenius activation energy effect on free convection about a permeable horizontal cylinder in porous media. Transport in porous media 128, 723–740 (2019). 27 https://www.sciencedirect.com/science/article/pii/B9781455731411500095 https://www.sciencedirect.com/science/article/pii/S0010218014004179 https://www.sciencedirect.com/science/article/pii/S0010218014004179 https://www.sciencedirect.com/science/article/pii/S1540748916302176 https://www.sciencedirect.com/science/article/pii/S1540748916302176 https://www.sciencedirect.com/science/article/pii/S1540748918303869 https://www.sciencedirect.com/science/article/pii/S1540748918303869 https://www.sciencedirect.com/science/article/pii/0142727X86900561 https://www.sciencedirect.com/science/article/pii/0142727X86900561 https://www.sciencedirect.com/science/article/pii/S008207849880146X https://www.sciencedirect.com/science/article/pii/S008207849880146X https://onlinelibrary.wiley.com/doi/10.1002/9781119975465.ch8 https://onlinelibrary.wiley.com/doi/10.1002/9781119975465.ch8 REFERENCES 44. Ohlemiller, T. Modeling of smoldering combustion propagation. Progress in Energy and Combustion Science 11, 277–310. ISSN: 0360-1285. https: //www.sciencedirect.com/science/article/pii/0360128585900048 (1985). 28 https://www.sciencedirect.com/science/article/pii/0360128585900048 https://www.sciencedirect.com/science/article/pii/0360128585900048 Chapter 3 Methodology 3.1 Modeling A 2D axisymmetric numerical model was developed using COMSOL Mul- tiphysics (version 5.4) to simulate a cylindrical reactor with a length L and a radius r. The geometry of the reactor is represented in dimensionless co- ordinates (r∗, z∗) as illustrated in Fig. 3.1. It is important to note that other axisymmetric geometries can also be simulated, such as conical and spherical shapes. The model incorporates different ways of regulating the airflow inside the reactor. Airflow can be driven by a pressure gradient, with zero pressure at the top (inlet) and negative pressure at the bottom (outlet), simulating a vacuum-induced smoldering reactor. Alternatively, airflow can be worked by forcing air at the inlet, maintaining zero pressure at the bottom. The model developed can simulate either a reactive porous bed or an inert porous bed. For a reactive porous bed, the ignition can be started at the top, because that is, the combustion front propagates downward, causing the height of the porous layer to decrease over time and creating a fluid domain in the upper 29 Methodology zone of the reactor. The temperature difference between the combustion front and the incoming cold airflow generates buoyancy forces that oppose the downward flow of air entering the reactor. Fig. 3.1 shows the boundary conditions for the two possible domains that can be created due to porous contraction (fluid and porous domains), as well as the constant boundary conditions with the external environment. In addition, most of the dimen- sional and dimensionless parameters are presented for each domain, which are responsible for governing the main physical behaviors within them. The displacement of the smoldering front causes the consumption of solid mass and gas generating. All equations related to the phenomena considered and used in the model are presented in a dimensionless form in Section 3.3. In this case, a two-dimensional axisymmetric approach was employed, along with dimensionless parameters for the spatial coordinates (r∗, z∗), time (t∗), velocity field (u∗ r , u∗ z ), pressure (p∗), temperature (T∗ s , T∗ f ), specific mass (ρ∗f ) and mass fractions for each phase (Y∗ s , Y∗ f ). Other assumptions can be considered: a two-temperature model for the porous bed, one for the gas phase and one for the solid phase (LTNE); the chemical reactions considered represent an oxidation process; a homogeneous porous bed with porosity and permeability constant in time and space. 3.2 Dimensionless group In light of the significant relevance of smoldering combustion within the scientific community, a general mathematical model utilizing dimensionless parameters has been developed primarily for this phenomenon. Although 30 Methodology Front Combustion Ax isy m m et ric Fluid Domain Porous Domain 𝑢𝑧 ∗ 𝑖𝑛 𝑢𝑟 ∗ 𝑖𝑛 𝜌∞ 𝑇∞ 𝑖𝑛 𝑌𝑓 ∗ 𝑢𝑧 ∗ 𝑜𝑢𝑡 𝑢𝑟 ∗ 𝑜𝑢𝑡 𝜌𝑓 ∗ 𝑜𝑢𝑡 𝑌𝑓 ∗ 𝑜𝑢𝑡 𝑌𝑓 ∗ 𝑤1 𝑢𝑧 ∗ 𝑤1 𝑢𝑟 ∗ 𝑤1 𝜌𝑓 ∗ 𝑤1 𝑌𝑓 ∗ 𝑤2 𝑢𝑧 ∗ 𝑤2 𝑢𝑟 ∗ 𝑤2 𝜌𝑓 ∗ 𝑤2 𝑃𝑓𝑟 ∗ 𝑌𝑓 ∗ 𝑓𝑟 𝑢𝑧 ∗ 𝑓𝑟 𝑢𝑟 ∗ 𝑓𝑟 𝜌𝑓 ∗ 𝑓𝑟 𝑇𝑓 ∗ 𝑓𝑟 𝑇𝑠 ∗ 𝑓𝑟 𝑇𝑓 ∗ 𝑇𝑠 ∗ 𝑌𝑠 ∗ 𝑌𝑓 ∗ W al l 1 W al l 2 Inlet Outlet 𝜆𝑠 𝜆𝑓 𝑈𝑟 ∗ 𝑈𝑧 ∗ 𝑑𝑝art 𝐾 𝑐𝑝𝑠 𝑐𝑝𝑓 𝑐𝑝𝑓 𝜌𝑓 ∗ 𝛼𝑠 𝛼𝑓 𝛼𝑓 𝜇𝑓 𝜇𝑓 𝜈𝑓 𝜈𝑓 𝑟∗ 𝑧∗ 𝐸 𝑅𝑠 𝑅 𝑅 𝐸 𝑢𝑟 ∗ 𝑢𝑧 ∗ 𝛽 𝑌𝑓 ∗ 𝑇𝑓 ∗ 𝜌𝑓 𝜆𝑓 𝑅𝑓 𝐷𝑓 𝐷𝑓 ∗ 𝑅𝑓 oxi . 𝑇∞ 𝑇∞ . . 𝑎𝑏𝑠 𝜌𝑠 𝑝∗ 𝑝∗ 𝐻𝑠 𝑌𝑠 ∗ 𝑤1 𝑌𝑠 ∗ 𝑤2 𝑌𝑠 ∗ 𝑜𝑢𝑡 𝑌𝑠 ∗ 𝑖𝑛 𝑌𝑠 ∗ 𝑓𝑟 𝑝∗ 𝑤2 𝑝∗ 𝑤1 𝑝∗ 𝑜𝑢𝑡 𝑝∗ 𝑖𝑛 𝜌∞ 𝑇∞ 𝑇𝑓 ∗ 𝑖𝑛 𝑇𝑓 ∗ 𝑤1 𝑇𝑓 ∗ 𝑤2 𝑇𝑓 ∗ 𝑜𝑢𝑡 𝜌𝑓 ∗ 𝑖𝑛 𝑇𝑠 ∗ 𝑖𝑛 𝑇𝑠 ∗ 𝑤1 𝑇𝑠 ∗ 𝑤2 𝑇𝑠 ∗ 𝑜𝑢𝑡 oxi oxi 𝑓 𝑠 𝐸 𝑓 Figure 3.1: Boundary conditions and crucial parameters involved in the general modeling. Developed by the authors. 31 Methodology other simplified processes can be simulate, such as the cooling of the porous bed in the axisymmetric reactor, this will be presented in Section 3.4.2. In what follows, the dimensionless groups proposed in this study are presented, along with the considerations made according to the characteristic scales of smoldering reactors. Figure 3.1 provides a visual representation of the physical phenomena that can influence the system, highlighting the large number of variables involved in this type of analysis. Therefore, adopting a dimensionless approach offers advantages in managing the complexity of the problem and allows to explore important physical parameters. To address the effects of porous bed contraction, a dimensionless time (Eq. 3.3) based on the total mass change over time (∆m) was proposed. This param- eter is associated with the reduction in bed height over time. Moreover, even in cases without combustion or significant temperature gradients, processes in- volving only heat transfer may still show mass variations due to evaporation, condensation, and diffusion. This supports the idea discussed earlier because the chosen dimensionless time parameter can also be applied to simplified processes in axisymmetric reactors. The longitudinal flow velocity (uz) is also related to this phenomenon and was incorporated into the dimensionless time parameter, as the flow velocity within the porous bed plays a key role in oxygen supply, solid mass consumption, and directly influences the reaction rate and experiment duration. Airflow is also important in processes without combustion, as it affects convective heat transfer, promotes mass transport, and contributes to phase changes. These aspects reinforce the relevance of including flow velocity in the dimensionless-time formulation, making the 32 Methodology framework applicable to a wider range of thermal and physical processes beyond smoldering combustion. r∗ = r dpart (3.1) z∗ = z dpart (3.2) t∗ = tρabs s uzd2 part ∆m (3.3) Radial and longitudinal velocities within the reactor are expressed by Eqs. 3.4 – 3.5. The lowercase parameter u refers to the Darcy velocity (average flow velocity), while the uppercase variable U represents the interstitial velocity (Eqs. 3.6 – 3.7). The latter was determined using the Dupuit–Forchheimer relation [1], given by: u = Uφ. u∗ r = urdpart ν f (3.4) u∗ z = uzdpart ν f (3.5) U∗ r = urdpart φν f (3.6) U∗ z = uzdpart φν f (3.7) 33 Methodology A local thermal non-equilibrium (LTNE) can be considered in the model. As the porous bed burns over time, the solid and fluid phase temperatures depend on parameters associated with the chemical reactions. Consequently, the activation energy of the solid material (Es) and fluid phase (E f ), and the ideal gas constant (R) were combined to define the dimensionless temperature employed in the model (Eqs. 3.8 - 3.9). This correlation aims to capture the influence of the combustion heat release on the modeled temperature field. When combustion is not involved, the dimensionless temperature can be simplified by disregarding the parameters related to chemical reactions (Eqs. 3.10 - 3.11). Then, without smoldering combustion involved in the case, the dimensionless equations presented in Section 3.3 will appear without the Arrhenius number, the source terms related to chemical reactions, and the source terms for the production or consumption of gases and solid materials. T∗ f = (︁ Tf − T∞ )︁ E f RT2 ∞ (3.8) T∗ s = (︁ Ts − T∞ )︁ Es RT2 ∞ (3.9) T∗ f = (︁ Tf − T∞ )︁ T∞ (3.10) T∗ s = (︁ Ts − T∞ )︁ T∞ (3.11) 34 Methodology The dimensionless pressure was defined based on the Bejan number, which quantifies frictional pressure losses in flows through porous media [2]. In this study, the pressure drop was evaluated over a characteristic length taken as the particle diameter of the porous bed (dpart), as shown in Eq. 3.12. The fluid phase density was normalized by the ambient density, as described in Eq. 3.13. p∗ = pd2 part µ f ν f (3.12) ρ∗f = ρ f ρ∞ (3.13) 3.3 Dimensionless equations Detailed considerations that following concepts showed chapter 1, chapter 2 and the initial sections of methodology regarding the mathematical modeling are presented below. Within an interstitial zone characterized by a reactive porous bed layer, oxidation of the solid phase leads to the generation of combustion gases depending on the reaction rate (Ṙoxi f ). Similarly, the mass fraction of the solid phase reduces over time in accordance with the reaction rates of the associated chemical processes. It is evident that in almost all the equations of the proposed model, the di- mensionless longitudinal velocity (u∗ z ) appears in transient terms. This reflects its direct role in the supply of oxygen necessary for smoldering combustion. The Particle Mass Fraction (m∗ part) also appears frequently, as it governs how 35 Methodology mass, momentum, and energy evolve over time, depending on the quantita- tive consumption of solid particles. The last parameter illustrates how the remaining solid mass influences, for example, the specific mass over time. The mass conservation for the fluid phase is expressed as in Eq. 3.14, following Miry et al [3]. φm∗ partu ∗ z ∂ρ∗f ∂t∗ + (︁ ρ∗f .▽ )︁ u∗ = v̇ f Ṙoxi f (3.14) The solid material remains almost stationary throughout the entire com- bustion process. Therefore, in the species transport equation for the solid domain (Eq. 3.15), the advective term is not considered. The source term associated with the Damköhler number (Das) appears in the equation and represents the intensity of the solid to fluid conversion through a chemical reaction relative to the gas flow velocity through the bed. ∂Y∗ s ∂t∗ = −Das (3.15) The porous medium is assumed to be homogeneous, and the effective diffusivity of the gas mixture is assumed to be constant in both time and space. Since the gas phase percolates through the interstices of the porous medium, the species transport equation for the fluid phase structurally includes tran- sient, advective, and diffusive terms (Eq.: 3.16). The Schmidt number appears in the diffusive term, following its classical meaning as defined in the litera- ture. 36 Methodology m∗ partu ∗ z ∂ [︁ ρ∗f Y ∗ f ]︁ ∂t∗ + [︃(︁ ρ∗f Y ∗ f )︁ .▽ ]︃ u∗ = Sc [︃ ▽2(︁ρ∗f Y ∗ f )︁]︃ + v̇ f Ṙoxi f (3.16) To ensure the ’dynamic coupling’ between pressures and velocities of the airflow, a mathematical model consisting of two momentum equations was developed: one for the fluid domain (Eq. 3.17) and the other for the porous domain (Eq. 3.18), (Brinkman equation, an extension of Darcy’s law). Since the proposed model addresses a reactor that is geometrically open at the top, natural convection effects occur and are accounted for as a source term in Eq. 3.17. Natural convection effects are expressed through the Grashof number, and the influence of chemical reactions on natural convection is captured via the Arrhenius number, which appears together with it in the source term in Eq. 3.17. This represents the influence of the fluid phase temperature on the buoyancy forces, since the Arrhenius number is related to the dimensionless temperature of the fluid phase, according to Eq. 3.8. m∗ partu ∗ z ∂u∗ ∂t∗ + (︁ u∗.▽ )︁ u∗ = −▽p∗ +▽2u∗ + Gr Arr (︁ T∗ f )︁ (3.17) m∗ partu ∗ z ∂U∗ ∂t∗ + [︃(︁ U∗.▽ )︁ U∗ ]︃ = −▽p∗ +▽2U∗ − 1 Da f u∗ (3.18) The specific mass and mass fraction of the fluid phase exhibit temporal and spatial variations. The behavior of the specific mass can be represented by the expression based on Equation 3.19, following the properties for the atmosphere air presented by [4, 5]. 37 Methodology ρ∗f = −3 · 10−7 · T∗ f 3 + 0.0001 · T∗ f 2 − 0.0152 · T∗ f + 1 (3.19) The energy equation for the fluid domain, in the upper zone of the reactor, is expressed as: Eq. 3.20. The Prandtl number appears in the diffusive term, following its classical meaning as defined in the literature. m∗ partu ∗ z ∂T∗ f ∂t∗ + [︃(︁ u∗.▽ )︁ T∗ f ]︃ = 1 Pr (︁ ▽2T∗ f )︁ (3.20) Inside the porous bed layer the Local Thermal Non-Equilibrium (LTNE) model is applied, with separate equations governing the fluid (Eq.: 3.21) and solid (Eq.: 3.22) phases. Chemical reactions occurring at the interstitial scale are primarily responsible for heating both the solid and fluid phases. Regardless of temperature, all physical bodies emit radiation. However, in close proximity to the combustion zone, radiation heat transfer predominantly occurs from the solid phase to the gas phase. Below the combustion zone, where the fluid flow is hotter than the solid phase, radiation heat transfer is primarily directed in the opposite direction. Due to fluid percolation through the porous medium, convective heat transfer also occurs. The direction of this convective heat transfer mirrors the behavior of radiation heat transfer. In Equation 3.22, different of the other equations before, the Thermal Peclet num- ber (Pe) appears in the transient term because the airflow velocity influences the heat extraction from the solid phase. The intensity of this effect is directly related to the amount of heat accumulated in the solid phase over time. 38 Methodology φm∗ partu ∗ z ∂T∗ f ∂t∗ + [︃(︁ U∗.▽ )︁ T∗ f ]︃ = φ 1 Pr (︁ ▽2T∗ f )︁ + τf Arr f Qstg f Ṙoxi f H f + Nuint Pr GF (︁ T∗ s − T∗ f )︁ + τf Arr f Prad s Qstg f [︃(︃ T∗ s Arr f + 1 )︃4 − (︃ T∗ f Arr f + 1 )︃4]︃ (3.21) (︁ 1 − φ )︁ m∗ part Pe ∂T∗ s ∂t∗ = (︁ 1 − φ )︁(︁ ▽2T∗ s )︁ + τs Arrs Qstg f Ṙoxi s Hs −BiintGF (︁ T∗ s − T∗ f )︁ − τs Arrs Prad s Qstg f [︃(︃ T∗ s Arrs + 1 )︃4 − (︃ T∗ f Arrs + 1 )︃4]︃ (3.22) The interstitial heat transfer coefficient (Γsg) between phases is based on an empirical Nusselt correlation (Nuint) as a function of Reynolds (Re) and Prandtl (Pr) numbers (Eq. 3.23) [6], and can be calculated using Eq. 3.24. The Reynolds number can be related to a dimensionless number proposed in this work, referred to as the dimensionless longitudinal velocity of the fluid phase (u∗ z ) (Eq. 3.5). Nuint = 2 (︃ 1 + 4 (︁ 1 − φ) φ )︃ + (1 − φ)0.5(︁u∗ z )︁0.6(︁Pr )︁ 1 3 (3.23) Γsg = Nuintλ f dpart (3.24) The general reaction rate (Ṙs) governs the consumption of the solid phase and the corresponding generation of the fluid phase, following the approach described by [3, 7]. 39 Methodology Ṙs = As · exp [︃ −Es Arrs RT∞T∗ s + Es ]︃ · Y∗ s · Y∗ f (3.25) The reaction rate of fluid generation used in equations 3.14, 3.16 and 3.21, denoted as Ṙoxi f , and the term of solid mass consumption term used in Equation 3.22, denoted as Ṙoxi s , represent the mass generated or consumed per unit volume per unit time in the respective phases. These terms are defined in Equations 3.26 and 3.27 [3], respectively. The parameter bO2 corresponds to the stoichiometric coefficient of oxygen, obtained from the chemical balance of the reaction between carbon and oxygen. Ṙoxi f = bo2φṘsρ abs s (3.26) Ṙoxi s = (1 − φ)Ṙsρ abs s (3.27) 3.4 Case studies To validate the entire methodology described in Sections 3.1, 3.2, and 3.3, it was proposed to perform a series of numerical simulations and compare the results with both experimental data from real conditions and numerical results reported by the literature on smoldering combustion. The following sections of the methodology present three case studies adapted for implementation in the proposed model. The level of complexity increases progressively from the first to the third case, aiming to demonstrate the robustness and versatility of the developed model. 40 Methodology • Case Study 1 – Combustion at the Fluid–Porous Interface: This is the simplest case, which involves a model under local thermal equilibrium (LTE) assumptions and steady-state flow equations. Its main purpose is to verify the model’s ability to capture flow recirculation caused by natural convection and to assess how such recirculations affect porous bed contraction and other physical phenomena. This analysis is aligned with the findings of Bittencourt (2023) [8]. • Case Study 2 – Cooling of a Porous Bed: The goal here is to calibrate the convective heat transfer between different phases. This case ex- tends the model to transient conditions and adopts a local thermal non- equilibrium (LTNE) approach for temperature fields. The experimental and numerical results of Martins (2008) [9] serve as a reference for this case. • Case Study 3 – Combustion in a Porous Bed: This final case aims to extend the model to simulate smoldering combustion as observed in reality. It is based on an experimental setup described by Martins (2008) [9], involving a cylindrical reactor entirely filled with a reactive porous bed. 3.4.1 Combustion at a fluid-porous interface As previously mentioned above, the software used to implement all the mathe- matical modeling proposed in Section 3.3 was COMSOL Multiphysics (Version 5.4). This first case study aims to analyze the behavior near the porous inter- face between distinct domains (fluid and porous). Bittencourt (2023) proposed 41 Methodology a steady-state numerical simulation of a smoldering combustion process using the same software, with the goal of capturing secondary flow recirculations observed in their experimental data and visualized during tests Bittencourt [8]. To better understand the case study, a vacuum-induced smoldering reactor with an internal diameter of 83 mm and a height of 300 mm was equipped with six inline thermocouples, strategically placed at different heights along the reactor axis. This arrangement enabled temperature measurements at multiple points within the chamber and allowed for monitoring of the com- bustion front’s path. A vacuum pump connected at the bottom of the reactor was responsible for creating a suction pressure to induce air flow through the reactor bed. More details of the physics simulated in this first case study are presented in Fig. 3.2. The author identified that the reactor geometry (open at the top) and the height of the porous layer (hp) influence the natural convection process above the porous interface during smoldering. Therefore, analyzing the dimensionless contraction of the porous bed was important, and this parameter was defined as h∗ = hp/L. Their methodology also employed a dimensionless approach using COM- SOL, similar to the one proposed in this work, but based on the classical equations provided by the software, such as Porous Media and Subsurface Flow (Brinkman Equations), Heat Transfer in Fluids, and Heat Transfer in Solids. Based on the dimensional analysis presented by Bittencourt (2023) [8], five dimensionless equations were considered to describe the problem. These equations correspond to mass conservation (Eq. 3.28); momentum conservation in the fluid (Eq. 3.29) and porous (Eq. 3.30) layers; and energy 42 Methodology Inlet Flux Air Suction Porous Bed Ax isy m m et ric Front Combustion Recirculation Zone Local Thermal Equilibrium Diffused Heat Diffused Heat Diffused Heat Figure 3.2: (a) Physical behavior of a vertical cylindrical - case 01. (b) Interstitial behavior of heat and flux - case 01. Developed by the authors. conservation in the fluid (Eq. 3.31) and porous (Eq. 3.32) layers. The dimen- sionless terms identified through this process are presented below, including classical dimensionless numbers commonly used in smoldering studies, such as the Prandtl, Grashof, and Darcy numbers. It is important to note that the author adopts a single-temperature model for the porous bed, which implies local thermal equilibrium (LTE). As shown in Eq. 3.32, this is evident from the fact that the temperature variable appears without any subscript that indicates the fluid or solid phase, being represented simply as T∗. ∇.u∗ = 0 (3.28) 43 Methodology (︁ u∗.∇ )︁ u∗ = −▽p∗ +∇2u∗ + Gr (︁ T∗)︁ (3.29) (︁ U∗.∇ )︁ U∗ = −▽p∗ +∇2U∗ − 1 Da f u∗ (3.30) (︁ u∗.∇ )︁ T∗ = 1 Pr (︁ ∇2T∗)︁ (3.31) (︁ U∗.∇ )︁ T∗ = 1 Prp (︁ ∇2T∗)︁+ q∗rad (3.32) In this work, the modeling of the permeable fluid-porous interface of the smoldering combustion reactor was proposed using a generic PDEs provided by COMSOL Multiphysics. This approach allows for greater flexibility in modeling equations in a dimensionless format. One of the mathematical inter- faces used to model the problem was the Coefficient Form PDE. This equation template provides a powerful general interface for specifying linear and non- linear equations, including the classical Poisson’s and Laplace structure. In the COMSOL interface, this PDE appears as the structure below Eq. 3.33: ea ∂2u ∂t2 + da ∂u ∂t +∇. (︁ − c∇u − αu + γ )︁ + β.u + au = f (3.33) 44 Methodology This equation contains terms that can be transformed into advection and conduction components, and it is also possible to include source terms. The parameter u is not a velocity vector in this case, but a generic vector that can be used to specify all independent variables. The coefficients ea, da, c, α, γ, β, a and f can be scalar parameters, vectors or tensors of different orders, depending on the number of independent variables chosen for modeling the problem and whether the phenomenon is isotropic or not. Another generic mathematical interface available in COMSOL is the Stabilized Convection- Diffusion Equation, which has the structure described below (Eq. 3.34). In this equation, fewer components are visible, and the parameter u is not a generic vector, but a unique independent variable. da ∂u ∂t +∇. (︁ − c∇u + αu )︁ + β.u + au = f (3.34) The boundary conditions used can include the Dirichlet Boundary Condi- tion, which allows for the definition of a constant value for a specific parameter at the edge of the domain, as well as the Flux/Source conditions, which ref- erence the Neumann Boundary Condition. The latter specifies the values that the derivative of a solution should take at the boundary of the domain. By applying tensorial calculations, it was possible to transform the PDE Eq. (3.33) in Eq. (3.28), Eq. (3.29), and Eq. (3.30), and through the same process, it was possible to transform the PDE Eq. (3.34) in Eq. (3.31) and Eq. (3.32). Therefore, the Coefficient Form PDE was designed to encompass the same structure as the Brinkman equation, meaning that through this PDE, it is possible to model governing equations related to the conservation of mass 45 Methodology and momentum. Furthermore, the Stabilized Convection-Diffusion Equation was designed to encompass the same structure as the Heat Transfer in Fluids and Solids, meaning that through this PDE, it is possible to model governing equations related to energy conservation in the fluid and porous bed domain. Because it is a circumferential geometry, one of the boundary conditions used to one edge of the reactor is axisymmetric (Fig. 3.3). Other boundary conditions used by the author of reference are those of a dimensionless con- stant temperature on the reactor walls, as well as at the fluid inlet and outlet. Furthermore, a constant dimensionless temperature is considered between the fluid and porous domains, which is called the dimensionless combustion front temperature T∗ h , and because it is always the highest temperature existing within the system, it receives a dimensionless value always equal to 1. Other boundary conditions related are: zero velocity on the wall, a prescript value of p∗in = 0 to the top of the reactor and p∗out = −Be to the bottom. In this case, the Bejan number is responsible for characterizing the induced airflow, which competes with free convection due to the shrinkage of the bed [2]. These boundary conditions, along with others, were implemented in the PDEs using Dirichlet Boundary Conditions. The locations of these conditions are shown in Fig. 3.3. All dimensionless parameters considered and their respective values are presented in Table 3.1, together with the initial conditions (Table 3.2) and the boundary conditions (Table 3.3). The tables present the dimensionless pa- rameters, initial conditions, and boundary conditions for the four simulations performed, considering porous bed contractions of 10%, 20%, 30%, and 40%, which correspond, respectively, to h∗ values of 0.9, 0.8, 0.7, and 0.6. 46 Methodology Front Combustion Ax isy m m et ric Fluid Domain Porous Domain 𝑢𝑧 ∗ 𝑖𝑛 𝑢𝑟 ∗ 𝑖𝑛 𝑇∞ 𝑢𝑧 ∗ 𝑜𝑢𝑡 𝑢𝑟 ∗ 𝑜𝑢𝑡 𝑢𝑧 ∗ 𝑤1 𝑢𝑟 ∗ 𝑤1 𝑢𝑧 ∗ 𝑤2 𝑢𝑟 ∗ 𝑤2 W al l 1 W al l 2 Inlet Outlet 𝑈𝑟 ∗ 𝑈𝑧 ∗ 𝐾 𝑐𝑝𝑓 𝑟∗ 𝑧∗ 𝑢𝑟 ∗ 𝑢𝑧 ∗ 𝛽 𝑇∞ 𝑇∞ . 𝑝∗ 𝑝∗ 𝑝∗ 𝑤2 𝑝∗ 𝑤1 𝑝∗ 𝑜𝑢𝑡 𝑝∗ 𝑖𝑛 𝑇∞ 𝑇ℎ ∗ 𝑝ℎ ∗ 𝑢𝑧 ∗ ℎ 𝑢𝑟 ∗ ℎ 𝑇𝑖𝑛 ∗ 𝑇𝑤1 ∗ 𝑇𝑤2 ∗ 𝑇𝑜𝑢𝑡 ∗ 𝛼 𝑔 𝜇 𝜈 𝐿 𝑇∗ 𝑇∗ ℎ𝑝 𝜎 𝜀 𝜌𝑓 𝜈 𝑘𝑚 𝛼𝑝 𝑇ℎ 𝐿 𝑇ℎ Figure 3.3: Representation of boundary conditions and crucial parameters involved in case 01. Developed by the authors. 47 Methodology Table 3.1: Value of dimensionless numbers. Results obtained to Bittencourt (2023) in a real experimental measure. Developed by the authors. h∗ Gr x [10−7] Pr Prp Be x [10−10] Da f x [109] 0.9 0.5 0.69 2.5 16.3 1.0 0.8 2.7 0.69 2.9 13.4 1.0 0.7 7.3 0.69 3.3 10.9 1.0 0.6 18.1 0.69 3.8 7.4 1.0 Table 3.2: Initial condition in numerical simulation - case 01. Nomenclature Describe Value u∗ z Longitudinal dimensionless velocity of fluid 0 u∗ r Radial dimensionless velocity of fluid 0 p∗ Dimensionless pressure of fluid 0 T∗ Dimensionless temperature of fluid 0 Table 3.3: Value of boundary conditions to numerical analyses. Results obtained to Bittencourt (2023) in a real experimental measure. Developed by the authors. h∗ T∗ in T∗ w1 T∗ w2 T∗ out 0.9 0.29 0.29 0.33 0.04 0.8 0.25 0.25 0.38 0.06 0.7 0.12 0.12 0.61 0.02 0.6 0.13 0.13 0.77 0.07 3.4.2 Cooling of a porous bed The mathematical model based on dimensionless parameters was developed primarily to simulate smoldering combustion. However, it can also be ap- plied to other simplified processes, such as the cooling of a porous bed in a axisymmetric reactor, as conducted by Martins (2008) [9], provided that small 48 Methodology adaptations are considered in the model. The experimental setup conducted by the author of reference features a vertical cylindrical with an internal di- ameter of 91 mm and a height of 300 mm. The reactor is equipped with high-precision instrumentation. Six in-line thermocouples (T1, T2, T3, T10, T11, T12), each with a diameter of 0.96 mm, are strategically placed at different heights along the reactor axis: z = 0, 45, 90, 180, 225, and 270 mm (measured from top to bottom). This arrangement enables temperature measurements at multiple points within the device. A total of 2340 g of a 3.6/96.4 wt char- coal/sand mixture were introduced into the cylindrical. The sand particles used in the mixture varied in size, ranging from 315–500 m, 500–1000 m, and 1–2 mm. The charcoal particles were ground to a size between 500–1000 m. To ensure a homogeneous mixture of charcoal and sand, a mortar prepa- ration method was used. The mixture was initially wetted and thoroughly mixed before being placed in the device. It was then left to dry overnight to remove any residual moisture within the porous bed. Initially, the dry mixture of charcoal and sand was heated to 68.2 °C in an oven. However, for this numerical simulation, the porous bed is assumed to be entirely composed of sand. This assumption is justified by the fact that sand constitutes more than 95% of the mixture in the experimental setup. Subsequently, the top of the cell was exposed to forced cold air, as illustrated in Fig. 3.4. As cold air percolated through the porous bed, it was cooled to ambient temperature (approximately 19°C). The equations used to model this case include Eqs. 3.14, 3.18 – 3.19, 3.21 - 3.24. It is important to note that the specific source terms in some equations were disregarded. For example, terms related to chemical 49 Methodology reactions and radiative heat transfer were omitted from the energy equations (Eqs. 3.21 and 3.22), as the objective of case 02 is to calibrate the model’s cooling process based on experimental data. Consequently, source terms for convective heat transfer between the solid and gas phases are considered to ensure the model accurately reflects experimental conditions. Radial heat loss was also considered. Equations 3.15 and 3.16 were not applied, as the objective does not involve the evaluating of chemical species transport. The specific mass of the fluid phase was modeled according to Equation 3.19. In line with the proposed simplifications, Equations 3.17 and 3.20 were excluded, as the reactor is entirely filled with a porous bed. Arrhenius numbers (Arrs and Arr f ) were not included in the dimensionless temperature group because the experiment was conducted at low temperatures, where activation energy is not significant. With this assumption, Eqs. 3.8 and 3.9 take a simplified form, excluding both the activation energy and the ideal gas constant, as expressed in the Eqs. 3.10 and 3.11. The other dimensionless parameters described in Section 3.2 follow their standard formulations. 50 Methodology Air Forced Outlet Flux Radial H eat Loss Heated Porous Bed