Heat transfer modelling in Discrete Element Method (DEM)-based simulations of thermal processes: Theory and model development

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Abstract

Over the past decade, DEM-based simulation has become a promising alternative to physical measurements of thermal particulate systems. Despite their rapid advancement and successful applications to a wide range of industrial processes, a comprehensive review of the theory that underpins the thermal DEM-based simulations is yet to be conducted. This work presents a critical and in-depth review of all major thermal models and heat transfer mechanisms pertinent to DEM-based simulations. Other critical aspects such as boundary conditions and particle body temperature distribution that were often overlooked are also summarised and discussed, aiming to provide a clear path for the development of robust thermal DEM-based models.

The quasi-analytical solution based on the Hertzian contact theory proves classic and remains the main method to solving the conduction of static contacts. Recent attempts have been mainly directed towards improving the calculation of conduction through collisional contacts and the thin wedge of interstitial fluid between particles. Empirical correlations that were developed before 1981 remain predominant in calculating the fluid-particle convection coefficient. Though more accurate, the discrete models of radiation that rely on the solution of view factors amongst individual particles have been applied much less than the continuum models due to the significant computational overhead. Generally, previous efforts have led to the construction of a solid framework of thermal DEM-based models. Significant work is required to improve existing or develop new heat transfer sub-models, particularly those for accurate and efficient modelling of conduction and radiation in particle-laden systems.

Introduction

Thermal/reacting particulate systems are commonly encountered in industrial processes as diverse as coal gasification, iron and steelmaking, drug manufacturing, energy generation and storage, fuel production, upgrading and decarbonization, and granular material processing including mixing, drying, granulation and coating. In these processes, particles are either the reactants themselves (e.g. [1], [2], [3], [4]), or the medium that is used to enhance the heat transfer or chemical reaction rates (e.g. [5,6]). The main mechanisms that make the involvement of particles so preferable for achieving high system operation efficiency include: (i) large interfacial area available for heat and mass transfer; (ii) high relative velocity between the fluid phase and discrete particles followed by the high convective heat transfer; and (iii) a third heat transfer mechanism, namely particle radiation, taking place. However, the presence of particles also introduces significant complexities and uncertainties of physics to the particulate system [7]. A good understanding of the theory and mechanisms that underlie the heat transfer characteristics in thermal/reacting particulate processes is largely lacking which however is pivotal to the design and operation of relevant industrial processes.

According to the third law of thermodynamics or the Clausius statement, heat, a form of energy, flows throughout the system driven by the temperature difference. Heat transfer occurs mainly through three mechanisms, namely, conduction, convection and radiation. Heat might be also simultaneously generated, in the context of particulate systems, often through chemical reactions [6,8], phase change [9] and the energy dissipation mechanisms [10] due to friction amongst particles and kinetic energy loss in the event of collisions. Al-Arkawazi [10] calculated the amount of heat energy due to different mechanisms in packed and fluidised beds and found the heat generated through the energy dissipation mechanisms due to friction and collision only adds up to a tiny portion (approaching zero after particles are fluidised) compared to the heat transferred through the three main mechanisms, hence has been neglected in the majority of previous studies. Heat generation due to chemical reactions and/or phase change that occur in certain application scenarios is not covered in the present work. Accordingly, this review specifically centres on the theory and mechanisms of heat transfer in thermal particulate systems.

Despite the significant progress achieved in the measuring techniques of heat transfer, e.g. the electrically heated single sphere buried in the unheated packing [11,12], laser flash method [13,14], mass transfer analogy and simultaneous heat and mass transfer [15] and the regenerative heating technique [16], it proves extremely challenging (if not impossible) to fully understand the heat transfer characteristics of particulate systems via the experimental approach alone. For this reason, the numerical approach based on Discrete Element Method (DEM) has become increasingly popular as it provides detailed local and transient information on flow and heat transfer at the particle scale. However, due to the complexities of the thermal particulate systems in terms of time- and temperature-dependent flow structures and material properties, new time and length scales introduced by particle interactions, high nonlinearity and anisotropy in interphase heat and mass transfer, and high heterogeneities in stress and particle contact mechanics, it is challenging to develop a robust thermal DEM-based model that is capable of addressing all major problems associated with heat transfer and suits different application scenarios.

Generally, numerical models developed for investigating the particulate systems could be categorised based on two main frameworks, namely Eulerian–Eulerian and Eulerian–Lagrangian. Models that were built based upon the Eulerian–Eulerian framework mainly include the mixture model and the two-fluid model (TFM). This category of models essentially treats both the fluid phase and solid particles as the continuum. The approach is computationally preferable and can simulate the flow and heat transfer of a large number of particles in actual processes [17], [18], [19], [20], [21], [22], [23], [24]. For applications in which mesoscopic features are of subtle importance, this approach suffices. However, the treatment of the solid phase as the continuum disobeys the discrete nature of solid particles, thus limited by the local homogeneity assumption and the difficulties associated with the construction of constitutive equations for the stress tensor and effective thermal conductivity of the solid phase. More importantly, the Eulerian–Eulerian approach cannot reveal information at the particle-scale such as temperature, intraparticle and particle surface temperature distribution, particle contact, local void fraction or local fluid-solid flow structure. All the information, however, is vital to the study of flow and heat transfer in thermal/reacting particulate systems.

Models that were built in the Eulerian–Lagrangian framework mainly include Direct Numerical Simulation (DNS)-DEM [25], Computational Fluid Dynamics (CFD)-DEM [26], and lattice Boltzmann Method (LBM)-DEM [27]. This category of models completely overcomes the abovementioned drawbacks of Eulerian–Eulerian based models. Indeed, it has been extensively proven that the DEM-based simulation is a promising approach for simulating the heat transfer and fluid flow in particulate systems [28,29]. For instance, by explicitly solving the particle dynamics of individual particles, the challenge associated with particle contact heterogeneities induced by stress chains is directly addressed, and the transient distribution of particle temperature is resolved at the mesoscale [30]. However, as individual particles are tracked and the flow and heat transfer related to each particle are resolved, the DEM-based simulation is computationally demanding for solving the solid phase. Moreover, models of DNS and LBM themselves entail high computational expense in solving the fluid phase. As a result, simulations using a DEM-based model were often limited to small systems that contain a small number of particles (e.g., thousands of particles). However, in recent years, with the ample availability of high-computational power accompanied with advanced parallel-computation algorithms (based on MPI, OpenMP and GPU) [6,7,[31], [32], [33], the DEM-based models (in particular CFD-DEM) have been applied successfully to a wide range of thermal/reacting industrial processes to undergo detailed analysis of the chemo-physics where millions of particles are present.

DEM-based models that involve heat transfer are also referred to as thermal DEM-based models. Despite the success of their broad industrial applications (e.g. [34], [35], [36]), a comprehensive review of the thermal models developed and/or applied in previous thermal DEM-based simulations is yet to be conducted except for a few reviews that have centred on specific topics, e.g., heat transfer of nanofluids in porous media by Xu et al. [37], DNS of mass, momentum and heat transfer in dense gas-solid flows by Deen et al. [25], and CFD simulation of dense particulate reaction systems by Zhong et al. [34]. This work places the review focus on the theory underpinning thermal DEM-based simulations in terms of thermal models and heat transfer mechanisms. Before presenting the details, a general description of the development and state-of-the-art of these thermal models is given as follows.

  • Conduction

Batchelor & Brien [38] proposed the classic equation that is an approximately analytical solution of conductive heat transfer when two particles are in contact. The equation is applicable for static contacts when ksrc/kfrp ≫ 1 where rc is the contact radius and rp is the rigid particle radius. ks and kf denote the thermal conductivities of solid particles and the fluid, respectively. For moving particles, Soo [39] applied the elasticity theory to evaluate the contact area and contact duration which were then quantitatively examined by Sun and Chen [40] who calculated the conductive heat transfer when two moving particles collided. The formula was subsequently adapted by Zhou et al. [41] for easy implementation in DEM-based simulations. In conventional DEM-based models, the unrealistic properties of particle material were used to allow for the acceptable simulation turnaround time. This, however, leads to the unrealistic contact area and contact duration, hence incorrect or, to be more specific, overestimated conductive heat exchange. Zhou et al. [42] later proposed a correction factor to rectify the incorrect collisional conduction calculation. However, this correction factor only accounts for the overestimated contact area [35,42]. Morris et al. [43] derived two physically-based terms to correct both the contact area and contact time, but the time correction was analytical derived for binary collisions, thus is limited to relatively dilute systems. Instead of using correction factors Patil et al. [44] calculated the correct integrated amount of heat transfer during one collision using the real particle properties and input it into DEM simulations as a packet of energy at the beginning of the collision. However, the model was confronted by other difficulties, e.g. unable to conveniently consider other contact events including the rolling or resting motion of particle clusters on the wall.

For scenarios where conduction through the interstitial fluid in between two interacting particles is significant, i.e. when ksrc/kfrp ≫ 1 is not satisfied, Rong&Horio [45] proposed a surrounding layer method that determines when and how to consider the indirect conduction through the interstitial fluid; the method was later slightly modified by Musser [46] for easy implementation. In this method, the conductive heat transfer through the body of solid particles is not considered as part of the indirect conduction paths. Rather differently, Cheng et al. [47] introduced the concept of Voronoi polyhedron and proposed two sub-models (Models A and B) that calculate the indirect particle-fluid-particle conduction through both the body of solid particles and the interstitial fluid. Recently, the Models A and B of Cheng et al. [47] have been modified by Gan et al. [48] and Chen et al. [49], respectively, for modelling polydisperse particulate systems. Vargas & McCarthy [50] extended the model of Batchelor & Brien [38] by simply adding the thermal conductance through the interstitial fluid for the application to granular materials. Also, Tsory et al. [51] proposed a simplified method by dividing the particle contact into solid-solid contact and air gaps and calculated the conductive heat transfer by summing up conduction through these two parts. For scenarios with ks/kf < 1, Cheng et al. [52] just recently proposed a similar approach to the concept of Voronoi polyhedrons, i.e. the Delaunay tessellation for calculating the conduction through stagnant fluid in void spaces of packed beds.

  • Convection

Compared to conduction and radiation, heat transfer through convection has been relatively better understood in particulate systems. Recent efforts towards improving the existing correlations or proposing new correlations are rather scarce. The classic model of Ranz & Marshall [53] proposed based on the evaporation experiments of water droplets has been widely applied in DEM-based simulations. This model also becomes the foundation of many succedent models (e.g. [54], [55], [56]). Generally, the models can be grouped based on their applications to systems of different solid concentrations into dense system models (packed bed, rotary drums), less dense system models (fluidised bed) and dilute system models (pneumatic conveying). Gnielinski [57,58] proposed a semi-empirical model based on the experimental data available in the literature for packed pebble particles. Similar works have been published since then with different correlations proposed for predicting the convective heat transfer coefficient in packed bed reactors but for different flow conditions (or Reynolds numbers) [15,59,60]. After verifying the correlation of Gunn [61] that proves accurate only for systems with the solid concentration less than 0.3, Tavassoli [62] modified Gunn's correlation [61] for the entire spectrum of solid concentration (from 0 to 0.65). Bandrowski & Kaczmarzyk [63] proposed a correlation for the pneumatic conveying system but covering a limited range of Reynolds number (from 180 to 1800). Li and Mason [64] modified the correlation of Bandrowski & Kaczmarzyk [63] for all possible Reynolds numbers in the operation of pneumatic transport systems. Li and Mason [64] therefore suggested users re-evaluate the exponent of their correlation and relate it to solid concentration and particle size, or particle Reynolds number. Li and Mason [64]’s work was followed by Al-Arkawazi [65] who recently conducted experiments and modified the exponent for dense particulate systems.

  • Radiation

Heat transfer through radiation is relatively more challenging to be fully considered in particulate systems compared to conduction and convection. Models available in the public domain could be categorised into i) the continuum approach and ii) the discrete approach. The former employs a radiation temperature for a particle which can be obtained through solving the field of radiative intensity [66], [67], [68] of the particle-fluid mixture. However, this radiation temperature was more commonly treated as an averaged surrounding temperature of the particle for the sake of computation efficiency [35,42,[66], [67], [68], [69], [70], [71], [72], [73], [74].

The discrete approach considers the radiative heat transfer amongst individual particles based on the solution of view factors [69]. Evidently, the discrete approach is more delicate to calculate the radiative heat transfer, yet the determination of view factors incurs significant computational overhead. Previous efforts are directed mainly towards accurately calculating the view factors [49,69,[75], [76], [77], [78]. Wu et al. [78] presented three sub-models for calculating radiative events within different ranges, namely, microscopic range, short-range and long-range, respectively. Models of the short-range radiation are based on the concept of Voronoi polyhedron and different methods (e.g. [49,69,78]) have been developed to calculate the view factors between two interacting particles. For the long-range radiation model, it is straightforward to calculate the view factor between unit particles without blockage. However, it is extremely challenging to determine the view factor between a pair of particles with blockage. To this end, different methods [79], [80], [81], [82], [83] have been proposed for the calculation of view factors, but largely limited to static particulate systems due to the inhibitive computational cost. Very recently, Forgber & Radl [84] proposed an inherently simple yet efficient method to calculating the view factor for dynamic particle-laden systems.

  • Other aspects relating to the thermal models of DEM-based simulations

Apart from the above thermal models for resolving the main heat transfer mechanisms, other aspects also play an important role in developing a robust thermal DEM-based model. Generally, these aspects require due care when different particle materials are simulated. Specifically, for thermally thick particles with a large Biot number (Bi ≫ 0.1), the assumption of isothermal particles does no longer hold. Both one-dimensional (1D) and three-dimensional (3D) models have been proposed to resolve the temperature distribution inside individual particles [85], [86], [87], [88], [89], [90], [91], [92], [93]. Also, for these thermally thick particles, the temperature at the particle surface is nonuniform [90,91,[94], [95], [96]. The influence of the non-uniform particle surface temperature distribution on the convective heat transfer was considered by multiplying the conventional heat transfer coefficient by a distribution function [90,91,[94], [95], [96], [97]. On the opposite, for thermally thin particles Oschmann&Kruggel-Emden [90,91] reported a significant discrepancy between predicted results and experimental measurements when the temperature distribution within the reactor wall was ignored. Further, Hrenya's research group [43,[98], [99], [100] proposed a new wall boundary condition with the constant heat flux, which was considered more practical and accurate in many industrial processes. All these aspects have been proven pivotal to the fidelity of a thermal DEM-based model after the extensive comparison with the experimental data.

The primary objective of this study is to provide a critical and in-depth review of the theory and all major thermal models relating to heat transfer mechanisms in the literature of thermal DEM-based simulations. Different models have been detailed, grouped and compared, with a view to providing modellers with the key information to selecting the ideal model for simulating a specific problem. The relative importance of main heat transfer mechanisms in different application scenarios has been examined and discussed to emphasise the dominant mechanisms as well as justify the negligence of trivial mechanisms for a specific application. The important roles of other aspects in developing a robust thermal DEM-model have also been demonstrated with examples and comparisons. Lastly, recommendations for future work are given to state the shortcomings of existing thermal models and shed light on the research scope towards which future efforts should be directed.

Section snippets

Description of particle flow dynamics

Since DEM-based simulation is a popular and active research area, the detailed description of the governing equations and the discussion of the model's capabilities, shortcomings and applications can be readily found in the literature, e.g. as summarised in [25,27,34,[101], [102], [103]. The governing equations of the fluid flow (if there is any) can also be found in the literature, thus not given here for brevity.

In particulate systems, a single particle is interacting with neighbouring

Heat transfer mechanisms and thermal models

In thermal particulate systems, the three main heat transfer mechanisms (conduction, convection and radiation) take place. Specifically, the heat transfer might involve the following mechanisms [117],

  • Conduction through the fluid in the gap between two-point contacting or non-contacting particles;

  • Conduction through the fluid between two area-contacting particles;

  • Conduction through the contact area between two area-contacting particles;

  • Conduction through the fluid in void spaces;

  • Convection

Intraparticle temperature distribution

In the majority of previous studies, the lumped capacitance approximation that assumes a homogeneous temperature distribution throughout each solid particle was often deployed. As clarified above, this simplification is only valid for thermally thin particles that have a small Biot number (Bi = h/(ks/dp) < 0.1) and a large Fourier number (Fo=αst/dp2>0.1), achieved by choosing materials with the high thermal conductivity and/or a small particle size. For cases with a large Bi and a small Fo,

Heat transfer modelling in previous DEM-based simulations

Major previous DEM-based simulations, in particular CFD-DEM simulations, that involve heat transfer have been reviewed [10,35,36,[42], [43], [44], [49], [50], [51],[66], [67], [68],[70], [71], [72], [73], [74],[84], [85], [86],[90], [91], [92],96,[98], [99], [100],119,120,137,141,[144], [145], [146],148,[150], [151], [152], 154,[189], [190], [191], [192] (as listed in Table S2 of the Supplementary Material). Zhu et al. [101] reported that the number of publications using DEM to simulate the

Conclusions

In this work, the thermal models and heat transfer mechanisms underpinning DEM-based simulations of thermal particulate systems were reviewed. Other important aspects relating to the development of thermal DEM-based models were also summarised and discussed. Major findings are given as follows.

Conduction between two particles was found significant only when they were close to each other. Indirect conduction was simulated mainly using the surrounding layer method and the Voronoi polyhedron-based

Declaration of Competing Interest

None.

Acknowledgments

The authors wish to acknowledge the financial support by The Australian Coal Industry’s Research Program (ACARP) C26004.

Dr Zhengbiao Peng’s principal area of expertise lies in areas of particle technology, fluid mechanics, and heat transfer and chemical reactions in multiphase processes. Other areas of his expertise include in-house codes of DEM-based modelling of fluid-solid flow, heat transfer and chemical reactions, in-house codes of computational fluid dynamics (CFD) modelling of thermal/reacting processes, interfacial phenomena, colloidal science, ice nucleation, microfluidics, parallel computation and

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    Dr Zhengbiao Peng’s principal area of expertise lies in areas of particle technology, fluid mechanics, and heat transfer and chemical reactions in multiphase processes. Other areas of his expertise include in-house codes of DEM-based modelling of fluid-solid flow, heat transfer and chemical reactions, in-house codes of computational fluid dynamics (CFD) modelling of thermal/reacting processes, interfacial phenomena, colloidal science, ice nucleation, microfluidics, parallel computation and multi-scale modelling of actual industrial problems. Dr Peng received his PhD (2009) in Chemical Engineering from Southeast University, China. Upon completion of his PhD studies, Dr Peng commenced his academic research career in the School of Engineering at The University of Newcastle (Australia). He has published over 80 scholarly articles and has been involved in many research projects (over $9 million) funded by both Australian government and industry.

    A/Prof Elham Doroodchi’s research is underpinned by the two general areas of fluid mechanics and particle technology with the main focus being on the fundamental and applied research into multiphase systems. Specifically, the research has focused on hydrodynamics of fluidized beds, and solid particle, droplet and bubble motion in fluid. The research carried out by A/Prof Doroodchi has involved a combination of novel and conventional experimental measurement techniques (e.g. particle image velocimetry) and theoretical modelling (e.g. computational fluid dynamics). In all cases, the fundamental understanding of the interaction between the phases is obtained under conditions relevant to the actual industrial-scale and subsequently has been applied to the engineering design of multiphase systems such as particle classifiers, separators and mixers. A/Prof Doroodchi has attracted in excess of $17 million in research grants and has published over 140 scholarly articles. She has also supervised 14 PhD projects to completion.

    Prof. Behdad Moghtaderi’s research theme is “Thermo-Fluid Engineering” encompassing applications in the general field of energy and the environment. The focus of his research is development of technologies suitable for direct/indirect minimisation of greenhouse gas emissions, particularly in application areas, such as: Ventilation Air Methane (VAM), renewable energy systems, advanced low emission coal technologies, hydrogen powered micro-energy systems, and energy efficiency. Since joining The University of Newcastle in 1999, Prof Moghtaderi has attracted in excess of $59 million in research funding. He is a co-inventor of the GRANEXTM heat engine which is being marketed internationally by Granite Power Pty Ltd.

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