Quantifying the impact of rigid interparticle structures on heat transfer in granular materials using networks

https://doi.org/10.1016/j.ijheatmasstransfer.2019.118514Get rights and content

Highlights

  • Contributions of grain contacts and near-contacts to thermal conductivity are analysed.

  • Local clustering coefficients can quantify the rigidity of granular materials.

  • Rigidity and particle shape variables correlate well with thermal conductivity.

Abstract

Coordination number can be used to quantify the particle connectivity and deformability of a granular material. However, it is a local feature of particles at the microscale, and the use of an average coordination number does not allow for full characterization of the microstructural variation in the granular material. Mesoscale structures can be used to overcome this limitation: triangular-like structures at the mesoscale tend to be rigid, whereas square-like structures tend to be deformable. However, the effect of these structures on heat transfer has not been studied in deforming granular materials. A better understating of how microstructure variation affects effective thermal conductivity is necessary. This work constructs contact networks representing the granular materials with particles as nodes and linking neighbouring nodes with edges that represent particle contacts. Then, ‘3-cycles’ (i.e., a triangular structure) and ‘clustering coefficients’ are extracted from the contact network. As contact thermal conductance is vital to heat transfer and affected by particle shape, microscale three-dimensional particle shape descriptors are also calculated. To compute the effective thermal conductivity of the granular assembly, a thermal network model is established by adding ‘near-contact’ edges to the contact network and assigning a thermal conductance to each edge. The results show that mesoscale local clustering coefficients can indicate the rigidity of granular materials and, together with particle shape descriptors, can be used to predict well the effective thermal conductivity of granular materials under deformation.

Introduction

Compaction is one of the simplest ways to improve the ground bearing capacity. It also has the potential to enhance the heat transfer of the ground in shallow geothermal energy systems because the interparticle contact areas and the number of interparticle contacts may increase while pore spaces shrink during compaction. Heat transfer in any materials occurs because of conduction, convection and radiation. Since convection is important due to fluid currents [1] and radiation becomes significant when the temperature is greater than 1000 K [2], [3], conduction usually contributes the most strongly to heat transfer in dry granular materials [1], [4]. The heat conduction depends on the thermal conductivity of solid particles [1], the interparticle contact conductance [1], [5], [6], [7], [8], [9] and the structure of particle packings [2], [10]. As the rigidity/deformability of granular materials is related to their microstructures [11], a better understating of how the microstructure variation affects the effective thermal conductivity (λeff) is necessary.

The coordination number has a strong relationship with mechanical stability [12] and the jamming transition [13], [14] in granular materials. However, the coordination number is a microscale variable describing the connection of an individual particle to others. The often-used average coordination number cannot fully capture the spatial variation of the microstructure of granular materials. An order characteristic can also indicate the packing structure by measuring the rotational symmetry of particles [15]. However, it required complex calculation and was applied to sphere packings in the study. According to rigidity theory, a triangular structure tends to resist more deformation than a quadrilateral structure under an external loading (Fig. 1). However, the effect of interparticle triangular structures on λeff has not been studied in deforming granular materials.

Complex network theory can quantify the structure of a complex system and it has been successfully applied to represent civil infrastructure systems [16], [17], [18]. As granular materials are also complex systems [19], complex network theory has also been used to investigate the mechanical behaviour [11], [20] and pore connectivity [21] in the granular materials. However, it has not been used to study heat transfer in granular materials. A granular material could be simplified as a contact network in which a node is assigned to each particle and an edge is created when two neighbouring particles are in contact. Various mesoscale structural features can be obtained by calculating the number of n-cycles using complex network theory [20]. A ‘cycle’ is a loop that begins and ends at the same node, so 3-cylce is a triangle, 4-cycle is quadrangle and 5-cycle is pentagon. A 3-cycle is the smallest arrangement of particles formed by 3 neighbouring particles in contact [22]. These 3-cycle structures are more persistent and stable than n-cycle of higher orders (n > 3) during deformation of granular materials [11]. 3-cycles have a crucial role in rigidity because they can frustrate rotation and provide lateral support to surrounding particles even in three-dimensional (3D) analyses [23], [24]. Rivier (2006) showed that odd circuits (3-cycle is an odd circuit) are sufficient to ensure stability in 3D [23]. Mesoscale clustering coefficients can also be extracted from the contact network to measure the density of 3-cycles (triangles). Compared with the coordination number, which only provides information on a single node, the mesoscale 3-cycle and clustering coefficients have the advantage of containing information about more than one node without comprising the entire network. Hence, investigating the relationship between mesoscale rigidity features and λeff can potentially improve our knowledge of heat transfer in deforming granular materials.

In addition to the microstructure (rigidity) of the packings that can be characterized by the 3-cycle or cluster coefficients, particle contact thermal conductance is also important in the overall heat conduction [25]. In dry materials, the contact conductance is believed to be affected by particle shape [26], [27], as particle shape affects both the contact number and contact area [1], [28]. Therefore, three-dimensional particle shape descriptors are employed here to study the variation in λeff.

To extract the ‘3-cycle’ and particle shape descriptors of granular materials, their internal microstructural information should be acquired. High-resolution X-ray computed tomography (CT) techniques applied to granular materials can generate sequential CT images at a certain interval (resolution) [29], [30], [31]. Based on the images, the particle geometrical information and connectivity can be extracted using imaging postprocessing techniques. The geometry of the granular materials can also be reconstructed and numerical simulations can be undertaken to estimate their λeff. Finite element simulation (FEM) is an available method to compute the λeff but it is time-consuming because fine meshes are required to discretize the interparticle contacts and the interface between solid and pore phases. It usually overestimates λeff due to oversmoothing the interparticle contact areas [32], [33] and the lack of consideration of particle surface roughness [32]. Alternatively, network models [34], [35], [36] can discretely represent particle packings and calculate the heat transfer through interparticle contacts (real contacts) and small gaps between particles (near-contacts). However, very few thermal network models are available for nonsphere packings. The thermal conductance network model (TCNM) [37] developed by our team extended the application to packings of irregular (i.e., nonspherical) particles.

This article aims to find the relationship between the deformability of granular materials or rigidity and the λeff of granular materials using network techniques. Five granular materials with different particle shapes were scanned using CT techniques under different loadings. For each material at each level of compaction, four smaller subsamples were selected to (i) construct contact networks to calculate the number of mesoscale 3-cycle and clustering coefficients to characterize the rigidity of granular materials, (ii) construct thermal conductance network models (TCNMs) to calculate λeff, and (iii) compute the shape descriptors of individual particles. The λeff calculated from TCNMs were compared to those from FEM and experiments. Then, multiscale parameters were used to analyze the reasons underlying the λeff variation in deforming materials.

Section snippets

Materials

Five granular materials were used in this work. The pictures in the upper row of Fig. 2 show that the selected materials have different particle shapes. The round glass beads were made of silica and have a silver coating. The Ottawa sand was sieved following ASTM standard C778 [38] to achieve particles retained between sieve No. 20 (0.60 mm) and No. 30 (0.85 mm). Particles in both Ottawa sand and Angular sand are mainly made of quartz, but the former are more rounded. Crushed schist A is made

Network construction

Two types of networks are constructed in this work. Contact networks are constructed to acquire the 3-cycles and cluster coefficients using complex network theory. Thermal networks are extensions of the contact networks that also consider near-contacts as edges (Fig. 3) and it can be used to calculate the λeff by adding thermal conductance at the edges.

As summarized in Fig. 3, a sequence of CT images with a representative element volume is cropped from the scanned sample and the image noise is

Effective thermal conductivity comparisons

For each material shown in Fig. 2 under no pressure, four subsamples with a dimension of 4.5 by 4.5 by 4.5 mm from random locations within the sample were selected to check the homogeneity of the sample. Their λeff were calculated by both FEM and TCNM, as shown in Fig. 11. Experimental measurements from the literature [32], [34] and our laboratory are also included. The porosity of the experimental results is the mean value of the four subsamples in FEM and TCNM.

Fig. 11 illustrates that the λeff

Conclusions

This work investigated the impact of microstructure variation on effective thermal conductivity. A thermal conductance network model (TCNM) was used to calculate the effective thermal conductivity λeff of granular materials based on CT images. By comparing the results with those from FEM and experimental measurements, the TCNM was found to be robust and without as much overestimation as FEM when calculating λeff. Since TCNM is derived from the thermal network by adding thermal conductance at

Declaration of Competing Interest

The authors declared that there is no conflict of interest.

Acknowledgements

This research was undertaken in the Imaging and Medical Beam Line (IMBL) at the Australian Synchrotron, Victoria, Australia. The authors would like to acknowledge Dr Anton Maksimenko and the other beam scientists at Australian Synchrotron for their support during our experiments. The authors also thank Dr Tabassm Afshar and Dr Xiuxiu Miao for their support in collecting the CT images. The first author thanks The University of Melbourne for offering the Melbourne Research Scholarship.

References (73)

  • T.S. Yun et al.

    Three-dimensional random network model for thermal conductivity in particulate materials

    Comput. Geotech.

    (2010)
  • O. Birkholz et al.

    Modeling the effective conductivity of the solid and the pore phase in granular materials using resistor networks

    Powder Technol.

    (2019)
  • H. Taylor et al.

    A new method to identify void constrictions in micro-CT images of sand

    Comput. Geotech.

    (2015)
  • J. Fonseca et al.

    Non-invasive characterization of particle morphology of natural sands

    Soils Found.

    (2012)
  • J. Sundberg et al.

    Estimation of thermal conductivity and its spatial variability in igneous rocks from in situ density logging

    Int. J. Rock Mech. Min. Sci.

    (2009)
  • G.A. Narsilio et al.

    Upscaling of Navier-Stokes equations in porous media: theoretical, numerical and experimental approach

    Comput. Geotech.

    (2009)
  • M. Moscardini et al.

    Discrete element method for effective thermal conductivity of packed pebbles accounting for the Smoluchowski effect

    Fusion Eng. Des.

    (2018)
  • W. Dai et al.

    Influence of gas pressure on the effective thermal conductivity of ceramic breeder pebble beds

    Fusion Eng. Des.

    (2017)
  • T.S. Yun et al.

    Fundamental study of thermal conduction in dry soils

    Granular Matter

    (2008)
  • U. El Shamy et al.

    Discrete element method study on effect of shear-induced anisotropy on thermal conductivity of granular soils

    Int. J. Geomech.

    (2013)
  • C. Argento et al.

    Thermal conductivity of granular media

    Powders & grains

    (1993)
  • R. Askari et al.

    Thermal conduction in deforming isotropic and anisotropic granular porous media with rough grain surface

    Transp. Porous Media

    (2018)
  • B. Aduda

    Effective thermal conductivity of loose particulate systems

    J. Mater. Sci.

    (1996)
  • M. Gangadhara Rao et al.

    A generalized relationship to estimate thermal resistivity of soils

    Can. Geotech. J.

    (1999)
  • J. Côté et al.

    Thermal conductivity of base-course materials

    Can. Geotech. J.

    (2005)
  • L. Fletcher

    Recent developments in contact conductance heat transfer

    J. Heat Transfer

    (1988)
  • L. Papadopoulos et al.

    Network analysis of particles and grains

    J. Complex Networks

    (2018)
  • J. Scott

    Social network analysis

    Sociology

    (1988)
  • A.J. Liu et al.

    The jamming scenario—an introduction and outlook

  • G. Fu et al.

    A spatial network model for civil infrastructure system development

    Comput.-Aided Civ. Infrastruct. Eng.

    (2016)
  • S. Argyroudis et al.

    Systemic seismic risk assessment of road networks considering interactions with the built environment

    Comput.-Aided Civ. Infrastruct. Eng.

    (2015)
  • A. Bozza et al.

    Alternative resilience indices for city ecosystems subjected to natural hazards

    Comput.-Aided Civ. Infrastruct. Eng.

    (2017)
  • H.M. Jaeger et al.

    Does the granular matter?

    Proc. Natl. Acad. Sci.

    (2000)
  • M.E. Newman

    The structure and function of complex networks

    SIAM Rev.

    (2003)
  • J.H. van der Linden et al.

    Machine learning framework for analysis of transport through complex networks in porous, granular media: a focus on permeability

    Phys. Rev. E

    (2016)
  • A.G. Smart et al.

    Evolving loop structure in gradually tilted two-dimensional granular packings

    Phys. Rev. E

    (2008)
  • Cited by (21)

    • A graph-theory based directed network feature for thermal anisotropy

      2022, International Journal of Heat and Mass Transfer
    • Estimation of thermal conductivity of cemented sands using thermal network models

      2022, Journal of Rock Mechanics and Geotechnical Engineering
    • Cyclic thermo-mechanical performance of granular beds: Effect of elastoplasticity

      2021, Powder Technology
      Citation Excerpt :

      Based on different assumptions on the heat flux pathways, some grain-scale analytical solutions have been developed, such as a gas film model [22], Voronoi cell based models [10,23,24] and the Batchelor and O'Brien analytical solution [25], to evaluate the inter-particle thermal resistance. Recently, combined with other techniques, e.g., machine learning, the thermal DEM has shown the potential to be applied to multi-scale investigations [26,27]. Most of the existing models for describing interparticle thermal conductance, however, focused on the elastic mechanical contacts for the sake of simplicity, leading to underestimate ETC and ignore the irreversible behaviour due to possible plastic particle deformation during cyclic loading.

    View all citing articles on Scopus
    View full text