Packing Characteristics and Heat Transfer Performance of Non-Spherical Particles for Concentrated Solar Power Applications

Abstract

Concentrated solar power (CSP) technology relies on thermal energy storage to extend operating hours, making the selection of heat storage media crucial for system efficiency. Bauxite powder, known for its availability and high-temperature stability, emerges as a potential alternative to conventional materials in CSP systems. This study employed the discrete element method to investigate the influence of particle shape on the packing and heat transfer characteristics of non-spherical particles. The research focused on assessing the impact of particle sphericity by comparing spherical particles with non-spherical shapes, including ellipsoids and cylinders, and exploring the effect of varying the aspect ratio (AR) of the cylindrical particles. Particle sphericity significantly influenced packing morphology, with the cylindrical particles exhibiting distinct structural patterns that were absent in the ellipsoidal particles, and strongly affected heat transfer, as observed in the average temperature variations within the packed bed over time. The cylinders with higher aspect ratios demonstrated enhanced heat transfer rates, driven by the increased contact area and coordination numbers, despite their predominant misalignment with the heat flux direction. These insights are valuable for optimizing thermal energy storage media in CSP systems.

1. Introduction

Concentrated solar power (CSP) technology offers a novel approach to energy generation by harnessing concentrated sunlight to heat thermal storage media and subsequently converting this thermal energy into electricity. Thermal energy storage media are vital in facilitating energy transfer [1,2]. Thermal energy storage operates under high temperatures and pressures. However, conventional storage media materials often fail to meet these strict requirements and are costly [3,4,5]. Thus, choosing the appropriate heat storage media is crucial for designing a system that stores energy efficiently and can operate under high processing conditions. For example, molten salts are well-recognized for their thermal storage capabilities, but their practical application is hindered by limitations such as freezing at low temperatures (<200 °C), dissociation at high temperatures (>800 °C), and susceptibility to high-temperature corrosion [6]. Granular materials offer a promising, cost-effective alternative to molten salts for thermal energy storage within CSP systems due to their superior performance in demanding operating conditions [7]. Bauxite powder, characterized by its affordability and favorable thermal properties, including high solar absorptivity, emerges as a potential candidate among granular storage media [8]. However, the irregular morphology of bauxite particles necessitates a comprehensive investigation into the behavior of non-spherical particles.

Most solar power research has focused on fine-grained materials, primarily spherical in shape, which often require substantial production costs. In contrast, naturally extracted granular material particles tend to have irregular shapes [9]. It is widely recognized that the packing structure of the granular bed significantly affects the heat storage efficiency [10,11,12]. In CSP systems, key components such as receivers, thermal storage units, and moving bed heat exchangers rely on slowly moving or stationary packed particle beds. Even in moving beds, the relatively slow particle motion allows for their approximation as packed beds during short time intervals. The heat transfer mechanism in powder-based CSP systems primarily relies on conduction and radiation, with a minimal contribution from convection due to the absence of forced gas flow through particle beds in thermal storage units and moving bed heat exchangers. Radiation heat transfer within powder beds, particularly those containing non-spherical particles, is a complex phenomenon influenced by factors such as particle shape, size distribution, and packing arrangement [5,13]. The irregular geometries of non-spherical particles introduce variability in radiative interactions, complicating the accurate modeling and quantification of radiation heat transfer in these systems. In the present study, we focused on conduction as an essential mode of heat transfer, with particle contact areas and orientations playing critical roles in determining the overall thermal resistance and system efficiency. Conduction through particle contacts is the primary heat transfer mechanism, especially in conditions where the particle proximity enhances solid-to-solid conduction. Moreover, as discussed by Su [14], the effective thermal conductivity within the bed is significantly influenced by the pellet number density and porosity, both of which have distinct effects on temperature distribution patterns. Moghaddam et al. [12] investigated the impact of various particle geometries, including spheres, rings, and cylinders, on the wall-to-bed heat transfer. They found that the effective radial thermal conductivity was higher for the packings of cylinders and Raschig rings compared to the packings of spheres.

The experimental evaluation of particle packing structures and heat transfer under high-temperature conditions presents significant difficulties [13]. Given these challenges, a numerical analysis emerges as a suitable approach for exploring the relationship between packing structure and heat transfer within granular materials. Few numerical methods can generate the compacts of non-spherical particles and assess their morphological, mechanical, and thermal properties. Finite Element Method (FEM) simulations, while capable of analyzing complex microstructural details and particle shape influences on heat transfer at small scales, often require fine discretization, leading to substantial computational costs, especially when dealing with large-scale granular assemblies. Additionally, FEM’s sensitivity to mesh quality, boundary conditions, and limitations in modeling discrete systems makes particle-based methods like discrete element method (DEM) more suitable for simulating granular assemblies [14]. DEM, introduced by Cundall and Strack [15], is particularly well-suited for simulating the dense packing and intricate microstructure of granular assemblies [16]. DEM models are primarily based on contact interactions between particles, typically derived from the geometric overlap of two particles [17]. The heat transfer between particles is generally represented as solid-to-solid conduction [18]. Consequently, DEM emerges as a robust methodology for simulating both the dynamic behavior of individual particles and the associated heat transfer phenomena within granular systems. For example, Yarrington et al. [19] conducted a comprehensive DEM study on the impact of elevated temperatures on particle flow behavior. While their work acknowledged the effect of temperature on granular flow dynamics, the study’s limitations, including the use of spherical particles and simplified thermal modeling, precluded a comprehensive understanding of the underlying mechanisms. Thus, the present study aims to investigate the influence of irregular particle shapes on heat transfer in the packed beds by applying DEM simulations.

Recent advancements in DEM simulations have enabled the accurate representation of non-spherical particle geometries through the utilization of superquadric shapes, significantly enhancing the modeling capabilities [20]. Wei et al. [21] employed this approach to study the packing and heating of oblate and prolate ellipsoidal particles, while Korkerd et al. [22] investigated the thermal behavior of non-spherical biomass particles in a bubbling fluidized bed. However, these studies primarily focused on convective particle–fluid heat transfer and paid less attention to the conductive heat transfer between non-spherical particles, which is crucial for CSP applications.

Only a limited number of studies have explored the influence of non-spherical particle shape on compact packing structures, which directly impact inter-particle heat transfer. Studies focusing on ellipsoids, cylinders, and tetrahedra have highlighted the correlation between particle shape, packing density, and inter-particle contacts [23,24,25]. Zou and Yu [23] verified that the packing structure of non-spherical particles differs significantly from that of spherical particles. Elongated particles, such as ellipsoids, have been shown to achieve higher packing densities in random arrangements compared to the optimal packing of spheres (face-centered cubic) [24]. The influence of particle shape complexity on packing structure is further evidenced by the work of Zhao et al. [25], who demonstrated the impact of particle eccentricity on coordination number distribution for tetrahedral particles. Furthermore, Gerhardter et al. [26] found that approximating non-spherical particles as spherical in the calculation of particle temperature leads to significant discrepancies compared to experimental data, emphasizing the necessity for the accurate numerical modeling of non-spherical particles. Therefore, it is essential to study the packing structure properties of non-spherical particles and compare them with spherical ones to understand their relationship to heat transfer behavior, particularly in terms of thermal conductivity.

Recent studies have placed significant emphasis on cylindrical particles to evaluate the effect of their spatial orientation on packing characteristics. For instance, Negotiates et al. [27] and Zhang et al. [28] investigated the distribution of cylindrical particles’ orientation angle, and Zhao et al. [29] evaluated the effect of the aspect ratio of monodisperse cylindrical particles on compaction. However, while the bulk properties of non-spherical particles are widely studied, individual particle-to-particle contacts are often overlooked in the literature. This is despite the fact that the concept of contact area, which is closely related to the type of contact between particles, is commonly used in engineering applications, particularly in heat transfer analysis.

Several studies [30,31,32] have employed the contact identification methodology proposed by Kodam et al. [33] and extended by Guo et al. [34] to characterize cylinder-to-cylinder contact configurations. However, these investigations have not comprehensively explored the correlation between contact type, packing structure, and subsequent heat transfer behavior. This research gap needs to be addressed to gain a comprehensive understanding of heat transfer in systems involving cylindrical particles.

The objectives of the present study are to investigate the packing structures of spherical and non-spherical particles such as ellipsoids and cylinders and to elucidate their influence on heat transfer characteristics.

The present study is divided into two parts. The first part investigates the effect of particle sphericity on the packing structure characteristics and heat transfer rate of non-spherical particles, specifically ellipsoids and cylinders, compared to spherical particles. The second part focuses on the impact of aspect ratio on the packing and thermal behavior of cylindrical particles, with particular attention to their spatial orientation and contact patterns.

2. Materials and Methods

2.1. DEM Contact Model

The DEM employs a discrete particle approach, wherein the translational motion of each particle is governed by Newton’s second law:

$$m{\ddot{\mathbf{X}}}_C=\mathbf{F}\left({\dot{\mathbf{X}}}_C,{\mathbf{X}}_C,t\right),$$

(1)

$${\dot{\mathbf{X}}}_C=\mathbf{U},$$

(2)

where $m$ is the particle mass, ${\mathbf{X}}_C$ is the particle center position, ${\dot{\mathbf{X}}}_C$ and ${\ddot{\mathbf{X}}}_C$ are the first and second derivatives denoting velocity and acceleration, respectively, $t$ is the time, $\mathbf{U}$ is the particle velocity and $\mathbf{F}$ is the total force acting on the particle.

The rotational particle motion is described by the following equation:

$$\dot{\mathbf{L}}=\mathrm{T},$$

(3)

where $\dot{\mathbf{L}}=\mathbf{I}·\mathbf{\Omega }$ represents the angular momentum of particle $i$, $\mathbf{I}$ relates to the tensor of inertia, and $\mathrm{T}$ denotes the total torque acting on particle $i$ with respect to its gravity center.

The force $\mathbf{F}$ acting on particle i results from a combination of contact forces and external forces such as gravity:

$$\mathbf{F}=\sum _{j\ne i}{\mathbf{F}}_{ij}+{\mathbf{F}}_{ext},$$

(4)

where ${\mathbf{F}}_{ij}$ is the contact force between particles $i$ and $j$, and ${\mathbf{F}}_{ext}$ is the external force. The contact model adopts the soft-particle method, allowing particles to slightly overlap each other. The contact force, ${\mathbf{F}}_{ij}$, is evaluated based on the overlapping distance, ${\delta }_{ij}^n$, defined as

$${\delta }_{ij}^n=R_i+R_j-‖{\mathbf{X}}_{Ci}-{\mathbf{X}}_{Cj}‖,$$

(5)

where $R_i$ and $R_j$ are the vectors from the centers of particles $i$ and $j$ to the point of contact, respectively.

The contact force consists of two components: the normal force, ${\mathbf{F}}_{ij}^n$, and the tangential force, ${\mathbf{F}}_{ij}^t$:

$${\mathbf{F}}_{ij}={\mathbf{F}}_{ij}^n+{\mathbf{F}}_{ij}^t,$$

(6)

The normal component of the contact force influences translational particle motion, while the tangential component induces rotational motion and generates torque. Both normal and tangential contact forces are modeled as a combination of repulsive (spring) and dissipative (damping) components:

$${\mathbf{F}}_{ij}^n=k_{ij}^n{\mathit{\delta }}_{ij}^n-{\gamma }_{ij}^n{\mathbf{U}}_{ij}^n,$$

(7)

$${\mathbf{F}}_{ij}^t=k_{ij}^t{\mathit{\delta }}_{ij}^t-{\gamma }_{ij}^t{\mathbf{U}}_{ij}^t,$$

(8)

where $k_{ij}^n$ and $k_{ij}^t$ are the elastic coefficients, while ${\gamma }_{ij}^n$ and ${\gamma }_{ij}^t$ are the dissipative coefficients of normal and tangential forces, respectively. ${\mathbf{U}}_{ij}^n$ and ${\mathbf{U}}_{ij}^t$ denote the normal and tangential components of relative velocity, respectively, and ${\mathit{\delta }}_{ij}^n$ and ${\mathit{\delta }}_{ij}^t$ represent the normal and the tangential overlaps, respectively.

The Hertz–Mindlin contact force model is employed to calculate contact forces between particles. The normal force is described by Hertz’s theory [35], while the tangential force follows Mindlin’s theory [36]. The elastic and dissipative coefficients between pairs of particles $i$ and $j$ are represented as follows [37]:

$$k_n={\displaystyle \frac{4}{3}}{Y}^{*}\sqrt{R^{*}{\delta }_n},$$

(9)

$${\gamma }_n=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{S_n{m}^{*}}\ge 0,$$

(10)

$$k_t={8G}^{*}\sqrt{R^{*}{\delta }_n},$$

(11)

$${\gamma }_t=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{S_t{m}^{*}}\ge 0.$$

(12)

Here, $m^{*}$ and $R^{*}$ denote the effective mass and radius, respectively, and they can be calculated as

$$R^{*}={\displaystyle \frac{R_i{R}_j}{R_i{+R}_j}},$$

(13)

$$m^{*}={\displaystyle \frac{m_i{m}_j}{m_i{+m}_j}},$$

(14)

The components $\beta$, $S_n$ and $S_t$ are estimated as

$$\beta ={\displaystyle \frac{\ln \left(e\right)}{\sqrt{{ln}^2\left(e\right)+{\pi }^2}}}\le 0,$$

(15)

$$S_n={2Y}^{*}\sqrt{R^{*}{\delta }_n},$$

(16)

$$S_t={8G}^{*}\sqrt{R^{*}{\delta }_n},$$

(17)

where $e$ is the restitution coefficient, $Y^{*}$ is the effective Young’s modulus, and $G^{*}$ is the effective shear modulus. These properties depend on Poisson’s coefficients of particles $i$ and $j$, and ${\nu }_i$ and ${\nu }_j$, respectively:

$$G^{*}={\left({\displaystyle \frac{1-{\nu }_i^2}{G_i}}+{\displaystyle \frac{1-{\nu }_j^2}{G_j}}\right)}^{-1},$$

(18)

$$Y^{*}={\left({\displaystyle \frac{1-{\nu }_i^2}{Y_i}}+{\displaystyle \frac{1-{\nu }_j^2}{Y_j}}\right)}^{-1}.$$

(19)

2.2. DEM Contact Detection for Superquadrics

A wide range of non-spherical shapes, including ellipsoids, cuboids, spheroids, and cylinders, can be accurately approximated using a superquadric function defined by five parameters [38]:

$$f\left(x,y,z\right)={\left({\left|{\displaystyle \frac{x}{a}}\right|}^{n_2}+{\left|{\displaystyle \frac{y}{b}}\right|}^{n_2}\right)}^{{\displaystyle \frac{n_1}{n_2}}}+{\left|{\displaystyle \frac{z}{c}}\right|}^{n_1}-1=0,$$

(20)

where $n_1$ and $n_2$ represent the blockiness indices referring to shape sharpness, and $a, b$, and $c$ denote the particle half-lengths along $x, y$, and $z$ axes, respectively. Spherical and ellipsoidal shapes can be generated using equal blockiness indices, while setting $n_1>2$ and $n_2=2$ will extend the shape of the particle to cylindrical. If $f\left(x,y,z\right)$ is positive, the point is located inside the particle; if $f\left(x,y,z\right)$ is negative, the point is outside the particle; and if $f\left(x,y,z\right)=0$, the point lies on the particle surface.

Determining the contact force and point of contact for non-spherical particles is complicated due to their spatial orientation. The accurate determination of a particle’s orientation is essential. The spatial orientation of non-spherical particles is typically represented by the rotation of the particle’s coordinate vectors ${\mathit{E}}_x={(\mathrm{1,0},0)}^T, {\mathit{E}}_y={(\mathrm{0,1},0)}^T$, and ${\mathit{E}}_z={(\mathrm{0,0},1)}^T$ from the global reference frame to the corresponding coordinate vectors $\{{\widehat{\mathit{E}}}_x,{\widehat{\mathit{E}}}_y,{\widehat{\mathit{E}}}_z\}$ in the local reference frame [37]. This rotation can be evaluated using quaternions $\mathit{q}$ based on Euler angles and can be constructed from the unit axis, $\mathit{e}$, and the rotation angle, $\alpha$:

$$\mathit{q}={\left(q_0,q_1,q_2,q_3\right)}^T=\cos \left(\alpha /2\right)+\mathit{e}\sin \left(\alpha /2\right),$$

(21)

The rotation matrix is defined by $\mathit{Q}·{\mathit{E}}_x={\widehat{\mathit{E}}}_x,\mathit{Q}·{\mathit{E}}_y={\widehat{\mathit{E}}}_y,\mathit{Q}·{\mathit{E}}_z={\widehat{\mathit{E}}}_z$ and $\mathit{Q}=\mathit{Q}\left(\mathit{q}\right)$ is based on the quaternions:

$$\mathit{Q}=\left(\begin{array}{ccc}1-2\left(q_2^2+q_3^2\right)& 2\left(q_1{q}_2-q_0{q}_3\right)& 2\left(q_1{q}_3+q_0{q}_2\right)\\ 2\left(q_1{q}_2+q_0{q}_3\right)& 1-2\left(q_1^2+q_3^2\right)& 2\left(q_2{q}_3-q_0{q}_1\right)\\ 2\left(q_1{q}_3-q_0{q}_2\right)& 2\left(q_2{q}_3+q_0{q}_1\right)& 1-2\left(q_1^2+q_2^2\right)\end{array}\right),$$

(22)

Particle orientation is updated at each time step using the following expression:

$$\dot{\mathit{q}}={\displaystyle \frac{1}{2}}\mathit{q} ⃘\mathit{\omega }=\left\{\begin{array}{c}{\dot{q}}_0={\displaystyle \frac{1}{2}}\left(-q_1{\omega }_x-q_2{\omega }_y-q_3{\omega }_z\right)\\ \begin{array}{c}{\dot{q}}_1={\displaystyle \frac{1}{2}}\left(q_0{\omega }_x+q_2{\omega }_z-q_3{\omega }_y\right)\\ \begin{array}{c}{\dot{q}}_2={\displaystyle \frac{1}{2}}\left(q_0{\omega }_y+q_3{\omega }_x-q_1{\omega }_z\right)\\ {\dot{q}}_3={\displaystyle \frac{1}{2}}\left(q_0{\omega }_z+q_1{\omega }_y-q_2{\omega }_x\right)\end{array}\end{array}\end{array}\right.,$$

(23)

where $\mathit{\omega }={\left({\omega }_x,{\omega }_y,{\omega }_z\right)}^T={\mathit{Q}}^{-1}\mathrm{\Omega }$ is the angular velocity in the local reference frame and $\mathrm{\Omega }$ is the angular velocity in the global reference frame.

The translational and rotational operations are usually used to define particle orientation in a global reference frame. The shape function $F$ of a superquadric particle located at the ${\mathit{X}}_C$ point is expressed in the global reference frame by applying the following rotation matrix:

$$F\left(\mathit{X}\right)=f\left({\mathit{Q}}^T·\left(\mathit{X}-{\mathit{X}}_C\right)\right).$$

(24)

The contact detection algorithm utilizes the gradient and Hessian matrix of the shape function to identify contact points. The gradient of the shape function calculated at a point $\mathit{x}={\left(x,y,z\right)}^T$ in the local frame is expressed as

$$\left\{\begin{array}{c}{f}_x^{\prime }={\displaystyle \frac{n_1}{a}}{\left|{\displaystyle \frac{x}{a}}\right|}^{n_2-1}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-1}sign\left(x\right)\\ f_y^{\prime }={\displaystyle \frac{n_1}{b}}{\left|{\displaystyle \frac{y}{b}}\right|}^{n_2-1}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-1}sign\left(y\right)\\ f_z^{\prime }={\displaystyle \frac{n_1}{c}}{\left|{\displaystyle \frac{z}{c}}\right|}^{n_1-1}sign\left(z\right)\end{array}\right.,$$

(25)

The Hessian matrix is defined as

$$\left\{\begin{array}{c}{f}_{xx}^{″}={\displaystyle \frac{1}{a^2}}\left(n_1\left(n_2-1\right)\right){\left|{\displaystyle \frac{x}{a}}\right|}^{n_2-2}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-1}+{\displaystyle \frac{1}{a^2}}\left(n_1-n_2\right)n_1{\left|{\displaystyle \frac{x}{a}}\right|}^{2n_2-2}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-2}\\ f_{yy}^{″}={\displaystyle \frac{1}{b^2}}\left(n_1\left(n_2-1\right)\right){\left|{\displaystyle \frac{y}{b}}\right|}^{n_2-2}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-1}+{\displaystyle \frac{1}{b^2}}\left(n_1-n_2\right)n_1{\left|{\displaystyle \frac{y}{b}}\right|}^{2n_2-2}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-2}\\ f_{xy}^{″}={\displaystyle \frac{1}{ab}}\left(n_1-n_2\right)n_1{\left|{\displaystyle \frac{x}{a}}\right|}^{n_2-1}{\left|{\displaystyle \frac{y}{b}}\right|}^{n_2-1}{\nu }^{{\displaystyle \frac{n_1}{n_2}}-2}sign\left(xy\right)\\ f_{zz}^{″}={\displaystyle \frac{1}{c^2}}\left(n_1\left(n_1-1\right)\right){\left|{\displaystyle \frac{z}{c}}\right|}^{n_1-2}\\ f_{xy}^{″}=f_{yx}^{″}, f_{yz}^{″}=f_{zy}^{″}=f_{xz}^{″}=f_{zx}^{″}=0 \end{array}\right.,$$

(26)

where the parameter $\nu$ is calculated as $\nu ={\left|{\displaystyle \frac{x}{a}}\right|}^{n_2}+{\left|{\displaystyle \frac{y}{b}}\right|}^{n_2}$.

2.3. Material Properties and Particle Geometry Characteristics

The material characteristics and particle density are based on the data from Baumann and Zunft [4] for sintered bauxite, as shown in

Table 1. Material properties and DEM settings.

	Units	Value
Material density	$\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$	2.780
Young’s modulus	$\mathrm{M}\mathrm{P}\mathrm{a}$	10
Poisson ratio	-	$0.3$
Thermal capacity	$\mathrm{J}/\mathrm{K}$	1275
Bulk thermal conductivity	$\mathrm{W}/\mathrm{m}\mathrm{K}$	200
Time step	s	$1\times 10^{-5}$
Gravity	$\mathrm{m}/{\mathrm{s}}^2$	9.81

. These parameters are similar to those used in our previous work [39]. The superquadric shape approximated using Equation (20) was applied to form equal-volume spherical, ellipsoidal, and cylindrical particles (see

Table 2. Geometric characteristics of superquadric particles.

Particle Shape	$\mathbf{Length}, \mathbf{m}\mathbf{m}$	$\mathbf{Diameter}, \mathbf{m}\mathbf{m}$	$\mathbf{Sphericity} \left(\mathit{\psi }\right)$	AR	n1	n2
Spherical	3.434	3.434	1.000	1.000	2	2
Ellipsoidal	4.500	3.000	0.974	1.500	2	2
Cylindrical 1	3.000	3.000	0.874	1.000	8	2
Cylindrical 2	3.931	2.621	0.859	1.500	8	2
Cylindrical 3	4.762	2.381	0.832	2.000	8	2
Cylindrical 4	5.526	2.210	0.805	2.500	8	2
Cylindrical 5	6.240	2.080	0.779	3.000	8	2

). The aspect ratio $AR$ is defined as the ratio of length $L$ to diameter $D$ of particle $AR={\displaystyle \frac{L}{D}}$, while sphericity is calculated using the equation developed by Waddel [40]:

$$\psi ={\displaystyle \frac{{\pi }^{1/3}{\left(6V_p\right)}^{2/3}}{A_p}},$$

(27)

where $V_p$ is the particle volume and $A_p$ is the surface area of the particle.

2.4. Modeling of Packed Bed and Heat Conduction

Aspherix-GUI (6.0.0) software [41] was employed to generate particle shapes such as spherical, ellipsoidal and cylindrical, packing of particles and heating of packed beds of particles. The first part of the study focused on investigating the effects of particle sphericity and comparing non-spherical particles to spherical ones. For this purpose, samples of spherical, ellipsoidal, and cylindrical particles with an AR of 1 were used. The second part of the study investigated the effects of varying aspect ratios of cylindrical particles. Both packing and heat transfer in packed bed simulations were performed for each part of the study.

The particle assemblies were packed under gravity applied along the z-axis. The simulation domain was defined as a 700 mm × 700 mm × 1500 mm box containing 8000 particles. To generate the powder bed, a block region was established at the top of the container. Within this region, particles were randomly oriented and positioned. Subsequently, these particles were allowed to fall freely under the influence of gravity into the container. This simulation setup closely replicates the poured packing conditions commonly encountered in industrial applications. Each simulation for each particle type was conducted five times, altering the randomness in particle placement and orientation. The averages of these five simulations were calculated, and the standard deviations were used to assess variability. Error bars were included in the relevant figures to reflect this variability.

The next step involved simulating heat transfer in the packed bed of particles. This study investigated how variations in particle shape, specifically sphericity, and aspect ratio affect the rate of heat transfer in packed beds through the mechanism of heat conduction. The variation of particle temperature with time is described by [42]

$$m_i{c}_{p,i}{\displaystyle \frac{d{T}_{p,i}}{dt}}=\sum _j{Q}_{ij},$$

(28)

where $m_i$, $c_{p,i}$, and $T_{p,i}$ are the mass, specific thermal capacity, and temperature of particle $i$, respectively, and $Q_{ij}$ is the heat transfer rate between particles $i$ and $j$.

The rate of heat transfer by conduction between particles $i$ and $j$, $Q_{ij}$, is given as

$${\dot{Q}}_{ij}=h_{c,ij}\left(T_{p,j}-T_{p,i}\right),$$

(29)

The heat transfer coefficient, $h_{c,ij}$, is defined as

$$h_{c,ij}={\displaystyle \frac{4k_{pi}{k}_{pj}}{k_{pi}+k_{pj}}} {\left({\displaystyle \frac{A_{contact,ij}}{\pi }}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$

(30)

where $k_{pi}$ and $k_{pj}$ are the thermal conductivities of particles $i$ and $j$, respectively, and $A_{contact,ij}$ is the contact area between particles $i$ and $j$, which depends on the extent of the overlap between particles. In the case of a circular contact area and assuming equal thermal conductivities for the particles i and j ($k_{pi}{=k}_{pj}=k_p$), Equation (30) can be transformed as follows:

$$h_{c,ij}=2·k_p·r_c.$$

(31)

Equation (31) coincides with the formula given in reference [18]. Here, $r_c$ denotes the radius of the circular contact area. The temperature is assumed to be uniform throughout the particle.

The DEM was employed to model the heat transfer phenomena within densely packed beds of particles with spherical, ellipsoidal, and cylindrical morphologies, generated in a preceding stage of simulations. To enhance the computational efficiency, the thermal conductivity of the material was adjusted from 2 $\mathrm{W}/\mathrm{m}\mathrm{K}$ [4] to 200 $\mathrm{W}/\mathrm{m}\mathrm{K}$ across all samples, as summarized in Table 1. Yu et al. [43] confirmed that changing the particle conductivity does not affect the heat transfer results.

A constant temperature heat source of 973 K was applied to the base of the packed bed, while adiabatic boundary conditions were imposed on the lateral and upper surfaces, as shown in Figure 1. The initial particle temperature was set at 298 K.

2.5. Packing Structure Analysis

The packing structure analysis included calculating the Radial Distribution Function (RDF), coordination number (CN), contact area, and force chain network using the DEM packing simulation results. The RDF provides a quantitative measure of particle spatial arrangement by determining the probability of locating a neighboring particle at a specified distance from a reference particle. Peaks in the RDF indicate the preferred inter-particle distances. The CN represents the number of contacting particles for each particle and is obtained from the DEM calculations as well as the contact area for each particle’s pair contact. The force chain network identifies the particles that are in direct contact with mutual compressive forces, along with the directions and magnitudes of these forces [44].

The voxelization method was used to examine the local packing structure of particle compacts, utilizing the superquadric parameters and particle center positions derived from DEM results, following the methodology developed by Pola et al. [45]. It involves discretizing the domain into cubic volume elements (voxels) along the x, y, and z directions, and then identifying whether each voxel lies inside or outside the particle. For this purpose, the voxel center coordinates were transformed into the particle’s local reference frame to enable a subsequent analysis using the superquadric shape equation.

2.6. Heat Transfer Evaluation

The heat transfer rate was assessed by monitoring the temporal evolution of the average layer temperature within three 15 mm thick sections of the packed bed (see Figure 1). In packed beds, heat transfer efficiency is strongly influenced by packing morphology, which varies along the bed height. To better understand these variations and their impact on heat transfer, we focused on the average layer temperature and its correlation with the total contact area across the layers. This approach provided deeper insights into how the morphological changes affected the heat transfer efficiency, which may be overlooked when only considering overall heat transfer.

The packed bed was divided into five layers to investigate the variation of the total contact area with bed height. To minimize wall effects, the analysis was confined to the central region of the packed bed, excluding the particle layers of appropriate thickness from each side of the box. The system comprised five distinct layers, based on the characteristic particle length, $L_c$, which is the average length of all the particle types investigated in the present study: namely, spherical, ellipsoidal, and cylindrical particles with different ARs. As an example, Layer 1 was defined as follows:

$$z \in \left(2\times L_c, 3.6\times L_c\right),$$

(32)

where z denotes the particle position along the z-axis and $L_c=4.5 \mathrm{m}\mathrm{m}$.

Figure 2 illustrates a schematic representation of the five-layer packed bed model employed in this study. Particles excluded from the analysis to mitigate wall effects are depicted as light orange cylinders. The red dashed line denotes the boundaries of the analyzed bed region.

In the present study, the particle orientation angle was defined as the angle between the global reference frame’s z-axis, $z_G$, and the particle’s local z-axis, $z_{cyl}$ (see Figure 3a). To identify the contact types between cylindrical particles, the shape of a cylindrical particle was defined by three sides: the face, edge, and band (see Figure 3b).

The contact detection criteria for main contact types between cylinders, such as face–face, face–edge, face–band, edge–band, edge–edge, and band–band (skewed and parallel), were implemented based on the method described by Kodam et al. [33]. Additionally, a specialized contact model developed by Guo et al. [34] was incorporated to account for the contacts characterized by a greater axial overlap than face–edge contacts and greater radial overlap than band–edge contacts, resulting in larger contact forces and a correspondingly larger contact area.

3. Results & Discussion

3.1. Comparison of Non-Spherical and Spherical Particles by Particle Sphericity

3.1.1. Packing Structure Analysis

The packing structure of packed beds of spherical and non-spherical particles, such as ellipsoids and cylinders with AR = 1.0, were analyzed in our previous work [39]. The CN analysis revealed that non-spherical particles exhibited higher coordination numbers than their spherical counterparts. This observation was attributed to the irregular shapes and diverse spatial orientations of the non-spherical particles. Furthermore, a correlation was established between lower particle sphericity and increased packing density, as also evidenced by the RDF analysis. The RDF analysis revealed distinct structural characteristics corresponding to different particle shapes. The spherical particles exhibited predominantly short-range order, the ellipsoidal particles showed an absence of noticeable structural order at both short- and long-range scales, while the cylindrical particles demonstrated both short-range and long-range order. The evaluation of local packing fraction by the voxelization method demonstrated that the non-spherical particles were less affected by wall effects than the spherical particles and exhibited a more uniform distribution throughout the bed, leading to denser packed beds. These findings provide a foundational basis for the present study and highlight the importance of exploring the impact of non-spherical particle packing structures on heat transfer in packed beds.

3.1.2. Heat Transfer Evaluation

Figure 4 presents the temperature distribution within packed beds composed of spherical, ellipsoidal, and cylindrical particles at heating times of 1 and 5 s. The impact of non-spherical particle irregularities (with sphericity values of 0.974 and AR = 1.5 for the ellipsoids and 0.874 and AR = 1.0 for the cylinders, which are slightly different from the ideal spherical particles) became pronounced during heating over time.

The analysis of the temporal evolution of the average layer temperature indicated that the most significant temperature changes occurred within the bottom layer (Layer 1) for all the particle types. The non-spherical particles exhibited distinct temperature distribution patterns compared to the spherical particles, even within this layer, suggesting enhanced heat transfer rates (Figure 5a). The heat transfer rate in the spherical particle compacts decreased with height, resulting in longer heating times compared to the non-spherical particle samples. Despite the lower particle sphericity of the cylindrical particles, their heat transfer rate was competitive with that of the ellipsoidal particles. To gain a more detailed understanding of the impact of the particle sphericity of non-spherical particles on heat transfer, additional analyses considering factors such as the contact area, particle orientation angle, and force chain network should be considered.

Figure 6 clearly shows that the total contact area of the spherical particles was significantly lower in all the layers compared to the non-spherical particles. The reduced sphericity of the non-spherical particles led to a substantial increase in the contact area, exceeding that of the spherical particles by more than 100 times. The spherical particles exhibited the least variation in the total contact area across the bed height, aligning with previous findings [46]. Conversely, all the particle shapes demonstrated a consistent decrease in the total contact area from the bed bottom to the top, suggesting a reduction in particle–particle contact forces with increasing bed height. This observation underscores the need for further investigation into the contact force distribution within the packed bed.

The force chain was constructed for the packed beds of spherical, ellipsoidal, and cylindrical particles, as shown in Figure 7. The force chain associated with the spherical particles exhibited greater overall strength compared to the non-spherical particles, as indicated by the magnitude of the contact force. However, the force chain did not directly relate to the contact area. The authors of another study [47] demonstrated that the magnitude of the elastic force is not correlated with the contact area in non-conservative systems involving both spherical and non-spherical particles. This lack of correlation was also evident in our findings. While the spherical particles exhibited stronger force chains, the ellipsoidal and cylindrical particles demonstrated significantly larger average contact areas, despite experiencing lower force magnitudes.

The heat transfer within packed beds is influenced not only by contact area but also by particle orientation, resulting in anisotropic thermal behavior, as reported by Govender et al. [46]. Our findings showed that the mechanical interlocking of the non-spherical particles in densely packed arrangements led to a wide range of particle orientations, as illustrated in Figure 8. The unimodal distribution of the particle orientation angle for the ellipsoids exhibited a predominant peak, centered at approximately 90 degrees, indicating a preferential alignment perpendicular to the heat flux due to their elongated shape. In contrast, the cylindrical particles with an AR = 1 displayed a range of orientation angles, with three distinct peaks at 60°, 100°, and 130°, consistent with the observations of Gan et al. [32]. The observed particle orientation trends can be attributed to the poured packing method employed in our DEM simulations. This method, characterized by the free-fall of particles from the top of the container, resulted in the particles adopting the most stable orientations upon contact with the existing bed. The random and disordered particle arrangement in the generated bed closely resembled the packing structures of the moving beds encountered in CSP applications. The multimodal distribution of the cylinder orientation angles suggests the need for further investigation into the types of contacts between cylinders to better understand particle orientation and its impact on heat transfer.

3.2. Comparison of Cylindrical Particles with Different Aspect Ratios

3.2.1. Packing Structure Analysis

Figure 9a illustrates the coordination number distributions for cylindrical particles of various aspect ratios. As the AR increased, the CN distribution broadened, indicating a greater variability in the number of contacting neighbors. Additionally, a rightward shift in the distribution was observed, signifying an increase in particles with higher CNs. This trend was particularly pronounced for the AR values of 2.5 and 3.0, suggesting a potential enhancement in heat transfer due to increased contact points. Figure 9b illustrates the variation of the average coordination number with the aspect ratio for cylindrical particles. As the aspect ratio increased, the average CN rose steadily, indicating a greater number of contact points between particles. This increase in the coordination number suggests enhanced particle interactions, which play a crucial role in improving heat transfer efficiency within the packed bed. Although CN is a crucial parameter in understanding heat transfer efficiency, a comprehensive analysis of packing structure should also consider the particle orientations and contact interactions, which contribute significantly to the overall heat transfer.

Figure 10 presents the RDF diagrams, featuring a primary peak centered at approximately a one-cylinder diameter for all the particle samples. As the AR increased, the distance between the first and second RDF peaks widened. These findings corroborate previous studies [30,31], indicating a more uniform short-range particle orientation for lower-AR samples and a pronounced long-range order for higher-AR samples.

For the cylindrical particles with aspect ratios ranging from 1.0 to 3.0, the first RDF peak primarily corresponded to band–band contacts (both parallel and skewed configurations). In the specific case of AR = 1, a detailed analysis revealed that the first RDF peak encompassed both face–band and parallel band–band contacts (Figure 11a,b), while the second peak predominantly corresponded to band–edge contacts (Figure 11c), with a smaller contribution from face–edge contacts (Figure 11d).

As the distance from the center of the particle increased, the RDF showed a significant drop in intensity, with no peaks eventually occurring, which signifies the absence of a long-range structure. Thus, based on the RDFs for all the packed beds of cylinders, there were instances of local packing structures being characterized by the prevalence of certain contact types. Although the RDF offers insights into the packing structure, it does not directly correlate with packing density. Therefore, a local packing fraction analysis is required for a more accurate approximation of packing density.

Figure 12 illustrates the local packing fraction of samples across the x, y, and z planes. In the horizontal x and y planes, a consistent trend was observed: higher ARs generally corresponded to lower packing fractions, except for the highest AR, which showed a significant increase in packing fraction. This phenomenon is attributed to the elongated shape of the longest cylinders, facilitating the better filling of free space and the formation of ordered particle clusters [48]. The analysis of the packing fraction distribution in the z plane revealed a correlation between particle heap formation and packing density (see Figure 12c). While the AR = 3.0 sample exhibited the highest heap, indicating a potentially lower packing density in the vertical direction, the AR = 2.5 sample demonstrated the densest packing and the least heap formation, suggesting a more complex relationship between particle shape and packing behavior in the vertical dimension.

3.2.2. Heat Transfer Evaluation

The observed packing fraction trends are expected to have a significant impact on heat transfer rates. Figure 13 depicts the temperature distribution throughout the beds of cylindrical particles over time. Applying a heat source from the bottom of the packed bed resulted in a gradual increase in the particle temperature along the vertical z-direction. The cylindrical particles with higher ARs exhibited enhanced heat transfer rates, as also evidenced by the temporal variations in the average layer temperature, depicted in Figure 14. The first layer demonstrated a distinct temperature behavior compared to the second and third layers. During the initial 20-second period, particles with lower ARs heated up more rapidly than those with higher ARs, with the exception of AR = 3. However, this trend reversed subsequently, as particles with higher ARs began to heat up faster, aligning with the behavior observed in the higher layers. The observed phenomenon can be attributed to the wall effect on heat transfer rates, given that the first layer was closest to the heat source and the particles were in direct contact with the wall.

The temperature distribution within the bed of cylinders exhibited greater uniformity for AR = 2.5 compared to AR = 3.0 within the 300–400 K range (Figure 13b). This observation suggests that a higher packing density can enhance heat transfer in the bed of cylindrical particles. To gain deeper insights into the effect of packing structure morphology on anisotropic heat transfer in non-spherical particle beds, an analysis of particle orientation trends is further presented.

The heat transfer rate was influenced by both heat flux and thermal contact resistance. The heat flux was conducted vertically, with the thermal resistance primarily governed by the contact resistance between particles. Figure 15 illustrates the distribution of the particle orientation angles for the packed beds of cylindrical particles with various ARs. Notably, the cylindrical particles showed an increasing tendency to align perpendicularly to the vertical axis as their AR increased. A greater deviation of particles from the vertical direction led to an increased thermal contact resistance. However, our results indicate that despite the increased thermal contact resistance of cylinders with higher ARs, they exhibited greater heat transfer rates (see Figure 14). This discrepancy suggests that another critical parameter, such as contact area, needs to be assessed to fully understand the heat transfer mechanism in packed beds of non-spherical particles.

Figure 16 demonstrates a positive correlation between the total contact area and the aspect ratio for cylindrical particles, attributed to both the increased particle surface area and the higher coordination numbers at larger AR values. Notably, a consistent decrease in the total contact area was observed from the bottom to the top of the packed bed for all the cylindrical samples.

The particles generated using the superquadric approach inherently possessed edges, which play a critical role in heat transfer by conduction. As the aspect ratio of these cylindrical particles increased, the contact scenarios shifted from edge contacts, characterized by the smallest median contact areas (see

Table 3. Median contact area for each contact type for cylindrical particles with different AR.

	Area per Contact, mm2
Contact Types	AR = 1.0	AR = 1.5	AR = 2.0	AR = 2.5	AR = 3.0
Band–band skewed	6.76 ± 0.01	11.40 ± 0.02	16.43 ± 0.03	21.15 ± 1.70	27.82 ± 0.02
Band–edge	6.13 ± 0.02	9.22 ± 0.06	14.28 ± 0.04	19.68 ± 0.07	25.24 ± 0.05
Band–band parallel	7.04 ± 0.01	11.95 ± 0.02	17.40 ± 0.05	23.18 ± 0.05	29.78 ± 0.10
Edge–edge	5.48 ± 0.08	8.11 ± 0.04	11.91 ± 0.09	16.47 ± 0.09	21.11 ± 0.12
Special	5.00 ± 0.04	7.84 ± 0.02	11.29 ± 0.06	15.08 ± 0.14	19.56 ± 0.55
Face–edge	6.09 ± 0.02	5.61 ± 0.06	6.65 ± 0.12	7.77 ± 0.17	9.14 ± 0.25

) and consequently higher thermal resistance, to band–band contacts, which exhibit larger median contact areas and, therefore, a lower thermal resistance. This demonstrates that the efficiency of heat transfer by conduction is closely related to the median contact area between particles.

A significant increase in the median contact area per contact was observed across all the contact types with the increasing AR of the cylindrical particles (Table 3). Figure 17 highlights the prevalence of band–edge and band–band skewed contacts for the higher AR values. These and other contact types are illustrated in Figure 18. The cylindrical particles with an AR of 3.0 exhibited notably larger median contact areas, reaching 27.82 mm² for band–band skewed and 25.24 mm² for band–edge contacts, representing a fourfold increase compared to AR 1.0 particles. This enhancement is attributed to the combined effects of the increased surface area and a greater tendency toward horizontal alignment (see Figure 15).

The densification mechanism for the elongated cylindrical particles was attributed to their limited degrees of freedom, promoting a preferential horizontal alignment, as seen in Figure 13b. This orientation facilitated the formation of planar linkages dominated by band–band skewed, band–band parallel, and band–edge contacts (see Figure 17). Furthermore, the dominant horizontal alignment, reflected in the narrow orientation angle distribution (Figure 15), influenced the heat transfer efficiency. Although these particles were aligned against the heat flux direction, the increased contact area between the cylinders reduced the thermal resistance. Additionally, the higher number of contact points between particles (as indicated by the arithmetic average coordination number in Figure 9b) enhanced the heat transfer rate. As a result, for the cylinders with higher ARs, the heat transfer rate increased, leading to a higher average layer temperature, as shown in Figure 14.

Conversely, the cylinders with the lowest AR, having an equilateral shape, tended to contact each other with lower coordination numbers and smaller contact areas across all the contact types. The cylindrical particles with an AR of 1.0 exhibited the lowest total and median contact areas compared to other particle geometries, as evidenced in Figure 16 and Table 3, respectively. They also demonstrated significant diversity in their contact types, consistent with the particle orientation angle analysis. The varied contact types and relatively small contact areas associated with some configurations are expected to hinder the overall heat transfer efficiency.

4. Conclusions

Concentrated solar power technology, which relies on thermal energy storage media, benefits from using cost-effective bauxite powder. The non-spherical nature of the powder granular material was assessed in terms of its packing characteristics and heat transfer capability.

The following findings can be drawn:

• Differences in particle sphericity among spherical, ellipsoidal, and cylindrical particles significantly affect the morphology of the packing structure. Cylindrical particles show both short- and long-range structural patterns, while ellipsoidal particles lack these structures.
• The particle sphericity of non-spherical particles notably impacts heat transfer, as observed in the average temperature changes within the packed bed over time.
• Cylinders with higher ARs exhibit increased contact areas and coordination numbers. With the increasing AR, band–band skewed and band–edge contacts become more prevalent, significantly increasing the median contact area.
• Although higher-aspect ratio cylinders predominantly orient horizontally, opposing the heat flux direction, their larger contact areas and the increased number of contact points reduce the thermal resistance, leading to an enhanced heat transfer rate.

In conclusion, this comprehensive analysis provides insights into the intricate packing structures and contact interactions within the non-spherical particle beds, facilitating a deeper understanding of their impact on heat transfer phenomena. These findings contribute to optimizing packing structures in powder-based thermal energy storage media for CSP applications.