# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 3 Simulation of Aggregate Formation  
In the present work, the effective thermal conductivity of biphasic systems was calculated. Particularly, heat transport in particle systems and granular porous media was simulated. All the structures encompass a number of spheres within a cubic cavity. Algorithms based on Monte Carlo techniques were used to represent the particulate systems and granular porous media. Equally sized spheres were assumed, and the structures were periodic in every direction. The algorithm allowed random placement of the spheres over the working domain. The first, simple, case that was considered was that of granular media that were made up of spheres with no contact between them [26]. An example of a granular medium with porosity 0.7 is presented in Figure 3.  
In addition, structures with aggregates of particles were simulated. In this case, every particle was in contact with at least another one. One of the determining parameters for the morphology of particle aggregates is their fractal dimension (\(d_{f}\)) [27; 28]. For an aggregate consisting of particles, the fractal dimension (\(d_{f}\)) relates the number of particles in the aggregate with its gyroscopic radius (\(R_{g}\)), through: \(\sim\left(R_{g}/r\right)^{d_{f}}\), where \(r\) is the radius of the particles. As a basic rule, one expects that the higher the fractal dimension, the more cohesive the aggregate.  
For the representation of the aggregates, the well-known DLA [29] and the BA method [30] were used here, as well as modifications of these models, to represent denser aggregates. The DLA model describes the appropriate phenomena for the formation of fractal structures [29]. An important parameter of this method is the mean free path of the Brownian diffusion of particles [30]. If the mean free path is smaller than the size of the aggregate, the morphological characteristics of the clusters are the same and the fractal dimension is approximately \(d_{f}\sim 1.9\). In the extreme case that the mean free path remains always larger than the aggregate size, the BA model is defined. The resulting aggregates are denser than those of the DLA model [30; 31]. In these cases, every particle was assumed to cease its motion as soon as it gets in contact with another one. Extension of these models allows the particle to move along the surface of the cluster in order to find a suitable second contact point (DLA 2p, BA 2p) [30]. This method results in more compact aggregates. In Figure 4, the mean fractal dimension of 50 particles forming a single aggregate is presented along with its standard derivation for the DLA, BA, and their 2p counterpart aggregation methods. A visualization of aggregates resulting from each method is also shown accordingly.  
Figure 2: (**a**) Initial geometry, (**b**) computational grid, and (**c**) integration domains around each node.  
The heat transport equation is solved in each of the aforementioned structures. The heat transfer is caused by the temperature difference imposed on the walls along the vertical axis. The rest of the boundaries are subject to periodic conditions.  
Figure 4: Mean fractal dimension (\(d_{f}\)) of 50 particles forming a single aggregate with the Diffusion Limited Aggregation (DLA) and Ballistic Aggregation (BA) methods, and their two-point contact (2p) counterparts, with their standard deviation, along with visualization of the corresponding aggregates.  
Figure 3: Reconstruction of granular porous medium with solid volume fraction 0.3.