# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 5 Results and Discussion
### Effect of Aggregation on Thermal Conductivity  
The convergence of the MLPG method in aggregate systems is shown in Figure 11 for two distinct structures resulting from convectional DLA and from DLA with two-point contact (DLA 2p) methods. In both methods the particle number was 20 and the volume fraction was \(f_{p}=0.05\), whereas their fractal dimension was 1.9 and 2.5, respectively. The insets of the figures show the percentage error  
Figure 10: Effective thermal conductivity of a granular porous media as a function of the porosity for (**a**) \(\mathrm{k_{r}}=13\), (**b**) \(\mathrm{k_{r}}=317\). (**c**) Effective thermal conductivity of the granular medium as a function of the conductivity ratio. Circles: experimental data, Green lines: simulation results. Blue, Yellow and Red: Maxwell, DEM and SC models, respectively. Abbreviations; SC: self-consistent approximation, DEM: differential effective medium approximation MLPG: meshless local Petrov-Galerkin, exper: experimental data.  
versus the total number of nodes, assuming as base value the conductivity that was obtained for the maximum number of nodes. For conductivity ratio below unity (\(\mathrm{k_{r}}=0.01\)), convergence with errors less than 1% was achievable with coarse grids, having a few thousand nodes (\(\sim\)30 k) only, Figure 11a. In contrast, the cases of high thermal conductivity of the aggregates compared to that of the bulk (\(\mathrm{k_{r}}=100\)) were computationally demanding, Figure 11b. For coarse grids (30 k nodes) the error was between 10 and 20%, and the convergence depended on the fractal dimension. The lower the fractal dimension, the denser the grid required. For one million (\(10^{6}\)) nodes, the error for a dense aggregate (\(\mathrm{d_{f}}=2.5\)) was below 1%. For the same convergence, the DLA aggregate required 1.7 million nodes. This referred to a grid of 60 nodes per dimension in the bulk, locally refined at 240 nodes near the interfaces. Using this as a reference for grid-independent solutions, the effective conductivity dependence on the fractal dimension for 40 aggregated systems, 10 for each of the methods (DLA, BA, DLA 2p, BA 2p) is depicted in Figure 12 for a wide range of conductivity ratio values. For \(\mathrm{k_{r}}=0.01\) or \(\mathrm{k_{r}}=0.1\) the effective conductivity was practically independent of the fractal dimension of the particles, Figure 12a,b. The divergence of the results was less than 1%. However, when the conductivity of the particles was higher than that of the bulk, aggregation acts positively for the conduction. Indeed, it appears that the smaller the fractal dimension, the higher the thermal conductivity. The increase was small (\(\sim\)1%) for the cases of \(\mathrm{k_{r}}=10\), though for strongly conductive particles (\(\mathrm{k_{r}}=100\)) a notable increase of 10% was observed. Moreover, it is noted that aggregates with the same fractal dimension resulting from different methods practically have the same effective conductivity.  
Figure 11: Effective thermal conductivity as a function of the total number of nodes for two different aggregates. Red lines: \(\mathrm{d_{f}}=2.5\). Black lines: \(\mathrm{d_{f}}=1.9\). (**a**) \(\mathrm{k_{r}}=0.01\). (**b**) \(\mathrm{k_{r}}=100\). Insets: the percentage error. Abbreviation; DLA: diffusion limited aggregation.