## 3 Results and Discussion
### Parametric Analysis
#### 3.2.2 Effect of Porosity, Particle Size and Fluid Velocity  
Simulations were performed with distinct porosities, particle diameter, and velocities for all the aspect ratios. The HTF inlet velocity was tested from 0.01 to 0.07 m/s, while the porosity was varied from 0.2 to 0.5. Results from simulations with distinct porosities and aspect ratios are shown in Figure 7a-c for the charging, and Figure 7d-f for the discharging, cycles. It can be seen that the outlet temperature for the air continuously increases during the charging period up to almost 120 \({}^{\circ}\)C, which makes the inlet and outlet temperature differential no more than 1 \({}^{\circ}\)C. It is also visible that when using greater fluid velocities (and consequently larger mass flows), temperature raise is much faster becoming practically independent from the porosity of the bed and the aspect ratio. The same can be said for the discharging cycle, in which the drop in outlet temperature is equally fast. Having in mind the obtained outcomes, it is possible to calculate the total energy stored in the system. The evaluations show that about 540 GJ can be stored on the bed with a porosity of 0.2, and about 350 GJ on the one with 0.35, while the intermediate porosity leads to a total of 450 GJ. The idea is that this heat that was stored can be withdrawn from the TES system in a cold season, i.e., when demand is excessive, and be employed in a wide range of potential applications. Figure 8 shows that, by increasing porosity in the system, a direct decrease in total energy stored is encountered, mainly due to the reduction of mass for a domain with constant total volume. On the other hand, aspect ratio and rock particle size are not seen to have significant effects in the stored/extracted capacity, which is the reason why Figure 8 only shows results for one case of each. It is also worth noting that, even though the total energy stored in the system slightly increases for higher mass flow rates, lower velocities can provide a longer charging/discharging period for the TES system. Finally, it is possible to conclude that the total heat storing capacity of the system is virtually the same for all the distinct aspect ratios and Reynolds tested, which leads us to believe that such capacity is nearly entirely dependent on the mass of the storage media. Figure 6: Velocity profiles for the proposed systems with a porosity of 0.2 and inlet fluid velocity of 0.05 m/s on the domains with aspect ratios of: (a) 1.0, (b) 1.2, and (c) 1.35. Reprinted from Ref. [39]. _Appl. Sci._**2020**, _10_, _7771_  
**Figure 7. Outlet air temperatures (\({}^{\circ}\)C) in charging mode (left column) and discharging (right column) versus time for different aspect ratios and porosities: (a) and (d) \(\varepsilon\) = 0.2; (b) and (e) \(\varepsilon\) = 0.35; and (c) and (f) \(\varepsilon\) = 0.5. Reprinted from Ref. [39]. **To evaluate the impact of these parameters on the pressure drop through the bed, the Reynolds number of the flow was varied from 100 to 750 and, through numerical calculations, the pressure drop across the system was calculated numerically. The results that can be seen in Figure 9 show the development of pressure gradient across the bed with Reynolds number. In addition, Figure 10 shows contours for maximum pressure gradient across the system against aspect ratio and inlet velocity for a constant particle size of 1.0 m. Figure 9 shows that, even though pressure drop increases with velocity for all scenarios, the growth is very subtle for higher porosities, being even an order of magnitude lower than that for a low porosity (and consequently low permeability). The same happens for the slight increase in pressure difference seen in larger aspect ratios, which only becomes significant at lower porosities. Perhaps this effect can be more clearly identified in Figure 10, as it shows the maximum pressure gradient contours growing rapidly with velocity, as opposed to \(\gamma\), but being considerably low except for the lowest porosity.