## 8 Modeling-Based Optimization of Thermal Management with Graphene-Enhanced PCMs  
Computer simulation of thermal management systems for battery packs provides valuable information for materials and system optimization. We corroborated our experimental results with the numerical solution defined by the heat diffusion equation for transient conductive heat transfer in solids and parametric data of the empirical battery design. These parameters along with the measured specific heat and thermal conductivity properties of the synthesized composites were used to model the conductive heat flow in the Li-ion battery pack using COMSOL's computer simulation software package. The constructed three-dimensional (3D) model facilitated the analysis of the six-cylinder battery pack in different media. The six batteries, 18.4 mm in diameter, were evenly distributed within a medium represented as a solid cylinder, 70 mm in diameter, all of which were enclosed within a 1 mm thick aluminum sheath. Due to the battery pack's simple construction, an extra coarse mesh of free tetrahedrals was used for the solid cylinders, heat conductive medium within which the solid cylinders were encased, and aluminum sheath. The transient conductive heat transfer equation was solved to determine the temperature rise inside and outside of the battery pack [29]. In all simulation runs only the material characteristics of the medium that fill the space among the battery cylinders were modified. The different media included air, conventional paraffin PCM, and graphene-enhanced PCM. For conventional paraffin PCM without graphene we used thermal conductivity \(K=0.25\) W/mK, mass density 900 kg/m\({}^{3}\), and heat capacity 2500 J/kgK. The simulation results included a transient analysis of temperature _vs._ time for any specific point inside the 3D modeled battery pack at any given time. Specific locations of the data analyzed correspond to the physical placement of thermocouples in the empirical test setup. Empirical data from the thermocouples were compared to these simulation results.  
Figure 6 illustrates the Li-ion battery pack used in both empirical test setups and modeling simulations. Included are the simulated temperature profiles of four different cases that correspond to  
actual experimental setups. In the case of air (_i.e._, no PCM) serving as the medium between battery cylinders, the cylinders reach their maximum temperature at 333 K. Upon introduction of conventional paraffin PCM as the medium the maximum temperature of the cylinders drops to 320 K. However, the best performing thermal management of these battery cylinders comes from the use of hybrid graphene-PCM medium with maximum temperatures of 310-315 K resulting also in a more uniform temperature profile. One should note here that substantial reduction of the temperature of the battery cylinders is achieved even at small loading fractions of graphene.  
Figure 6: Schematic of the battery pack and simulated temperature rise. The structure parameters correspond to the experimentally tested batteries. Modeling based optimization of the materials and pack parameters are essential for practical applications of graphene-enhanced PCMs in energy storage technologies.