## 3 Method
### Grid Independence and the Code Validation  
The first stage of computational fluid dynamics (CFD) research, called the grid independence study, is crucial because it assesses the impact of different mesh densities on phase transition events in a certain geometric arrangement. Quadrilateral elements are used to generate the mesh for heat transfer simulations, since this model is a better choice to improve the accuracy and enhance the heat transfer representation. The primary objective of this experiment was to observe the progression of phase change events taking place inside the heat transfer fluid (HTF) under a consistent input flow rate of \(1\ \mathrm{m/s}\). Three different mesh densities, with element counts of 22,701, 24,032, and 26,088, were carefully selected to cover a range of computational resolutions, as shown in Figure 2. After conducting a thorough analysis of the complete grid ensemble, it was determined that the sequence of phase transition remained uniform across all element numbers. This observation demonstrates grid independence, indicating that the calculated results were not significantly affected by the choice of mesh density. The mesh with the lowest element count (22,701) was carefully chosen for subsequent simulations since it had a faster computing time. This choice ensured that computational efficiency was maximized while maintaining accuracy standards. This methodological approach improved the dependability of the computational results and emphasizes the importance of careful grid selection in CFD calculations.  
The liquid fraction is calculated during the simulation using Equation (7), which is applied in each step or iteration of the process. Initially set to zero at the beginning of the simulation, the liquid fraction progressively increases as time advances. This increase signifies the transition of the PCM from a solid to a liquid state. As more heat is absorbed and the PCM reaches its melting point, the liquid fraction approaches and eventually reaches 1, indicating that the PCM has wholly melted and transformed into a liquid state. A rigorous validation method was rigorously carried out to enhance the trustworthiness of the numerical simulation framework. The simulation code was enhanced with additional features, allowing for the simulation of complicated geometric structures and varying liquid amounts. These improvements were made to synchronize the computational framework with the influential research conducted by [PERSON] and [PERSON] [11], which examined the intricacies of fluid flow and heat transport during the phase transition of a PCM in a vertical cylindrical tube. After conducting a thorough comparison analysis, the results produced by the improved simulation code were compared with [PERSON] and [PERSON]'s findings. This comparison showed a significant agreement between the two datasets, as shown in Figure 3. The comparison between the slightly different computing methodologies and model settings showed very small differences, which highlights the correctness and strength of the improved simulation code. Interestingly, the enhanced simulation code was able to achieve a full phase change in just 28 h, which is a shorter duration than the 24 h in [PERSON] and [PERSON]'s work. However, there was a remarkable similarity in the progression of liquid percentage between the two studies. The liquid portion reached an impressive 98% within 24 h in both [PERSON] and [PERSON]'s study and the current simulation, highlighting the accuracy and dependability of the improved numerical framework in accurately representing the complex dynamics of phase transition processes. The validation method served to enhance the credibility of the computational findings and showcases the simulation code's competence in properly replicating real-world occurrences.  
Figure 2: Grid independency test velocity = 1 m/s.