Coupled Heat Transfer and Phase Change in a Porous Nanofluid-Enhanced Cold Thermal Energy Storage System: An Adaptive Mesh-Based Numerical Study

Cold energy storage systems are essential for managing energy supply and demand, especially in applications like refrigeration, air conditioning, and food preservation. Phase change materials (PCMs) are integral to many of these advanced thermal storage solutions, as they can absorb and release significant amounts of latent heat during their freezing and melting processes. This property makes PCMs highly effective for storing cold energy, particularly during off-peak hours when energy costs are lower. However, a major limitation of PCMs is their low thermal conductivity [1], [2], [3]. To tackle this challenge, researchers have investigated the incorporation of nanoparticle-enhanced phase change materials (NEPCMs). By embedding high-conductivity nanoparticles into the PCM matrix, the overall heat transfer capacity is meaningfully boosted. These nanomaterials not only minimize thermal resistance but also promote a more uniform temperature distribution across the storage medium. As a result, the cold energy storage system becomes more responsive and efficient, providing reliable cooling performance even under fluctuating thermal loads [4], [5]. In addition to nanoparticle additives, integrating porous metal foams within the storage container has proven to be highly effective. These highly conductive structures offer extended heat transfer surfaces and create multiple pathways for thermal conduction, significantly accelerating the solidification process. This is particularly important in cold storage systems, where fast and uniform freezing is crucial for preserving the quality and safety of perishable goods. The inclusion of porous foam materials not only enhances the overall thermal conductivity of the system but also reduces temperature gradients across the PCM, ensuring faster response times and more efficient utilization of the stored cold energy [6], [7]. By synergistically integrating NEPCM technology with metallic porous foam structures, cold storage systems can effectively overcome the thermal limitations typically associated with traditional PCMs. This hybrid enhancement approach leads to more efficient, compact, and high-performance thermal storage units, crucial for advancing sustainable energy management in contemporary cooling applications [7], [8], [9]. Lizcano et al. [10] experimentally attached PCM slabs to a solar system, testing the setup under conditions from two different countries. Their results revealed that integrating paraffin into the solar panel led to an increase in electrical performance of approximately 1.6% in Italy and 2.5% in the Netherlands.

Shi et al. [11] examined the incorporation of graphite nanoplatelets and graphene as supporting materials for PCM to overcome leakage problems. They reported that paraffin blended with exfoliated graphite nanoplatelets remained stable only up to 340 K. Díaz et al. [12] investigated paraffin-based passive cooling for panels and found that integrating phase change materials (PCMs) boosted annual electricity generation by 5.8%. Ibrahim et al. [13] integrated various finned containers at the bottom of the panel to enhance the system's performance. Their findings demonstrated that this modified system led to a 3.9% boost in electrical power, highlighting the effectiveness of the finned container in improving system efficiency. Nishad et al. [14] studied PV thermal management with paraffin composites and found they lowered module temperature by 27 K, improving overall efficiency.

Despite significant progress in enhancing phase change material (PCM)-based cold thermal storage systems, several previous articles have focused either on using nanoparticles or on applying metal foams and extended surfaces separately to improve heat transfer. However, the simultaneous integration of ternary hybrid nanoparticles ($\mathrm{Al}_2 \mathrm{O}_3-\mathrm{TiO}_2-\mathrm{Ag}$) with metal foam structures inside wavy containers equipped with rectangular fins remains largely unexplored. This combined approach offers a new pathway to significantly intensify conductive heat transfer, which is the dominant mechanism during solidification. Moreover, few studies have numerically examined such hybrid systems under realistic conditions while accounting for adaptive meshing and minimizing computational complexity by justifiably neglecting the negligible liquid flow during phase change. The current study addresses this research gap by developing a robust numerical model using the Galerkin method with adaptive grid refinement to capture the transient thermal behavior accurately. This study offers new perspectives on optimizing solidification by examining nanoparticle concentration and foam porosity. The results deliver practical guidelines for developing advanced cold storage systems, marking a significant improvement over traditional PCM enhancement methods.

This work investigates the solidification behavior inside a wavy-walled container fitted with rectangular fins, as depicted in Figure 1. Two main strategies were applied to speed up solidification: adding $\mathrm{Al}_2 \mathrm{O}_3-\mathrm{TiO}_2-\mathrm{Ag}$ nanoparticles to the water and embedding metal foam in the container. Both approaches primarily improve thermal conduction, which is the dominant heat transfer mechanism governing the solidification process in this system. Due to the negligible velocity of the liquid phase during solidification, convective effects were considered insignificant and therefore omitted from the simulation model. The resulting mathematical formulation involves two scalar transport equations, which were solved using the Galerkin method combined with an adaptive mesh strategy to enhance numerical accuracy and efficiently capture steep gradients near the solidification front. The study systematically investigates the effects of two critical parameters—namely, the volume fraction of the nanoparticles ($\phi$) and the porosity of the metal foam ($\gamma$)—to understand their influence on the solidification dynamics under various operating conditions. Based on the simplifying assumptions, the final model consists of the following two governing equations [15]:

The thermophysical properties of the nanomaterial, assuming a homogeneous mixture, can be calculated using the following formulas as reported in the literature [16], [17], [18]:

In this study, the governing equations were solved using a custom code developed in FLEX PDE software, which employs the Galerkin method for numerical approximation. One of the key strengths of FLEX PDE is its capability to implement an adaptive mesh refinement technique, allowing the grid to dynamically adjust and concentrate computational elements in regions with steep gradients, such as near the solidification front. Additionally, the software utilizes an unsteady implicit time-stepping scheme with automatic time-step control, ensuring numerical stability and efficiency throughout the transient simulation. The application of FLEX PDE for modeling cold storage systems was first pioneered by Sheikholeslami [15], whose work established the foundational numerical framework used in the current research. This prior study demonstrated the advantages of combining adaptive meshing with implicit discretization methods to accurately capture the complex behaviors of phase change processes, thus validating the approach adopted in the present investigation.

This study presents a detailed numerical simulation of PCM solidification in a wavy container with rectangular fins for thermal optimization. This unique design is specifically tailored to accelerate the freezing process, a critical aspect in cold thermal energy storage systems. To further enhance thermal conductivity, water was enriched with a ternary hybrid nanofluid consisting of aluminum oxide ($\mathrm{Al}_2 \mathrm{O}_3$), titanium dioxide ($\mathrm{TiO}_2$), and silver (Ag) nanoparticles. Additionally, the container was embedded with a high-conductivity metal foam to amplify the heat transfer through conduction, which dominates the solidification mechanism in such configurations. Given the limited movement within the domain during the phase change, the velocity of the liquid phase remains minimal and its contribution to the flow dynamics was deemed negligible, allowing for simplification in the simulation by omitting these terms. The final mathematical model incorporates two scalar transport equations, which were numerically solved. To improve computational accuracy, an adaptive grid refinement technique was employed.

The thermophysical properties of the hybrid nanofluid were estimated using the homogeneous mixture model, offering a realistic approximation of the effective behavior of the ternary nanocomposite. The investigation focused on two critical parameters that influence the solidification performance. Multiple case studies were conducted to understand how these parameters affect the freezing front and overall heat transfer enhancement. This work introduces a novel combination of advanced heat transfer techniques—wavy-finned container design, ternary nanofluids, and porous metal enhancement—to accelerate solidification, which remains a major bottleneck in many thermal energy storage systems. The outcomes of this research contribute significantly to the design of more efficient and compact PCM-based cooling and storage units, addressing an important gap in the current literature related to hybrid-enhanced solidification processes.

Figure 2 illustrates the mesh configuration inside the container. As time progresses, the area of denser mesh shifts to follow the movement of the solidification front. This adaptive refinement is necessary because the region around the solid front experiences steep gradients in temperature and phase change variables, requiring a higher concentration of elements for accurate resolution. The use of an adaptive mesh technique thus ensures enhanced precision in capturing these critical variations during the simulation. Figure 3 presents the validation results, comparing the current simulation data with previously published results [15]. The close agreement between the two datasets confirms the reliability and accuracy of the present numerical model in simulating the cold storage process. This validation provides confidence that the modeling approach is robust and suitable for analyzing phase change and heat transfer phenomena in similar systems.

Figure 4, Figure 5, and Figure 6 illustrate the system’s behavior under three different scenarios, highlighting the effects of the two enhancement techniques. Over time, the solidification front gradually progresses from the cold boundary surfaces toward the adiabatic walls. The shape of the isotherms closely follows the geometry of the container, reflecting the dominant conduction heat transfer mechanism. When ternary nanoparticles are dispersed in the water, the freezing time decreases from 303.76 seconds to 263.85 seconds, primarily due to the increased thermal conductivity of the nanofluid. However, the most significant improvement is observed when metal foam is incorporated into the container, reducing the freezing time drastically to 46.95 seconds. This rapid solidification is attributed to the uniform and highly efficient conduction pathway established by the porous metal foam structure, which significantly accelerates heat transfer throughout the domain. Overall, the presence of metal foam leads to the fastest freezing process by enhancing thermal conduction more effectively than nanoparticles alone.

Figure 7 shows the progression of the solidification front throughout the freezing process. It is evident that the advancement speed of the solid front significantly increases when metal foam is embedded within the container. A similar, albeit less pronounced, enhancement is observed when ternary nanoparticles are dispersed in the base fluid (water). The incorporation of metal foam facilitates faster heat conduction through the domain, accelerating the phase change process and enabling the solidification front to propagate more rapidly. Comparatively, the influence of metal foam on the progression of the solid front is found to be approximately 6.26 times more effective than that of the nanoparticles. This result highlights the dominant role of the porous metallic structure in strengthening the conduction mechanism, thus making it a far more efficient strategy for enhancing freezing performance in cold thermal energy storage systems.

Figure 8 and Figure 9 present the evolution of key scalar quantities—namely, the average solid fraction and temperature—as functions of time during the freezing process. The graphs clearly illustrate a steady increase in the solid fraction over time, indicating the progressive solidification of the phase change material. In contrast, the temperature profile shows a declining trend, which is expected as heat is continuously extracted from the domain. As the liquid phase diminishes, the system releases latent heat, leading to a noticeable reduction in thermal energy over time. This process is more efficient when enhanced by advanced thermal techniques. For instance, increasing the volume fraction of ternary nanoparticles results in a moderate increase in the solid fraction due to improved thermal conductivity. However, a much more significant impact is observed when the porosity of the metal foam is reduced, indicating that tighter metallic structures provide better conduction pathways and enhance the overall freezing rate. Furthermore, the presence of metal foam in the container notably lowers the temperature of the domain, reflecting its role in rapidly conducting heat away from the material and accelerating the solidification process.

These findings confirm that while nanoparticle enhancement contributes positively to heat transfer, the structural characteristics of the porous medium—especially low porosity metal foam—play a more dominant role in governing the freezing behavior of the cold storage system.

Figure 10 illustrates the effect of different thermal enhancement techniques on the total freezing time within the cold energy storage system. The comparison clearly highlights how each modification—namely, the use of ternary nanoparticles and metal foam—impacts the speed of the solidification process. Replacing water with a ternary nanoparticle-based nanofluid reduces freezing time by about 13.14%. This improvement results from the higher thermal conductivity of the nanofluid, which accelerates heat removal during the phase change. The nanoparticles enhance the conduction pathways, enabling more efficient extraction of latent heat. Introducing metal foam into the container yields a far more substantial reduction in freezing time. The metal foam’s high thermal conductivity and large surface area form multiple conduction paths, reducing solidification time by 82.2%. This highlights the crucial role of the porous metallic structure in speeding up the freezing front. In contrast, the base case with pure water and no metal foam exhibits the slowest solidification, requiring the longest time to complete. However, by combining both techniques—ternary nanofluid and metal foam—the system achieves the most significant reduction in freezing time, shortening the process by nearly 84.54% compared to the baseline. This synergistic effect demonstrates how the simultaneous enhancement of thermal conductivity (via nanoparticles) and conduction surface area (via metal foam) can dramatically boost the overall efficiency of the storage system. In summary, the data in Figure 10 clearly confirm that while nanoparticle addition offers a moderate improvement, the integration of metal foam is a much more impactful technique. When used together, these enhancements significantly shorten the freezing time, rendering the system highly efficient and ideal for advanced cold storage applications.

This study presents a comprehensive numerical analysis of PCM solidification within a wavy container fitted with rectangular fins. The process is accelerated using a ternary hybrid nanofluid as the base fluid, while metal foam is incorporated to enhance conductive heat transfer. Since solidification is primarily governed by conduction and the liquid motion was minimal, flow-related terms were omitted from the simulation. The model was developed using two scalar transport equations and solved with the Galerkin method, incorporating an adaptive grid to improve numerical accuracy. The combination of these advanced strategies demonstrates the potential to significantly enhance the solidification rate, offering a promising solution for improving the performance. The simulation results clearly demonstrate the significant influence of both ternary nanoparticles and metal foam on accelerating the solidification process within the cold storage unit. As time progresses, a natural decline in latent heat occurs due to the shrinking volume of the liquid phase, while the solid fraction shows a steady increase—indicating the progressive phase change from liquid to solid. This behavior is accompanied by a corresponding decrease in temperature, as reflected in the temperature distribution curves. Among all the scenarios studied, the base case using pure water without any enhancements exhibited the longest freezing duration. However, by dispersing ternary nanoparticles into the water, the total freezing time decreased from 303.76 seconds to 263.85 seconds, representing a reduction of approximately 13.14%. This improvement is attributed to the increased effective thermal conductivity of the nanofluid, which enhances conductive heat transfer during the freezing process. The introduction of metal foam into the storage domain has an even more substantial impact. When used alone, the metal foam reduced the freezing time dramatically to 46.95 seconds—an improvement of over 82%. Furthermore, when both techniques were applied simultaneously (nanoparticles and metal foam), the freezing time decreased by as much as 84.54%, showcasing a highly synergistic enhancement of the system’s thermal performance. Additionally, the temperature across the domain was found to drop more rapidly and uniformly in the presence of metal foam, indicating its strong role in improving thermal conductivity and distributing the cooling effect efficiently. While increasing the nanoparticle volume fraction yields a modest improvement in the solid fraction over time, a much more pronounced enhancement is observed when the porosity of the metal foam is reduced, confirming that foam structure plays a dominant role in freezing dynamics. In conclusion, the integration of ternary nanomaterials and porous metal foam represents a powerful strategy to significantly accelerate freezing in cold energy storage systems, offering potential applications in high-efficiency thermal management and sustainable refrigeration systems.