Solidification Behavior of a Cold Energy Storage System with Metal Foam and Ternary Hybrid Nanofluids

Phase change materials (PCMs) are widely recognized for their effectiveness in thermal energy storage systems. Among these, the solidification process plays a vital role in cold energy storage systems, where the efficient removal of heat directly influences the system’s performance and response time. However, the inherently low thermal conductivity of most PCMs limits their effectiveness, especially in applications requiring rapid energy exchange. To overcome this limitation, researchers have introduced thermally conductive nanoparticles into the base PCM. The dispersion of nanoparticles enhances the thermal conductivity of the mixture, thereby expediting the freezing [1], [2]. At the same time, incorporating metal foam or porous structures into the PCM matrix has become an effective method for enhancing thermal performance. These configurations create a highly conductive thermal network that significantly improves conduction-dominated heat transfer within the storage unit, resulting in more rapid and uniform solidification. The combined use of nanoparticles and porous metal foam offers a synergistic effect, enabling faster phase transitions and higher energy storage efficiency [3], [4], [5]. In recent years, numerous innovative designs and composite materials have been introduced by researchers to enhance the efficiency [6], [7]. Additionally, advanced computational techniques have been developed to model complex transient thermal behavior more accurately [7], [8]. Öztop et al. [9] investigated heat transfer within a square enclosure filled with paraffin. Their study examined how variations in Grashof numbers (Gr) and nanoparticle concentrations influenced melting performance. The results demonstrated that elevating both the nanoparticle concentration and the Grashof number significantly strengthened thermal convection, leading to an enhancement in heat storage capacity of up to 35.80%.

Amidu et al. [10] developed a numerical model to analyze a two-phase nanomaterial inside a triplex tube. Their findings revealed that a uniform distribution of nanoparticles enhanced heat transfer; however, higher nanoparticle concentrations led to a 50% decline in melting efficiency during the second cycle, attributed to sedimentation effects. Qin [11] conducted a simulation of the freezing process of a CuO–water nanofluid inside a rectangular inner cylinder. The study examined how the nanoparticle size and the amplitude of the outer wall influenced the freezing duration. It was found that an outer wall amplitude of 0.3 led to a reduction in freezing time by 5.82%. Wang et al. [12] investigated the melting characteristics of a rectangular latent heat unit, which was partially embedded with metal foam and filled with paraffin. Their findings revealed that incorporating porous metal foam significantly accelerated the melting process, reducing melting time by up to 69.2%. Conversely, decreasing the metal foam mass by 75% led to a substantial increase in melting time, extending it by 263%. Younsi and Naji [13] performed numerical simulations to assess the thermal performance of brick walls incorporating PCM for use in building applications. Their results showed that adding 20–30% PCM could lower peak indoor temperatures by as much as 3°C and smooth out daily temperature variations. Additionally, increasing the PCM content further reduced energy usage, thereby improving the building’s overall energy efficiency.

Despite the extensive use of PCMs in cold energy storage, the low thermal conductivity of these materials continues to limit the rate of heat removal during solidification. This issue becomes more pronounced in systems where a short freezing time and stable thermal response are required. Previous studies have examined the use of nanoparticles or porous structures to improve heat transfer, but these approaches are often considered separately and mostly in simplified geometries. Their combined effect in storage units with more complex configurations has received less attention.

The present study considers a cold energy storage unit with a complex-shaped enclosure, in which porous metal foam is introduced together with a ternary hybrid nanofluid based on Ag–TiO$_2$–Al$_2$O$_3$ nanoparticles dispersed in water. The solidification process is described using a transient latent heat formulation, and the governing equation is solved by the Galerkin finite element method with adaptive mesh refinement. The analysis focuses on how the nanoparticle volume fraction and the foam porosity influence the freezing time, temperature field, solid fraction, and stored energy during the process. The results are discussed in terms of their implications for the thermal response of cold energy storage systems used in refrigeration and related applications.

This study focused on the thermal energy storage performance of a complex-shaped container filled with metal foam, designed specifically for cold energy storage applications (illustrated in Figure 1). The investigation centered on the solidification process within this study, which was numerically analyzed using the Galerkin method. To improve the accuracy of the numerical simulations, an adaptive mesh refinement technique was employed, enabling a denser grid around critical areas like the solidification front. To accelerate the solidification and improve heat transfer, two enhancement techniques were integrated into the model. First, the base fluid (water) was improved by adding ternary nanoparticles ($\phi$) to enhance the effective thermal conductivity of PCM. Second, the container was packed with metal foam, which serves to augment the conduction pathways and overall heat transfer performance within the storage unit. Owing to the relatively low velocity of the liquid phase during solidification, convective effects—particularly buoyancy forces—were considered negligible and thus omitted from the momentum equations. Consequently, the energy equation was solved incorporating a transient latent heat source term to accurately capture the phase change dynamics. Furthermore, the study systematically evaluated the influence of key parameters such as the volume fraction of ternary nanoparticles and the porosity of the metal foam ($\gamma$) on the freezing behavior. Various simulations explored how these factors affect the rate of solidification and overall thermal performance.

To accurately simulate the system, the following governing equations must be taken into account, as detailed in the studies [14], [15], [16], [17], [18], [19]:

The scalar variables in the above equations were evaluated at each computational node during the solution process. Before solving these equations, the thermophysical properties of the ternary nanomaterial were determined using established models, which were calculated based on established models as described in references [16], [17], [18], [19]:

In the current formulation, the velocity-related terms were neglected due to their minimal influence and the dominance of conduction heat transfer during the solidification process. This simplification allowed focusing on the primary heat transfer mechanisms without compromising accuracy. An adaptive grid refinement method was utilized. This adaptive meshing capability was a key feature of the FlexPDE software used in this study, which fundamentally relied on the Galerkin finite element method.

The foundation for applying this numerical approach was laid by Sheikholeslami [15], [16], who pioneered the application of the Galerkin method combined with adaptive meshing for a variety of heat transfer and phase change problems. His work demonstrated the robustness and accuracy of this method in capturing intricate phase change behaviors, particularly in complex configurations. Building on this foundation, the present study used these advanced numerical techniques to accurately simulate the solidification process within a metal-foam-enhanced energy storage container, thereby providing a reliable framework for optimizing thermal performance in such systems.

This study offers a new strategy for cold energy storage through the use of a complex-shaped container embedded with metal foam, aimed at significantly improving the efficiency of the solidification process. To accelerate the solidification process, the study introduces two synergistic enhancement techniques aimed at improving the conduction-dominated heat transfer regime. Firstly, a ternary hybrid nanofluid is dispersed in water to boost the effective thermal conductivity of the phase change medium. Secondly, the use of metal foam within the container further enhances heat transfer by augmenting the surface area. Given the negligible fluid velocity in the system—due to extremely low natural convection—the momentum equations are simplified by omitting the buoyancy terms. As a result, the energy equation incorporates a latent heat source term and is solved using the Galerkin method alongside an implicit time discretization scheme, ensuring numerical stability and accuracy in transient thermal simulations. Parametric studies were conducted to evaluate the influence of two critical factors: the volume fraction of ternary nanoparticles and the porosity of the metal foam structure. These simulations provide valuable insights into how material enhancements and structural characteristics impact the freezing rate and thermal response of the system.

To ensure higher accuracy in capturing the solidification front, a non-uniform adaptive mesh refinement strategy was employed in the present numerical model. A denser grid was generated in regions close to the advancing ice front, where temperature gradients are steep and phase change dynamics are most significant. This adaptive meshing approach enables the model to resolve local phenomena more precisely without excessively increasing the overall computational cost. Figure 2 illustrates how the mesh configuration evolves with time, demonstrating the dynamic refinement near the freezing interface.

To validate the reliability of the developed numerical code, a benchmark comparison was conducted against previously published results for a well-established geometry. Figure 3 presents a comparison between the present simulation results and those reported in the study [16]. The close agreement between the two datasets verifies the accuracy and reliability of the adopted method. This successful validation provides confidence in applying the same numerical framework to analyze the complex geometry and thermal enhancement techniques presented in the current study.

Figure 4, Figure 5, and Figure 6 illustrate the freezing behavior of the cold energy storage unit using contour plots. The freezing initiates at the cold wall, progressing gradually toward the adiabatic boundary. As the solid phase volume grows, the average temperature within the container decreases accordingly. This characteristic is particularly evident in the base case without enhancement techniques, where natural convection is negligible, and heat transfer is primarily governed by conduction. Introducing ternary hybrid nanoparticles (Ag–TiO$_2$–Al$_2$O$_3$) into the PCM mixture results in a noticeable reduction in total freezing time—from 2767 seconds (base case) to 2403.59 seconds. When porous metal foam is embedded within the domain, the freezing process accelerates dramatically, with the total solidification time reduced to approximately 423.34 seconds. In the porous region, heat conduction is greatly enhanced, leading to a more uniform and faster freezing front. Overall, the presence of metal foam produces the most significant improvement in solidification behavior, followed by the addition of nanoparticles. The combination of both techniques offers an optimal solution for improving freezing performance in cold energy storage systems. From a system perspective, the reduction in solidification time corresponds to a faster thermal response and a shorter charging period for the storage unit.

Figure 7, Figure 8 and Figure 9 present the temporal evolution of the ice front, temperature distribution, solid fraction, and energy content within the cold energy storage unit under various enhancement scenarios. Figure 7 illustrates the progression of the ice front over time for different configurations. The results clearly show that the use of both enhancement techniques—namely, ternary hybrid nanoparticles and porous metal foam—leads to the fastest advancement of the ice front due to the significantly improved heat conduction. Notably, the impact of the porous foam on the ice front movement is far more substantial than that of the nanoparticles alone. Quantitatively, the rate of solidification achieved with metal foam is approximately 6.27 times greater than that achieved through nanoparticle addition alone. Figure 8 and Figure 9 depict the variations of scalar quantities, such as the solid fraction and thermal energy, as functions of time. This solidification leads to a corresponding decrease in the system’s total thermal energy, primarily due to the declining volume of the liquid phase, which stores more energy than the solid phase. Incorporating porous metal foam not only speeds up heat extraction but also promotes a faster and more uniform temperature decline throughout the entire domain. This, in turn, results in a faster increase in solid fraction, which eventually reaches a value of unity, indicating complete solidification. On the other hand, the addition of ternary nanoparticles produces a smaller but still beneficial reduction in temperature and enhances the freezing process. However, its impact remains modest compared to that of the metal foam.

The duration of solidification is a key factor in assessing the efficiency and overall performance of a cold thermal energy storage system. Figure 10 presents a comparative analysis of the total freezing time under different configurations. Among all cases examined, the slowest solidification process occurs when pure water is used as PCM without any thermal enhancement, such as metal foam. In contrast, the fastest freezing is achieved when both ternary hybrid nanoparticles (Ag–TiO$_2$–Al$_2$O$_3$) and porous metal foam are incorporated into the storage domain. The addition of ternary nanoparticles enhances the thermal conductivity of the base fluid, thereby accelerating heat extraction during the solidification process. This leads to a reduction in the total freezing time by approximately 13.13% compared to the base case. However, the improvement is more pronounced with the inclusion of metal foam, which significantly boosts the conduction heat transfer pathway within the PCM. The use of metal foam alone shortens the required solidification time by around 82.39%, demonstrating its dominant role in enhancing thermal performance. When both enhancement strategies—ternary nanoparticles and metal foam—are simultaneously applied, their synergistic effect results in the most substantial improvement. The total freezing time decreases by approximately 84.7% relative to the pure water case without foam. This significant reduction highlights the effectiveness of combining nanoparticle-induced conductivity enhancement with the extended conductive network provided by the porous structure.

The solidification process in a cold energy storage unit with a complex-shaped enclosure has been examined using a Galerkin finite element formulation with adaptive mesh refinement. The analysis considered the combined influence of porous metal foam and a ternary hybrid nanofluid (Ag–TiO$_2$–Al$_2$O$_3$ dispersed in water) under conduction-dominated conditions, where fluid motion remains weak during phase change.

The results show that solidification initiates at the cold boundary and gradually progresses toward the adiabatic wall, which represents the last region to freeze. As the solid phase develops, the average temperature within the storage domain decreases accordingly. The introduction of metal foam leads to a marked reduction in freezing time, with a decrease of about 82.39% relative to the base case. In comparison, the use of ternary hybrid nanoparticles alone results in a reduction of approximately 13.13%. When both approaches are applied together, the total freezing time is reduced by nearly 84.7%.

A comparison of the different configurations indicates that the effect of metal foam on heat transfer is considerably stronger than that of nanoparticle addition. The presence of the porous structure provides continuous conductive pathways across the domain, which promotes faster heat removal and a more uniform advance of the solidification front. In contrast, the contribution of nanoparticles remains limited to moderate changes in effective thermal conductivity. The freezing rate obtained with metal foam is about 6.27 times higher than that achieved with nanoparticles alone.

These observations indicate that the structure of the storage medium plays a dominant role in controlling the solidification process. The results may be useful in the design of cold energy storage systems where a shorter charging period and a faster thermal response are required.