As the global transition to sustainable energy gains momentum, solar panels have emerged as linchpins in the pursuit of clean and renewable power sources [1]. However, the efficiency of solar panels is intricately tied to a formidable challenge–the impact of elevated operating temperatures on overall performance [2]. The quest for optimal thermal management strategies has thus become an imperative facet of advancing solar energy technologies [3].

In addressing this challenge, conventional cooling methods such as air cooling and passive techniques have shown limitations, particularly as solar technologies evolve with increased power densities [4]. To meet the demands of contemporary solar systems, innovative and efficient cooling strategies are essential [5]. In this context, nanofluids, which are colloidal suspensions of nanoparticles such as alumina (Al₂O₃), titanium dioxide (TiO₂), copper (Cu), silicon dioxide (SiO₂), Graphene and so on, in a base fluid, have emerged as promising candidates for revolutionizing heat transfer fluid technologies. With their enhanced thermophysical properties, such as increased thermal conductivity and superior heat transfer capacity, nanofluids offer an innovative and efficient solution to overcome challenges associated with heat management in various industrial and energy applications. Their use could not only improve the efficiency of heat transfer systems but also contribute to a significant reduction in energy consumption and operational costs [6,7].

This study positions itself at the dynamic crossroads of nanotechnology and solar energy, aiming to explore and harness the potential of nanofluids as advanced heat transfer fluids tailored explicitly for the cooling of solar panels [8]. The interplay between temperature and efficiency in photovoltaic systems underscores the pivotal role that effective cooling mechanisms play in enhancing overall energy conversion efficiency [9].

The nanofluids under scrutiny in this study are hybrid in nature, incorporating nanoparticles of alumina (Al₂O₃), titanium dioxide (TiO₂), and copper (Cu) into a water base fluid. Our primary focus is to unravel the thermophysical properties of these hybrid nanofluids, with a specific emphasis on thermal conductivity and specific heat. This work serves as a pre-study for our subsequent research, which will concentrate on photovoltaic thermal applications. Recent research has demonstrated that hybrid nanofluids, combining different types of nanoparticles, exhibit superior thermal properties compared to their single-nanoparticle counterparts [10]. This observation reinforces their relevance in addressing the thermal challenges inherent in solar panel applications.

Understanding how these properties evolve across varying volumetric concentrations for each nanoparticle type is pivotal to our research objectives. This study specifically investigates concentrations ranging from 0% to 4% for each type of nanoparticle, as well as mixtures reaching up to 10% total volumetric concentration. Importantly, our model maintains a constant temperature and nanoparticle diameter throughout these concentrations, ensuring a focused exploration of their impact on thermal properties.

In terms of our unique contribution, this study not only extends the theoretical understanding of hybrid nanofluid behavior but also offers practical insights for the design and optimization of nanofluids in real-world solar panel applications. Our developed model has undergone verification against experimental data available in the literature, demonstrating a commendable agreement with established results. This verification process enhances the reliability and applicability of our model to real-world scenarios.

Hybrid nanofluids, an innovative subclass within the nanofluid spectrum, were employed in this study. Comprising a combination of two or more distinct nanoparticles, the choice of nanoparticle types and their relative proportions was of paramount importance for influencing thermophysical properties [11]. Despite the promising potential, observations from collective research indicated instances where thermal properties of hybrid nanofluids experienced a decline compared to conventional counterparts [12]. The incorporation of nanoparticles into the base fluid induces alterations in thermophysical properties (Fig. 1) such as thermal conductivity, viscosity, and specific heat, impacting heat transfer through convection [13]. Various nanomaterials modify their parameters to varying extents, and factors such as nanoparticle concentration, purity level, shape, and size significantly alter thermophysical properties [14].

The paramount thermophysical characteristic governing the performance of any working fluid within a thermal system is its thermal conductivity. This critical property in hybrid nanofluids is intricately influenced by a multitude of variables, encompassing the type and composition of the base fluid, the specificities of the nanoparticles in terms of type, combination, and composition, as well as temperature, volume concentration, particle size, and shape [15].

In earlier investigations, efforts were made to estimate the thermal conductivity of hybrid nanofluids by adapting modified versions of models initially designed for individual nanofluids as presented in Table 1. This intricate formulation takes into account diverse elements, including particle volume fraction, temperature, particle shape (cylindrical, spherical…), pH value, nanoparticle diameter, and the influence of Brownian motion resulting from the erratic movement of particles [16]. A comprehensive understanding of these interconnected factors is indispensable for the accurate modeling and prediction of thermal conductivity in hybrid nanofluids, ultimately enhancing the efficiency of thermal systems [17].

In the context of this study, anchored in the results and discussion section, the values of thermal conductivity and specific heat are listed in Table 2. This table highlights the properties of both the base fluid and nanoparticles used in this research. The investigation focuses on the influence of nanoparticle volume fraction at various concentrations, relying on a specialized computer modeling program. This program incorporates heat transfer equations, particularly emphasizing thermal conductivity and specific heat, for nanofluids, including hybrid nanofluids such as TiO₂/Al₂O₃-Water, TiO₂/Cu-Water, and Cu/Al₂O₃-Water. The graphical representation of results takes into account multiple factors. In this section, we discuss the outcomes derived from comparing the ratios of thermal conductivity and specific heat among hybrid nanofluids (TiO₂/Al₂O₃-Water, TiO₂/Cu-Water, and Cu/Al₂O₃-Water). This comparison relied on well-established models, including Maxwell and the Xuan and Roetzel model, considering the volume fraction range from 0% to 4% (0≤φ≤4%) and 0% to 10% (0≤φ≤10%) .

According to Fig. 2, the obtained results reveal significant variations in the increase of thermal conductivity for the different studied hybrid nanofluids. For the TiO₂/Al₂O₃-Water hybrid nanofluid, an increase of 1.59% was observed as the volume fraction increased. This slight increment may indicate a moderate reactivity of this specific nanoparticle mixture, suggesting a less pronounced influence on thermal conductivity. In contrast, the TiO₂/Cu-Water hybrid nanofluid exhibited a more pronounced increase of 2.065%, signaling a more substantial response to the increase in volume fraction. This substantial improvement could be attributed to synergistic interactions between TiO₂ and Cu nanoparticles, contributing to an enhanced thermal conductivity. On the other hand, the Cu/Al₂O₃-Water hybrid nanofluid showed a more modest increase of 0.46%, indicating a relatively weaker reactivity of this particular combination. These findings underscore the dependence of the thermal performance of hybrid nanofluids not only on the volume fraction but also on the specific interactions between the employed nanoparticles.

Based on the obtained results Fig. 3, the analysis reveals further insights into the thermal conductivity variations of the studied hybrid nanofluids. For the TiO₂/Al₂O₃-Water hybrid nanofluid, there was a notable increase of 3.87% as the volume fraction ranged. This significant enhancement suggests a heightened reactivity of this nanoparticle combination, potentially leading to improved thermal conductivity. Similarly, the TiO₂/Cu-Water hybrid nanofluid exhibited a substantial increase of 5.04%, indicating a robust response to the changing volume fraction. This noteworthy improvement could be attributed to synergistic interactions between TiO₂ and Cu nanoparticles, contributing to an amplified enhancement in thermal conductivity. On the other hand, the Cu/Al₂O₃-Water hybrid nanofluid showed a moderate increase of 1.12%, indicating a comparatively lower reactivity for this specific combination. These findings further underscore the impact of volume fraction changes on the thermal performance of hybrid nanofluids, emphasizing the role of nanoparticle interactions in influencing thermal conductivity.

In summary, the analysis of thermal conductivity variations across different volume fraction ranges highlights nuanced behaviors within the studied hybrid nanofluids. For the initial range of 0% to 4%, subtle yet discernible changes were observed, with the TiO₂/Cu-Water hybrid nanofluid standing out with a noteworthy increase of 2.065%. As the volume fraction expanded to 10%, more pronounced enhancements were evident, particularly in the TiO₂/Al₂O₃-Water and TiO₂/Cu-Water hybrids, showcasing increases of 3.87% and 5.04%, respectively. These results underscore the dynamic interplay between nanoparticle combinations and volume fraction, elucidating the potential for tailored adjustments to optimize thermal conductivity in specific applications. The findings contribute valuable insights for the strategic design of hybrid nanofluids, paving the way for enhanced thermal performance in various heat transfer systems.

The results obtained for specific heat at the volume fraction ranging from 0% to 4% (Fig. 4) reveal subtle variations in the performance of hybrid nanofluids. The TiO₂/Al₂O₃-Water hybrid nanofluid exhibited a slight increase in specific heat, reaching 0.078%. Similarly, TiO₂/Cu-Water showed an increment of 0.29%, while Cu/Al₂O₃-Water recorded an increase of 0.37%. These findings suggest moderate responses of specific heat to the increase in volume fraction for all studied hybrid nanofluids. It is noteworthy that these modest variations, although subtle, can have significant implications in applications where slight changes in specific heat are critical, emphasizing the influence of nanoparticle interactions on the thermal properties of hybrid nanofluids.

In Fig. 5, the trends underscore the continuation of the observed phenomenon, portraying a clear positive correlation between specific heat and the increasing volume fraction of nanoparticles in the hybrid nanofluids. The specific heat values for the TiO₂/Al₂O₃-Water, TiO₂/Cu-Water, and Cu/Al₂O₃-Water hybrid nanofluids at the extended volume fraction range of 0% to 10% further solidify this trend. Specifically, the TiO₂/Al₂O₃-Water hybrid nanofluid exhibited a specific heat increase of 0.20%, indicating a sustained alignment with the escalating volume fraction. In comparison, the TiO₂/Cu-Water hybrid nanofluid demonstrated a more pronounced increment of 0.79%, emphasizing a heightened positive correlation. Notably, the Cu/Al₂O₃-Water hybrid nanofluid displayed a significant increase of 1%, substantiating the consistent trend of rising specific heat with the continuous augmentation of nanoparticle volume fraction. These insights, as depicted in Figs. 4 and 5, further emphasize the intricate relationship between specific heat and volume fraction, offering valuable guidance for optimizing thermal properties in diverse applications.

Our numerical model has been previously validated through experimental studies conducted by Chakar et al. [10] for thermal conductivity and for specific heat by Çolak et al. [23], as presented in Figs. 6 and 7, respectively, maintaining the same parameters as the latter. Our analysis deduces that the experimental results exhibited good agreement with the values obtained from our numerical model.

In conclusion, our comprehensive examination of the thermal properties of hybrid nanofluids has provided valuable insights into their performance across various volume fractions. The observed substantial variations in thermal conductivity and specific heat as the volume fraction increases underscore the dynamic nature of these nanofluids. Notably, the TiO₂/Cu-Water hybrid nanofluid emerged as a standout performer, exhibiting an impressive 5.04% increase in thermal conductivity at a volume fraction of 10%. Similarly, the Cu/Al₂O₃-Water hybrid nanofluid demonstrated a notable 1% increase in specific heat within the same volume fraction range. These key quantitative results highlight the significant improvements in thermal properties achieved through the use of hybrid nanofluids. Considering the application-specific context of solar panel cooling, the choice of the optimal hybrid nanofluid hinges on a multitude of factors, encompassing operational constraints and specific thermal requisites. Our results strongly advocate for the TiO₂/Cu-Water hybrid nanofluid as a preferable option, primarily due to its superior thermal conductivity performance. Nevertheless, it is imperative to acknowledge that decisions should factor in additional considerations, such as long-term stability and cost implications, which could influence the ultimate selection of the optimal hybrid nanofluid for enhancing the cooling efficiency of solar panels.

These findings not only contribute to the fundamental understanding of nanofluid behavior but also present promising prospects for targeted applications that demand heightened thermal performance. Beyond solar panel cooling, the improvements in thermal conductivity and specific heat of hybrid nanofluids have potential impacts on other applications such as electronics cooling and automotive systems. As the field continues to evolve, the optimization of nanofluid formulations based on specific volume fractions holds considerable potential for advancing thermal management solutions in diverse technological applications, thereby contributing to the broader field of thermal management and renewable energy.