Experimental and explainable machine learning approach on thermal conductivity and viscosity of water based graphene oxide based mono and hybrid nanofluids

Characterization

Figure 1a to c show field emission SEM images of the nanoparticles (Fig. 2). Analysis of Fig. 3a and b indicates that both SiO₂ and TiO₂ nanoparticles have a spherical shape and tend to aggregate due to Van der Waals forces. This spherical form is beneficial because it improves the contact surface area between the nanoparticles and the base fluid, hence improving heat transfer efficiency. Additionally, the uniformity of spherical particles helps reduce fluid resistance, promoting smoother flow dynamics in nanofluids. In contrast, Fig. 1c illustrates that GO nanoparticles have a layered or sheet-like structure. Unlike spherical particles, the sheet-like structure of GO may increase the viscosity of the nanofluid, potentially affecting its flow characteristics. However, the high surface area of these GO nanosheets can enhance the stability and thermal conductivity of the nanofluid, offsetting the potential viscosity increase. Furthermore, the spherical shape of metal oxide nanoparticles helps lower the viscosity of the nanofluids in comparison with irregular structures like GO. This makes SiO₂ and TiO₂ more suitable for applications where it is crucial to maintain low viscosity while improving heat transfer. Combining these materials can provide a balance between enhanced thermal conductivity and manageable viscosity levels in hybrid nanofluid systems.

The nanoparticles were characterized using a Bruker Advance diffractometer with Cu-Kα radiation over a 2θ range of 10° to 60°, at a 0.02° step size. The crystallite size was calculated using the Scherrer formula. XRD analysis of TiO₂ (Fig. 3a) shows pure anatase phases, confirming no impurities and a mean nanoparticle size of 40 nm. In XRD analysis presented in Fig. 3b, GO typically shows a peak around 2θ = 10–15° due to the increased interlayer spacing caused by oxygen functional groups. This peak confirms the successful oxidation of graphite into GO.

Nanofluid stability

The zeta potential is an important parameter in determining the dispersed nanoparticles stability in nanofluids. A zeta potential value exceeding ± 30 mV indicates stable colloidal dispersions.

In this study, the mass ratio of Polyvinyl pyrrolidone (PVP) to GO, and to hybrid nanofluids, was kept constant and matched the weight of GO to ensure consistent experimental conditions for accurate comparisons. Stability analysis using the Zetasizer (Malvern Instruments, UK) revealed that zeta potential values were consistent both immediately after preparation and after 25 days, demonstrating the high stability of the nanofluids shown in Fig. 3. These results highlight the effective dispersion of nanoparticles, which is essential for enhancing heat transfer and ensuring long-term stability and performance.

Viscosity

Figure 4a–d illustrate how the viscosity of nanofluids varies with concentration and temperature. Viscosity rises with increased concentration but decreases as temperature increases. The highest and lowest viscosity ratio (µ _nf/µ _bf) observed for GO nanofluid is 2.77 and 1.38 at 30 and 60 °C for 1 and 0.1 vol%, correspondingly. The relationship between temperature and viscosity in nanofluids is governed by a complex interplay of molecular dynamics and nanoparticle behavior. As thermal energy increases, it catalyzes more vigorous molecular motion, weakening the cohesive forces between fluid molecules. This diminution of intermolecular attraction facilitates easier relative movement, manifesting as reduced viscosity and enhanced flow characteristics. Concurrently, elevated temperatures amplify the Brownian motion of suspended nanoparticles, causing them to collide more frequently with fluid molecules and disrupt local fluid structures. This disruption further attenuates fluid viscosity by impeding the formation of transient molecular networks that contribute to flow resistance.

However, the influence of nanoparticles on viscosity is concentration-dependent and non-linear. At higher concentrations, nanoparticles exhibit increased propensity for inter-particle interactions, potentially leading to the formation of temporary or permanent agglomerates. These larger structures can significantly impede fluid flow, counteracting the viscosity-reducing effects of temperature. Moreover, the surface properties of nanoparticles, such as charge distribution and functional groups, play a crucial role in determining their aggregation behavior and subsequent impact on fluid dynamics.

The presence of nanoparticles also introduces additional mechanisms that modulate viscosity. For instance, the formation of a nanolayer of fluid molecules around each particle can alter local fluid properties, creating regions of modified viscosity that influence bulk fluid behavior. Furthermore, the shape anisotropy of certain nanoparticles, such as GO flakes, can induce orientation-dependent effects on fluid flow, adding another layer of complexity to the viscosity profile of nanofluids at varying temperatures and shear rates.

Figure 4d contrasts at various concentrations and temperatures the viscosity of hybrid and mono nanofluids. Whereas SiO₂ nanofluids display the lowest viscosity, GO nanofluids have the greatest. The plate-like structure and large surface area of GO hinder fluid movement, increasing its viscosity. On the other hand, the smaller surface area and the spherical shape of SiO₂ facilitate easier fluid flow, resulting in lower viscosity. TiO₂ nanofluids have higher viscosity than SiO₂, likely due to their smaller particle size and higher density. GO-TiO₂ nanofluids display higher viscosity compared to GO-SiO₂, as TiO₂’s smaller particles and SiO₂’s shape promote smoother flow. Hybrid nanofluids, such as GO-SiO₂ or GO-TiO₂, generally exhibit higher viscosity than mono nanofluids due to increased particle interactions, larger particle clusters, and potential synergistic effects between different nanoparticles, leading to greater flow resistance. While adding GO raises viscosity, mixing it with SiO₂ or TiO₂, which have lower surface areas, reduces the viscosity of GO in hybrid formulations.

Thermal conductivity

Figure 5 (a–c) illustrate the thermal conductivity of both hybrid and mono nanofluids, which increases with temperature and concentration. The maximum and minimum thermal conductivity amplification of 1.52 and 1.09 noticed for GO nanofluid at 60 and 30 °C for 1 and 0.1 vol%, respectively, compared to the water.

The thermal conductivity of nanofluids exhibits a complex relationship with temperature and nanoparticle concentration, driven by several interacting mechanisms. As temperature rises, the intensified Brownian motion of nanoparticles leads to improved dispersion and reduced agglomeration, creating a more uniform distribution of heat-conducting elements throughout the fluid. This enhanced particle mobility facilitates more frequent and energetic collisions between nanoparticles and fluid molecules, establishing numerous transient nanoscale heat transfer bridges that significantly boost overall thermal conductivity. The augmented thermal energy at elevated temperatures also promotes the formation of more robust and extensive percolation networks among nanoparticles, particularly in fluids with higher concentrations. These networks serve as preferential pathways for rapid heat propagation, markedly enhancing the fluid’s thermal transport capabilities. Furthermore, the temperature-induced reduction in fluid viscosity allows for more efficient heat transfer at the nanoparticle-fluid interface, as the decreased resistance to molecular motion enables swifter energy exchange.

Increasing nanoparticle concentration introduces additional heat conduction pathways, amplifying the fluid’s thermal conductivity. However, this relationship is non-linear and exhibits a critical threshold. Beyond a certain concentration, particle crowding can lead to the formation of larger aggregates, which paradoxically may impede heat flow by reducing the effective surface area for heat exchange and disrupting the continuity of thermal pathways. The synergistic interplay between temperature and concentration effects on thermal conductivity is particularly noteworthy. Higher temperatures can mitigate the negative impacts of increased concentration by enhancing particle dispersion and preventing excessive agglomeration, thus maintaining optimal heat transfer conditions even at elevated particle loadings.

It’s crucial to consider the role of nanoparticle material properties in this context. Materials with high intrinsic thermal conductivity, such as graphene or carbon nanotubes, can yield disproportionate increases in fluid thermal conductivity even at relatively low concentrations. The aspect ratio and surface chemistry of nanoparticles also significantly influence their dispersion behavior and interfacial thermal resistance, further modulating the temperature and concentration-dependent thermal conductivity enhancements in nanofluids.

Figure 5d compares the TC of hybrid and mono NFs. At 1.0 vol% and 60 °C, the highest thermal conductivity ratios for SiO₂, TiO₂, GO-SiO₂, GO-TiO₂, and GO nanofluids are 1.17, 1.23, 1.39, 1.43, and 1.52 respectively.

GO demonstrates the most significant TC enhancement owing to its inherent higher TC and layered structure, while SiO₂, with lower thermal conductivity and surface area, shows the least. Hybrid nanofluids generally outperform mono nanofluids (except for GO) due to synergistic effects that combine the properties of different nanoparticles to create more efficient heat transfer pathways. Various nanoparticles improve thermal conductivity through enhanced phonon transport, better dispersion, and reduced agglomeration. Additives also help improve dispersion, which enhances the thermal conductivity of hybrids. GO-TiO₂ exhibits higher thermal conductivity than GO-SiO₂, due to TiO₂’s smaller size and higher thermal conductivity. The improvement in thermal conductivity for nanofluids depends on particle size, shape, dispersion, and inter-particle interactions, which are essential for effective thermal management.

Data pre-analysis

The data gathered in lab-based testin phase was evaluated for correlation among the data columns. The correlation matrix offers important new perspectives on the interactions among the variables. Beginning with nanoparticle concentration, it shows a quite significant positive association with the viscosity ratio (0.97), meaning that the viscosity ratio also grows practically proportionately as the nanoparticle concentration rises (Fig. 6a). This makes sense as the inclusion of nanoparticles usually raises the viscosity of the nanofluid. Likewise, the concentration displays a strong positive association with the thermal conductivity ratio (0.82), implying that increasing nanoparticle concentrations enhance the thermal conductivity of the fluid. Conversely, temperature (T) has no effect on concentration (0) since in this scenario the concentration of nanoparticles is independent of temperature. With the TC ratio (0.54), it does, however, exhibit a modest positive correlation, meaning that higher temperatures usually improve the TC of the nanofluids. By contrast, the temperature exhibits a mild negative association with the viscosity ratio (-0.13), meaning that rising temperature somewhat lowers viscosity, a normal behavior for fluids where viscosity drops with heating.

At last, the viscosity ratio and TC ratio show a quite strong positive connection (0.74), implying that in the nanofluid these two characteristics are connected. A rise in one generally follows a rise in the other as the nanoparticles raise both viscosity and heat conductivity. Understanding how changing one feature could affect the total thermal performance of the nanofluid depends on this interdependence. The scatter plot (pair plot) depicted in Fig. 6b visually shows the associations between several dataset variables. Whereas the off-diagonal parts display scatter plots, indicating the correlations between pairs of variables, each diagonal element provides a histogram for a single variable. This plot supports the results of the correlation matrix shown in Fig. 6a. While temperature has a secondary influence on thermal conductivity more than viscosity, the concentration of nanoparticles has a very significant effect on both thermal conductivity and viscosity ratios. The image demonstrates that concentration drives the behavior of the system mostly; temperature has a relatively moderate influence.

Thermal conductivity ratio models

The TC ratio prediction models were developed by employing three modern ML approaches namely Random Forest (RF), Gradient Boosting (GB), and Decision Tree (DT). The models were then used for statistic-based evaluation and comparison. Table 2 shows the values of mean squared error (MSE), coefficient of determinants (R²), and mean absolute percentage error (MAPE) in each case. With a low Train MSE of 0.0001 and Test MSE of 0.0002 the RF model (Fig. 7a) exhibits good predictive ability. Train R² of 0.9860 and Test R² of 0.9575 show that the model generalizes effectively to unknown data, therefore attesting to great accuracy. Reflecting little prediction errors, the MAPE is likewise rather low, with 0.90% for training and 1.04% for testing. With a Train MSE of 0.0001 and Train R² of 0.9874 indicating outstanding fit the GB model (Fig. 7b) performs similarly on the training set. On the other hand, the Test MSE is somewhat higher at 0.0003 and the Test R² falls to 0.9202, therefore indicating less generalization than the Random Forest. With a MAPE for the test set of 1.42%, greater than that of Random Forest, this model clearly generates more prediction errors in unseen data. Reflecting a quite strong match to the training data, the DT model has a similar Train MSE of 0.0001 and a high Train R² of 0.9876. With a higher Test MSE of 0.0006 and Test R² of 0.8500, its performance on the test set does, however, clearly deteriorates. This suggests that the model suffers with generalizing. Furthermore, supporting that this model generates more errors while testing than Random Forest and Gradient Boosting is the test MAPE of 1.91%. While Decision Tree (Fig. 7c) displays symptoms of overfitting and less dependable predictions on the test data, Random Forest offers the best balance between training and test performance followed by Gradient Boosting.

VST model prediction

Three contemporary ML techniques RF, GB, and DT were used in development of the VST ratio prediction models. After that, statistical-based assessment and comparison made advantage of the models. Table 3 lists in each scenario MAPE, R², and MSE. With their identical Train MSE values and high Train R² values of about 0.97, the performance metrics for the VST ratio models reveal consistent outcomes across all three models: Random Forest, Gradient Boosting, and Decision Tree during training. This implies that every model may reasonably capture the fundamental trends in the training data. The test R² values of the models on the test data expose some variations in their generalizing capacity. With Test R² values of 0.9405 and 0.9637 respectively, the Random Forest and Gradient Boosting models show equivalent generalization. This suggests great forecasting ability despite on unavailability of data. Conversely, the Decision Tree model exhibits a somewhat lower Test R² of 0.9217, which reflects a decreased capacity to sustain accuracy on the test set, thereby maybe indicating a higher vulnerability to overfitting in compared to the ensemble-based models.

These trends are supported by the test data’s Mean Absolute Percentage Error (MAPE). While Gradient Boosting and Decision Tree show better error percentages, Random Forest has the lowest Test MAPE, indicating more exact predictions. This trend implies that, presumably because they have lower overfitting relative to the single-tree Decision Tree model, ensemble models especially Random Forest, handlers the volatility in the data better. Finally, all models perform well during training; with Gradient Boosting closely behind, the Random Forest model offers the best trade-off between training performance and generalization to test data. Though still useful, the Decision Tree model shows more overfitting and poor accuracy in test data predictions (Fig. 8).

eXplainable machine learning using SHapley additive exPlanations analysis

The ML approached employed in last section provided excellent results. However, these are black box methods and stakeholders may not how these models predicted or how much was the contribution of each feature involved. By pointing out the most important factors influencing TC and VST ratio predictions, explainable machine learning can offer understanding of the decision-making process of the model. Understanding feature importance and interpreting model outputs helps researchers identify causes of overfitting or errors, therefore strengthening model dependability and guiding optimization for maximum performance.

Employing SHAP values, the Fig. 9a offers a thorough examination of the relative significance of two predictors—nanoparticle concentration (Conc., Vol.%) and temperature (T, °C) on the output of the model. The first plot a bar chart of mean SHAP values showcases how mean SHAP value of nanoparticle concentration exceeds that of temperature (0.0452). This implies that, relative to temperature, concentration contributes more generally to the predictions of the model. The output shows a more significant influence from the SHAP values for concentration, so changes in concentration greatly affect the response of the model.

The bee swarm plot (Fig. 9b) expands on the distribution and range of SHAP values for every predictor. With a higher concentration of SHAP values in the positive range, nanoparticle concentration exhibits a wider distribution whereby both high and low values can either positively or negatively affect the model result. This suggests that generally the expected value rises with increasing concentration. On the other hand, temperature shows a smaller range of SHAP values; most of the values are near zero, thereby suggesting it has a more restricted influence on the prediction of the model. Though less significantly than concentration, high-temperature values nevertheless help the output in some beneficial manner. This implies that, with temperature having a minor but still significant impact, concentration dominates in determining the predictions of the model.

In the case of viscosity ration model, the effects of nanoparticle concentration (Conc., Vol.%) and temperature is depicted in the Fig. 10a. The bar chart shows that concentration, with a substantially higher mean SHAP value (0.3125) than temperature (0.0439), clearly influences the projections of the model. The size of the SHAP value of concentration suggests that variations in this variable cause more significant variations in the viscosity ratio prediction. With a general trend demonstrating that larger concentrations (shown by the red dots) greatly improve the model output, nanoparticle concentration shows a wide range of SHAP values both positive and negative in the bee swarm plot (Fig. 10b). On the other hand, the small range of the SHAP values for temperature indicates their reduced influence on the model since they cluster at zero. High temperatures still somewhat influence the results of the model, but less clearly. Consequently, in this model the concentration of nanoparticles still mostly determines the viscosity ratio.