Gradient Descent-Based Prediction of Heat-Transmission Rate of Engine Oil-Based Hybrid Nanofluid over Trapezoidal and Rectangular Fins for Sustainable Energy Systems

1  Introduction

Fin is referred to as a thin, elongated, extended structure attached to a solid surface. Engineers realised in the early 20th century that heat dissipation through convection may be improved by expanding a solid surface’s surface area. Fins are widely employed in automotive, electronics, aerospace, steel, metallurgy and renewable energy industries to enhance heat transfer. The choice of fin type, size, shape, and spacing depends on the specific heat transfer requirements of the system and the constraints of the application. Fins with the best designs are in more demand. Aziz and Fang [1] examined temperature distribution in longitudinal fin focusing on trapezoidal, rectangular, and concave profiles. Han and Peng [2] investigated thermal management in a moving fin considering a radiative and convective environment. Pavithra et al. [3] analysed the temperature distribution in the dovetail longitudinal fin considering radiation and hybrid nanofluid. Riasat et al. [4] examined the Darcy model of internal heat generation in a radial fin. Luo et al. [5] used a deep generative model to analyse thermal performance in a fin. Abd-Elmonem et al. [6] analysed the heat transport via a fin in a vertical pipe using a machine learning technique. Rehman et al. [7] using ANN predicted improving buoyancy results in convective heat transfer in a T-shaped fin. Arshad [8] reported that the addition of a tree-shaped fin reduced heat sink temperature by 8%. Sowmya et al. [9] investigated thermal distribution in a rectangular fin incorporating magnetic and radiation. Basha et al. [10] investigated entropy generation and heat transport in nanofluid flow across a square enclosure fitted with a fin and estimated the optimal transport using machine learning technology. The development of extended surface technology has led to the replacement of conventional solid fin structures with porous ones. Conventional uses for heat transmission in porous media include solar collectors, heat exchangers, and reactor cooling. Kiwan and Al-Nimr [11] were the ones who first suggested using porous fins by presenting the Darcy model. They took up a comparative study between porous and conventional fins and noticed that the porous fin shows better performance in heat transfer. Ahmad et al. [12] applied an ANN model to analyse the optimal thermal dispersion considering a porous triangular moving fin by including radiation, surface temperature, and Peclet number parameters. Hu et al. [13] addressed thermal characteristics using structured porous fins. The study reported that structured porous fins (SPFs) significantly enhanced phase change material (PCM) heat transfer and achieved a 62% decrease in melting time. Alotaibi et al. [14] applied ANN and the ISPH method to analyse the optimal heat transfer performance. Their study reported that 15% of the temperature was reduced due to the addition of a triangular porous fin. Nandy and Balasubramanian [15] observed that a porous wavy fin in a microchannel improved thermos-hydraulic performance due to the use of design C at Re of 300.

The angle of inclination has a significant impact on fin performance. In particular, in natural convection, it influences the boundary layer growth, flow structure and the heat transfer coefficient. While in forced convection, it impacts the flow distribution and associated pressure drop. He et al. [16] asserted that by improving the inclination angle, the average heat transfer and heat flux of a solid-perforated spiral fin rose by 15.5%. Li et al. [17] demonstrated that micro-finned wall position and inclination angle play a crucial role in vapour-liquid distribution, heat transfer and flow dynamics. Dhaoui et al. [18] revealed that optimising fin angle remarkably enhances heat transfer and overall effectiveness of solar stills. Komathi et al. [19] carried out an analysis of heat transfer in an inclined wetted moving fin and reported that fin temperature declined with increasing wet porous, radiation, inclination angle and convective parameter. Zhong et al. [20] took up a detailed investigation on heat transfer considering four different fins, namely curved, vertical, serpentine and inclined fins. They noticed that heat transfer efficiency was more effective for the curved fin, and the serpentine fin exhibited outstanding heat transfer. Chen et al. [21] from 20 to 180 W, the bending heat type’s heat transfer is greatly improved by the observed small inclination angle. Choi and Eastman [22] first showed that the deferment of nanoparticles in a working fluid, like oil, ethylene glycol, and water, can greatly enhance their thermal properties and heat transfer performance. Owing to these advantages, nanofluids are increasingly used in heat transfer applications in commercial and engineering industries. In recent years, attention has shifted towards hybrid and ternary nanofluids. Bahiraei et al. [23] conducted an experimental study and described a hybrid nanofluid’s thermal characteristics, which involves the amalgamation of two distinct nanoparticles in a base fluid. Research confirms that hybrid nanofluid outperforms mono nanofluid with enhanced rheological characteristics. Ternary hybrid extends this concept by incorporating three distinct nanoparticles. Varatharaj et al. [24] analysed the impact of first-order slip, porous media, ternary hybrid nanofluid flow across a permeable stretching surface via joule heating and viscous dissipation. Mishra et al. [25] examined how slip affected a ternary hybrid nanofluid passing across a permeable plate. Gul et al. [26] elucidated that the incorporation of Au,Fe2O3,SWCNT Nanoparticles in Casson fluid enhances heat transfer. Anjum et al. [27] analysed Casson ternary hybrid nanofluid enhanced thermal performance in drug delivery. Hussain [28] investigated ternary hybrid nanofluid flow in biomedical applications. Jahan and Nasrin [29] found that the highest heat transfer in a microchannel heat exchanger was produced by combining graphene, boron nitride, and carbon nanotubes in a ratio of 4/3:1/3:1/3. Hemmat et al. [30] investigated experimentally how a ternary hybrid nanofluid can improve heat transfer.

A computational investigation of Cattaneo-Christov double diffusion and mixed convection effects in a non-Darcian Sutterby nanofluid was conducted by Rehman et al. [31] employing multi-objective optimization through Response Surface Methodology (RSM). Using a modified Buongiorno’s model, Wang et al. [32] examined the heat and mass transport of an Ag–H2O nano-thin film flowing over a porous medium. Rehman et al. [33] used response surface methods to study the impact of heat radiation and magnetohydrodynamics on shear-thinning Williamson nanofluids with stability analysis. Oscillatory convective gear-generalised differential quadrature analysis was studied by Xia et al. [34], Darcy-Forchheimer and Lorentz quadratic drag forces in second-grade fluids’ Taylor-Couette flows.

Novelty of the Present Study:

Despite extensive research on fin problems, inclined rectangular and triangular porous fins wetted with hybrid nanofluid remain largely unexplored with this combination. Therefore, because applications for sustainable energy systems are in high demand, this study aims to address the thermal performance of inclined porous rectangular and triangular porous fins wetted with a hybrid nanofluid (Engine Oil + Graphene Oxide (GO) + Molybdenum Disulfide (MOS2)) through machine learning gradient descent optimization technique forecasting the rate of heat transfer in rectangular and trapezoidal fins, the difference between expected and actual values is negligible, shows the precise thermal performance estimation for various fin geometries.

2  Mathematical Formulation

The interplay of conduction, convection, and radiation is used to analyse heat transmission in a fin. A differential control volume technique is used to move heat internally along the x-direction. The temperature gradient from the base temperature (Tb) causes the heat flux (qx) to change. Heat is lost through convection and radiation as it enters the surrounding environment after passing through the fin. The geometrical characteristics of the fin, including its thickness (tb), width (w), and length (L), are important in determining how heat is transferred and distributed. As internal conduction counteracts the heat loss at the surface, the temperature drops along the x-direction. Whereas radiative heat loss happens through thermal radiation to the surrounding environment at temperature (Ta), convective heat loss transfers energy to the surrounding fluid. The fin’s ensuing temperature gradient shows how internal heat conduction and outward heat dissipation are balanced. Comprehending these principles is essential for maximising thermal management in engineering systems, guaranteeing efficient heat dissipation for enhanced stability and performance.

The governing differential equation for this issue is provided by Khan et al. [35]

ddx[t∗dTdx]−2gk(ρβ)hnf(ρCp)hnfsin⁡τ(T−Ta)2khnfμhnf−2σε(T4−Ta4)khnf−…2haifg(1−ϕ)(ω−ωa)(T−Ta)khnfCpLe23(Tb−Ta)−2ha(1−ϕ)(T−Ta)khnf(Tb−Ta)=0(1)

where, Hybrid nanofluid is denoted by the subscript hnf, C stands for specific heat capacity at constant pressure, ϕ for porosity, β for thermal expansion coefficient, k for thermal diffusivity, μ for effective kinematic viscosity, and ρ for effective mass density.

The following provides the corresponding boundary conditions:

T=Tbatx=L(2)

dTdx=0atx=0.(3)

The linearization of the T4 components as a function of temperature is possible with the Rosseland approximation, i.e., T4=4Tα3T−3Tα4.

Now consider the dimensionless quantities:

X=xL,θ=T−T∞Tb−T∞.(4)

Eq. (5) is obtained after solving by substituting Eq. (4) into Eq. (1)

d2θdX2−CdθdX−CXd2θdX2−((ρCp)hnf(ρCp)f)((ρβ)hnf(ρβ)f)(μfμhnf)(kfkhnf)Shsin⁡τθ2−…Nr(kfkhnf)θ−(kfkhnf)m2θ2=0(5)

Following the non-depersonalization process, the boundary conditions turn into

θ=1atX=0(6)

dθdX=0atX=1(7)

In the present study, normalization parameters are:

Nr=4σεT∞3L2kft is the radiation parameter,

m0=2haL2(1−ϕ)kftb, m1=2haL2(1−ϕ)ifgb2kftbCpLe2/3, m2=m0+m1 is the wet porosity parameter,

Sh=DaRakf(Lt)2 Is the porosity parameter,

(x)=tb−δ(xL) is the local semi-finish thickness,

ω−ωa=b2(T−Ta).

For ease of simplification,

Let a=Nr(kfkhnf)

b=((ρCp)hnf(ρCp)f)((ρβ)hnf(ρβ)f)(μfμhnf)(kfkhnf)Shsin⁡(τ)+(kfkhnf)m2.

Now the equation becomes,

d2θdX2−CdθdX−CXd2θdX2−bθ2−aθ=0(8)

Fin’s heat transfer rate by using the study of Khan et al. [35]:

Q=−khnfkfθ′(0)

3  Methodology of the Present Study

3.1 Lobatto IIIa Collocation Method

•   The given boundary conditions are then turned into a first-order system of differential equations, formulating the boundary value problem.

θ=f1,θ′=f2BoundaryConditionsfa(1)=1fb(2)=0

•   The first mesh is created on the solution interval, and a good initial guess of the solution is provided.

InitialGuess={0.1,0.1}

•   The ordinary differential equation (ODE) system is expressed as a function, and the boundary conditions are expressed as a separate boundary condition function.

•   The Lobatto IIIa collocation approach is an internal approach used in the bvp5c solver of MATLAB to impose accuracy at collocation points.

•   Tolerance error is 10−3, Step size is 0.01 and computational time is 5 s.

•   The solver automatically optimises the mesh to meet error tolerances, and the solver is numerically stable.

3.2 Gradient Descent Machine Learning

•   Input: Observe and preprocess the input information of important physical parameters.

•   Influence of Features: Research the effect of parameters on physical quantities of the target.

•   Model Training: Finding a minimum loss of a machine learning algorithm using gradient descent. The Cost function is J(θ0,θ1)=12n∑i=1n(θ0+θ1xi−yi)2.

J(θ0,θ1)=CostFunction,θ0=Intercept,θ1=Slope,xi=Input,yi=ActualOutput,(θ0+θ1xi)=Predicted Outputθ0+θ1xi−yi=ResidueError,(θ0+θ1xi−yi)2=SquareError

•   Analysis: Producing the predictions of linear fits and determining their performance using such measures as mean squared error (MSE) and root mean squared error (RMSE).

4  Results and Discussion

An inclined longitudinal porous fin wetted with a hybrid nanofluid (Engine Oil Graphene Oxide Molybdenum Disulfide (MOS2)) is studied numerically for both rectangular and trapezoidal fins. Temperature profiles θ(X) is analysed for rectangular and trapezoidal fins by altering the radiative parameter, porosity parameter, wet porous parameter, and angle of inclination. The thermophysical property relations are shown in Table 1, the thermophysical properties of the base fluid engine oil and the nanoparticles are listed in Table 2 and Table 3 displays the change in heat transfer rate of the porous rectangular and trapezoidal wet fin with hybrid nanofluid as determined by the numerical model and Gradient Descent algorithm for various scales of governing parameters (T,Sh,Nr and m2). Heat transfer rate shows a substantial enhancement with an increase in the degree of inclination (T), parameter for wet porous (m2), porosity parameter (Sh) and radiative parameter (Nr) for both rectangular and trapezoidal fins. Among the two geometries, the trapezoidal fin exhibits superior heat transfer performance compared to the rectangular fin, making it an ideal choice for advanced thermal management applications. From the obtained values, it is evident that the observed variations between both the approaches, gradient descent and numerical, are very minimal, typically below 1%–1.5%, confirming the reliability and accuracy of the Optimization technique. These observations confirm the accuracy, reliability, and physical consistency of the proposed model in forecasting the trapezoidal and rectangular porous fins’ thermal response. The accuracy and dependability of the current study are demonstrated by Table 4, which compares, under specific conditions, the findings of the current investigation with those of the previous study. Fig. 1 shows the applications of the current research, Fig. 2 shows a schematic representation of the trapezoidal and rectangular fin, Fig. 3 shows how the Gradient Descent Algorithm works, and Fig. 4 shows the methodology used in this study.

4.1 Temperature Profiles for Various Parameters and Error Analysis

From Fig. 5 it is evident that as the wet porous parameter (m2) improves from 0.2 to 0.8, both rectangular and trapezoidal fins exhibit faster deterioration in temperature along the fin length. Rectangular fin retains a higher temperature compared to a trapezoidal one, indicating a lower heat transfer rate. Thermal performance is better in trapezoidal due to its tapered geometry and the efficient surface area utilisation. Fig. 6 illustrates the absolute error for the variation in wet porous parameter (m2) on temperature θ(X). Both fin types exhibit similar error trends. From the plot, it is evident that the absolute error is very small; most of the error increase in m2 is less than 10−8. Even the largest error is less than 10−5. Such a low magnitude error confirms the model is numerically stable and robust for engineering applications. Fig. 7 illustrates that as the porosity parameter (Sh) improves from 0.1 to 1.3, the temperature profiles θ(X) decrease along the fin length. Higher porosity strengthens the fluid-solid interaction and thereby supports the convective heat transfer. Consequently, fin dissipates heat more effectively throughout its length. Rectangular fin consistently preserves a higher temperature compared to trapezoidal fin, exhibiting tapered geometry endorses greater heat transfer. Fig. 8 demonstrates the absolute error for the corresponding variation in Sh on temperature θ(X). The absolute error varies from 10−12 from the base to 10−8 at the tip of the fin, which is extremely small. Even the largest error value is below 10−8, displaying computational model remains numerically accurate and reliable. The trapezoidal fin exhibits slightly higher absolute error compared with the rectangular fin because of its geometry, having a variable cross-sectional area. Fig. 9 illustrates the gradual loss of heat as a result of radiation and conduction throughout the length of the fin for different values of radiative parameter (Nr) from 0.5 to 2. Compared to the trapezoidal fin, the rectangular fin has a higher temperature, because of its shape, the trapezoidal fin benefits more. Fig. 10 portrays the absolute error for the temperature distribution θ(X) for the corresponding variation in Nr. The absolute error is relatively small, ranging from 10−12 to 10−6 for all the cases. This confirms the precision of the numerical model. Fig. 11 depicts the variation θ(X) along the fin length for different inclination angle values (τ) for both trapezoidal and rectangular fins. As T∗ increases from 300 to 900, θ(X) declines along the fin length. Fins at larger inclination angles dissipate more heat and lower fin temperature. The trapezoidal fin, due to its tapered profile, consistently exhibits lower values θ(X) than a rectangular fin. Fig. 12 represents the absolute error for the temperature distribution θ(X) for the corresponding variation in T∗. Extremely low magnitude error (below 10−7) confirms robustness and high accuracy of the computational approach. The rectangular fin exhibits consistently low absolute error compared to the trapezoidal fin.

Figure 5: Plot of m2vs.θ(X)

Figure 6: Absolute error of θ(X) for m2

Figure 7: Plot of Shvs.θ(X)

Figure 8: Absolute error of θ(X) for Sh

Figure 9: Plot of Nrvs.θ(X)

Figure 10: Absolute error of θ(X) for Nr

Figure 11: Plot of τvs.θ(X)

Figure 12: Absolute error of θ(X) for τ

4.2 Surface and Contour Plots

The surface and contour plots in Figs. 13 and 14 explain the decline in temperature θ(X) for the corresponding increase in fin length and angle of inclination (τ) for the rectangular fin, due to conductive-convective heat transfer. The consistent contours and even gradients validate the numerical model’s accuracy and stability. The results are useful for designing heat exchangers, electronic cooling systems, solar thermal devices, and radiators, where the proper selection of inclination angle and fin geometry can substantially augment heat transfer. Figs. 15 and 16 illustrate the decrease in temperature θ(X) with an increment in wet porous parameter (m2) and fin length for the rectangular fin. This decline corresponds to enhanced convective heat transfer within the porous medium. The obtained results are useful in the design of moisture-based cooling systems and porous heat exchangers, where optimized porosity and fin geometry can substantially enhance energy efficiency and thermal performance. A noticeable temperature θ(X) decline is observed due to the combined increase in fin length and radiative parameter (Nr) for the rectangular fin from Figs. 17 and 18. Stronger Nr corresponds to greater heat loss from the fin surface. The obtained results are significant for the design of high temperature fins, radiators, space thermal control systems, solar absorbers etc. where radiation has a dominant role in overall heat transfer. Figs. 19 and 20 demonstrate a progressive decrease in temperature θ(X) for the rectangular fin with the improving porosity parameter (Sh) and the fin length. This is because higher porosity enhances convective heat transfer within the fin material by allowing greater hybrid fluid penetration and interaction. Practically the results are valuable for optimizing the design of catalytic converters, electronic cooling modules and porous heat exchangers. The surface and contour plots, Figs. 21 and 22 depict a decline in temperature θ(X) with the corresponding increase in fin length and angle of inclination (τ) for the trapezoidal fin. This trend arises due to conductive-convective heat transfer. The results suggest that appropriate adjustment of fin geometry and inclination can improve electronic cooling systems, compact heat exchangers, and solar air heaters’ thermal efficiency. Figs. 23 and 24 represent the temperature distribution θ(X) is decreasing progressively in a trapezoidal fin due to the combined rise in wet porous parameter (m2) and the fin length. Higher values of m2 enhance convective heat transfer within the porous medium thereby promote improved heat dissipation from the fin’s surface. From Figs. 25 and 26, it is evident that the temperature distribution θ(X) declines in trapezoidal fin with the rise in both radiation and fin length. Rise in Nr facilitates greater energy dissipation from the fin surface. The result is applicable to engineering applications such heat exchangers, gas turbines, and air conditioning, electronic component and solar thermal collectors where radiation plays a vital role in thermal management. Figs. 27 and 28 demonstrate a progressive decrease in temperature θ(X) for the trapezoidal fin with the improving porosity parameter (Sh) and the fin length. An increase in porosity substantially promotes fluid permeability within the fin material; consequently, convective heat transfer is improved. These findings are extremely pertinent to engineering applications like small cooling devices and porous heat exchangers, among others.

Figure 13: Surface plot of θ for τ in rectangular fin case

Figure 14: Contour plot of θ for τ in rectangular fin case

Figure 15: Surface plot of θ for m2 in rectangular fin case

Figure 16: Contour plot of θ for m2 in rectangular fin case

Figure 17: Surface plot of θ for Nr in rectangular fin case

Figure 18: Contour plot of θ for Nr in rectangular fin case

Figure 19: Surface plot of θ for Sh in rectangular fin case

Figure 20: Contour plot of θ for Sh in rectangular fin case

Figure 21: Surface plot of θ for τ in trapezoidal fin case

Figure 22: Contour plot of θ for τ in trapezoidal fin case

Figure 23: Surface plot of θ for m2 in trapezoidal fin case

Figure 24: Contour plot of θ for m2 in trapezoidal fin case

Figure 25: Surface plot of θ for Nr in trapezoidal fin case

Figure 26: Contour plot of θ for Nr in trapezoidal fin case

Figure 27: Surface plot of θ for Sh in trapezoidal fin case

Figure 28: Contour plot of θ for Sh in trapezoidal fin case

4.3 Gradient Descent Optimization

The outcomes of the optimisation algorithm based on the Gradient Descent method.

Fig. 29 presents a linear fit plot of m2 for the rectangular fin and demonstrates a good correlation with R2 = 0.9970 with a low RMSE of 0.006882. This is a sign that there is great congruence between the predicted and observed values, and therefore, the fitted model was high in accuracy and reliability. Fig. 30 illustrates the linear fit plot of m2 for the trapezoidal fin, showing an excellent correlation with R2=0.9969 and a low RMSE = 0.007199. The close alignment between truth and fitted values confirms the high precision and reliability of the proposed model for trapezoidal fin performance prediction. The linear fit plot of Nr is shown in Fig. 31 for the rectangular fin, a high linear relationship with R2 = 0.9958 and a low RMSE of 0.01764 shows outcomes with good agreement between the truth values and the predictions. The linear fit plot of Nr is represented in Fig. 32 in the case of the trapezoidal fin, showing a very good linear correlation with R2 = 0.9957 and a low RMSE of 0.01838. The fitted data are almost the same assures the validity and precision of the model for forecasting the trapezoidal fin’s thermal performance. The plot of the linear fit of Sh is indicated in Fig. 33 for the rectangular fin, which shows a perfect correlation with R2 = 1 and an extremely low RMSE of 3.2517e−06. This signifies that there is an ideal linear relationship between which proves the outstanding accuracy and stability of the model for the rectangular fin thermal analysis. The linear fit plot of Sh for the trapezoidal fin is plotted in Fig. 34, and the linear relationship with R2 = 1 and a very low RMSE of 3.507e−06. This shows an excellent fit between the anticipated and real data, demonstrating the model’s accuracy and dependability for the trapezoidal fin thermal analysis. Fig. 35 depicts the linear regression graph of τ very strong correlation is observed with R2 = 0.9934 and minimum RMSE = 0.003322. The fact that the actual data and the fitted data confirm the validity and application of the linear model in estimating thermal behaviour in a rectangular fin. Fig. 36 illustrates the linear fit plot of τ with a strong correlation in the trapezoidal fin case with R2 = 0.9934 and a very low RMSE of 1.656e−05. The high ability of the proposed linear model with a good fit guarantees accurate prediction of the trapezoidal fin’s thermal performance. This indicates a strong linear dependence of the temperature distribution θ(X) on the governing parameters with ranges (π6≤τ≤π2), (0.2≤m2≤0.8), (0.1≤Sh≤1.3), and (0.5≤Nr≤2) for both trapezoidal and rectangular fin.

Figure 29: Fit Plot of m2 for rectangular fin

Figure 30: Fit Plot of m2 for trapezoidal fin

Figure 31: Fit Plot of Nr for rectangular fin

Figure 32: Fit Plot of Nr for trapezoidal fin

Figure 33: Fit Plot of Sh for rectangular fin

Figure 34: Fit Plot of Sh for trapezoidal fin

Figure 35: Fit Plot of τ for rectangular fin

Figure 36: Fit Plot of τ for trapezoidal fin

4.4 Sensitivity Analysis

A variance-based technique for global sensitivity analysis, Sobol sensitivity analysis calculates the degree to which each input parameter (τ,m2,Sh,Nr) and its interactions affect the overall variance of a model’s output (Heat Transfer). The total effect indices are used to account for the direct and indirect influence of a parameter, and the total output variance is broken down into fractions attributable to individual inputs (first-order indices) and combinations of inputs (higher-order indices). This approach helps discover important drivers in a complicated system by determining which parameters have the most influence. The results of the Rectangular Fin case’s Sobel Sensitivity Analysis are shown in Fig. 37 with the first-order (S1) and total-order (ST) indices with the 95% confidence interval. The ST values of 1, 2, and 4 are very high (around 0.96–0.98), and this proves that the impact is overall strong, whereas the ST of Input 3 is rather high (S1 = 0.72), which means that it significantly influences the output. The Sobel Sensitivity Analysis results for the Trapezoidal Fin case are shown in Fig. 38 in colour-enhanced bars. The ST indices (pink) of the inputs of 1, 2 and 4 are close to 0.97–0.98, representing the overall influence, whereas the highest S1 is found with the input of 3 (0.75), representing the strongest direct impact on the model output.

Figure 37: Sensitivity analysis plot for rectangular fin case

Figure 38: Sensitivity analysis plot for trapezoidal fin cas

4.5 Stability Analysis and Grid Independence Test

Fig. 39 shows that the results for both the medium and fine meshes agree in the rectangular and trapezoidal cases, indicating that the current study meets the criteria for grid independence testing. Fig. 40 depicts a linear rise in heat transfer with the parameter Nr, which means a predictable and stable relationship. The fitted line Q=0.450+0.486Nr comes close to the data points, and all values lie within specification limits and hence, good numerical stability for the rectangular fin case. Fig. 41 represents a sharp linear growth in heat transfer with the parameter Nr passing through the fitted line, Q=0.470+0.501Nr forecasts are close to the data points, and they are within the specification limits, meaning that the trapezoidal fin case displays stable, reliable numeric behaviour.

Figure 39: Grid independence test

Figure 40: Numerical stability analysis for the rectangular fin

Figure 41: Numerical stability analysis for the trapezoidal fin

5  Conclusion

The thermal behaviour of a wetted hybrid nanofluid with inclined trapezoidal and rectangular porous fins was investigated in this work in MATLAB software with the Lobatto IIIa collocation method using the BVP5C solver to examine the impact of the inclination angle (τ), parameter for wet porous (m2), porosity parameter (Sh) and radiative parameter (Nr) for both rectangular and trapezoidal fins, and Gradient Descent optimization applied to predict the heat transfer rate in rectangular and trapezoidal fins scenarios.

Key Findings of the Present Study:

❖   The temperature decreases progressively with the rise in the inclination angle (τ), parameter for wet porous (m2), porosity parameter (Sh) and radiative parameter (Nr), indicating substantial heat dissipation.

❖   Trapezoidal fin consistently exhibits better heat transfer mechanism than rectangular fin due to its tapered geometry.

❖   Due to the increased permeability and solid-fluid interaction within the fin, the wet porous parameter and porosity significantly lower the temperature of the hybrid nanofluid.

❖   Integrating both radiative and convective heat transfer leads to effective thermal performance.

❖   The absolute error remains extremely small 10−12–10−6, validating that the adapted numerical method is highly accurate and stable.

❖   The linear regression results reveal excellent correlation between fitted and computed data, with R2 values are (≈1) confirming the consistency and reliability of the model.

Limitations of the Present Study:

❖   The present study examined a hybrid nanofluid only

❖   Comparison of results with experimental study

❖   In the present study, limited convective and radiative effects on heat transfer were considered.

❖   This study is limited to steady-state heat transfer

Future Scope:

To simplify the model, it was assumed that there was a perfect stationary state with homogeneous throughout flotation of the hybrid nanofluid. In the future, the aggregation, sedimentation and stability of nanoparticles will be taken into consideration with the aim of making the model have more practical considerations.

Acknowledgement: Not applicable.

Funding Statement: This research was supported by the “Regional Innovation System & Education (RISE)” through the Seoul RISE Center, funded by the Ministry of Education (MOE) and the Seoul Metropolitan Government (2025-RISE-01-027-04).

Author Contributions: Maddina Dinesh Kumar: Writing—original draft, Methodology, Resources, Software, Visualization, Validation. S. U. Mamatha: Writing—original draft, Methodology, Investigation, Formal analysis. Khalid Masood: Validation, Conceptualization, Methodology, Formal analysis. Nehad Ali Shah: Investigation, Supervision, Formal analysis, Methodology, Conceptualisation. Se-Jin Yook: Writing—review & editing, Software, Supervision, Project administration. Maddina Dinesh Kumar and Nehad Ali Shah contributed equally to this work and are co-first authors. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Data will be available with the corresponding author on reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Nomenclature

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