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.

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 (T_b) 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] (1)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)=0where, 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: (2)T=Tbatx=L (3)dTdx=0atx=0.

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: (4)X=xL,θ=T−T∞Tb−T∞.

Eq. (5) is obtained after solving by substituting Eq. (4) into Eq. (1) (5)d2θdX2−CdθdX−CXd2θdX2−((ρCp)hnf(ρCp)f)((ρβ)hnf(ρβ)f)(μfμhnf)(kfkhnf)Shsin⁡τθ2−…Nr(kfkhnf)θ−(kfkhnf)m2θ2=0

Following the non-depersonalization process, the boundary conditions turn into (6)θ=1atX=0 (7)dθdX=0atX=1

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, (8)d2θdX2−CdθdX−CXd2θdX2−bθ2−aθ=0

Fin’s heat transfer rate by using the study of Khan et al. [35]: Q=−khnfkfθ′(0)

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.

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.

Limitations of the Present Study: ❖

The present study examined a hybrid nanofluid only

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.