A hybrid procedure FLT-HPM was proposed in this study, by combining the homotopy perturbation method (HPM) with Fourier transform and Laplace transform which aimed to find an approximate analytical solution to the problem of two-dimensional transient natural convection in a horizontal cylindrical concentric annulus bounded by two isothermal surfaces. The effect of the Grashof number, Prandtl number, and the radius ratio on fluid flow (air) and heat transfer with different values awreas discussed. Moreover, the velocity distributions and the mean Nusselt numbers were studied, and the Nusselt numbers were used to represent local and general heat transfer rates. Finally, the convergence of FLT-HPM was tested theoretically through the proof of some theorems. In addition, these theorems were applied to the results of the new solutions obtained using FLT-HPM.

Helical or spiral coiled heat exchangers, prevalent in industries such as power generation, heat recovery systems, the food sector, and various plant processes, exhibit potential for performance enhancement through optimal fluid selection. Notably, nanofluids, distinguished by their superior thermophysical properties, including enhanced thermal conductivity, viscosity, and convective heat transfer coefficient (HTC), are considered viable candidates. In this study, the thermo-physical attributes of helical coil heat exchangers (HCHEs), when subjected to nanofluids, were meticulously examined. During the design phase, Creo parametric design software was employed to refine the geometric configuration, subsequently enhancing fluid flow dynamics, thereby yielding a design improvement for the HCHE. Subsequent computational fluid dynamics (CFD) simulations of the heat exchanger were conducted via the ANSYS CFX program. A CuO/water nanofluid, at a 1% volume fraction, served as the basis for the CFD analysis, incorporating the Re-Normalisation Group ($k-\varepsilon$) turbulence model. From these simulations, zones exhibiting elevated temperature and pressure were discerned. It was observed that the wall HTC value for the CuO/water mixture surpassed that of pure water by 10.01%. Concurrently, the Nusselt number, when the CuO/water nanofluid was employed, escalated by 6.8% in comparison to utilizing water alone. However, it should be noted that a 5.43% increment in the pressure drop was recorded for the CuO/water nanofluid in contrast to pure water.

In the quest to design a robust model for microwave heating systems with symmetrical octagonal tube cavities (MWHSO), a fuzzy-based approach, specifically the Takagi Sugeno Fuzzy Model, was explored to capture the dynamics of the heating process. To achieve this, the mathematical model was adaptively adjusted according to varying input conditions through the utilization of fuzzy logic. Input data were sourced from two magnetrons, with the system outputs derived from measurements acquired from five temperature sensors placed on the heated object. For performance evaluation, the Root Mean Square Error (RMSE) was employed. A comparison was drawn with the autoregressive model with exogenous variable (ARX), a traditional approach wherein the system's mathematical model remains static. Simulation studies were conducted, treating every probe measurement across all dataset validations as distinct cases. It was found that the T-S Fuzzy model surpassed the ARX40 in performance in 33 of the total cases, accounting for 92.49%. The most notable performance of the fuzzy-based approach was observed at a 180-Watt power input, recording an average RMSE of 0.00574 across the five sensors. In contrast, the ARX-based model registered an RMSE of 0.01256. These findings suggest that the fuzzy-based modeling approach presents a compelling alternative for representing the dynamic heating processes within MWHSO.

This investigation elucidates the intertwined effects of magnetic fields and porous media on the flow of nanofluids towards a stretching sheet, contemplating variable viscosity and convective boundary conditions. A nanofluid model, incorporating the influences of thermophoresis and Brownian motion, is adopted. Via judicious transformations, the fundamental governing coupled non-linear partial differential equations are condensed, and the consequent transformed equations are numerically resolved employing the Finite Element Method (FEM). Paramount emphasis is accorded to parameters embodying notable physical significance, inclusive of the Prandtl number (Pr), Hartmann number, Lewis number (Le), Brownian motion number (Nb), thermophoresis number (Nt), and permeability parameter. The numerical results acquired, as particular instances of the aforementioned study, are found to be congruent with previously reported findings, substantiating the accuracy and reliability of the proposed methodology. A thorough examination of the collective impact of the selected parameters on flow and heat transfer characteristics has been systematically undertaken, revealing intricate dependencies and fostering a deeper understanding of the complex phenomenon under consideration. This study, hence, paves a pathway towards bolstering the comprehension of flow mechanics in porous media under the influence of magnetic fields, contributing valuable insights to the overarching field of fluid dynamics in nano-engineering applications.

In this study, a numerical investigation into heat transfer and entropy generation characteristics using confined-slot jet impingement was conducted. Comparisons were drawn between the heat transfer and entropy generation attributes of two wing ribs positioned on the heated impinging target surface and those of a rib-less surface. The influences of variations in the spacing between the stagnation point and the rib (B) of (10-30 mm), ranging from 10 to 30 mm, rib heights (A) between 0.5 to 2 mm, and a Reynolds number of the jet (Re) between 3000 to 8000 on fluid flow, heat transfer, and entropy generation were elucidated. Employing the Finite Volume Method (FVM) managed the continuity, momentum, and energy equations in adherence to the principles of the SIMPLE methodology. Results revealed that the Nusselt number $(\overline{N u})$, pressure drop, and total entropy $\left(\bar{S}_{\text {total }}\right)$ escalated in accordance with Re and A. Conversely, they diminished with reduced spacing from the stagnation point to B. Notably, a superior heat transfer rate was observed when employing a target plate integrated with wing ribs in contrast to a rib-less configuration. Performance Evaluation Criteria (PEC) values were noted to augment with rib height increment. It is demonstrated that the PEC increases as A increases. Also, the lower value of PEC equals 1.044 at A of 2 mm, B of 10 mm, and Re of 8000, while the higher value of the PEC equals 1.68 at A of 2 mm, B of 10 mm, and Re of 3000. The findings suggest that slot-Jet impingement complemented by wing ribs plays a pivotal role in enhancing the cooling efficiency of electronic devices.