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The purpose of this paper is to present a numerical analysis of the flow and heat transfer in a tube with a wire coil insert. A second law analysis of the results is accounted for, in order to assess the local and overall entropy generation in relation with the increased pressure drop and convective heat transfer. A wire coil with p/D=1.25 and e/D=0.076 is selected as insert device. A Reynolds number range between 100 and 1,000 is investigated, which corresponds to the typical operating regimes in the risers of liquid solar collectors. Different wall heat fluxes and inclination angles allow to analyze the potential impact of mixed convection in the presence of tube inserts.

Three-dimensional numerical simulations are performed using a finite-volume solver, assuming laminar flow conditions. Pure water and a mixture of water and propylene-glycol (20 percent) are used as working fluids, with temperature-dependent properties. Fanning friction factor, Nusselt number and local entropy generation results are obtained in the fully developed region.

The friction factor results are successfully compared with a well-known experimental correlation for wire coil inserts. The earlier onset of transition is devised at Re > 300. Nusselt number augmentations between 2.5- and 6-fold are reported with respect to the smooth tube. The mixed convection regime encountered in the smooth tube for the operating conditions investigated is canceled in the wire coiled tube, owing to the opposed effect of the swirl flow induced and the bouyancy forces. Frictional, heat transfer and overall entropy generation rates are computed locally in the fully developed region, allowing to relate these results with the flow structures in the mixed convection smooth tube and in the wire coiled tube. A threefold decrease in the entropy generation rate is reported for tubes with wire coil inserts.

An holistic understanding of the heat transfer enhancement in tubes with wire coil inserts is provided through the analysis of the flow pattern, Fanning friction factor, Nusselt number and local entropy generation rates. The reduced entropy generation in the enhanced tube serves as a performance criteria to confirm the positive effect of wire coil inserts in heat transfer for the operating regime under investigation, in spite of the increased pressure drop.

This study aims to numerically scrutinize the entropy generation minimization and mixed convective heat transfer of multi-walled carbon nanotubes–Fe₃O₄/water hybrid nanofluid flow in a lid-driven square enclosure with heat generation in the presence of a porous layer on inner surfaces, considering local thermal non-equilibrium (LTNE) approach and the non-Darcy flow model.

The dimensionless governing equations for hybrid nanofluid and solid phases are solved by applying the finite volume method and semi-implicit method for pressure-linked equations algorithm.

The roles of the internal heat generation in the porous layer, LTNE model and nanoparticles volume fraction on mixed convection phenomenon and entropy generation are introduced for lid-driven cavity hybrid nanofluid flow. Based on the investigation of entropy generation and heat transfer, the minimum total entropy generation and average Nusselt numbers are found at 1 ≤ Ri ≤ 10 where the effect of the forced and free convection flow directions being opposite each other is very significant. When considering various nanoparticle volume fractions, it becomes evident that the minimum entropy generation occurs in the case of φ = 0.1%. The outcomes of LTNE number reveal the operating parameters in which thermal equilibrium occurs between hybrid nanofluid and solid phases.

The analysis of entropy generation under various shear and buoyancy forces plays a significant role in the suitable thermal design and optimization of mixed convective heat transfer applications. This research significantly contributes to the optimization of design and the advancement of innovative solutions across diverse engineering disciplines, such as packed-bed thermal energy storage and thermal insulation.

This study aims to numerically investigate the thermal-hydrodynamic performance of silicon oxide/water nanofluid laminar flow in the heat sink miniature channel with different fin cross-sections. The effect of the fin cross-section including semi-circular, rectangular and quadrant in two directions of flat and curved, and channel substrate materials of steel, aluminum, copper and titanium were examined. Finally, the analysis of thermal and frictional entropy generation in different channels is performed.

According to the numerical results, the highest heat transfer coefficients belong to the rectangular, quadrant 2, quadrant 1 and semi-circular fins compared to the channel without fin is 38.65%, 29.94%, 27.45% and 17.1%, respectively. Also, the highest performance evaluation criteria belong to the rectangular and quadrant 2 fins, which have 1.35 and 1.29, respectively. Based on the thermal conductivity of the substrate material, the best material is copper. According to the results of entropy analysis, the reduction of thermal irreversibility of the channel with rectangular, quadrant 1, quadrant 2 and semi-circular compared to non-finned channel is equal to 72%, 57%, 63% and 48%, respectively.

The rectangular and quadrant 2 fins are the best fins and the copper substrate material is the best material to reduce the entropy generation.

The silicon oxide/water nanofluid flow in the heat sink miniature channel with various fin shapes and the curvature angle against the fluid flow was simulated to increase the heat transfer performance. The whole test section is simulated in three-dimensional. Different channel materials have been investigated to find the best channel substrate material.

The main purpose of this numerical study is to study on entropy generation in natural convection of nanofluid in a wavy cavity using a single-phase nanofluid model.

The cavity is heated non-uniformly from the wavy wall and cooled from the right side while it is insulated from the horizontal walls. The physical domain of the problem is transformed into a rectangular geometry in the computational domain using an algebraic coordinate transformation by introducing new independent variables ξ and η. The governing dimensionless partial differential equations with corresponding initially and boundary conditions were numerically solved by the finite difference method of the second-order accuracy. The governing parameters are Rayleigh number (Ra = 1000-100000), Prandtl number (Pr = 6.82), solid volume fraction parameter of nanoparticles (φ = 0.0-0.05), aspect ratio parameter (A = 1), undulation number (κ = 1-3), wavy contraction ratio (b = 0.1-0.3) and dimensionless time (τ = 0-0.27).

It is found that the average Bejan number is an increasing function of nanoparticle volume fraction and a decreasing function of the Rayleigh number, undulation number and wavy contraction ratio. Also, an insertion of nanoparticles leads to an attenuation of convective flow and enhancement of heat transfer.

The originality of this work is to analyze the entropy generation in natural convection within a wavy nanofluid cavity using single-phase nanofluid model. The results would benefit scientists and engineers to become familiar with the flow behaviour of such nanofluids, and will be a way to predict the properties of this flow for the possibility of using nanofluids in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

This study aims to analyze the impact of fractional derivatives on heat transfer and entropy generation during transient free convection inside various complex porous enclosures, such as triangle, L-shape and square-containing wavy surfaces. These porous enclosures are saturated with Cu-water nanofluid and subjected to the influence of a uniform magnetic field.

In the present study, Darcy’s model is used for the momentum transport equation in the porous matrix. Additionally, the Caputo time fractional derivative is introduced in the energy equation to assess the heat transfer phenomenon. Furthermore, the total entropy generation has been computed by combining the entropy generation due to fluid friction (S_ff), heat transfer (S_ht) and magnetic field (S_mf). The complete mathematical model is further simulated using the penalty finite element method, and the Caputo time derivative term is approximated using the L1 scheme. The study is conducted for various ranges of the Rayleigh number (102≤Ra≤104), Hartmann number (0≤Ha≤20) and fractional order parameter (0<α<1) with respect to time.

It has been observed that the fractional order parameter α governs the characteristics of entropy generation and heat transfer within the selected range of parameters. The Bejan number associated with heat transfer (Be_ht), fluid friction (Be_ff) and magnetic field (Be_mf) further demonstrate the dominance of flow irreversibilities. It becomes evident that the initial evolution state of streamlines, isotherms and local entropy varies according to the choice of α. Additionally, increasing Ra values from 10² to 10⁴ shows that the heat transfer rate increases by 123.8% for a square wavy enclosure, 7.4% for a triangle enclosure and 69.6% for an L-shape enclosure. Moreover, an increase in the value of Ha leads to a reduction in heat transfer rates and entropy generation. In this case, Bemf→1 shows the dominance of the magnetic field irreversibility in the total entropy generation.

Recently, fractional-order models have been widely used to express numerous physical phenomena, such as anomalous diffusion and dispersion in complex viscoelastic porous media. These models offer a more accurate representation of physical reality that classical models fail to capture; this is why they find a broad range of applications in science and engineering.

The fractional derivative model is used to illustrate the flow pattern, heat transfer and entropy-generating characteristics under the influence of a magnetic field. Furthermore, to the best of the author’s knowledge, a fractional-derivative-based mathematical model for the entropy generation phenomenon in complex porous enclosures has not been previously developed or studied.

The purpose of this paper is to review previous research studies on mathematical models for entropy generation in the magnetohydrodynamics (MHD) flow of nanofluids. In addition, the influence of various parameters on the velocity profiles, temperature profiles and entropy generation was studied. Furthermore, the numerical methods used to solve the model equations were summarized. The underlying purpose was to understand the research gap and develop a research agenda.

This paper reviews 141 journal articles published between 2010 and 2022 on topics related to mathematical models used to assess the impacts of various parameters on the entropy generation, heat transfer and velocity of the MHD flow of nanofluids.

This review clarifies the application of entropy generation mathematical models, identifies areas for future research and provides necessary information for future research in the development of efficient thermodynamic systems. It is hoped that this review paper can provide a basis for further research on the irreversibility of nanofluids flowing through different channels in the development of efficient thermodynamic systems.

Entropy generation analysis and minimization constitute effective approaches for improving the performance of thermodynamic systems. A comprehensive review of the effects of various parameters on entropy generation was performed in this study.

The purpose of this paper is to investigate conjugate heat transfer of natural convection and entropy generation of nanofluid in the presence of external magnetic field via numerical approach in an inclined square cavity enclosure.

Control volume finite volume method with collocated arrangement of grids was used for discretization of continuity, momentum, solid and fluid energy equations. Rhie and Chow interpolation technique was applied to avoid checkerboard problem in pressure field and the well-established SIMPLE algorithm was followed to deal with the pressure and velocity coupling. The cavity is filled with water and nanoparticles of the aluminum oxide (Al₂O₃). This study has been conducted for the certain pertinent parameters of the volume fraction of nanoparticle (φ = 0–0.08), the angle of inclination (ϴ = 0°–330°), the Ra number (Ra = 103–108), the solid to fluid conductivity ratio (k_sf = 1–400), the Ha number (Ha = 0–80) and the wall thickness ratio (δ/L = 0–0.3).

The results indicate that averaged Nu number increases by approximately 9% by increasing volume fraction from 0.0 to 0.08. Nu increases with an increasing inclination angle to 40° and decreases abruptly in 90° because of the formation of two weaker vorticity with opposite circulation pattern intensifying the density of isotherm curves in a vertical direction. Nu increases sharply with increasing Ra more than 105. Nu also augments almost 67% by increasing k_sf = 1 to k_sf = 50 and remains constant by increasing k_sf more than 50. Nu number reduction is almost 72% with a variation of wall thickness ratio from d/L = 0 to 0.3. Entropy generation because of fluid flow, magnetic field and heat transfer reduces linearly almost 30%, 19% and 16% by increasing volume fraction, respectively. With increasing k_sf, entropy generation because of fluid flow, magnetic field and heat transfer increases asymptotically, but Bejan number decreases.

A brief review of conducted research studies in nanofluid flow and heat transfer reveals that the effect of wall thermal inertia was not investigated in MHD natural convection of nanofluids in an inclined enclosure. The aim of the present study is to analyze conjugate heat transfer in an inclined cavity filled with water and Al₂O₃.

The aim of this paper is to investigate performance improvements of a monolithic solid oxide fuel cell geometry through an entropy generation analysis.

The analysis of entropy generation rates makes it possible to identify the phenomena that cause the main irreversibilities in the fuel cell, to understand their causes and to propose changes in the design and operation of the system. The various contributions to entropy generation are analyzed separately in order to identify which geometrical parameters should be considered as the independent variables in the optimization procedure. The local entropy generation rates are obtained through 3D numerical calculations, which account for the heat, mass, momentum, species and current transport. The system is then optimized in order to minimize the overall entropy generation and increase efficiency.

In the optimized geometry, the power density is increased by about 10 per cent compared to typical designs. In addition, a 20 per cent reduction in the fuel cell volume can be achieved with less than a 1 per cent reduction in the power density with respect to the optimal design.

The physical model is based on a simple composition of the reactants, which also implies that no chemical reactions (water gas shift, methane steam reforming, etc.) take place in the fuel cell. Nevertheless, the entire procedure could be applied in the case of different gas compositions.

Entropy generation analysis allows one to identify the geometrical parameters that are expected to play important roles in the optimization process and thus to reduce the free independent variables that have to be considered. This information may also be used for design improvement purposes.

In this paper, entropy generation analysis is used for a multi‐physics problem that involves various irreversible terms, with the double use of this physical quantity: as a guide to select the most relevant design geometrical quantities to be modified and as objective function to be minimized in the optimization process.

The purpose of this paper is to study the influence of magnetic field on entropy generation and natural convection inside an enclosure filled with a hybrid nanofluid and having a conducting wavy solid block. Also, the effect of fluid–solid thermal conductivity ratio is investigated.

The governing equations that are formulated in the dimensionless form are discretized via finite volume method. The velocity–pressure coupling is assured by the SIMPLE algorithm. Heat transfer balance is used to verify the convergence. The validation of the numerical results was performed by comparing qualitatively and quantitatively the results with previously published investigations.

The results indicate that the magnetic field and the conductivity ratio of the wavy solid block can significantly affect the dynamic and thermal field and, consequently, the heat transfer rate and entropy generation because of heat transfer, fluid friction and magnetic force.

To the best of the authors’ knowledge, the present numerical study is the first attempt to use hybrid nanofluid for studying the entropy generation because of magnetohydrodynamic natural convective flow in a square cavity with the presence of a wavy circular conductive cylinder. Irreversibilities due to magnetic effect are taken into account. The effect of fluid–solid thermal conductivity ratio is considered.

The purpose of this paper is to study the effects of using different Brownian models on natural and mixed convection fluid flow and heat transfer inside the square enclosure filled with the AlOOH–water nanofluid.

Due to fulfill of this demand, five different models for the effective thermal conductivity and viscosity of the nanofluid are considered. The following results are presented for the Ra=10⁷ to 10¹⁰ and Ri=0.01 to 100, whereas the volume fraction of the nanoparticles is varied from φ = 0.01 to 0.04.

According to the obtained results, increasing of Rayleigh number and reduction of Richardson number leads to the higher values of the average Nusselt number and entropy generation. Also, it is realized that, variation trend of the average Nusselt number and entropy generation in all cases is increasing by growing the volume fraction. It is found that the obtained average Nusselt numbers and entropy generations with Koo and Kleinstreuer are the highest among all the studied cases, and it is followed by Patel, Vajjha and Das, Corcione and Maxwell–Brinkman models, respectively.

Based on the results of present investigation, the Nusselt number difference predicted between the Maxwell–Brinkman model (as constant-property model) and Koo and Kleinstreuer model is about 7.84 per cent at 0.01 per cent volume fraction and 5.47 per cent at 0.04 per cent volume fraction for the Rayleigh number equal to 107. The entropy generation difference predicted between the two above studied model is about 8.05 per cent at 0.01 per cent volume fraction and 5.86 per cent at 0.04 per cent volume fraction for the Rayleigh number equal to 107. It is observed that using constant-property model has a significant difference in the obtained results with the results of other variable-property models.