Thermo-hydrodynamic analysis and exergy destruction minimization in a three- dimensional corrugated

Abstract

Keywords: Heat Transfer; Hydrodynamic Behavior; Magnetic Field; Numerical Modeling; Ferrofluid; Exergy Destruction

Introduction

Recent academic research has increasingly focused on innovative methods to increase heat transfer through convection phenomena. Techniques such as the application of a magnetic field, the incorporation of corrugated walls, and the use of nanofluids have been explored. These advancements are recognized for their potential to significantly improve the efficiency of various systems, including solar panels and electronic cooling devices. By optimizing heat transfer processes, these methods contribute to more effective thermal management, leading to enhanced performance, reduced energy consumption, and increased longevity of technological and industrial applications. Corrugated channels are extensively used in plate heat exchangers which are required in the fields of air-conditioning, chemical reactors, thermal power plants, etc. The presence of corrugations in the geometry of these channels promotes turbulence and increases the surface area, leading to improved thermal performance. Such equipments have been extensively studied in recent years. Varol and Oztop [1] demonstrated that corrugated geometries can enhance heat transfer under free convection in a wavy enclosure filled with a single fluid. Additionally, Ahmed [2] reported that corrugated surfaces enhance the heat resistance of enclosures. According to a study by Nasrin et al. [3], using corrugated surfaces instead of flat surfaces results in an average Nusselt number that is 14.7% greater. The impact of corrugated channels on thermal transfer performance was also explored by Ajeel et al. via both numerical and experimental methods [4,5]. Zhang et al. [6] demonstrated that introducing overall curvature in conventional wavy channels improves heat transfer when the wave amplitudes are set to 0 mm, 0.40 mm, and 0.80 mm. However, as the amplitude increases, the positive effect of the curvature on heat transfer diminishes. Tian et al. [7] reported that channels with varying wave patterns generally outperform those with uniform wave patterns in terms of thermal performance. Liu and Chen performed numerical analyses to examine the impact of waveform channels, porous layer characteristics, cooling-wall subcooling, and inlet air mass fractions on flow and condensation at different inlet velocities. These findings, supported by experimental data, reveal that the vortex distribution and air accumulation play crucial roles in influencing the condensation heat transfer [8]. Sheikholeslami et al. [9] conducted an analysis of a photovoltaic thermal unit integrated with a Thermoelectric

Generator (TEG) module and a nanofluid filter. These findings revealed that increasing the velocity of the nanofluid filter significantly enhanced the electrical performance of the system. With the addition of a reflector, the electrical efficiency improved by 37.7%. Sheikholeslami and Khalili [10] introduced a cutting-edge design for a photovoltaic thermal system that features a nanofluid splitter. This trailblazing study is the first worldwide study to assess the spectral behaviour of a system and employ nanofluids to enhance irradiation, ensuring that the bandgap wavelength effectively reaches the silicon layer. Roy et al. [11] conducted a numerical study on laminar flow heat transfer using Al3O3/ethylene glycol and Al3O3/water nanofluids in a radial flow system. Their research revealed a notable increase in the heat transfer rate. They also reported that the wall shear stress increased with both the concentration of nanoparticles and the Reynolds number. Liang Zhang et al. [12] introduced SiO2 nanoparticles with a size of 15 nm into water. The results demonstrated that the convective heat transfer coefficient of the nanofluids increased by 36.8% compared with that of pure water at the same Reynolds number. The literature includes numerous numerical studies on nanofluids [13-19]. Sheikholeslami et al. [20] reported that the temperature gradient increases with the volume fraction of Fe3O4 and the Rayleigh number. The literature includes numerous numerical studies on Fe3O4/water [21-29], which have been employed to increase the heat transfer. Sheikholeslami [30] discussed an innovative numerical approach demonstrating nanofluid magnetohydrodynamic (MHD) flow through a porous enclosure. Additionally, a numerical approach was investigated to analyse the thermal behavior of alumina nanofluids in a duct. A neural network was employed to estimate the heat transfer rate. Sheikholeslami and Ellahi [31] investigated the presence of a magnetic field in a nanofluid flow within a three-dimensional coordinate system. Their study revealed that the presence of the magnetic field increases the resistive (drag) force and reduces the convection current. Lajvardi et al. investigated forced convection heat transfer using a 5% solid volume concentration of Fe3O4/water ferrofluid flowing over a hot tube under a constant magnetic field. Their findings revealed that, in the absence of the magnetic field, the use of Fe3O4 nanoparticles as a dispersed phase in water does not improve the convective heat tranfer in the laminar flow regime [32]. Qiang et al. [33] reported that as the magnetic field is strengthened parallel to the temperature

gradient, the thermal conductivity of magnetic nanofluids increases. This phenomenon occurs because particle chains form within magnetic nanofluids along the direction of the temperature gradient. These chains establish a highly efficient, low-resistance pathway for energy transmission when the magnetic field aligns with the gradient. Ghufran et al. [34] investigated the forced convection heat transfer of a Fe3O4/water nanofluid in a circular tube under the influence of both constant and alternating magnetic fields. The static magnetic field negatively impacted the convective heat transfer capacity of the magnetic nanofluid, whereas the use of an alternating magnetic field increased the heat transfer rate by up to 27.6%. In their study, Fadaei et al. [35] reported that applying a magnetic field intensifies fluid mixing along the pipe length, consequently increasing the Nusselt number. This enhancement underscores the effectiveness of magnetic fields in augmenting heat transfer characteristics within the fluid. Olayemi et al. [36] reported that ferromagnetic Fe3O4 nanofluids demonstrate better thermal conductivity than ferromagnetic Mn-ZnFe2O4 nanoparticles do. Petrini et al. showed through numerical simulations that the convective heat transfer rate significantly varies depending on the position and orientation of the magnets. They reported that with an optimal magnet configuration, the average Nusselt number increases by more than 51% compared with that under calm flow conditions [37]. Many other studies exist in the literature on the effects of magnetic fields on flow structure and heat transfer [38-41]. All these studies have been published recently and report heat transfer enhancement when the magnetic field is properly oriented. The effect of a magnetic field on nanofluid entropy and exergy has also been reported in many recently published works. Sheikholeslami [42] conducted a comprehensive study on the entropy and exergy analysis of nanofluid

flow through a porous medium under the influence of the Lorentz force. He incorporated the non-Darcy model to simulate the flow dynamics. The results revealed that the Bejan number decreases as the permeability of the porous medium decreases.Tabarhoseini and Sheikholeslami [43] connected the concepts of entropy generation and thermal analysis by developing a mathematical model to study nanofluid flow within the geometry of an evacuated tube. Similar research [44–48] has been conducted on various multiphysics issues. One can conclude from the above review that magnetic fields play a crucial role in many scientific and engineering fields, yet, to the best of our knowledge, the impact of a uniform magnetic field on convective heat transfer in a three-dimensional, triangle-shaped, corrugated channel filled with ferrofluid Fe3O4-water has not been thoroughly studied. In our research, we chose this specific channel geometry to address this gap and to understand the influence of the magnetic field on the thermodynamic and hydrodynamic behaviours of a ferrofluid. This study therefore explores the effects of variations in magnetic field strength (B), particle volume fraction (ϕ), and Reynolds number (Re) on flow dynamics, pressure, entropy generation, and heat transfer of the ferrofluid Fe3O4-water.

Problem Description

Figure 1 schematically illustrates the channel model under examination. The model consists of a three-dimensional channel with a width of 50 mm (L1). It features two heights: a maximum of 16 mm and a minimum of 10 mm. The channel incorporates both a corrugated wall and a flat wall. Each adiabatic flat wall extends 40 mm (L2) before and after the corrugated section, with an axial pitch length (e) of 20 mm. The top and bottom corrugated walls are subjected

to a constant heat flux of 50 kW/m², whereas the flat walls are thermally insulated. A uniform magnetic field is applied vertically across the channel. As depicted in Figure 1, this constant magnetic field is generated by an electromagnet. The channel contains an Fe3O4-water nanofluid, which is characterized as incompressible and Newtonian. The flow is three-dimensional and laminar. Table 1 provides a summary of the physical characteristics of the magnetite nanoparticles. The governing equations are derived on the basis of the following assumptions: • The flow is three-dimensional. • The ferrofluid behaves as a laminar and incompressible fluid. • The base fluid and Fe3O4 nanoparticles are treated as a single phase. • The ferrofluid flow follows a Newtonian model. • The Fe3O4 nanoparticles are assumed to be spherical. • The magnetic field is applied in the Y direction. The equations are formulated as follows: Continuity Equation (1) Momentum Equation (2)

(8) The magnetic susceptibility, denoted χm, is expressed as follows [51]: (9) Equations 7 and 9 can be used to calculate the magnetic force, as follows [52]: (10) The electrical system is located in the electromagnet air gap, which only has a magnetic field applied in the Y direction [49]. As a consequence, can be expressed in this way: (11) In the current investigation, the characteristics of pure water were considered to be temperature dependent. Consequently, the models described in reference [53] have been employed: (12) (13)

Energy Equation (3) Where is the magnetic volume force, also referred to as the Kelvin body force, and is computed via the following formula: (4) Maxwell’s equations enable the representation of the magnetic field through equations (5) and (6) [50]. (5) (6) The following relationship connects M and H:

(14) Additionally, the following correlations were found to be appropriate for determining the characteristics of the ferrofluid [54]: (15) (16) (17) (18)

(7) With the following definition of the magnetization vector [51]:

Formula for Calculating the Flow Nusselt Number The definition of the local Nusselt number (NuL)is as follows [55]:

Tm and Tw stand for the mean temperature of the fluid and the local temperature of the wall, respectively. The average Nusselt number (Nuave) can be obtained by calculating the average value of the local Nusselt numbers in the top and bottom corrugated walls. (20) Where A is the corrugated walls area. Formula for Calculating the Entropy Production and Exergy Destruction The local entropy production is calculated using the following formula [56]: S = Sht+Sff

Where Sht and Sff represent the entropy produced by heat transport and fluid friction, respectively. (22)

The exergy destruction was calculated using the following formula [57]: (24) Where Sgen is the global entropy production and is obtained using the formula: (25)

Flow Boundary Conditions Inlet Conditions - The ferrofluid enters at a uniform velocity Uin, calculated from the Reynolds number using the formula: Where dh is the hydraulic diameter Sentry being the inlet section and Pwetted its wetted perimeter. - The inlet temperature is uniform and equal to 298 K. Channel (Boundary) Conditions - A constant heat flux of 50 kW/m² is applied to the upper and lower corrugated channels. - The non-corrugated walls are insulated. Outlet Conditions - Pressure outlet conditions were applied at the outlet. Numerical Modelling, Grid Independence, and Numerical Method Validation The flow and heat transfer governing equations were numerically solved using the ANSYS FLUENT solver, which employs a finite volume approach. User-defined functions (UDFs) were developed and integrated into the program’s code to incorporate magnetic effects. The velocity‒pressure relationship was managed using a coupled algorithm, which in ANSYS Fluent, simultaneously solves the momentum and continuity equations to ensure accurate and efficient calculation of the interdependent velocity and pressure fields, whereas the convective term was discretized using the second-order upwind method. Residual levels less than 10-6 were considered. Six sets of grid systems were evaluated to determine the most suitable mesh that ensures good accuracy. The results, shown in Table 2, depict the Nu variation as a function of the Re for pure water. The various cases did not significantly differ from each other. Therefore, for this study, a grid size of 500,000 was selected. Figure 2 displays the generated grid. The present results have been validated through two comparisons. The initial validation involved a comparison with earlier research by Ahmed et al. [58], as depicted in Figure 3, which displays the average Nusselt number for

Table 2. The number of elements used for mesh independence Number of nodes

pure water at various Reynolds numbers. For the second validation, the experimental data from Ashjaee et al. [59] were used for comparison. Figure 4 compares the local heat transfer coefficient corresponding to a Reynolds number (Re) of 600 at B = 0G and B = 400G. The comparisons, depicted in Figures 3 and 4, show a strong agreement between the current results and the benchmark values.

Results And Discussion

In this study, we examine the effect of a magnetic field on the forced convection heat transfer of a ferrofluid in a channel with a corrugated geometry. We present the results by means of the velocity contours, the temperature contours, the entropy contours, the streamlines, the Nusselt number, the pressure drop, and exergy. Additionally, this

Figure 3. Comparison of the Nu number with the results in the literature.

Figure 4. Comparison of the local heat transfer coefficient with the results in the literature.

investigation considers the effects of a 2% volume fraction of nanoparticles and of the Reynolds number Re. Velocity Contours and Axial Velocity Analysis Figures 5a-d show the velocity contours in multiple cross-sections along the flow direction and in the symmetry plane at φ = 2% and Re = 200 for B = 0G, B = 100G, B = 200G, and B = 300G; while figure 5e shows the variation of the velocity along the centerline of the symmetry plane for B = 0G and B = 300G. For a given magnetic field and within the entrance region there is formation of the dynamic boundary layer with velocities in the core region becoming more and more greater than those near the channel walls. As soon as the fluid reaches the corrugated region the magnetic force acts in the opposite direction of the flow and therefore the fluid decelerates, it then starts accelerating at the exit of the corrugated region. The boundary layer reaches its quasi-developed state at the outlet of the channel. Figure 5e clearly shows that for a given magnetic field, the ferrofluid centerline velocity decreases when it crosses the corrugated region. As the magnetic field strength increases, the centerline velocity in the corrugated region decreases. However, in the region preceding the corrugated walls, where there is no magnetic field, the velocity curves coincide with each other. Temperature Contours and Axial Temperature Analysis The contours of the temperature along the flow direction and the symmetry plane in various cross-sections at φ = 2% and Re = 200 for the different values of the magnetic field: B = 0G, B = 100G, B = 200G, and B = 300G, are shown in figures 6a-d, while figure 6e shows the variation of the

temperature along the centerline of the symmetry plane for B = 0G and B = 300G. As can be noticed in figures 6a-d, for a given magnetic field and within the entrance region the temperature remains constant. In the corrugated channel the fluid is heated and therefore there is formation of the thermal boundary layer with temperatures in the near wall region becoming more and more greater than those in the core region. As soon as the fluid reaches the exit of the corrugated region the fluid ceases to be heated and therefore conserves the same quasi developed profile throughout the exit flat region of the channel. As for figure 6e, one can observe that an increase of the magnetic field from 0G to 300G provokes an increase of the transfer of heat by convection and consequently the value of the temperature along the centerline of the symmetry plane increases. Figures 6a-d also show that for a given Reynolds number, the thermal layer thickening implies an increase in heat transfer and therefore in the temperature gradients near the corrugated wall. This is particularly noticeable in cases B = 200G and B = 300G, where the effect is more significant than in cases B = 0G and B = 100G. Entropy Contours Analysis For the different values of the magnetic field: B = 0G, B = 100G, B = 200G, and B = 300G, Figures 7 a-d illustrate the local entropy generation contours at φ = 2% and Re = 200 in various cross-sections along the flow direction and the symmetry plane. Close to the corrugated wall, where the temperature gradients and velocity fluctuations are prominent, entropy generation (Sg) is concentrated. However, entropy creation diminishes when moving towards the channel axis from the corrugated walls because of decreased velocity

Figure 5a-d. The behaviour of the ferrofluid with and without the magnetic field for φ = 2% and Re = 200 and for the different values of the magnetic field: (a) B = 0G, (b) B = 100G, (c) B = 200G, and (d) B = 300G.

Figure 5e. The velocity distribution along the axis of the channel.

Figure 6 a-d. Temperature contours at φ = 2% and Re = 200 for the different values of the magnetic field: (a) B = 0G, (b) B = 100G, (c) B = 200G, and (d) B = 300G.

Figure 6e. The temperature distribution along the axis of the channel for Re=200.

and temperature gradients. Furthermore, with an increase in the magnetic field, it appears that entropy generation decreases. This fact is particularly evident in the B = 200G and B = 300G cases, indicating a decrease in energy loss.

The streamlines for cases B = 0G, B = 100G, B = 200G, and B = 300G at x = 110 mm, φ = 2%, and Re = 200 are shown in Figures 10 a-d. These figures clearly show that as the magnetic field increases, its influence on the nanofluid becomes more pronounced. The number of eddies increases in the nanofluid; case B = 200G shows a high number of eddies, and case B = 300G shows an even greater number of eddies.

Streamlines Contours Analysis The streamlines comparison in the symmetry plane at φ = 2% for the cases: Re = 200, Re = 400, Re = 600, Re = 900, and Re = 1200, and for B = 0G and B = 300G, is shown in Figures 8 a-e. As can be observed, eddies are more pronounced at higher Reynolds numbers, whether in the presence or absence of a magnetic field. Furthermore, the impact of the magnetic field is more significant at lower Re; eddies at Re = 200 and Re = 400 are clearly visible when a magnetic field is present. The streamlines for: B = 0G, B = 100G, B = 200G, and B = 300G, are depicted in Figures 9 a-d at φ = 2% and Re = 200. These figures show that an increase in the magnetic field causes more eddies to appear in the corrugated cavities. In the case of B = 300G, the eddies are more noticeable than in the remaining cases.

Thermal Analysis The average Nusselt number for the pure water case (i.e., φ = 0%) and the ferrofluid case (φ = 2%) when B = 0G, B = 100G, B = 200G, and B = 300G are given in Figure 11 and Table 3 for various Reynolds numbers. The figure shows that as Re increases, the average Nusselt number also increases. Furthermore, because Fe3O4 has a greater thermal conductivity than pure water, its use leads to a greater value of Nu. Additionally, Nu increases in the corrugated channel due to the presence of the magnetic field, which indicates that the heat transfer is enhanced. In the case of B = 300G, this effect is more noticeable at Re = 200 and Re =

Figure 7 a-d. Local entropy generation contours at φ = 2% and Re = 200 for the cases: (a) B = 0G, (b) B = 100G, (c) B = 200G, and (d) B = 300G.

e) Figure 8 a-e. Streamlines comparison in the symmetry plane at φ= 2% for B = 0G and B = 300G for the cases: (a) Re = 200, (b) Re = 400, (c) Re = 600, (d) Re = 900, and (e) Re = 1200.

400. It follows that at lower Re, the magnetic field appears to have a greater influence on heat transfer. The same behavior of the average Nusselt number (Nuave) as a function of

Reynolds number (Re) and magnetic field was observed by Mehrez and El Cafsi [60] in their study of heat transfer in a horizontal channel. They reported an increase of the

Figure 9 a-d. Streamlines for cases (a) B = 0G, (b) B = 100G, (c) B = 200G, and (d) B = 300G in the symmetry plane at φ = 2% and Re = 200.

Figure 10 a-d. Streamlines for examples (a) B = 0G, (b) B = 100G, (c) B = 200G, and (d) B = 300G at x = 110 mm, φ = 2%, and Re = 200.

Figure 11. Average Nusselt number for different Re values in the case of pure water (φ = 0%) and ferrofluid (φ = 2%) for B = 0G, B = 100G, B = 200G, and B = 300G.

average Nusselt number as the magnetic field strength or the Reynolds number increase. Figure 12 and Table 4 show the variations of the average temperature of the corrugated wall under different Reynolds numbers for both the pure water case and the cases B = 0G, B = 100G, B = 200G, and B = 300G at a concentration of φ = 2%. The average temperature in the corrugated wall decreases with increasing Reynolds number when the Fe3O4-water nanofluid is used. Additionally, as the magnetic field strength is increased, the temperature further decreased. This explains the rise in Nu in Figure 11, indicating an improvement in heat transfer. The average temperature curves also show that the greatest temperature drops occur when the Reynolds number is increased from Re = 200 to Re = 400. Hydrodynamic Analysis Figure 13 shows the pressure drop for various Reynolds numbers in the cases of pure water and B = 0G, B = 100G, B = 200G, and B = 300G at φ = 2%. The pressure drop increases

as Re increases. However, the addition of nanoparticles further increased the pressure drop. When a magnetic field is present in the corrugated channel, a slight decrease in the pressure drop is observed as the strength of the magnetic field increases. A similar trend in the evolution of pressure drop (Δp) as a function of the Reynolds number (Re) and the magnetic field was observed by Bezaatpour and Goharkhah [49] in their study on the effects of a uniform external magnetic field and porous fins on convective heat transfer and pressure drop. Exergy Destruction Analysis Figures 14-16 show the variation of the thermal exergy destruction, the frictional exergy destruction, and the total exergy destruction with different Reynolds numbers for the cases of pure water and B = 0G, B = 100G, B = 200G, and B = 300G at φ = 2%. These figures illustrate that the exergy destruction decreases as Re increases and when the magnetic field in the corrugated channel increases. Accordingly, the use of a magnetic field reduces energy losses, and its

Figure 12. The average temperature in the corrugated wall versus Re for the case of pure water and the cases B = 0G, B = 100G, B = 200G, and B = 300G at φ = 2%.

Figure 13. Variation in the pressure drop with Re for the pure water case and the B = 0G, B = 100G, B = 200G, and B = 300G cases at φ = 2%.

effect becomes more noticeable when the Reynolds number is lower than 400. This explains the improved heat transport in the corrugated channel. Figure 14 shows that the thermal exergy destruction in the corrugated channel decreases as both the Re and magnetic field strengths increase, especially in the cases of Re = 200 and Re = 400 with B = 300G. According to Figure 15, as Re increases, the value of frictional exergy destruction also increases. In the presence or absence of the magnetic field, the curves are nearly identical.

Since thermal exergy destruction dominates frictional exergy destruction, figure 16 and Table 5 show that the total exergy destruction decreases as Re increases in the presence of a magnetic field. This finding indicates that applying the magnetic field reduces the total energy losses and that the influence of the magnetic field becomes more noticeable at Reynolds numbers lower than 400. This can be considered as a second factor for the enhancement of heat transfer in the corrugated channel besides the high thermal conductivity of the ferrofluid, which is the main factor.

Figure 14. Thermal exergy destruction with Re for the pure water case and the B = 0G, B = 100G, B = 200G, and B = 300G cases at φ = 2%.

Figure 15. Frictional exergy destruction with Re for the case of pure water and the cases B = 0G, B = 100G, B = 200G, and B = 300G at φ = 2%.

Figure 16. Total exergy destruction with respect to Re for the pure water case and the B = 0G, B = 100G, B = 200G, and B = 300G cases at φ = 2%.

Conclusion

This research presents an in-depth, original numerical investigation of the impact of a uniform magnetic field on ferrofluid flow and heat transfer in triangular corrugated channels. The channel walls at the top and bottom are subjected to a constant heat flux. The numerical findings lead us to several conclusions: • Owing to the entrainment of ferrofluid in the cavities of the base channel, the magnetic field has a more pronounced effect on heat transfer at Reynolds numbers less than 400. • Entropy generation decreases with increasing magnetic field, as is more evident in the B = 200G and B = 300G cases. • When a magnetic field is present, eddies become apparent at Re = 200 and Re = 400, particularly when B = 300G is applied. • According to our study, at low Reynolds numbers (Re = 200), the Nusselt number increases by more than 80% when B = 300G.

The corrugated-wall average temperature drop is greater for the low Reynolds number values. The pressure drop is not significantly impacted by the magnetic field compared with when it is absent in the triangular corrugated channel. The thermal exergy destruction decreases as both the Reynolds number and the magnetic field strengths increase, especially in the cases of Re = 200 and Re = 400 with B = 300G. The frictional exergy destruction increases as both the Reynolds number and the magnetic field strength increase. The total exergy destruction decreases as both the Reynolds number and the magnetic field strengths increase, especially in the cases of Re = 200 and Re = 400 with B = 300 G, despite an increase in frictional exergy destruction in the triangular corrugated channel. The minimum total exergy destruction is achieved at Re = 1200, B = 300G, and φ = 2%. The thermal exergy destruction and total exergy destruction in the case of B = 300G decrease by 37% compared to water.

Nomenclature

Magnetic flux density (N/A.m) Specific heat capacity (J/kg.K) Body magnetic force (N/ m3) Intensity of magnetic field (A/m) Local heat transfer coefficient (W/ m2.K) Thermal conductivity (W/m.K) Magnetization (A/m) Average Nusselt number Pressure (Pa) Reynolds number Entropy (J/K) Average Temperature (K) Coordinates in Cartesian space (m) x, y, and z directions’ velocities (m/s)

Greek symbols μ Dynamic viscosity (kg/m.s) ρ Density (kg/ m3) φ Volume fraction of particles χ0 Reference magnetic susceptibility χm Magnetic susceptibility µ0 Free space permeability (N/A2) Subscripts f fluid nf nanofluid np nanoparticle

Artificial intelligence was not used in the preparation of the article.

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