Abstract
The aim of this research work is to study the performance of hybrid eclectically conducting nanofluid (MWCNT-Fe3O4) with entropy optimization impacts using the CVFEM approach. 3D and contour plots are two types of outcomes that extend the discrepancy of analyzed variables. The Newtonian liquid is presumed as a testing fluid, and the non-Darcy model is employed to simulate the permeable domain. The geometry has two adiabatic walls and one wavy hot wall with unvarying flux, and the other is kept at a constant temperature. The impact of Brownian motion in the hybrid nanofluid (MWCNT-Fe3O4) is considered. The irreversibility effect is also taken into account. This effect acts against the performance of the system and reduces the efficiency of the system. To solve the final equations with higher accuracy, we utilized the method of CVFEM. The validation of the results obtained through CVFEM is clearly and graphically presented with the available literature. The outputs of the simulation showed that there is an inverse relation between permeability and the Bejan Number, and the boundary layer thickness augments with intensification in Lorentz force. These variations in the form of streamline, isotherm, and various contour plots have been reported at and for different values of Da, Ha, and
Introduction
Nanofluids have been investigated for the last decade with the giant potential to enrich the efficiency of the heat transfer features. Though, well-dispersed nanoparticles for nanofluids are very much expected to improve mechanical performance in many industries. The use of nanoparticles in engineering and technology has provided remarkable achievements, especially in heat transfer augmentation. The applications of these nanoparticles within the porous enclosure in the occurrence of heat transfer analysis are widely investigated by many researchers [1, 2]. Some of these applications are solar collectors, fuel cells, cooling of electronic accessories (on a small and large scale), and drying process. The augmentation of the heat transfer inside the square enclosure is not only related to the de-purification of the fluid inside but also related to the shape and nature of the walls of the square. For this purpose, Ghasemi et al. [3] analyzed the natural convective flow of the MHD fluid flow in the x-direction inside the square cavity. They reported that with a larger Rayleigh number and smaller values of the Hartmann number the heat transfer enhances. This enhancement may or may not occur when the nanofluid is added which affects the performance of the Hartmann and Rayleigh numbers. The thermal analysis of the nanofluid inside an inclined enclosure is reported by Abbassi [4]. The geometry was chosen in such a way that heat flows from left to right, where the right wall chosen was cool. The impact of the inclination angle and the aspect ratio has been studied in detail. The aspect ratio impact increases or decreases the Nusselt number due to its larger or smaller values. The influence of the magnetic parameter by the nanofluid flow inside an enclosure by considering the Brownian motion is reported by Mahmoudi et al. [5] and Ghasemi and Aminossadati [6]. The Al2O3 and CuO nanoparticles in a water-based mixture through a trapezium shape enclosure are numerically analyzed by Saleh et al. [7]. They constituted a new empirical formula for the Nusselt number and reported that CuO-water nanofluid has greater thermal conductivity as compared to that of Al2O3-water. The finite element analysis of the CuO-water nanofluid through a parallelogram shape enclosure is analyzed by Mustafa and Hussein [8]. The top wall of the enclosure was chosen open through which the thermally heated fluid flows. The walls are heated in such a way that the inclined walls are kept cool while partial heating to the enclosure is provided from the bottom. The impacts of the inclination angles of the cold walls, Rayleigh number, a source of heat in addition to the concentration of the nanofluid are reported in detail. They described that for the betterment of the heat transmission from the source of heat should be nearer to the left wall that is inclined at 60°. In their work, they showed that heat flux shows an increasing trend by the larger values of the Rayleigh parameter.
The entropy, as well as the influence of various nanoparticles (Cu, TiO2, Al2O3, etc.) through different complex enclosures, are reported by Cho et al. [9]. They showed the impact of motion of the surface with its length and Rayleigh parameter over the heat transmission and fluid flow. They observed that the generation of entropy is very low for Cu nanoparticles as compared to the other nanoparticles chosen in the study. On the other hand, the transfer of heat has the maximum bound in the performance of heat transfer. The effects of the inclined magnetic field, as well as the internal heat source inside an enclosure for the natural convective flow of the Al2O3 nanofluid through an enclosure, are analyzed by Sadeghi et al. [10]. The enclosure was chosen in such a way that it has one cold wavy wall with a trapezium heat source. They numerically investigated the system for the influence of Rayleigh and Hartmann parameters together with the shape factor and the location of the heat source for the analysis of the fluid as well as the heat transfer. They reported that the magnetic parameter reduces the transfer of heat in the natural convection case when it acts horizontally. The saturation of nanofluids is recently studied with various models (Darcy model, LTE, Darcy–Brinkmann model, LTNE, etc.) by many researchers [11, 12]. In the same way, Rashed and Ahmed [13] numerically studied the internal source impact on the MHD nanofluid through a porous enclosure. Izadi et al. [14] analyzed the variable magnetic field effect for the fluid flows through semi-circular heated cylinders numerically. They used the non-equilibrium model for the hybrid nanofluid to fill in the gap. They reported the impacts of various important parameters like Rayleigh number, porosity coefficient, magnetic source, and the Hartmann number over the state variables. The Cu-Al2O3 hybrid nanofluid inside a wavy enclosure with the porous medium is analyzed by Kadhim et al. [15]. The Darcy–Brinkmann model is applied and the finite element method is used for solution purposes. They reported that Al2O3–water nanofluid impact over the heat flux is not very effective as compared to that of the Cu–water nanofluid. Apart from this, heat flux acts as a function of the Darcy and Rayleigh number. A similar analysis for the Ag-MgO–water nanofluid through a square enclosure is reported by Mehryan et al. [15]. More interesting results on square cavity can be found in [16–18].
The study of different geometries within an enclosure has been reported by many researchers. For instance, the inner body moving (in all directions) has been reported in [19–21]. Yoon et al. [22] studied the flow between two circular cylinders that were immersed in an enclosure. The flow had taken natural convective and the analysis of the flow pattern is investigated with the numerical scheme for various parameters over the state variables. Kefayati and Tang [23] reported the flow through the two different shape cylinders placed within a square enclosure by using LBM. The enclosure was chosen inclined and the impact of the inclination angle together with the inner body position and the Rayleigh number is analyzed for all the states variables. Rectangular, elliptical, and circular shape bodies within an enclosure filled with nanofluid by considering the natural convective flow are studied by Roy [24]. He showed that the impact of the circular shapes body inside the cavity is enhancing the transfer of heat effectively as compared to the other shapes considered in the study. Wang et al. [25] interrogated the vertical motion of the circular shaped object in an enclosure filled with nanofluid having transient convection. They showed that the position of the object highly influences the nanofluid and the heat transmission. Bhowmick et al. [26] considered the cylinders in pairs inside the porous enclosure and analyzed the magnetic effect and entropy generation for the natural convective flow inside the enclosure. They found that the heat augments with rising distance between the immersed cylinders. In the same way, Alsabery et al. [27] investigated the square shaped body effect inside an enclosure for the transfer of heat by considering the two-phase model. Despite its applications, geometries like, U, H, C, and I shape for the natural convective flow have not been widely studied by researchers [28–30]. One such study is reported by Armaghani et al. [32] by considering the shape porous enclosure filled with nanofluid. They concluded the impact of the magnetic field for the heat transfer analysis. Similar geometry is considered by Ma et al. [33] for the analysis of the multi-square nanofluid flow. For the analysis of the flow inside the square enclosure by considering various impacts of the nanofluid flow has been studied through innovative methods [34, 35]. The lattice Boltzmann equations for the three-dimensional flow are analyzed by Succi et al. [36]. In their work, they used Darcy’s law and analyzed the results for variations of the Reynolds number. Heat transfer analysis with the effect of a magnetic field for innumerable nanofluids with different properties in different geometries can be found in [37–47]. Recently, Rashidi et al. [41, 42] discussed the entropy generation by analyzing the stagnation point flow and the circular tube heat exchanger by utilizing the nanofluids. A more detailed survey on the entropy generation by using various approaches has been reported by Abbas et al. [48–59]. Thus, according to the literature, Table 1 presents the properties of hybrid MWCNT and Fe3O4. More development of simulating approaches were demonstrated in recent articles [60–71].
From the aforementioned discussion, it is clear that the entropy generation inside a square enclosure has not been reported yet. In this work, we will investigate the hybrid nanofluid (MWCNT- Fe3O4) flow inside a permeable enclosure (tank). The influence of the magnetic field and entropy generation has been taken into account for the analysis of the heat transfer. The geometry is chosen in such a way that the two walls are adiabatic, and one wall is wavy. For simulation purposes, an advanced computational technique CVFEM is used. Through this technique, various parameters like, Hartmann number, Reynolds number, and volume fraction of the nanofluid have been studied in detail. The results are validated with the available literature through graph.
Physical and Mathematical Model
Geometric Model
Assume a permeable geometry having one hot wavy wall (through which a uniform flux passes) and two horizontal adiabatic walls, while the rest of the wall (perpendicular) is kept at constant temperature Tc as shown in Figure 1. The testing fluid for the non-Darcy model through this geometry considered is Newtonian. The hybrid nanofluid is utilized and estimated for the correlation properties as per the procedure described in the Ref. [48]. The magnetic field is applied at an angle and it is presumed that the induced field produced is ignored. For the problem solution and utilization, the advanced numerical procedure CVFEM is used [49].
FIGURE 1
Porous zone with considering heat flux.
Governing Equations
For the assumptions and flow pattern through Figure 1, the basic model equations are constituted as follows [20–25]:
Since the vorticity effect is affecting the flow, for this, we assume the following formulation in this work:
The formulae describing the nanofluid and nanomaterial characteristics are outlined as follows [18–22]:
To reduce the complexity, and transform the given PDEs system into a dimensional form, we assume the following important transformations [20, 26]:
Using Eq. 6 in the system of PDEs (1)–(4), we have [20–26]
The dimensionless constants are defined as follows [20–24]:
Nusselt Number
For the hot body placed inside the enclosure, we use the following mathematical form for the computational purposes of the Nusselt number [20, 31]:where
In the same way, the Nusselt number for the multiple bodies can be computed from the sum of all the Nusselt numbers for each hot body placed inside the enclosure.
Entropy Optimization
Entropy generation implies energy waste; henceforth, reducing entropy production is often a prime aim. It can be applied to recognize the source of wastefulness in any system and deliver opportunities for device or process design developments. The second Law of thermodynamics is used to scrutinize energy creating, changing, and consuming systems from a theoretical point of view. Because of irreversibility in the thermal dynamic process, entropy generation causes energy dissipation. Whenever heat and mass transfer are involved, entropy production minimization techniques are used in numerous sectors. The main goal is to save as much energy as possible while improving thermodynamic performance.
The entropy generated inside the enclosure is expressed in the following functional forms [43–45]:
Numerical Technique and Validation of Results
Sheikholeslami [49] introduced a new technique Control Volume Finite Element Method (CVFEM) for the computation of numerical results. The consistency and reliability of this method has been reported in various references for different problems. In this study we have used the CVFEM procedure for the computation of the numerical results in which residual ranges up to 10−5. The results are obtained here are validated with Khanafer [72] shown in Figure 2. It is clear that as the values of increases from 0.018 up to 0.11 our results approaching towards the already available results, and at both the results are the same. The dimensionless temperature show an increase at the initial and intermediate values of , while at the larger values of this effect is very small. Both the results show a good agreement.
FIGURE 2
Code verification according to [31].
Results and Discussion
In the occurrence of a uniform magnetic field and irreversibility effect the heat transmission analysis has been studied through CVFEM. The validation of the results obtained is discussed in the above section. In this study we have used the range for Rayleigh number, for Hartmann number at and and for the computation of the Nusselt number and the Bejan number
The variations in the Bejan number caused by the irreversibility effect under the impact of the Rayleigh parameter are plotted through isotherms and streamlines in Figures 3A–G. From these diagrams; it is clear that the nanofluid inside the enclosure is affected by the buoyancy and Lorentz forces. Figures 3A, B, consisting of four subplots, in the form of isotherms and streamlines during the nanofluid convective flow in the presence and absence of radiation effect for various values of the Rayleigh number (103 and 105). Graphs are plotted in the form of two columns; whereas column one graphs represent the convective flow in the absence of the radiation effect, while on the other hand, the second column show the radiation effect as well as the convection effect. The buoyancy forces are directly linked with the Raleigh number, as a result, the increasing values of show less strength in the contours as compared to the smaller values of Ra as shown in Figures 3A, B. It is observed that larger or smaller values of (associated with buoyancy forces) cause to decline in the temperature of the hot wall, that further makes the inner (center) temperature inside the enclosure approximately uniform. This effect of uniformity inside the enclosure is more strengthened when the thermal radiations are included for the nanofluid flow. The smaller values of causing the heat to flow fast as compared to the higher values as clear from the isotherm shapes. Physically, is the function of Pr and Gr; when declines the product of Pr and Gr also declines, and as a result, the Prandtl number jumps that further enhances the heat transfer. The stream functions show its maximum value at the high intensity of the cavity. When the natural convection becomes stronger the distortion in the isotherms becomes more powerful. Therefore, it is clear that is a function of the stream that directly affects the nanofluid as exposed in the figure. The convection rate of the current is increasing with larger values of , due to the efficacy of the buoyancy forces over the viscous forces. Since we have considered the circulation impact in this work, this effect pushing the cold fluid underneath and the hot fluid came to the rescue of the empty spaces, and as a result, the transfer of heat becomes dominant. In Figures 3C–F the irreversibility effect is plotted. In these plots, the main two forces affecting the irreversibility are the Lorentz and buoyancy forces. The natural convective flow is dominating at higher values of the buoyancy forces. On the other hand, the Lorentz forces are acting against the natural convection at higher values. In the ratio one of these two forces will be effective. The buoyancy forces are very effective in the case while the Lorentz forces become dominant when . The increasing values of for all the values of make the transfer pure conductive. We know that is the function of Pr and Gr, when declines the Pr or Gr will jump up and as a result the conduction will be higher. This effect causes due to the buoyancy and Lorentz forces; that interact and the fluid flow becomes dominant over the convection rate. The impact of the higher values of the Rayleigh number pushes for the pure convection rate. Physically, to slow down the impact of the convection rate a higher magnetic parameter is required. It is due to the physical significance of the magnetic parameter that disturbs the flow patterns, and further causes the reduction in convection in the heat transfer case. In addition to this declines the stream function strength. Since is in direct relation with the magnetic field strength and inversely related to the dynamic viscosity and as a result the higher values of and a small amount of the Rayleigh number the direction elongated wall totally represents the kinetic energy. As the values of increase, new smaller circles are produced and the already available cells elongate on the y-axis. From this physically we can say that the kinetic energy goes up as the values of increase.
FIGURE 3
Contour of streamlines, isotherms, irreversibility, and the Bejan Number changes at
The variations in the Bejan number caused by the irreversibility effect under the impact of the Hartmann number with respect to the magnetic field influence on fluid flow and heat transfer are plotted through isotherms and streamlines in Figures 4A–G. The impact of the Hartmann number shows diverse shapes inside the body reported in Figure 4A. It is clear that the higher values of the Hartmann number reduce the strength of the nanofluid flow. The Isotherm effect of the Hartmann number on nanofluid flow is shown in Figure 4B. It shows that a higher value of Ha enhances the hybrid nanofluid flow motion. The effect of the Hartmann number on entropy generation is presented in Figures 3C–F. Graphs are plotted in the form of two columns, where different effects are observed for different values, and the results are clear from the graphs displayed. The entropy behavior with the effect of the Hartmann number is observed that entropy optimization is decreasing with higher values of the Hartmann number. The effect of the Bejan number weakens with a higher value of Ha, which is due to the reducing behavior of entropy.
FIGURE 4
Contour of streamlines, isotherms, irreversibility, and the Bejan Number changes at
The variations in the Bejan number caused by the irreversibility effect under the impact of the Darcy number and are plotted through isotherms and streamlines in Figures 5A–G at . The Streamlines and isotherms that are plotted show that the strength of vortexes augments with enhancing penetrability. Actually, the greater porous enclosure porousness permits the easy hybrid nanofluid movement, and hence the thermal energy transfer enhances. Darcy’s number is indirect relation to the permeability of the medium and is inversely related to the square of the characteristic length. This functional relation physically interprets that increasing the dimensionless value of Darcy’s number leads to an obvious increment in the permeability of the porous medium, which helps nanofluid to penetrate into the left layer leading to an increase in the convection heat transfer mode; because the conductive mode is dominant at low Darcy number. Also, it is obvious from the shape of the streamlines in the left layer as there are no inner cells that reflect low fluid flow strength. The fluid temperature drops with increasing medium porosity as displayed with isotherms plots. The entropy generation impact on Da is shown in different figures by their respective contours. It is observed that an increase in permeability enhances the entropy in different phases.
FIGURE 5
Contour of streamlines, isotherms, irreversibility, and the Bejan Number changes at
Figure 6 shows the local Nusselt number variation with the increasing distance along the y-direction at different values of Ra, Da, and Ha. This figure consists of three subfigures. The lowest value of and higher values of it display concentric circles. When and jumps up to , these circles become flattened on both of the surfaces. It is clear that the presence of heat transfer augments the thermal energy transfer during the convective nanofluid motion. The subfigure shows that the value of the local Nusselt number is higher for larger values of Ra, as one moves away from the hotter wall. These relations are shown as
FIGURE 6
Nuave values for different cases.
Figure 7 shows the Bejan number variation at different values of log(Ra) = 4, Da = 50 and Ha = 5. This figure consists of three subfigures. From these figures, it is observed that the higher Rayleigh number reduces the Bejan number, while there is no effect on Darcy’s number Da. In the second case, it is observed that the higher values of the Hartmann number enhance the Bejan number, while it reduces with Darcy’s number Da. In the third case, the Bejan number reduces with the enhancement of Ha at constant Ha.
FIGURE 7
Be values for different cases.
Conclusion
The main outcomes of this work are concluded as follows:
1. The heat transfer is enhanced with the increasing values of and nanofluid loading
2. The fluid flow is at its peak point when the inner object is placed near the down wall of the enclosure, but this effect is valid up to some threshold number
3. The entropy effect is mainly affected by the two forces: Lorentz and buoyancy
4. The buoyancy forces are very effective in the case , while Lorentz forces become dominant when
5. The increasing values of the magnetic parameter disturb the flow pattern that further affects the flow of the fluid in the heat transfer case
6. The Nusselt number variations along the y-axis near the hot wall enhance the heat transfer for the increasing values of
7. Furthermore, the Nusselt number vary directly with the larger values of and inversely to
8. The impact of the magnetic parameter inside the enclosure is fully influenced by the Rayleigh number, Nusselt number, and Hartmann number to fully control the convection
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This project was financed by Lucian Blaga University of Sibiu and Hasso Plattner Foundation research grants LBUS-IRG-2021-07.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Summary
Keywords
nanoparticle, entropy, heat transfer, porous, simulation
Citation
Shah Z, Ullah A, Musa A, Vrinceanu N, Ferrandiz Bou S, Iqbal S and Deebani W (2022) Entropy Optimization and Thermal Behavior of a Porous System With Considering Hybrid Nanofluid. Front. Phys. 10:929463. doi: 10.3389/fphy.2022.929463
Received
26 April 2022
Accepted
16 May 2022
Published
01 July 2022
Volume
10 - 2022
Edited by
Muhammad Mubashir Bhatti, Shandong University of Science and Technology, China
Reviewed by
Mohammad Rahimi Gorji, Ghent University, Belgium
Mohammad Mehdi Rashidi, Tongji University, China
Munawer Abbas, University of Baltistan, Pakistan
Updates
Copyright
© 2022 Shah, Ullah, Musa, Vrinceanu, Ferrandiz Bou, Iqbal and Deebani.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Narcisa Vrinceanu, vrinceanu.narcisai@ulbsibiu.ro
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
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