Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al2O3 Hybrid Nanofluid

The present paper aims to analyze the thermal convective heat transport and generated irreversibility of water-Cu-Al2O3 hybrid nanosuspension in an odd-shaped cavity. The side walls are adiabatic, and the internal and external borders of the enclosure are isothermally kept at high and low temperatures of Th and Tc, respectively. The control equations based on conservation laws are formulated in dimensionless form and worked out employing the Galerkin finite element technique. The outcomes are demonstrated using streamlines, isothermal lines, heatlines, isolines of Bejan number, as well as the rate of generated entropy and the Nusselt number. Impacts of the Rayleigh number, the hybrid nanoparticles concentration (φhnf), the volume fraction of the Cu nanoparticles to φhnf ratio (φr), width ratio (WR) have been surveyed and discussed. The results show that, for all magnitudes of Rayleigh numbers, increasing nanoparticles concentration intensifies the rate of entropy generation. Moreover, for high Rayleigh numbers, increasing WR enhances the rate of heat transport.


Introduction
The interest in the field of heat transfer has increased considerably recently. While device performance and heat generation have gone up, the sizes of the components have shrunk. For instance, computer chips that are made today are at least several orders of magnitude smaller in size as compared to the past. Thus, the generated heat per surface area is growing continuously. Thus, novel designs with improved heat transfer capabilities are demanded to cool devices that produce large amounts of heat.
Heat transfer in enclosed geometries with various initial and boundary conditions have been a hot topic in the past few decades. This is because these types of geometries have been utilized extensively in real-life applications, such as the cooling of electronic devices, thermal design of buildings, lubrication and drying applications, furnace and nuclear reactors, biochemical, and food processing. Nonetheless, rectangular and square lid-driven cavities have been studied more because of their simplicity and applicability. As such, a large portion of convective heat transport in cavities' studies has been devoted to rectangular geometries.
Among all the commonly used approaches to enhance heat transfer efficiency, including the passive and active methods, the dispersion of nano-sized particles of highly The vertical (or horizontal) length of the hot (or cold) surfaces is L (or L-W). The width of the cavity is W and can vary to have different width ratios (W/L). The temperature of the interior walls, shown in red in Figure 1, is isothermally kept constant at Th, and the external walls, depicted in blue, are at the temperature of Tc. The remaining borders are assumed to be adequately insulated. It is assumed that the circulation is steady, incompressible, laminar, and the radiation has no significant effect on the process.

Governing Equations
Under the above-mentioned assumptions, the control equations for the hybrid nanoliquid, based on conservation laws, are [8,27]: Continuity Equation: Momentum Equation: Energy Equation: ( ) ℎ ( + ) = ℎ ( where u and v are the velocity components in x and y directions, T is the temperature of the hybrid nanofluid, g is the gravity and p is the pressure. The corresponding boundary conditions are

Governing Equations
Under the above-mentioned assumptions, the control equations for the hybrid nanoliquid, based on conservation laws, are [8,27]: Continuity Equation: ∂u ∂x

Momentum Equation:
ρ hn f u ∂u ∂x Energy Equation: (ρc) hn f u ∂T ∂x where u and v are the velocity components in x and y directions, T is the temperature of the hybrid nanofluid, g is the gravity and p is the pressure. The corresponding boundary conditions are T = T h at the inner walls T = T c at the outer walls ∂T ∂n = 0 at the rest of the surfaces u = v = 0 at all solid surfaces (5)

Thermophysical Characteristics
In Equations (1)-(4), ρ hnf , (ρc) hnf , β hnf and µ hnf denote the density, thermal capacitance, heat expansion coefficient and dynamic viscosity of the nanoliquid which could be calculated from the below relations [28]: Here, ϕ hn f = ϕ Al 2 O 3 + ϕ Cu is the volume fraction of the hybrid nanofluid. Moreover, the subscripts f, Cu, and Al 2 O 3 refer to the base fluid, Cu, and Al 2 O 3 nanoparticles' properties, respectively. The physical characteristics of the hybrid nano-sized particles and water (as the host fluid) are gathered in Table 1. Table 1. Thermal characteristics of Cu and Al 2 O 3 nanoparticles [29].

∂U ∂X
where Rayleigh and Prandtl numbers are defined as The boundary conditions (5) in non-dimensional form become θ = 1 at the inner walls θ = 0 at the outer walls ∂θ ∂N = 0 at the rest of the surfaces U = V = 0 at all solid surfaces (17)

Rate of Heat Transfer
The parameter of interest in this research is the rate of energy transference from the hot border, which can be quantified by the average Nusselt number Nu avg as:

Entropy Generation
The strength of generated entropy is expressed as: The first term in the above equation is the energy transference component of entropy production. The second one indicates that the portion of generated entropy caused by friction of the hybrid nanofluid. The generated entropy could be transformed into the non-dimensional form using the below parameters [30,31]: Equation (15) in the non-dimensional form reads as: where Here χ is called the irreversibility function and can be defined as: The ratio of generated entropy induced by energy transference to the total entropy production is called the Bejan number, which is It is apparent that for Be > 0.5 (Be < 0.5), the irreversibility of the energy transference (fluid friction) governs the total entropy production.

Heat Function
The xand y-derivatives of the heat function equal the energy flux components (j ,x , j ,y ) and for the hybrid nanofluid read as [32]: , the dimensionless form of the above equations is obtained as: With a cross derivation, the dimensionless presentation of the heat function could be calculated as the following: The boundary conditions for the above equation can be obtained by integrating over the Equations (23) or (24) and imposing the no-slip condition (U = V = 0). In addition, the heat lines on the bottom wall can be arbitrary set to zero [32]: on the bottom wall: on the inner walls: on the outer walls: on the right adiabatic wall: The employed numerical method, details of the grid check and validations are discussed in the Appendices A and B.

Results and Discussion
In this section, different parameters including Rayleigh number (Ra = 10 3 , 10 4 and 10 5 ), width ratio (WR = W/L = 0.2, 0.3 and 0.4), the concentration of the hybrid nanoparticles (0.0 ≤ φ hnf ≤ 0.05) and the concentration of Cu to φ hnf ratio (0.0 ≤ φ r = φ Cu /φ hnf ≤ 1.0) have been surveyed and discussed. Moreover, the Prandtl number Pr = 6.2 is considered to be constant in all simulations. Figure 2 shows the isotherms, heat lines, and streamlines in the cavity filled with hybrid nanosuspension for different Ra and WR values. Figure 2a represents the isotherms, which as seen, the isotherms are layered in the horizontal direction at Ra = 10 3 . The layered arrangements of the isotherms confirm that conduction energy transference is the dominant mode of heat transport in the chamber. It should be noted that the isotherms signify the strength of the conduction heat transfer; for instance, it can be observed from Figure 2 that the conduction energy transference has larger values near the walls (especially the Symmetry 2021, 13, 122 7 of 17 hot walls). Conversely, the heat lines at Ra = 10 3 and WR = 0.2 show the perpendicular lines toward isotherms, and heat lines in this figure represent the routes of the energy transference from the hot border to the cold border with the conduction mechanism. Furthermore, the first row of Figure 2c demonstrates a vortex zone in the vertical part of the chamber, which represents a slight fluid velocity in this region. By increasing the WR from 0.2 to 0.3 at the same Ra (second row), the isotherms tend to separate as a result of conduction energy transference reduction. In this new state, the maximum value of the heat line is reduced from 9 to 5.5, and this reduction shows the lower energy transport strength in the cavity. Therefore, it can be concluded that the rate of energy transport from the hot border to the cold border decreases if the wall width increases when the conduction is the dominant phenomenon.
cles (0.0 ≤ ϕhnf ≤ 0.05) and the concentration of Cu to ϕhnf ratio (0.0 ≤ ϕr = ϕCu/ϕhnf ≤ 1.0) have been surveyed and discussed. Moreover, the Prandtl number Pr = 6.2 is considered to be constant in all simulations. Figure 2 shows the isotherms, heat lines, and streamlines in the cavity filled with hybrid nanosuspension for different Ra and WR values. Figure 2a represents the isotherms, which as seen, the isotherms are layered in the horizontal direction at Ra = 10 3 . The layered arrangements of the isotherms confirm that conduction energy transference is the dominant mode of heat transport in the chamber. It should be noted that the isotherms signify the strength of the conduction heat transfer; for instance, it can be observed from Figure 2 that the conduction energy transference has larger values near the walls (especially the hot walls). Conversely, the heat lines at Ra = 10 3 and WR = 0.2 show the perpendicular lines toward isotherms, and heat lines in this figure represent the routes of the energy transference from the hot border to the cold border with the conduction mechanism. Furthermore, the first row of Figure 2c demonstrates a vortex zone in the vertical part of the chamber, which represents a slight fluid velocity in this region. By increasing the WR from 0.2 to 0.3 at the same Ra (second row), the isotherms tend to separate as a result of conduction energy transference reduction. In this new state, the maximum value of the heat line is reduced from 9 to 5.5, and this reduction shows the lower energy transport strength in the cavity. Therefore, it can be concluded that the rate of energy transport from the hot border to the cold border decreases if the wall width increases when the conduction is the dominant phenomenon. The uniformity of isotherms is gone when the Ra number is increased to 10 5 at the same WR = 0.3 (third row), especially in the horizontal zone of the chamber. It depicts that the conduction energy transference is mainly restricted to the walls, and Ra increment leads to increasing buoyancy force. Moreover, the streamlines show different vortices at Ra = 10 5 and WR = 0.3, and these vortices are started to evolve in a horizontal direction, and the routes of heat transfer are mainly similar to streamlines. Therefore, it can be concluded; the free convective energy transference now is the dominant mode in the cavity. In the last line of Figure 2, the effect of WR is depicted at the high value of Ra. By increasing the WR from 0.3 to 0.4 at the Ra = 10 5 , the conduction effect on heat transfer is weakened, even in the cavity's vertical part. Furthermore, the stronger vortexes and more extensive heat lines distribution, especially in the horizontal part, show the dominance convection heat transfer, which is boosted by increasing the wall width. It can be found from Figure  2 that the maximum value of the heat lines is raised in the high Ra when the wall width is increased. Therefore, increasing wall width has a direct influence on energy transference at the high Ra. Figure 2d shows the isentropic lines as the Be number is changed in the range between 0 and 1. It can be observed from the figures that the lines, with larger values, are close to walls. It shows in these areas the thermal entropy has mainly the same values as total entropy. In addition, some lines with lower values of Be are emerged in the The uniformity of isotherms is gone when the Ra number is increased to 10 5 at the same WR = 0.3 (third row), especially in the horizontal zone of the chamber. It depicts that the conduction energy transference is mainly restricted to the walls, and Ra increment leads to increasing buoyancy force. Moreover, the streamlines show different vortices at Ra = 10 5 and WR = 0.3, and these vortices are started to evolve in a horizontal direction, and the routes of heat transfer are mainly similar to streamlines. Therefore, it can be concluded; the free convective energy transference now is the dominant mode in the cavity. In the last line of Figure 2, the effect of WR is depicted at the high value of Ra. By increasing the WR from 0.3 to 0.4 at the Ra = 10 5 , the conduction effect on heat transfer is weakened, even in the cavity's vertical part. Furthermore, the stronger vortexes and more extensive heat lines distribution, especially in the horizontal part, show the dominance convection heat transfer, which is boosted by increasing the wall width. It can be found from Figure 2 that the maximum value of the heat lines is raised in the high Ra when the wall width is increased. Therefore, increasing wall width has a direct influence on energy transference at the high Ra. Figure 2d shows the isentropic lines as the Be number is changed in the range between 0 and 1. It can be observed from the figures that the lines, with larger values, are close to walls. It shows in these areas the thermal entropy has mainly the same values as total entropy. In addition, some lines with lower values of Be are emerged in the middle of vertical cavity by increasing Ra and WR that show the effect of viscous entropy in these areas. Furthermore, these lines extend in horizontal part when the convective energy transport is the dominant mode in the chamber. Figure 3 shows the alteration of average Nu values against the total amount of nanoparticle volume fractions, with an equal proportion of Cu and Al 2 O 3 in the water-based fluid. It can be obtained from Figure 3a, that the Nu avg has a more considerable value in the higher Ra as a result of natural convection increment, which was discussed in Figure 2. On the other hand, Figure 3a shows that the Nu avg rises when a higher concentration of solid particles presents in the medium. The investigated nanoparticles in this research have a high heat transfer conduction coefficient, and the presence of these materials can enhance the conduction phenomena. For this reason, the same slops are observed in Figure 3a, and this represents that the nanoparticles have the same and constant effect in different Ra values. Figure 3b shows that the Be number goes down when the Ra number is increased. For high Ra, the energy transport is enhanced. Therefore, the generated entropy due to energy transference increases, but it can be found from Figure 3b that the generated entropy caused by friction of the hybrid nanofluid has a higher value increment compared to S th that leads to Be number reduction. However, nanoparticle volume fractions have no influence on the Be number in any values of the Ra (see Figure 3b). After increasing the nanoparticle volume fractions, heat transfer entropy and overall entropy both increased, so Be remained almost constant.  Figure 4a reveals that the percentage of Cu nanoparticle volume fractions has no considerable effect on the Nuavg. Both Cu and Al2O3 have a similar influence on hybrid nanofluid at Ra = 10 3 . It can be seen from this Figure that the Nuavg reduces considerably when the wall width is increased from 0.2 to 0.4. As previously described at Ra = 10 3 , the conduction energy transference is the dominant regime; therefore, it is evident that the Nuavg decreases according to the Fourier law when the WR increases. This reduction in heat transfer rate leads to entropy decrement, generated by heat transfer, and the Be reduction in Figure 4b is caused by this Sth alterations. Similar to the Nuavg, the percentage of the Cu nanoparticle volume fractions has no considerable effect on Be number, because it does not affect the rate of energy transference at the low Ra; therefore, no entropy is generated by heat transfer and nanofluid frictions; then, Be remains constant at different values of ϕr.  Figure 4a reveals that the percentage of Cu nanoparticle volume fractions has no considerable effect on the Nuavg. Both Cu and Al 2 O 3 have a similar influence on hybrid nanofluid at Ra = 10 3 . It can be seen from this Figure that the Nu avg reduces considerably when the wall width is increased from 0.2 to 0.4. As previously described at Ra = 10 3 , the conduction energy transference is the dominant regime; therefore, it is evident that the Nu avg decreases according to the Fourier law when the WR increases. This reduction in heat transfer rate leads to entropy decrement, generated by heat transfer, and the Be reduction in Figure 4b is caused by this S th alterations. Similar to the Nu avg , the percentage of the Cu nanoparticle volume fractions has no considerable effect on Be number, because it does not affect the rate of energy transference at the low Ra; therefore, no entropy is generated by heat transfer and nanofluid frictions; then, Be remains constant at different values of φ r . erable effect on the Nuavg. Both Cu and Al2O3 have a similar influence on hybrid nanofluid at Ra = 10 3 . It can be seen from this Figure that the Nuavg reduces considerably when the wall width is increased from 0.2 to 0.4. As previously described at Ra = 10 3 , the conduction energy transference is the dominant regime; therefore, it is evident that the Nuavg decreases according to the Fourier law when the WR increases. This reduction in heat transfer rate leads to entropy decrement, generated by heat transfer, and the Be reduction in Figure 4b is caused by this Sth alterations. Similar to the Nuavg, the percentage of the Cu nanoparticle volume fractions has no considerable effect on Be number, because it does not affect the rate of energy transference at the low Ra; therefore, no entropy is generated by heat transfer and nanofluid frictions; then, Be remains constant at different values of ϕr.     Figure 5a shows that increasing the Cu particles in the hybrid nanofluid has a positive influence on energy transference at Ra = 10 5 , in opposite results, as obtained from Figure 4a. This phenomenon can be attributed to the larger conduction heat transfer coefficient of Cu compared to Al2O3 nanoparticles. By increasing the wall width from 0.2 to 0.3, Nuavg is increased, as shown in Figure 5a. At WR = 0.3, the nanofluid can be circulated more easily in the cavity as a result of buoyancy force at Ra = 10 5 . However, energy transference strength decreases considerably when WR increases to 0.4. Figure 5b shows similar results in Figure 4b, but for different reasons. The generated entropy caused by frictions increases when the wall width changes from 0.2 to 0.3 because of the nanofluid movement in the cavity. As for the previous results, Be number more depends on Sviscous than Sth at high Ra numbers. Therefore, a considerable reduction can be observed on Be number in Figure 5b when WR reaches 0.3. However, by increasing WR to 0.4, the convection effect is decreased (see Figure 5a); therefore, the nanofluid velocity decreases that leads to entropy reduction (Sviscous). In this state, the Sviscous tends to Sth value, and because of this, the Be does not change noticeably at WR = 0.4 compared to WR = 0.3. Figure 5b shows that the Cu nanoparticle volume fractions in hybrid nanofluid have no substantial effect on Be number as a result of convection mechanisms dominance at high Ra numbers. However, Figure 5b shows that Be slightly decreases with ϕr increment at WR = 0.2. These slight changes can be attributed to Cu density that has a larger value compared to Al2O3. The greater density at high Ra with a narrow wall width leads to increase entropy caused by friction factors and decrease Be number.  Figure 5a shows that increasing the Cu particles in the hybrid nanofluid has a positive influence on energy transference at Ra = 10 5 , in opposite results, as obtained from Figure 4a. This phenomenon can be attributed to the larger conduction heat transfer coefficient of Cu compared to Al 2 O 3 nanoparticles. By increasing the wall width from 0.2 to 0.3, Nu avg is increased, as shown in Figure 5a. At WR = 0.3, the nanofluid can be circulated more easily in the cavity as a result of buoyancy force at Ra = 10 5 . However, energy transference strength decreases considerably when WR increases to 0.4. Figure 5b shows similar results in Figure 4b, but for different reasons. The generated entropy caused by frictions increases when the wall width changes from 0.2 to 0.3 because of the nanofluid movement in the cavity. As for the previous results, Be number more depends on S viscous than S th at high Ra numbers. Therefore, a considerable reduction can be observed on Be number in Figure 5b when WR reaches 0.3. However, by increasing WR to 0.4, the convection effect is decreased (see Figure 5a); therefore, the nanofluid velocity decreases that leads to entropy reduction (S viscous ). In this state, the S viscous tends to S th value, and because of this, the Be does not change noticeably at WR = 0.4 compared to WR = 0.3. Figure 5b shows that the Cu nanoparticle volume fractions in hybrid nanofluid have no substantial effect on Be number as a result of convection mechanisms dominance at high Ra numbers. However, Figure 5b shows that Be slightly decreases with φ r increment at WR = 0.2. These slight changes can be attributed to Cu density that has a larger value compared to Al 2 O 3 . The greater density at high Ra with a narrow wall width leads to increase entropy caused by friction factors and decrease Be number. Figure 6 shows the influence of the Ra and the hybrid nanoparticles' concentration on the averaged of produced entropy. As previously mentioned, by increasing the Ra number, the buoyancy force intensifies, and hence, the average velocity and the rate of heat transfer in the enclosure increase (See Equations (22) and (23)). As a result, the velocity and temperature gradient are enhanced, and the generated entropy increases. As can be seen from Figure 6a,b, increasing the Rayleigh number has a considerable impact on both components of the produced entropy. It is evident that for low Rayleigh numbers, the friction share of entropy is almost feeble and the entropy generation is mainly controlled by the heat transfer mechanism. However, in high Rayleigh numbers, S viscous overtakes its counterpart, S th and the entropy generation induced by the friction of the hybrid nanofluid completely prevails. On the other hand, increasing the volume fraction of the hybrid nanoparticles, according to Equations (6) and (8) leads to the increment of the thermal conductivity and also the viscosity of the suspension. Hence, it slightly enhances the S th (See Equation (22)) and S viscous (See Equation (23)). Finally, Figure 6c represents the final result that the volume fraction of the hybrid nanoparticles, as well as the Ra number, have a direct influence on the total generated entropy in the cavity; however, the impact of the Rayleigh number (buoyancy force) is more substantial than the φ hnf . a direct influence on the total generated entropy in the cavity; however, the impact of the Rayleigh number (buoyancy force) is more substantial than the ϕhnf.

Conclusions
The thermo-gravitational convection of water-Cu-Al2O3 in an enclosed cavity was investigated. The enclosed cavity concluded two vertical and horizontal parts, and the heat was transferred from the left vertical border and bottom horizontal border toward other directions. The Cu and Al2O3 nanoparticles were chosen as composite nanoparticles. Then,

Conclusions
The thermo-gravitational convection of water-Cu-Al 2 O 3 in an enclosed cavity was investigated. The enclosed cavity concluded two vertical and horizontal parts, and the heat was transferred from the left vertical border and bottom horizontal border toward other directions. The Cu and Al 2 O 3 nanoparticles were chosen as composite nanoparticles. Then, the different volume fractions of hybrid nanoparticles and their proportion were investigated on the energy transference. Furthermore, the effect of Ra and wall width and their relationship with nanoparticle volume fractions were studied as two other important factors, which affect the energy transference and generated entropy in the chamber. The conclusions are: -Hybrid nanoparticles enhanced energy transport when the conduction mechanism was dominant. Conversely, they had no significant influence on convective transport; - The wall-width ratio (WR) is a parameter that can have a different influence on the energy transference rate in different conditions. Increasing the wall width led to a reduction of the energy transference rate at low Ra (10 3 ) owing to the dominant conduction heat transfer mechanism; -WR had a positive influence on energy transference at high Ra (10 5 ) when WR was increased from 0.2 to 0.3. However, a further increase of wall width reduced the heat transfer in the cavity when WR > 0.4, and therefore, an optimum wall width can enhance the heat transfer at high Ra; -At Ra = 10 3 , the nanoparticles of Cu and Al 2 O 3 had a similar effect on nanofluid in the range of φ hnf = 0-0.05, and they enhanced the strength of energy transference to be the same as each other. Conversely, Cu nanoparticles had a stronger impact on heat transfer compared to Al 2 O 3 in convection heat transfer at Ra = 10 5 ; -Ra and φ hnf both could enhance generated thermal and viscous entropy, however, Ra had a more intensive influence on generated entropy in the cavity. Data Availability Statement: Data will be available on request.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Numerical Approach and Grid Check
The set of non-linear, coupled, and non-dimensional equations discussed above, has been worked out with the Galerkin finite element technique. The method has been well explained in [33]. To replace the pressure terms in the equation of momentum, the penalty parameter is defined as [33,34]: The mass conservation equation will be satisfied automatically when χ is adequately high. By replacing the pressure terms with the above-defined penalty parameter, the momentum equations read: A basis function ζ k | N k=1 is used to expand all the existing variables (U, V, θ) within the computational domain: By applying the Galerkin finite element technique, the following set of residual equations will be received for the inner nodes: It should be noted that as the solutions of heat function and stream function equations could be obtained from post-processing, there is no need to solve these (and evaluate their residual equations) along with the other equations. Moreover, the process stops iterating As presented in Table A1, the average Nu and the total generated entropy obtained by five different grid densities are compared to ensure that the outcomes are grid-independent. Structured grids with equal length and height (∆x = ∆y) are used for grid check. The error percentage for Nu avg and the total generated entropy has been obtained as: It can be seen that the results of the employed case No. 3 with 7344 (∆x = ∆y = 1.0/120) have admissible accuracy and hence, has been employed in all of the simulations. In addition, a view of the employed gird (case 3) is depicted in Figure A1.

Appendix B. Validation of the Numerical Code
The validity of the developed finite element code should b accuracy and correctness of the results. As such, three different re chosen and their outcomes have been re-stimulated to guarante code are fine and error-free. In Figure A2, a comparison is perform lines and isotherms of this research with another one shown by [35], the buoyancy-driven flow of a simple flow has been addres ditions and geometry of this research are the same as the present patterns of streamlines and the isotherms are in quite a resembla that the developed code can accurately analyze the circulation an the cavity.

Appendix B. Validation of the Numerical Code
The validity of the developed finite element code should be checked to ensure the accuracy and correctness of the results. As such, three different research studies have been chosen and their outcomes have been re-stimulated to guarantee that all aspects of the code are fine and error-free. In Figure A2, a comparison is performed between the streamlines and isotherms of this research with another one shown by Nithiarasu et al. [35]. In [35], the buoyancy-driven flow of a simple flow has been addressed. The boundary conditions and geometry of this research are the same as the present one. It is evident that the patterns of streamlines and the isotherms are in quite a resemblance with [35], indicating that the developed code can accurately analyze the circulation and energy transference in the cavity.
In addition, to check that the equations governing the thermophysical characteristics of the nanofluid are appropriately implemented, the code has been further validated against a survey published by Kahveci [36] on the thermo-gravitational convection of nanosuspension in a chamber. In this research, the nano-sized particles are supposed to be easily fluidized, and hence, the solid-liquid two-phase flow of the nanoliquid is assumed to be homogenous and behaves as a single-phase fluid. The side walls in [36] are set to a temperature drop, and the upper and lower borders are adiabatic. The base fluid in work done by Kahveci [36] is water (Pr = 6.2), and the impact of dispersion of different nanoparticles (including Cu, Ag, CuO, Al 2 O 3 and TiO 2 ) on the circulation pattern and the rate of energy transference has been evaluated. Figure A3 depicts the comparison between the outcomes of Kahveci [36] and the present research for the dependency of the Nu avg on the concentration of Al 2 O 3 nano-sized particles. It is apparent that the obtained values of the Nu avg coincide with the data of Kahveci [36], representing a finite element code that can appropriately simulate the thermal convective energy transference of nanoliquids in a confined chamber.
As the final step, two more verifications have been performed to ensure that both entropy generation terms and the heat lines are correctly calculated. Figure A4 shows the isentropic lines and isolines of local Bejan number for the work done by Ilis [37] and the current study. In [37], the entropy generation resulting from free convective transport in a square chamber has been analyzed. The imposed boundary conditions are exactly the same as [36]. It is evident that both S gen and Be are correctly calculated, and the values of similar isolines are quite similar. Finally, the survey of Deng and Tang [38], shown in Figure A5, has been utilized to check the validity of the solved heat function equation. The employed boundary conditions in [38] are exactly the same as [36,37], except that a solid rectangular solid is placed in the center of the cavity, and an extra energy equation for the solid block needs to be solved. As seen, the heat lines are in acceptable agreement with those presented by Deng and Tang [38].
code are fine and error-free. In Figure A2, a comparison is performed between the streamlines and isotherms of this research with another one shown by Nithiarasu et al. [35]. In [35], the buoyancy-driven flow of a simple flow has been addressed. The boundary conditions and geometry of this research are the same as the present one. It is evident that the patterns of streamlines and the isotherms are in quite a resemblance with [35], indicating that the developed code can accurately analyze the circulation and energy transference in the cavity.  metry 2021, 13,122 In addition, to check that the equations governing the thermop of the nanofluid are appropriately implemented, the code has b against a survey published by Kahveci [36] on the thermo-gravi nanosuspension in a chamber. In this research, the nano-sized par be easily fluidized, and hence, the solid-liquid two-phase flow of sumed to be homogenous and behaves as a single-phase fluid. Th set to a temperature drop, and the upper and lower borders are ad in work done by Kahveci [36] is water (Pr = 6.2), and the impact of nanoparticles (including Cu, Ag, CuO, Al2O3 and TiO2) on the circu rate of energy transference has been evaluated. Figure A3 depicts th the outcomes of Kahveci [36] and the present research for the depe the concentration of Al2O3 nano-sized particles. It is apparent that the Nuavg coincide with the data of Kahveci [36], representing a fin can appropriately simulate the thermal convective energy transfer a confined chamber. Figure A3. Influence of Al2O3 nanoparticles volume fraction on the Nuavg (P Сomparison between the present work and outcomes of Kahveci [36].
As the final step, two more verifications have been performe entropy generation terms and the heat lines are correctly calculated Comparison between the present work and outcomes of Kahveci [36].