Correlations for Total Entropy Generation and Bejan Number for Free Convective Heat Transfer of an Eco-Friendly Nanofluid in a Rectangular Enclosure under Uniform Magnetic Field

In this paper, focusing on the study of entropy generation (EGN), the convection flow of an eco-friendly nanofluid (N-F) in a rectangular enclosure is studied numerically. The nanoparticles (N-Ps) used are silver N-P, which are obtained in an eco-friendly manner from natural materials. By suspending these N-Ps in an equal mixture of water and ethylene glycol (E-G), the N-F has been prepared. There are two constant-temperature triangular obstacles with height w and base H that are placed on the hot wall. There is a magnetic field (M-F) in the x-direction. To simulate the N-F flow, eco-friendly N-P relations are used, and the equations are solved using the volume control method and the SIMPLE algorithm. The variables include Rayleigh number (Ra), Hartmann number (Ha), H, W, and the volume fraction of silver N-Ps. The effect of these parameters is evaluated on the EGN and Bejan number (Be). Finally, a correlation is expressed for the EGN for a range of variables. The most important results of this paper demonstrate that the addition of silver eco-friendly N-Ps intensifies the EGN so that the addition of 3% of N-Ps enhances the EGN by 3.8%. An increment in the obstacle length reduces the Be barrier while increasing the Ha, which enhances the Be when the convection is strong. Increasing the height of the obstacle intensifies entropy generation.


Introduction
In recent years, researchers have paid more attention to nanotechnology. Nanotechnology has applications in many industries, including medicine, aerospace, military, construction, food, heat transfer, renewable energy, etc. [1][2][3][4][5]. One of the applications of nanoparticles (N-Ps) in manufacturing is the preparation of nanofluids (N-Fs). N-Fs are widely used in heat transfer industries [6][7][8][9][10]. One of the important applications of nanofluids is their use in closed enclosures, which are used to enhance the improvement of thermal and refrigeration equipment [11][12][13][14][15]. Ghasemi and Aminossadati [16] investigated the effect of N-Fs in a square enclosure. They changed the percentage of N-Ps in the base fluid and found that a higher volume percentage leads to more heat transfer in the enclosure. Additionally, a rise in the Rayleigh number (Ra) leads to an increase in the vortices' velocity, suggesting an increase in heat transfer. Among the various N-Ps, some researchers have

Problem Description
The two-dimensional enclosure consists of a rectangular enclosure, with dimensions as shown in Figure 1. The enclosure has two walls of high temperature and low temperature and two insulated walls. Two triangular blades are mounted on the bottom wall. The distance between the sides of the blades is equal. The enclosure is saturated with ecofriendly silver N-Ps dispersed in 50:50 water and E-G. It is noteworthy that the N-Ps used are eco-friendly and synthesized from the environment. The enclosure is under a M-F in the direction as shown in Figure 1.

Governing Equations and Boundary Condition
The non-dimensional equations for the N-F flow, including the continuity, the momentum, and the energy equations, for steady laminar flow and incompressible Newtonian N-F, are as follows. Additionally, the effects of viscosity loss, radiative heat transfer, and volumetric forces except the gravitational force are neglected [40]: where X and Y indices mean the first derivation with respect to these parameters and XX and YY indices represent the second derivation with respect to these parameters. Equations (2) and (3) are momentum equations in the x-and y-directions. In the momentum equation in the Y-direction, there is a buoyancy term due to the gravitational force. There is also a magnetic field force in this equation. Equation (5) is used to non-dimensionalize the equations. Additionally, in Equation (6), the definitions of the Prandtl, Ra, Ha, and EGN parameters are presented. Figure A1. A schematic of the enclosure.

Governing Equations and Boundary Condition
The non-dimensional equations for the N-F flow, including the continuity, the momentum, and the energy equations, for steady laminar flow and incompressible Newtonian N-F, are as follows. Additionally, the effects of viscosity loss, radiative heat transfer, and volumetric forces except the gravitational force are neglected [40]: where X and Y indices mean the first derivation with respect to these parameters and XX and YY indices represent the second derivation with respect to these parameters. Equations (2) and (3) are momentum equations in the xand y-directions. In the momentum equation in the Y-direction, there is a buoyancy term due to the gravitational force. There is also a magnetic field force in this equation. Equation (5) is used to non-dimensionalize the equations. Additionally, in Equation (6), the definitions of the Prandtl, Ra, Ha, and EGN parameters are presented.
To solve the governing equations, boundary conditions must be used. Figure A2 shows the non-dimensionalized boundary conditions.
To solve the governing equations, boundary conditions must be used. Figure 2 shows the non-dimensionalized boundary conditions.
The above equation consists of three terms. Temperature changes cause thermal EGN. Velocity changes cause frictional ENG, and velocity changes for different magnetic fields cause magnetic field ENG. The EGN in the enclosure is expressed using the equation below.
The Be is defined as follows:

N-F Properties Equations
Eco-friendly silver N-Ps are used to make silver N-Fs in a 50% volume aqueous solution E-G. These N-Ps are obtained from tea leaves in different stages. The following are the relationships that can be used for this N-F [46][47][48]: (11) ρ nf = (1 − φ)ρ f + φρ , (12)  The amount of EGN in the general case is defined as follows. The total EGN includes three types of EGN: thermal EGN, EGN due to fluid friction, and EGN due to M-F.
The above equation consists of three terms. Temperature changes cause thermal EGN. Velocity changes cause frictional ENG, and velocity changes for different magnetic fields cause magnetic field ENG. The EGN in the enclosure is expressed using the equation below.
The Be is defined as follows: Be a = S gen,Thermal S g ,

N-F Properties Equations
Eco-friendly silver N-Ps are used to make silver N-Fs in a 50% volume aqueous solution E-G. These N-Ps are obtained from tea leaves in different stages. The following are the relationships that can be used for this N-F [46][47][48]: Processes 2021, 9,1930 5 of 20 Correlations related to the thermal conductivity of N-F can be used for the ones with volume fractions up to 1%. Table A1 lists the thermophysical characteristics of E-G and silver N-Ps. Table A1. Thermophysical properties of water/E-G and Ag [18,47,[49][50][51][52][53].

Numerical Procedure
To solve dimensionless equations, the equations are first algebraized using the volume control method. Then, a home code is written in FORTRAN software to solve dimensionless equations using the SIMPLE algorithm with the help of boundary conditions. In this software, a structured mesh is generated. The convergence criterion to solve all equations, including continuity, momentum, and energy equations, is 10 −8 . Grid study and validation are performed, and the results are presented in the Appendix A.     Figure 4 reveals the effect of Ra and Ha on the temperature field for H = W = 0.2 and a volume fraction of 0.3%. In the case of low Ra, the isotherms do not change with the Ha and remain as parallel lines. However, for Ra = 10 5 , the variations of the Ha affect the isotherms and change their shape. The changes of the Ra also affect the temperature contours, changing it from parallel lines to overlapping ones.     3%. An enhancement in the Ra due to an increase in the velocity and an increment in the sudden changes in the temperature enhances the EGN. However, an increase in the Ha reduces the EGN due to the reduction in the velocity. Obviously, at Ra = 10 5 , where the velocity is higher, and the increase in Ha has a greater effect on the deceleration, the decrease in the EGN is lower. Thus, the maximum EGN corresponds to Ra = 10 5 in the absence of the M-F.  Figure A6 demonstrates the EGN for different Ra and Ha when W = 0.5, H = 0.7, and the volume fraction is 0.3%. An enhancement in the Ra due to an increase in the velocity and an increment in the sudden changes in the temperature enhances the EGN. However, an increase in the Ha reduces the EGN due to the reduction in the velocity. Obviously, at Ra = 10 5 , where the velocity is higher, and the increase in Ha has a greater effect on the deceleration, the decrease in the EGN is lower. Thus, the maximum EGN corresponds to Ra = 10 5 in the absence of the M-F.    Figure A7 shows the EGN for different Ra and various blade heights when Ha = 30, H = 0.7, and volume fraction is 0.3%. The trend of EGN changes with the Ra is an increasing trend. The EGN is enhanced with the height, which has less of an effect on EGN than the Ra. The enlargement of the blade causes sudden changes in temperature as well as velocity, producing irreversibility and an increase in EGN. Figure A8 shows the EGN for different Ra and various blade lengths when Ha = 30, W = 0.5, and the volume fraction is 0.3%. The increasing trend of EGN with the Ra is observed. The change in blade length, however, has different effects on EGN at different Ra. At low Ra, an increase in the blade length first increases and then decreases EGN. While, at Ra greater than 45,000, the increase in blade length prevents a constant enhancing trend in EN. At Ra = 10 5 , the EGN is enhanced with the blade length.        Figure A11 shows the EGN for different blade heights and lengths when Ra = 10 5 , Ha = 30, and the volume fraction is 0.3%. The variations of EGN with the blade height have an increasing trend so that for all blade lengths, increasing its height enhances the amount of EGN. An increment in the length of the blade initially enhances the EGN slightly and then reduces the EGN so that the EGN prevents the increasing trend to W = 0.4, and then the trend is reversed. The minimum EGN occurs at W = 1.2 and H = 0.2, while the maximum one occurs at W = 0.4 and H = 0.8. Figure A12 illustrates the Be for different Ra and Ha when W = 0.5 and H = 0.7, and the volume fraction is 0.3%. The Be has a constant decreasing trend with the Ra, while the Be is reduced and is enhanced with the Ha. This trend is more visible at low Ra. At high Ra, an increasing trend takes place for the Be. The minimum value of the Be occurs at Ra = 10 5 while its maximum occurs at Ra = 10 3 and Ha = 60. Processes 2021, 9, x FOR PEER REVIEW 11 of 21    Figure A13 demonstrates the Be for different Ra and various blade heights when H = 0.7 and Ha = 30, and the volume fraction is 0.3%. It can be seen that the variation of the Be is under a decreasing trend with the Ra, but changing the blade height has a different effect on the Be, so that for high and low Ra, changing the blade height has a different impact on the Be. For high Ra, an enhancement in the blade height first intensifies the Be and then reduces it, while at low Ra, an increment in the blade height reduces the Be. Figure A14 reveals the EGN for different Ra and various blade heights when H = 0.7 and Ha = 30, and volume fraction is 0.3%. The variations of the Be with the blade length have a constant decreasing trend so that an increase in the blade length reduces the amount of Be for all Ra. The declining trend of the Be is maintained with the Ra for various blade lengths. esses 2021, 9, x FOR PEER REVIEW 12 of Figure 10. EGN for different Ha and various blade lengths when Ra = 10 5 , W = 0.7, and the volume fraction is 0.3%. Figure 11 shows the EGN for different blade heights and lengths when Ra = 10 5 , = 30, and the volume fraction is 0.3%. The variations of EGN with the blade height ha an increasing trend so that for all blade lengths, increasing its height enhances the amou of EGN. An increment in the length of the blade initially enhances the EGN slightly a then reduces the EGN so that the EGN prevents the increasing trend to W = 0.4, and th the trend is reversed. The minimum EGN occurs at W = 1.2 and H = 0.2, while the ma mum one occurs at W = 0.4 and H = 0.8.  Table A2 presents the EGN for different volume percentages of silver eco-friendly N-Ps at Ra = 10 5 , Ha = 0, and W = H = 0.2. It is observed that an increment in the amount of silver eco-friendly N-Ps in the fluid enhances the amount of EGN. Improving the heat transfer by adding silver N-Ps causes the temperature changes in the enclosure to intensify, and as a result, the amount of EGN, especially thermal EGN, is enhanced. For instance, the addition of 0.5% silver N-Ps to a mixture of water and E-G results in an enhancement in the EGN by 6.2% compared to the base fluid. esses 2021, 9, x FOR PEER REVIEW 13 of Figure 11. EGN for different blade heights and lengths when Ra = 10 5 , Ha = 30, and the volume fraction is 0.3%. Figure 12 illustrates the Be for different Ra and Ha when W = 0.5 and H = 0.7, and t volume fraction is 0.3%. The Be has a constant decreasing trend with the Ra, while the is reduced and is enhanced with the Ha. This trend is more visible at low Ra. At high R an increasing trend takes place for the Be. The minimum value of the Be occurs at Ra = 1 while its maximum occurs at Ra = 10 3 and Ha = 60.    Figure 13 demonstrates the Be for different Ra and various blade heights when H 0.7 and Ha = 30, and the volume fraction is 0.3%. It can be seen that the variation of the is under a decreasing trend with the Ra, but changing the blade height has a differe effect on the Be, so that for high and low Ra, changing the blade height has a differe impact on the Be. For high Ra, an enhancement in the blade height first intensifies the and then reduces it, while at low Ra, an increment in the blade height reduces the Be.    Table 2 presents the EGN for different volume percentages of silver eco-friendly N-Ps at Ra = 10 5 , Ha = 0, and W = H = 0.2. It is observed that an increment in the amount of silver eco-friendly N-Ps in the fluid enhances the amount of EGN. Improving the heat transfer by adding silver N-Ps causes the temperature changes in the enclosure to intensify, and as a result, the amount of EGN, especially thermal EGN, is enhanced. For instance, the addition of 0.5% silver N-Ps to a mixture of water and E-G results in an enhancement in the EGN by 6.2% compared to the base fluid.  Equation (16) is a correlation to express the total EGN in terms of the Ra, the Ha, and the blade length and height.

Results and Discussion
It is noteworthy that an increment in ENG in different heat exchangers is undesirable. Enhancing ENG in the enclosure results in an enhancement of irreversibility. Increasing ENG ultimately leads to an intensification of energy loss. Thus, it is attempted to reduce the ENG in different devices so that the irreversibility is reduced, and the efficiency of the devices is enhanced. Using the above relation, it can be found how many variable parameters can have the minimum ENG in the enclosure.

Conclusions
In this paper, the EGN in a rectangular enclosure saturated with an eco-friendly N-F containing silver N-Ps dispersed in an equal mixture of water and E-G was evaluated. The

Appendix A
To perform the grid independence test, various studies were performed for different blade conditions. Finally, by examining the amount of EGN for different grids (Table A1), a grid resolution of 450 × 150 was selected for the simulations. The simulations are required to validate using previous numerical or experimental results. For this purpose, it is necessary to simulate one of the previously presented works in the same way and compare the results with the previous ones. Various validations are performed to validate the code written for the present study, leading to reasonable results. For example, one of these validations performed with the results of Pirmohammadi and Ghassemi [54] is shown in Figure A1. In this validation, the average Nu at different Ha is calculated. The findings of the current simulations are consistent with those published by Pirmohammadi and Ghassemi [54], as shown in the Figure A1.