**4. Results and Discussion**

**4. Results and Discussion** In this important part of the research, we present the results that show the impacts of Darcy's number (Da), Hartmann's number (Ha), Rayleigh's number (Ra) and the concentration of PCM nanoparticles on thermal activity and how the suspension (PCM In this important part of the research, we present the results that show the impacts of Darcy's number (Da), Hartmann's number (Ha), Rayleigh's number (Ra) and the concentration of PCM nanoparticles on thermal activity and how the suspension (PCM nanoparticles + water) moves inside the room.

nanoparticles + water) moves inside the room. We note a scientific fact that occurs in a fluid: when the fluid absorbs a quantity of thermal energy, its weight decreases and, therefore, begins to move upward as it shifts We note a scientific fact that occurs in a fluid: when the fluid absorbs a quantity of thermal energy, its weight decreases and, therefore, begins to move upward as it shifts downward. In contrast, the fluid moves downward as it loses heat energy. This kind of heat transfer accompanied by mass transfer is called Buoyancy-driven flow.

downward. In contrast, the fluid moves downward as it loses heat energy. This kind of heat transfer accompanied by mass transfer is called Buoyancy-driven flow. Figure 3 is devoted to understanding the effect of the Ra (between 10<sup>3</sup> and 10<sup>6</sup> Figure 3 is devoted to understanding the effect of the Ra (between 10<sup>3</sup> and 10<sup>6</sup> ) number on the studied system for Ha = 0, Da = 10−<sup>2</sup> and ϕ = 4%. This understanding is achieved by analyzing the contours of isotherms (dimensionless temperature), streamlines (line paths)

number on the studied system for Ha = 0, Da = 10−2 and φ = 4%. This understanding is achieved by analyzing the contours of isotherms (dimensionless temperature),

temperature (isotherms) shows a gradual temperature distribution from the hot fin toward the cold walls of the container. We note that the fluid layers adjacent to the fin

)

and phase-changing zone (heat capacity Cr). The dimensionless temperature (isotherms) shows a gradual temperature distribution from the hot fin toward the cold walls of the container. We note that the fluid layers adjacent to the fin have a high temperature. Then this distribution is centered over the fin in an upward direction. In contrast, cold fluid layers are centered near the lateral walls. It is noted that the intensity of this distribution increases in terms of the number Ra. The streamlines are exactly equivalent to the isotherm patterns, meaning that the fluid layers above the trapezoidal fin move toward the top, while the cold layers on the lateral edges of the chamber move downwards. Finally, we note the formation of a circular motion of the suspension flow. This movement is divided into symmetric parts, one on the right side with the same clockwise direction and the second on the left side in the anticlockwise direction. We also note that the intensity of the movement of the two vortices increases in terms of the number Ra, whereas the center of this recirculation zone moves upward as the value of Ra increases. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 10 of 17 have a high temperature. Then this distribution is centered over the fin in an upward direction. In contrast, cold fluid layers are centered near the lateral walls. It is noted that the intensity of this distribution increases in terms of the number Ra. The streamlines are exactly equivalent to the isotherm patterns, meaning that the fluid layers above the trapezoidal fin move toward the top, while the cold layers on the lateral edges of the chamber move downwards. Finally, we note the formation of a circular motion of the suspension flow. This movement is divided into symmetric parts, one on the right side with the same clockwise direction and the second on the left side in the anticlockwise direction. We also note that the intensity of the movement of the two vortices increases in terms of the number Ra, whereas the center of this recirculation zone moves upward as the value of Ra increases.

**Figure 3.** Ra number influence on streamlines, isotherms, and Cr for Da = 10<sup>−</sup>2, Ha = 0, and φ = 4%. **Figure 3.** Ra number influence on streamlines, isotherms, and Cr for Da = 10−<sup>2</sup> , Ha = 0, and ϕ = 4%.

When Ra = 106, the intensity of thermal buoyancy becomes very high, and we notice that the center of the vortex is divided into two parts, the first near the cold wall and the second near the hot fin. For the heat capacity contours (Cr), we notice an effect of the number Ra on the heat capacity (Cr). We notice that there is a line in the form of a semicircle around the trapezoidal fin for Ra = 103, then it is divided into two symmetric parts as the number Ra increases. The blue lines indicate the regions where the change of physical state occurs for PCM nanoparticles. We notice that the lines expand in terms of Ra because the movement of suspension particles increases in terms of Ra. In addition to this, it is noticed that the two heat capacity lines are thinner, close to both cold and hot walls, while they are thicker in the middle of the room. The greater the temperature When Ra = 10<sup>6</sup> , the intensity of thermal buoyancy becomes very high, and we notice that the center of the vortex is divided into two parts, the first near the cold wall and the second near the hot fin. For the heat capacity contours (Cr), we notice an effect of the number Ra on the heat capacity (Cr). We notice that there is a line in the form of a semicircle around the trapezoidal fin for Ra = 10<sup>3</sup> , then it is divided into two symmetric parts as the number Ra increases. The blue lines indicate the regions where the change of physical state occurs for PCM nanoparticles. We notice that the lines expand in terms of Ra because the movement of suspension particles increases in terms of Ra. In addition to this, it is noticed that the two heat capacity lines are thinner, close to both cold and hot walls, while they are thicker in the middle of the room. The greater the temperature gradient, the thinner the thickness of the equivalent heat capacity line for the zones where the change of physical state occurs. gradient, the thinner the thickness of the equivalent heat capacity line for the zones where the change of physical state occurs.

In order to understand the impact of the values of Hartmann number (between the values 0 to 100) on the studied system, Figure 4 illustrates this effect of Da = 10−<sup>2</sup> , Ra = 10<sup>5</sup> and ϕ = 4%. The scientific explanation for the influences of the Hartmann number is found by analyzing the contours of isotherms, streamlines and heat capacity (Cr). It is known that the magnetic field creates an electromagnetic force (Lorentz force) that hinders the movement of fluid particles, so we notice a gradual decline in isotherm distribution in terms of the number Ha. The streamlines also show a decrease in the intensity of vorticity by increasing the value of Ha. The contours of heat capacity (Cr) show that the blue lines of physical state change converge as the value of Ha increases because the velocity of the suspension particles decreases in terms of Ha. In order to understand the impact of the values of Hartmann number (between the values 0 to 100) on the studied system, Figure 4 illustrates this effect of Da = 10−2, Ra = 105 and φ = 4%. The scientific explanation for the influences of the Hartmann number is found by analyzing the contours of isotherms, streamlines and heat capacity (Cr). It is known that the magnetic field creates an electromagnetic force (Lorentz force) that hinders the movement of fluid particles, so we notice a gradual decline in isotherm distribution in terms of the number Ha. The streamlines also show a decrease in the intensity of vorticity by increasing the value of Ha. The contours of heat capacity (Cr) show that the blue lines of physical state change converge as the value of Ha increases because the velocity of the suspension particles decreases in terms of Ha.

**Figure 4.** Ha number influence on streamlines, isotherms, and Cr for Ra = 105, Da = 10−2, and φ = 4%. **Figure 4.** Ha number influence on streamlines, isotherms, and Cr for Ra = 10<sup>5</sup> , Da = 10−<sup>2</sup> , and ϕ = 4%.

The results of this work also include the effect of the medium permeability on the studied system, so Figure 5 illustrates the influences of the Darcy number (between Da = 10−2 and Da = 10−2) on each of the isotherms, heat capacity and streamlines for Ra = 105, Ha = 0, and φ = 4%. It is known that the permeability of the space is defined in terms of the Darcy number; that is, the higher the value of this number, the greater the permeability. Based on this, the isotherms show an expansion of the dimensionless temperature in terms of Darcy's number because the suspension movement becomes easier. On the other hand, the streamlines depict an increase in the intensity of the vortices and an increase in their sizes as the value of Da increases. Heat capacity contours are also influenced by the Darcy The results of this work also include the effect of the medium permeability on the studied system, so Figure 5 illustrates the influences of the Darcy number (between Da = 10−<sup>2</sup> and Da = 10−<sup>2</sup> ) on each of the isotherms, heat capacity and streamlines for Ra = 10<sup>5</sup> , Ha = 0, and ϕ = 4%. It is known that the permeability of the space is defined in terms of the Darcy number; that is, the higher the value of this number, the greater the permeability. Based on this, the isotherms show an expansion of the dimensionless temperature in terms of Darcy's number because the suspension movement becomes easier. On the other hand, the streamlines depict an increase in the intensity of the vortices and an increase in their sizes as the value of Da increases. Heat capacity contours are also influenced by the Darcy

number because the increase in the speed of the flow expands the blue lines of the regions with the change in the physical state of nano-encapsulated PCM. number because the increase in the speed of the flow expands the blue lines of the regions with the change in the physical state of nano-encapsulated PCM.

**Figure 5.** Da number influence on streamlines, isotherms, and Cr for Ra = 105, Ha = 0, and φ = 4%. **Figure 5.** Da number influence on streamlines, isotherms, and Cr for Ra = 10<sup>5</sup> , Ha = 0, and ϕ = 4%.

Figure 6 presents the developments of the mean values of the Nusselt number (Nu) of the hot trapezoidal fin in terms of Da, Ha, Ra and φ. Figure 6a is intended to show the effect of numbers Ha and Ra on Nu. For φ = 4% and Da = 10−2 we notice that the values of Nu increase with increasing values of Ra and decrease as the values of Ha increase, just as expected. That is, the higher the speed of the suspension particles, the better the convective heat transfer, and accordingly, the values of the number Nu increase. In addition to this, we notice that the increase in the values of Nu in terms of Ra gradually changes with the increase in the value of Ha. That is, as the value of Ha increases from 0 to 100, the evolution becomes nonlinear. Figure 6b is intended to present the influence of the nano-encapsulated PCM concentration and Ra on Nu. For Ha = 0 and Da = 10−2 we notice that in this studied space, there is no strong influence of the nano-encapsulated PCM concentration on Nu. The negligible effect of the ratio of NEPCM on Nu can be explained by the finite motion of the flow within the chamber. Figure 6c presents the effect of the Darcy number, i.e., the porosity of the medium and Ra number on Nu of the hot surfaces of trapezoidal fin for Ha = 0 and φ = 4%. We note an effective effect of the medium permeability (Da number) on Nu values. That is, the values of Nu increase with augmenting Da. The latter can be explained by the following: the expansion of the medium's permeability means a decrease in the flow obstruction, which allows for Figure 6 presents the developments of the mean values of the Nusselt number (Nu) of the hot trapezoidal fin in terms of Da, Ha, Ra and ϕ. Figure 6a is intended to show the effect of numbers Ha and Ra on Nu. For ϕ = 4% and Da = 10−<sup>2</sup> we notice that the values of Nu increase with increasing values of Ra and decrease as the values of Ha increase, just as expected. That is, the higher the speed of the suspension particles, the better the convective heat transfer, and accordingly, the values of the number Nu increase. In addition to this, we notice that the increase in the values of Nu in terms of Ra gradually changes with the increase in the value of Ha. That is, as the value of Ha increases from 0 to 100, the evolution becomes nonlinear. Figure 6b is intended to present the influence of the nano-encapsulated PCM concentration and Ra on Nu. For Ha = 0 and Da = 10−<sup>2</sup> we notice that in this studied space, there is no strong influence of the nano-encapsulated PCM concentration on Nu. The negligible effect of the ratio of NEPCM on Nu can be explained by the finite motion of the flow within the chamber. Figure 6c presents the effect of the Darcy number, i.e., the porosity of the medium and Ra number on Nu of the hot surfaces of trapezoidal fin for Ha = 0 and ϕ = 4%. We note an effective effect of the medium permeability (Da number) on Nu values. That is, the values of Nu increase with augmenting Da. The latter can be explained by the following: the expansion of the medium's permeability means a decrease in the flow obstruction, which allows for enhanced convective heat transfer.

enhanced convective heat transfer.

**Figure 6.** the effect of (**a**) Ha number, (**b**) nanoparticle volume fraction and (**c**) Da number on the Nu avg for different values of Ra. **Figure 6.** The effect of (**a**) Ha number, (**b**) nanoparticle volume fraction and (**c**) Da number on the Nu avg for different values of Ra.

In order to know the quality of the convective heat transfer along the hot surfaces of trapezoidal fin, Figure 7 presents the distribution of the local values of Nu in terms of Da, Ra, and Ha and φ. Figure 7a illustrates the effect of Da on the local distribution of Nu for Ra = 10<sup>5</sup> , φ = 4% and Ha = 0. We note that the local values of Nu on both sides of the trapezoidal fin are greater than the values in the middle of the fin because the velocity of suspension particles on both sides of the fin is greater. In addition, raising the number Da increases the local values of Nu number along the fin. Figure 7b is intended to display the local distribution of Nu number in terms of Ha for Ra = 10<sup>5</sup> , φ = 4% and Da = 10−2. All local values of Nu decrease with increasing Ha along the heated surfaces of the fin. Figure 7c is devoted to presenting the impact of the Ra number on local values of Nu along the heated fin for Ha = 0, φ = 4% and Da = 10−2. It is clearly demonstrated that raising Ra increases all local values of Nu along the heated surfaces. Figure 7d is devoted to showing the concentration's impact on the local Nu distribution for Ha = 0, Ra = 10<sup>5</sup> and Da = 10−2. We note that there is a very small effect of this element on the local values of Nu. In order to know the quality of the convective heat transfer along the hot surfaces of trapezoidal fin, Figure 7 presents the distribution of the local values of Nu in terms of Da, Ra, and Ha and ϕ. Figure 7a illustrates the effect of Da on the local distribution of Nu for Ra = 10<sup>5</sup> , ϕ = 4% and Ha = 0. We note that the local values of Nu on both sides of the trapezoidal fin are greater than the values in the middle of the fin because the velocity of suspension particles on both sides of the fin is greater. In addition, raising the number Da increases the local values of Nu number along the fin. Figure 7b is intended to display the local distribution of Nu number in terms of Ha for Ra = 10<sup>5</sup> , ϕ = 4% and Da = 10−<sup>2</sup> . All local values of Nu decrease with increasing Ha along the heated surfaces of the fin. Figure 7c is devoted to presenting the impact of the Ra number on local values of Nu along the heated fin for Ha = 0, ϕ = 4% and Da = 10−<sup>2</sup> . It is clearly demonstrated that raising Ra increases all local values of Nu along the heated surfaces. Figure 7d is devoted to showing the concentration's impact on the local Nu distribution for Ha = 0, Ra = 10<sup>5</sup> and Da = 10−<sup>2</sup> . We note that there is a very small effect of this element on the local values of Nu.

**Figure 7.** Effects of (**a**) Da number, (**b**) Ha number, (**c**) Ra number and (**d**) nanoparticle volume fraction on the local Nusselt number. **Figure 7.** Effects of (**a**) Da number, (**b**) Ha number, (**c**) Ra number and (**d**) nanoparticle volume fraction on the local Nusselt number.

#### **5. Conclusions 5. Conclusions**

Through this work, we were able to create a digital simulation for suspension inside a room in the form of an inversed T shape. The suspension consists of water and the elements of nano-encapsulated PCM. The chamber is permeable and has a compound bottom with a hot trapezoidal fin, while the lateral ends are cold. We sought, through the simulation results, to highlight the heat transfer of free convection form between hot and cold elements by using the suspension as a thermal conductor. The study was also carried out under the effect of the intensity of the magnetic field. Analyzing the results of this research, we concluded the following: Through this work, we were able to create a digital simulation for suspension inside a room in the form of an inversed T shape. The suspension consists of water and the elements of nano-encapsulated PCM. The chamber is permeable and has a compound bottom with a hot trapezoidal fin, while the lateral ends are cold. We sought, through the simulation results, to highlight the heat transfer of free convection form between hot and cold elements by using the suspension as a thermal conductor. The study was also carried out under the effect of the intensity of the magnetic field. Analyzing the results of this research, we concluded the following:


• The bar indicating the location of the change in the physical state of PCM elopements is characterized by two states: the first is the presence of a single band hovering around the heated surface for the low speed of the flow; the second case is characterized by the presence of two opposite bands, one on the right and the other on the left when the flow speed is high.

**Author Contributions:** Conceptualization, A.A. and A.M.; methodology, O.Y. and M.A.-K.; software, A.A.; validation, O.Y. and Z.D.; formal analysis, H.L.; investigation, H.L. and K.G.; resources, K.G. and O.Y.; data curation, A.M.; writing—original draft preparation, A.M. and H.L.; writing—review and editing, A.A., M.A.-K. and R.M.; visualization, A.M. and M.A.-K.; supervision, A.A.; project administration, A.A.; funding acquisition, M.A.-K. All authors have read and agreed to the published version of the manuscript.

**Funding:** Not applicable.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number (R.G.P.2/224/43). The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4331317DSR75). Authors also extend their appreciation to Mathematics Department at Khalifa University for supporting this work.

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