*4.2. Effects of Nanoparticle Volume Fraction*

Figure 4 shows the isotherms and streamlines for different nanoparticle volume fraction and Rayleigh number at *Da* = 10−<sup>2</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. The color scales on the left represent the dimensionless temperature and those on the right represent the dimensionless velocity, as in other sections of the article. From Figure 4, for *Ra* = 10<sup>3</sup> , the isotherms have a uniform distribution; this is because the buoyancy force is weak compared with the viscous force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is weak. For *Ra* = 10<sup>4</sup> , a slight thermal disturbance appeared, which indicates that the flow is enhanced and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For *Ra* = 10<sup>5</sup> , the isotherms are almost horizontally distributed, especially when *φ* = 0.9, which means that the natural convection heat transfer turns out to be more significant and the effect of volume fraction is more pronounced. With the increase of Rayleigh number, the streamlines become denser near the walls and the cell becomes bigger and has a tendency to move upward due to the enhanced buoyance force. In addition, for *Ra* = 10<sup>3</sup> and *Ra* = 10<sup>4</sup> , the effect of volume fraction on the streamlines is very weak, while for *Ra* = 10<sup>5</sup> , the effect is more pronounced. Figure 5 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in volume fraction leads to heat transfer enhancement for all considered Rayleigh numbers, and the effect of volume fraction is more pronounced when the Rayleigh number is high.

**Figure 3.** Effects of Brownian motion on the average Nusselt number (*Nu*avg) along the inner wall for different nanoparticle volume fractions (*ϕ*) and nanoparticle diameters (*d*sp) at *ε* = 0.5, *RR* = 2, and (**a**) *Ra* = 5 × 103, *Da* = 10<sup>−</sup>2, (**b**) *Ra* = 5 × 104, *Da* = 10<sup>−</sup>2, (**c**) *Ra* = 5 × 104, *Da* = 10<sup>−</sup>3. **Figure 3.** Effects of Brownian motion on the average Nusselt number (*Nu*avg) along the inner wall for different nanoparticle volume fractions (*φ*) and nanoparticle diameters (*d*sp) at *<sup>ε</sup>* = 0.5, *RR* = 2, and (**a**) *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>3</sup> , *Da* = 10−<sup>2</sup> , (**b**) *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , (**c**) *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>3</sup> .

Figure 4 shows the isotherms and streamlines for different nanoparticle volume fraction and Rayleigh number at *Da* = 10−2, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. The color scales on the left represent the dimensionless temperature and those on the right represent the dimensionless velocity, as in other sections of the article. From Figure 4, for *Ra* = 103, the isotherms have a uniform distribution; this is because the buoyancy force is weak compared with the viscous force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is weak. For *Ra* = 104, a slight thermal disturbance appeared, which indicates that the flow is enhanced and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For *Ra* = 105, the isotherms are almost horizontally distributed, especially when *ϕ* = 0.9, which means that the natural convection heat transfer turns out to be more significant and the effect of volume fraction is more pronounced. With the increase of Rayleigh number, the streamlines become denser near the walls and the cell becomes bigger and has a tendency to move upward due to the enhanced buoyance force. In addition, for *Ra* = 103 and *Ra* = 104, the effect of volume fraction on the streamlines is very weak, while for *Ra* = 105, the effect is more pronounced.

*4.2. Effects of Nanoparticle Volume Fraction* 

Figure 5 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in volume fraction leads to heat transfer enhancement for all considered Rayleigh numbers, and the

effect of volume fraction is more pronounced when the Rayleigh number is high.

**Figure 4.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle volume fractions (*ϕ*) and Rayleigh numbers (*Ra*) at *d*sp = 50 nm, *Da* = 0.01, *ε* = 0.5, and *RR* = 2. **Figure 4.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle volume fractions (*φ*) and Rayleigh numbers (*Ra*) at *d*sp = 50 nm, *Da* = 0.01, *ε* = 0.5, and *RR* = 2.

**Figure 5.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle volume fractions (*ϕ*) and Rayleigh numbers (*Ra*) at *Da* = 10<sup>−</sup>2, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. **Figure 5.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle volume fractions (*φ*) and Rayleigh numbers (*Ra*) at *Da* = 10−<sup>2</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2.

Figure 6 illustrates the isotherms and streamlines for different nanoparticle volume fraction and Darcy number at *d*sp = 50 nm, *Ra* = 105, *ε* = 0.5, and *RR* = 2. From Figure 6, for *Da* = 10−4, the isotherms have a uniform distribution due to the low permeability, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is slight. For *Da* = 10−3, due to the enhanced permeability, the flow in the annulus is strengthened, and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For *Da* = 10−2, the isotherms are almost horizontally distributed, especially when *ϕ* = 0.9, which means the natural convection heat transfer plays a significant role and the effect of volume fraction is more pronounced. With the increase of Darcy number, the streamlines become denser near the walls and the cells become bigger and have a tendency to move upward due to the enhanced flow. In addition, for *Da* = 10−4 and *Da* = 10−3, the effect of volume fraction on the streamlines is very weak, while for *Da* = 10−2, the effect is more pronounced. Figure 7 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Darcy numbers. The figure shows that an increase in volume fraction leads to heat transfer enhancement for all considered Darcy numbers and the effect of volume fraction is increased with the increase of Darcy number. Figure 6 illustrates the isotherms and streamlines for different nanoparticle volume fraction and Darcy number at *d*sp = 50 nm, *Ra* = 10<sup>5</sup> , *ε* = 0.5, and *RR* = 2. From Figure 6, for *Da* = 10−<sup>4</sup> , the isotherms have a uniform distribution due to the low permeability, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The effect of volume fraction on the isotherms is slight. For *Da* = 10−<sup>3</sup> , due to the enhanced permeability, the flow in the annulus is strengthened, and the transition from conduction to natural convection takes place. In this case, the effect of volume fraction becomes more important. For *Da* = 10−<sup>2</sup> , the isotherms are almost horizontally distributed, especially when *φ* = 0.9, which means the natural convection heat transfer plays a significant role and the effect of volume fraction is more pronounced. With the increase of Darcy number, the streamlines become denser near the walls and the cells become bigger and have a tendency to move upward due to the enhanced flow. In addition, for *Da* = 10−<sup>4</sup> and *Da* = 10−<sup>3</sup> , the effect of volume fraction on the streamlines is very weak, while for *Da* = 10−<sup>2</sup> , the effect is more pronounced. Figure 7 displays the effect of volume fraction on the overall heat transfer rate along the inner wall at different Darcy numbers. The figure shows that an increase in volume fraction leads to heat transfer enhancement for all considered Darcy numbers and the effect of volume fraction is increased with the increase of Darcy number.

Figure 8 presents the evolution of the local Nusselt number along the inner wall for different nanoparticle volume fractions at *Ra* = 105, *Da* = 10−2, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. The increase of *Nu*loc in the whole region means that the local heat transfer rate is enhanced. It can be found that the maximum *Nu*loc occurred at *γ* = 180°, which means the natural convection heat transfer is more intense in the bottom half of the inner wall. In addition, the heat transfer is enhanced with the increase of volume fraction. Figure 8 presents the evolution of the local Nusselt number along the inner wall for different nanoparticle volume fractions at *Ra* = 10<sup>5</sup> , *Da* = 10−<sup>2</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. The increase of *Nu*loc in the whole region means that the local heat transfer rate is enhanced. It can be found that the maximum *Nu*loc occurred at *γ* = 180◦ , which means the natural convection heat transfer is more intense in the bottom half of the inner wall. In addition, the heat transfer is enhanced with the increase of volume fraction.

**Figure 6.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle volume fractions (*ϕ*) and Darcy numbers (*Da*) at *Ra*= 105, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. **Figure 6.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle volume fractions (*φ*) and Darcy numbers (*Da*) at *Ra*= 10<sup>5</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2.

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**Figure 7.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle volume fractions (*ϕ*) and Darcy numbers (*Da*) at *Ra*= 105, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. **Figure 7.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle volume fractions (*φ*) and Darcy numbers (*Da*) at *Ra*= 10<sup>5</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. tions (*ϕ*) and Darcy numbers (*Da*) at *Ra*= 105, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2.

0° 60° 120° 180° 240° 300° 360°

γ **Figure 8.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle volume fractions (*ϕ*) at *Ra* = 105, *Da* = 10<sup>−</sup>2, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. **Figure 8.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle volume fractions (*φ*) at *Ra* = 10<sup>5</sup> , *Da* = 10−<sup>2</sup> , *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2.

#### **Figure 8.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle volume fractions (*ϕ*) at *Ra* = 105, *Da* = 10<sup>−</sup>2, *d*sp = 50 nm, *ε* = 0.5, and *RR* = 2. *4.3. Effects of Nanoparticle Diameter 4.3. Effects of Nanoparticle Diameter*

*4.3. Effects of Nanoparticle Diameter*  Figure 9 shows the isotherms and streamlines for different nanoparticle diameters and Rayleigh numbers at *Da* = 10−2, *ϕ* = 0.05, *ε* = 0.5 and *RR* = 2. From Figure 9, for *Ra* = 103, the isotherms have a uniform distribution due to the weak buoyancy force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The nanoparticle diameter has a weak effect on the isotherms. For *Ra* = 104, the natural convection heat transfer is strengthened due to the enhanced buoyancy force. The isotherms near the top half of the inner wall are disturbed, and the change of the isotherms distribution is more remarkable at *d*sp = 10 compared with *d*sp = 90. For *Ra* = 5 × 104, the isotherms are almost horizontally distributed, especially when *d*sp = 10, which means that the natural convection dominates the heat transfer and the effect of the nanoparticle diameter is more pronounced. In addition, for *Ra* = 103 and *Ra* = 104, the effect of nanoparticle diameter on the streamlines is very weak, while for *Ra* = 105, the effect is more pronounced, and the cell becomes bigger and has a tendency to move upward. Figure 10 displays the effect of nanoparticle diameter on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in the nanoparticle diameter leads to reduced heat transfer for all considered Rayleigh numbers. For *Ra* = 103, the effect of nanoparticle diameter is less pronounced. For *Ra* = 5 × 104, the effect of nanoparticle diameter is remarkable, especially when it is at a low value. For instance, *Nu*avg decreased by 11.79% from *d*sp = Figure 9 shows the isotherms and streamlines for different nanoparticle diameters and Rayleigh numbers at *Da* = 10−2, *ϕ* = 0.05, *ε* = 0.5 and *RR* = 2. From Figure 9, for *Ra* = 103, the isotherms have a uniform distribution due to the weak buoyancy force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The nanoparticle diameter has a weak effect on the isotherms. For *Ra* = 104, the natural convection heat transfer is strengthened due to the enhanced buoyancy force. The isotherms near the top half of the inner wall are disturbed, and the change of the isotherms distribution is more remarkable at *d*sp = 10 compared with *d*sp = 90. For *Ra* = 5 × 104, the isotherms are almost horizontally distributed, especially when *d*sp = 10, which means that the natural convection dominates the heat transfer and the effect of the nanoparticle diameter is more pronounced. In addition, for *Ra* = 103 and *Ra* = 104, the effect of nanoparticle diameter on the streamlines is very weak, while for *Ra* = 105, the effect is more pronounced, and the cell becomes bigger and has a tendency to move upward. Figure 10 displays the effect of nanoparticle diameter on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in the nanoparticle diameter leads to reduced heat transfer for all considered Rayleigh numbers. For *Ra* = 103, the effect of nanoparticle diameter is less pronounced. For *Ra* = 5 × 104, the effect of nanoparticle diameter is remarkable, especially when it is at a low value. For instance, *Nu*avg decreased by 11.79% from *d*sp = Figure 9 shows the isotherms and streamlines for different nanoparticle diameters and Rayleigh numbers at *Da* = 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5 and *RR* = 2. From Figure 9, for *Ra* = 10<sup>3</sup> , the isotherms have a uniform distribution due to the weak buoyancy force, and it indicates that the heat transfer in the annulus is dominated by thermal conduction. The nanoparticle diameter has a weak effect on the isotherms. For *Ra* = 10<sup>4</sup> , the natural convection heat transfer is strengthened due to the enhanced buoyancy force. The isotherms near the top half of the inner wall are disturbed, and the change of the isotherms distribution is more remarkable at *<sup>d</sup>*sp = 10 compared with *<sup>d</sup>*sp = 90. For *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , the isotherms are almost horizontally distributed, especially when *d*sp = 10, which means that the natural convection dominates the heat transfer and the effect of the nanoparticle diameter is more pronounced. In addition, for *Ra* = 10<sup>3</sup> and *Ra* = 10<sup>4</sup> , the effect of nanoparticle diameter on the streamlines is very weak, while for *Ra* = 10<sup>5</sup> , the effect is more pronounced, and the cell becomes bigger and has a tendency to move upward. Figure 10 displays the effect of nanoparticle diameter on the overall heat transfer rate along the inner wall at different Rayleigh numbers. The figure shows that an increase in the nanoparticle diameter leads to reduced heat transfer for all considered Rayleigh numbers. For *Ra* = 10<sup>3</sup> , the effect of nanoparticle diameter is less pronounced. For *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , the effect of nanoparticle diameter is remarkable, especially when it is at a low value. For instance, *Nu*avg decreased by 11.79% from *d*sp = 10 to *d*sp = 30 and decreased by 5.81% from *d*sp = 30 to *d*sp = 90 at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , while *Nu*avg decreased by 0.48% from *d*sp = 10 to *d*sp = 30 and decreased by 0.12% from *d*sp = 30 to *d*sp = 90 at *Ra* = 10<sup>4</sup> .

at *Ra* = 104.

10 to *d*sp = 30 and decreased by 5.81% from *d*sp = 30 to *d*sp = 90 at *Ra* = 5 × 104, while *Nu*avg decreased by 0.48% from *d*sp = 10 to *d*sp = 30 and decreased by 0.12% from *d*sp = 30 to *d*sp = 90

**Figure 9.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle diameters (*d*sp) and Rayleigh numbers (*Ra*) at *Da*= 10<sup>−</sup>2, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. **Figure 9.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle diameters (*d*sp) and Rayleigh numbers (*Ra*) at *Da*= 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2.

**Figure 10.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle diameters (*d*sp) and Rayleigh numbers (*Ra*) at *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. **Figure 10.** Average Nusselt number (*Nu*avg) of the inner wall for different nanoparticle diameters (*d*sp) and Rayleigh numbers (*Ra*) at *Da* = 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2.

Figure 11 illustrates the isotherms and streamlines for different nanoparticle diameter and Darcy numbers at *Ra* = 5 × 104, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. From Figure 11, for *Da* = 10−4, the isotherms have a uniform distribution due to the low permeability, which indicates that the flow is weak and thermal conduction plays a leading role. The effect of nanoparticle diameter on the heat transfer is weak. For *Da* = 10−3, due to the enhanced permeability, the natural convection heat transfer is strengthened. For *Da* = 10−2, the isotherms are almost horizontally distributed, especially when *d*sp = 10, which means that the natural convection heat transfer plays a significant role and the effect of nanoparticle diameter is more pronounced. In addition, for *Da* = 10−4 and *Da* = 10−3, the effect of nanoparticle diameter on the streamlines is very weak, while for *Da* = 10−2, the effect is more pronounced; the cell becomes bigger and has a tendency to move upward. Figure 12 displays the effect of the nanoparticle diameter on the overall heat transfer rate along the inner wall at different Darcy numbers. The figure shows that an increase in the nanoparticle diameter leads to heat transfer enhancement for all considered Darcy numbers, and it can be found that the effect of nanoparticle diameter is increased with the increase of the Darcy number. Figure 13 presents the evolution of the local Nusselt number along the inner wall for Figure 11 illustrates the isotherms and streamlines for different nanoparticle diameter and Darcy numbers at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2. From Figure 11, for *Da* = 10−<sup>4</sup> , the isotherms have a uniform distribution due to the low permeability, which indicates that the flow is weak and thermal conduction plays a leading role. The effect of nanoparticle diameter on the heat transfer is weak. For *Da* = 10−<sup>3</sup> , due to the enhanced permeability, the natural convection heat transfer is strengthened. For *Da* = 10−<sup>2</sup> , the isotherms are almost horizontally distributed, especially when *d*sp = 10, which means that the natural convection heat transfer plays a significant role and the effect of nanoparticle diameter is more pronounced. In addition, for *Da* = 10−<sup>4</sup> and *Da* = 10−<sup>3</sup> , the effect of nanoparticle diameter on the streamlines is very weak, while for *Da* = 10−<sup>2</sup> , the effect is more pronounced; the cell becomes bigger and has a tendency to move upward. Figure 12 displays the effect of the nanoparticle diameter on the overall heat transfer rate along the inner wall at different Darcy numbers. The figure shows that an increase in the nanoparticle diameter leads to heat transfer enhancement for all considered Darcy numbers, and it can be found that the effect of nanoparticle diameter is increased with the increase of the Darcy number.

different nanoparticle diameters at *Ra* = 105, *Da* = 10−2, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. It can be found that the local Nusselt number has a symmetrical distribution due to the symmetrical geometry and boundary conditions, and the maximum *Nu*loc occurred at *γ*= 180°, which means that the natural convection heat transfer is more intense in the bottom half of the inner wall. Furthermore, the local Nusselt number is increased with the decrease of nanoparticle diameter, which indicates that the nanoparticle diameter at high value will weaken the natural convection heat transfer. Figure 13 presents the evolution of the local Nusselt number along the inner wall for different nanoparticle diameters at *Ra* = 10<sup>5</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2. It can be found that the local Nusselt number has a symmetrical distribution due to the symmetrical geometry and boundary conditions, and the maximum *Nu*loc occurred at *γ*= 180◦ , which means that the natural convection heat transfer is more intense in the bottom half of the inner wall. Furthermore, the local Nusselt number is increased with the decrease of nanoparticle diameter, which indicates that the nanoparticle diameter at high value will weaken the natural convection heat transfer.

**Figure 11.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle diameters (*d*sp) and Darcy numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. **Figure 11.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different nanoparticle diameters (*d*sp) and Darcy numbers (*Da*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2.

3.6 3.8

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**Figure 12.** Average Nusselt number (*Nu*loc) of the inner wall for different nanoparticle diameters (*d*sp) and Darcy numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. **Figure 12.** Average Nusselt number (*Nu*loc) of the inner wall for different nanoparticle diameters (*d*sp) and Darcy numbers (*Da*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2. (*d*sp) and Darcy numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2.

γ **Figure 13.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle diameters (*d*sp) at *Ra* = 105, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. **Figure 13.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle diameters (*d*sp) at *Ra* = 10<sup>5</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5, and *RR* = 2.

#### **Figure 13.** Local Nusselt number (*Nu*loc) along the inner wall for different nanoparticle diameters (*d*sp) at *Ra* = 105, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *ε* = 0.5, and *RR* = 2. *4.4. Effects of Porosity 4.4. Effects of Porosity*

*4.4. Effects of Porosity*  Figure 14 shows the isotherms and streamlines for different porosity and Rayleigh numbers at *d*sp = 50 nm, *Da* = 10−2, and *RR* = 2. From Figure 14, for *Ra* = 103, the isotherms have a uniform distribution, and the isotherms are almost unchanged when the porosity increased from *ε* = 0.1 to *ε* = 0.9. This is because at low Rayleigh numbers, the buoyancy force is very weak, and the heat transfer in the annulus is dominated by thermal conduction. The porosity has a slight effect on the isotherms. For *Ra* = 104, the isotherms have a similar trend with that for *Ra* = 103 at *ε* = 0.1, while the isotherms have an obvious change at *ε* = 0.9. For *Ra* = 5 × 104, the natural convection heat transfer plays a leading role due to the enhanced flow, and the heat transfer is more intense at *ε* = 0.9 compared with *ε* = 0.1. The streamlines have a similar distribution, but some details are different. For *Ra* = 103, the streamlines have little change at *ε* = 0.1 and *ε* = 0.9. For *Ra* = 104 and *Ra* = 5 × 104, it can be found the cells become bigger and have a tendency to move upward when the porosity increases. Figure 15 displays the effect of porosity on the overall heat transfer rate along the inner wall at different Rayleigh numbers. It can be found that a continuous increase of Figure 14 shows the isotherms and streamlines for different porosity and Rayleigh numbers at *d*sp = 50 nm, *Da* = 10−2, and *RR* = 2. From Figure 14, for *Ra* = 103, the isotherms have a uniform distribution, and the isotherms are almost unchanged when the porosity increased from *ε* = 0.1 to *ε* = 0.9. This is because at low Rayleigh numbers, the buoyancy force is very weak, and the heat transfer in the annulus is dominated by thermal conduction. The porosity has a slight effect on the isotherms. For *Ra* = 104, the isotherms have a similar trend with that for *Ra* = 103 at *ε* = 0.1, while the isotherms have an obvious change at *ε* = 0.9. For *Ra* = 5 × 104, the natural convection heat transfer plays a leading role due to the enhanced flow, and the heat transfer is more intense at *ε* = 0.9 compared with *ε* = 0.1. The streamlines have a similar distribution, but some details are different. For *Ra* = 103, the streamlines have little change at *ε* = 0.1 and *ε* = 0.9. For *Ra* = 104 and *Ra* = 5 × 104, it can be found the cells become bigger and have a tendency to move upward when the porosity increases. Figure 15 displays the effect of porosity on the overall heat transfer rate along the inner wall at different Rayleigh numbers. It can be found that a continuous increase of the overall heat transfer rate occurs with the increase of porosity for all considered Ray-Figure 14 shows the isotherms and streamlines for different porosity and Rayleigh numbers at *d*sp = 50 nm, *Da* = 10−<sup>2</sup> , and *RR* = 2. From Figure 14, for *Ra* = 10<sup>3</sup> , the isotherms have a uniform distribution, and the isotherms are almost unchanged when the porosity increased from *ε* = 0.1 to *ε* = 0.9. This is because at low Rayleigh numbers, the buoyancy force is very weak, and the heat transfer in the annulus is dominated by thermal conduction. The porosity has a slight effect on the isotherms. For *Ra* = 10<sup>4</sup> , the isotherms have a similar trend with that for *Ra* = 10<sup>3</sup> at *ε* = 0.1, while the isotherms have an obvious change at *ε* = 0.9. For *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , the natural convection heat transfer plays a leading role due to the enhanced flow, and the heat transfer is more intense at *ε* = 0.9 compared with *ε* = 0.1. The streamlines have a similar distribution, but some details are different. For *Ra* = 10<sup>3</sup> , the streamlines have little change at *<sup>ε</sup>* = 0.1 and *<sup>ε</sup>* = 0.9. For *Ra* = 10<sup>4</sup> and *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , it can be found the cells become bigger and have a tendency to move upward when the porosity increases. Figure 15 displays the effect of porosity on the overall heat transfer rate along the inner wall at different Rayleigh numbers. It can be found that a continuous increase of the overall heat transfer rate occurs with the increase of porosity for all considered Rayleigh numbers and the effect of porosity is more pronounced at high Rayleigh numbers.

the overall heat transfer rate occurs with the increase of porosity for all considered Rayleigh numbers and the effect of porosity is more pronounced at high Rayleigh numbers.

leigh numbers and the effect of porosity is more pronounced at high Rayleigh numbers.

**Figure 14.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different porosity (*ε*) and Rayleigh numbers (*Ra*) at *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. **Figure 14.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different porosity (*ε*) and Rayleigh numbers (*Ra*) at *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2.

**Figure 15.** Average Nusselt number (*Nu*avg) of the inner wall for different porosity (*ε*) and Rayleigh numbers (*Ra*) at *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. **Figure 15.** Average Nusselt number (*Nu*avg) of the inner wall for different porosity (*ε*) and Rayleigh numbers (*Ra*) at *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2.

Figure 16 displays the isotherms and streamlines for different porosity and Darcy numbers at *Ra* = 5 × 104, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. From Figure 16, for *Da* = 10−4, the isotherms have a uniform distribution at *ε* = 0.1 and *ε* = 0.9; the reason is that the flow is constrained by the low permeability. In this case, porosity has little effect on the isotherms. For *Da* = 10−3, due to the enhanced permeability, the flow in the annulus is strengthened, and a disturbance occurs in the isotherms. For *Da* = 10−2, the isotherms at *ε* = 0.1 and *ε* = 0.9 have a remarkable difference. The isotherms are almost horizontally distributed at *ε* = 0.9, which means that the overall heat transfer is dominated by natural convection. For *Da* = 10−4, the streamlines are almost the same at *ε* = 0.1 and *ε* = 0.9. For *Da* = 10−3 and *Da* = 10−2, the cell becomes bigger and has a tendency to move upward when porosity increases from *ε* = 0.1 to *ε* = 0.9. Figure 17 displays the effect of porosity on the overall heat transfer rate along the inner wall at different Darcy number. The figure shows that an increase in porosity leads to heat transfer enhancement for all considered Darcy numbers, and the effect Figure 16 displays the isotherms and streamlines for different porosity and Darcy numbers at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2. From Figure 16, for *Da* = 10−<sup>4</sup> , the isotherms have a uniform distribution at *ε* = 0.1 and *ε* = 0.9; the reason is that the flow is constrained by the low permeability. In this case, porosity has little effect on the isotherms. For *Da* = 10−<sup>3</sup> , due to the enhanced permeability, the flow in the annulus is strengthened, and a disturbance occurs in the isotherms. For *Da* = 10−<sup>2</sup> , the isotherms at *ε* = 0.1 and *ε* = 0.9 have a remarkable difference. The isotherms are almost horizontally distributed at *ε* = 0.9, which means that the overall heat transfer is dominated by natural convection. For *Da* = 10−<sup>4</sup> , the streamlines are almost the same at *ε* = 0.1 and *ε* = 0.9. For *Da* = 10−<sup>3</sup> and *Da* = 10−<sup>2</sup> , the cell becomes bigger and has a tendency to move upward when porosity increases from *ε* = 0.1 to *ε* = 0.9. Figure 17 displays the effect of porosity on the overall heat transfer rate along the inner wall at different Darcy number. The figure shows that an increase in porosity leads to heat transfer enhancement for all considered Darcy numbers, and the effect of porosity is more pronounced at high Darcy numbers.

of porosity is more pronounced at high Darcy numbers. Figure 18 presents the evolution of the local Nusselt number along the inner wall for different porosity at *Ra* = 5 × 104, *Da* = 10−2, *ϕ* = 0.05, *ε* = 0.5, *d*sp = 50 nm, and *RR* = 2. The local Nusselt number has a symmetrical distribution due to the symmetrical geometry and boundary conditions, and the maximum *Nu*loc occurred at *γ* = 180°. Furthermore, the local Nusselt number is increased with the increase of porosity, so it can be concluded that Figure 18 presents the evolution of the local Nusselt number along the inner wall for different porosity at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *ε* = 0.5, *d*sp = 50 nm, and *RR* = 2. The local Nusselt number has a symmetrical distribution due to the symmetrical geometry and boundary conditions, and the maximum *Nu*loc occurred at *γ* = 180◦ . Furthermore, the local Nusselt number is increased with the increase of porosity, so it can be concluded that the porosity has a positive effect on the overall heat transfer rate.

the porosity has a positive effect on the overall heat transfer rate.

**Figure 16.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different porosity (*ε*) and Darcy numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. **Figure 16.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different porosity (*ε*) and Darcy numbers (*Da*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2.

3.4

*Nanomaterials* **2021**, *11*, x FOR PEER REVIEW 20 of 24

**Figure 17.** Average Nusselt number (*Nu*avg) of the inner wall for different porosity (*ε*) and Darcy numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. **Figure 17.** Average Nusselt number (*Nu*avg) of the inner wall for different porosity (*ε*) and Darcy numbers (*Da*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2. numbers (*Da*) at *Ra* = 5 × 104, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2.

0° 60° 120° 180° 240° 300° 360°

γ **Figure 18.** Local Nusselt number (*Nu*loc) along the inner wall for different porosity (*ε*) at *Ra* = 5 × 104, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. **Figure 18.** Local Nusselt number (*Nu*loc) along the inner wall for different porosity (*ε*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *RR* = 2.

#### **Figure 18.** Local Nusselt number (*Nu*loc) along the inner wall for different porosity (*ε*) at *Ra* = 5 × 104, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *RR* = 2. *4.5. Effects of Radius Ratio 4.5. Effects of Radius Ratio*

*4.5. Effects of Radius Ratio*  Figure 19 illustrates the isotherms and streamlines for different radius ratio at *Ra* = 5 × 104, *Da* = 10−2, *ϕ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5. The radius ratio is an important parameter that affects the fluid flows and natural convection heat transfer according the previous research [10–12]. From Figure 19, when the radius ratio increased from *RR* = 1.2 to *RR*= 6, the flow is strengthened, a main cell is formed in the middle of the annulus, and the cell becomes bigger and has a tendency to move upward. It is worth noting that the color scales on the right have a downward trend with the increase of *RR*, which is inverse to the above conclusion. In fact, the reason is that the change of *AR* leads to a change of the characteristic length (*r*o–*r*i). The isotherms distribution is changed from vertical to horizontal, which means that the heat transfer is dominated by natural convection. When *RR* = 1.2, multicellular flow structures are formed. Figure 20 displays the effect of radius ratio on the overall heat transfer rate along the inner wall. It can be found that heat transfer in the annulus is enhanced with the increase of radius ratio from *RR* = 1.3 to *RR* = 6, but a fall occurred in the Figure 19 illustrates the isotherms and streamlines for different radius ratio at *Ra* = 5 × 104, *Da* = 10−2, *ϕ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5. The radius ratio is an important parameter that affects the fluid flows and natural convection heat transfer according the previous research [10–12]. From Figure 19, when the radius ratio increased from *RR* = 1.2 to *RR*= 6, the flow is strengthened, a main cell is formed in the middle of the annulus, and the cell becomes bigger and has a tendency to move upward. It is worth noting that the color scales on the right have a downward trend with the increase of *RR*, which is inverse to the above conclusion. In fact, the reason is that the change of *AR* leads to a change of the characteristic length (*r*o–*r*i). The isotherms distribution is changed from vertical to horizontal, which means that the heat transfer is dominated by natural convection. When *RR* = 1.2, multicellular flow structures are formed. Figure 20 displays the effect of radius ratio on the overall heat transfer rate along the inner wall. It can be found that heat transfer in the annulus is enhanced with the increase of radius ratio from *RR* = 1.3 to *RR* = 6, but a fall occurred in the range of *RR* = 1.2 to *RR* = 1.3. A conclusion can be drawn that a bifurcation point exists when Figure <sup>19</sup> illustrates the isotherms and streamlines for different radius ratio at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5. The radius ratio is an important parameter that affects the fluid flows and natural convection heat transfer according the previous research [10–12]. From Figure 19, when the radius ratio increased from *RR* = 1.2 to *RR*= 6, the flow is strengthened, a main cell is formed in the middle of the annulus, and the cell becomes bigger and has a tendency to move upward. It is worth noting that the color scales on the right have a downward trend with the increase of *RR*, which is inverse to the above conclusion. In fact, the reason is that the change of *RR* leads to a change of the characteristic length (*r*o–*r*<sup>i</sup> ). The isotherms distribution is changed from vertical to horizontal, which means that the heat transfer is dominated by natural convection. When *RR* = 1.2, multicellular flow structures are formed. Figure 20 displays the effect of radius ratio on the overall heat transfer rate along the inner wall. It can be found that heat transfer in the annulus is enhanced with the increase of radius ratio from *RR* = 1.3 to *RR* = 6, but a fall occurred in the range of *RR* = 1.2 to *RR* = 1.3. A conclusion can be drawn that a bifurcation point exists when the radius ratios is in the range of *RR* = 1.2 to *RR* = 1.3 for the considered parameters.

range of *RR* = 1.2 to *RR* = 1.3. A conclusion can be drawn that a bifurcation point exists when the radius ratios is in the range of *RR* = 1.2 to *RR* = 1.3 for the considered parameters.

the radius ratios is in the range of *RR* = 1.2 to *RR* = 1.3 for the considered parameters.

**Figure 19.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different radius ratios (*RR*) at *Ra* = 5 × 104, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5. **Figure 19.** Isotherms (left) and streamlines (right) for Cu–water nanofluid for different radius ratios (*RR*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5.

1 2 3 4 5 6 7 8 9 10 *RR* **Figure 20.** Average Nusselt number (*Nu*avg) of the inner wall for different radius ratios (*RR*) at *Ra* = **Figure 20.** Average Nusselt number (*Nu*avg) of the inner wall for different radius ratios (*RR*) at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *Da* = 10−<sup>2</sup> , *φ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5.

#### **Figure 20.** Average Nusselt number (*Nu*avg) of the inner wall for different radius ratios (*RR*) at *Ra* = , *Da* = 10−2 **5. Conclusions**

5 × 10<sup>4</sup>

a nanofluid.

numbers are at high value.

tion point exists.

[51706213].

**Nomenclature**

*RR* radius ratio

*c<sup>p</sup>* thermal capacity, J/(kg × K)

5 × 104, *Da* = 10<sup>−</sup>2, *ϕ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5. **5. Conclusions** Natural convection heat transfer in a porous annulus filled with a Cu-Nanofluid has been investigated numerically. The effects of Brownian motion, solid volume fraction, na-Natural convection heat transfer in a porous annulus filled with a Cu-Nanofluid has been investigated numerically. The effects of Brownian motion, solid volume fraction, nanoparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, tem-

noparticle diameter, Rayleigh number, Darcy number, porosity on the flow pattern, temperature distribution, and heat transfer characteristics are discussed in detail. The follow-

The effect of Brownian motion becomes more remarkable with the increase of Rayleigh number and nanoparticle volume fraction, while it is less pronounced with the in-

(2) The increase of nanoparticle volume fraction results in an improvement of the overall heat transfer rate, and the effect is more remarkable when the Rayleigh and Darcy

(3) Increasing the nanoparticle diameter has a negative effect on the overall heat trans-

(4) The porosity affects the flow pattern, temperature distribution, and heat transfer rate. The flow motion is limited when the porosity is too low. The heat transfer rate is strengthened

(5) The radius ratio has a significant influence on the isotherms, streamlines, and heat transfer rate. The rate is greatly enhanced with the increase of radius ratio. Additionally, when the radius ratio is too low, multicellular flow structures are formed, and a bifurca-

**Author Contributions:** Conceptualization, Y.H. and M.L.; Methodology, L.Z.; Software, L.Z.; Validation, Y.H., L.Z. and M.L.; Formal Analysis, L.Z.; Investigation, Y.H.; Resources, Y.H.; Data Curation, L.Z.; Writing—Original Draft Preparation, L.Z.; Writing—Review & Editing, L.Z.; Visualization, L.Z.; Supervision, Y.H.; Project Administration, M.L.; Funding acquisition Administration,

**Funding:** This research was funded by National Nature Science Foundation of China, grant number

fer rate and the effect has a limit when the nanoparticle diameter reaches a high value.

with the increase of porosity, especially with high Rayleigh and Darcy numbers.

M.L. All authors have read and agreed to the published version of the manuscript.

**Data Availability Statement:** Data available in a publicly accessible repository.

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

crease of Darcy number and nanoparticle diameter.

, *ϕ* = 0.05, *d*sp = 50 nm, and *ε* = 0.5.

perature distribution, and heat transfer characteristics are discussed in detail. The following conclusions could be drawn:

(1) Brownian motion should be considered in the natural convection heat transfer of a nanofluid.

The effect of Brownian motion becomes more remarkable with the increase of Rayleigh number and nanoparticle volume fraction, while it is less pronounced with the increase of Darcy number and nanoparticle diameter.

(2) The increase of nanoparticle volume fraction results in an improvement of the overall heat transfer rate, and the effect is more remarkable when the Rayleigh and Darcy numbers are at high value.

(3) Increasing the nanoparticle diameter has a negative effect on the overall heat transfer rate and the effect has a limit when the nanoparticle diameter reaches a high value.

(4) The porosity affects the flow pattern, temperature distribution, and heat transfer rate. The flow motion is limited when the porosity is too low. The heat transfer rate is strengthened with the increase of porosity, especially with high Rayleigh and Darcy numbers.

(5) The radius ratio has a significant influence on the isotherms, streamlines, and heat transfer rate. The rate is greatly enhanced with the increase of radius ratio. Additionally, when the radius ratio is too low, multicellular flow structures are formed, and a bifurcation point exists.

**Author Contributions:** Conceptualization, Y.H. and M.L.; Methodology, L.Z.; Software, L.Z.; Validation, Y.H., L.Z. and M.L.; Formal Analysis, L.Z.; Investigation, Y.H.; Resources, Y.H.; Data Curation, L.Z.; Writing—Original Draft Preparation, L.Z.; Writing—Review & Editing, L.Z.; Visualization, L.Z.; Supervision, Y.H.; Project Administration, M.L.; Funding acquisition Administration, M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Nature Science Foundation of China, grant number [51706213].

**Data Availability Statement:** Data available in a publicly accessible repository.

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