*2.2. GFEM Approach*

The Galerkin finite element method handles the transformed coupled Equations (1)–(4), which include both the flow and heat transfer phenomena as well as the aforementioned boundary conditions. The governing equations' weak forms are given and discretized on a nonuniform structural grid. The results are then simulated using mathematical software. The process is described in full in [51,52]. The validation of the present code was obtained and is displayed in Figure 3 using the additional numerical results of Tan et al. [53]. We are confident in our results based on this figure.

**Figure 3.** Mesh grid and of the present model in comparison to [53].

### **Figure 3.** Mesh grid and of the present model in comparison to [53]. **3. Results and Discussion**

**3. Results and Discussion** In this section, we illustrate and interpret the results obtained from examining the melting effects on the flow of a suspension that contains the phase change material (PCM). Here, the worked fluid was Al2O3/n-octadecane paraffin and the flow area was a cylinder tube including cross-section fins. Features of the temperature, velocity, Bejan number, and liquid fraction were examined for various heating cases, namely, a cylinder with two horizontal wings, a cylinder with two vertical wings, a cylinder with four wings, and a cylinder with eight heated fins. The variations in the time of required ranged from 100 and 600 s and the values of the nanoparticles volume fraction were considered between = 0% and 8%. To present a comprehensive investigation, the average values of the liquid fraction , Bejan number , and the rate of the heat transfer over time are pre-In this section, we illustrate and interpret the results obtained from examining the melting effects on the flow of a suspension that contains the phase change material (PCM). Here, the worked fluid was Al2O3/n-octadecane paraffin and the flow area was a cylinder tube including cross-section fins. Features of the temperature, velocity, Bejan number, and liquid fraction were examined for various heating cases, namely, a cylinder with two horizontal wings, a cylinder with two vertical wings, a cylinder with four wings, and a cylinder with eight heated fins. The variations in the time of required ranged from 100 and 600 s and the values of the nanoparticles volume fraction were considered between *ϕ* = 0% and 8%. To present a comprehensive investigation, the average values of the liquid fraction *β*, Bejan number *Beavg*, and the rate of the heat transfer *Nuavg* over time are presented graphically for a wide range of the considered parameters. Additionally, the condition of the totally melted NEPCM (liquid fraction = 1) may be used to terminate the computations.

sented graphically for a wide range of the considered parameters. Additionally, the condition of the totally melted NEPCM (liquid fraction = 1) may be used to terminate the computations. Figure 4 displays the temperature, velocity, local Bejan number, and the liquid fraction for various cases of inner heating. Notably, the temperature features concentrated around the fins in all the cases, with a cold zone indicated near the bottom of the outer cylinder. These temperature distributions achieved their maximum values in case 4 (eight wings) pointing to a decrease in the aforementioned cold zone at the bottom. We also observed that the increase in number of wings reduced the temperature differences; hence, both the temperature gradients and heat transfer rate diminished. Additionally, a clear reduction in the velocity values was noted as the number of wings increased. Physically, the increase in the number of wings enhances the complexity in the flow area; hence, the flow resistance is augmented. In the same context, the features of the Bejan number showed that the increase in the number of heated fins reduced the gradients of Figure 4 displays the temperature, velocity, local Bejan number, and the liquid fraction for various cases of inner heating. Notably, the temperature features concentrated around the fins in all the cases, with a cold zone indicated near the bottom of the outer cylinder. These temperature distributions achieved their maximum values in case 4 (eight wings) pointing to a decrease in the aforementioned cold zone at the bottom. We also observed that the increase in number of wings reduced the temperature differences; hence, both the temperature gradients and heat transfer rate diminished. Additionally, a clear reduction in the velocity values was noted as the number of wings increased. Physically, the increase in the number of wings enhances the complexity in the flow area; hence, the flow resistance is augmented. In the same context, the features of the Bejan number showed that the increase in the number of heated fins reduced the gradients of the temperature; hence, the fluid friction irreversibility became dominant. Furthermore, the melting zone was seen in the upper half of the domain for all the considered cases as the increase in number of heated fins enhanced the melted zone.

the temperature; hence, the fluid friction irreversibility became dominant. Furthermore, the melting zone was seen in the upper half of the domain for all the considered cases as

the increase in number of heated fins enhanced the melted zone.

**Figure 4.** Temperature, velocity, Bejan number, and liquid fraction contour in different geometries. **Figure 4.** Temperature, velocity, Bejan number, and liquid fraction contour in different geometries.

The features of the temperature, velocity, local number, and local liquid fraction with variations in time are depicted in Figure 5. During these computations, case 3 used an inner cylinder with four wings. The results indicated that at the beginning of the cal‐ culations (small values of the time), the distributions of the temperature, velocity, and Bejan number occurred around the inner part heated, indicating a nonactive zone near the outer boundaries. Over time, the fluid started to carry and distribute the temperature throughout the whole domain. Therefore at *t* ൌ 600 s, a good thermal domain was ob‐ tained with a higher velocity rate near the bottom of the outer boundaries. Additionally, for higher time values, the fluid friction irreversibility near the bottom dominated com‐ pared to the heat transfer irreversibility. From the physical viewpoint, this behavior is due to the gradients of the velocity that enhance with time, resulting in an augmentation in fluid friction irreversibility. Furthermore, a mushy zone was observed within the full flow The features of the temperature, velocity, local *Be* number, and local liquid fraction with variations in time are depicted in Figure 5. During these computations, case 3 used an inner cylinder with four wings. The results indicated that at the beginning of the calculations (small values of the time), the distributions of the temperature, velocity, and Bejan number occurred around the inner part heated, indicating a nonactive zone near the outer boundaries. Over time, the fluid started to carry and distribute the temperature throughout the whole domain. Therefore at *t* = 600 s, a good thermal domain was obtained with a higher velocity rate near the bottom of the outer boundaries. Additionally, for higher time values, the fluid friction irreversibility near the bottom dominated compared to the heat transfer irreversibility. From the physical viewpoint, this behavior is due to the gradients of the velocity that enhance with time, resulting in an augmentation in fluid friction irreversibility. Furthermore, a mushy zone was observed within the full flow domain with increasing time.

domain with increasing time.

**Figure 5.** Temperature, velocity, Bejan number, and liquid fraction contour in different time steps. **Figure 5.** Temperature, velocity, Bejan number, and liquid fraction contour in different time steps.

Figure 6 shows the distributions of the temperature, velocity, local Bejan number, and liquid fraction under impacts of the volume fraction parameter . The inner heated cylinders with four wings were used in this case. We noted a low convective transport at higher values of due to the increase in the viscosity of the mixture. The results indi‐ cated diminishing velocity and temperature gradients with increasing . Additionally, the local Bejan number occurred around the wings instead of the bottom boundaries at low values of . Conversely, the increase in enhanced the mushy zone within the flow area until completely melted conditions were obtained at 0.04. Figure 6 shows the distributions of the temperature, velocity, local Bejan number, and liquid fraction under impacts of the volume fraction parameter *ϕ*. The inner heated cylinders with four wings were used in this case. We noted a low convective transport at higher values of *ϕ* due to the increase in the viscosity of the mixture. The results indicated diminishing velocity and temperature gradients with increasing *ϕ*. Additionally, the local Bejan number occurred around the wings instead of the bottom boundaries at low values of *ϕ*. Conversely, the increase in *ϕ* enhanced the mushy zone within the flow area until completely melted conditions were obtained at *ϕ* ≥ 0.04. *Nanomaterials* **2022**, *12*, x 9 of 14

**Figure 6.** Temperature, velocity, Bejan number, and liquid fraction contour of nanoparticles' con‐

Figures 7 and 8 illustrate the profiles of the average liquid fraction , average Bejan number ௩, and average Nusselt number ௩ under with different numbers of heated wings, time parameters, and volume fraction parameters. The results revealed that case 4, in which eight heated wings were assumed, produced the highest average liquid fraction values. However, the average Bejan and Nusselt numbers decreased as the num‐ ber of heated wings increased. Additionally, the average rate of heat transfer diminished as increased due to the decrease in the temperature gradients. Furthermore, higher values of caused the irreversibility of the heat transfer to be dominant compared to the fluid friction irreversibility. Finally, we observed that the increase in the volume fraction parameter enhanced the mushy zone and, hence, the average liquid fraction rose.

**numberFigure 6.** *Cont*.

**Bejan**

**Liquid**

**Fraction**

centrations.

**Temperature**

**Velocity**

*Nanomaterials* **2022**, *12*, x 9 of 14

= 0 **= 0.02 = 0.04 = 0.08**

**Figure 6.** Temperature, velocity, Bejan number, and liquid fraction contour of nanoparticles' con‐ **Figure 6.** Temperature, velocity, Bejan number, and liquid fraction contour of nanoparticles' concentrations.

centrations. Figures 7 and 8 illustrate the profiles of the average liquid fraction , average Bejan number ௩, and average Nusselt number ௩ under with different numbers of heated wings, time parameters, and volume fraction parameters. The results revealed that case 4, in which eight heated wings were assumed, produced the highest average liquid fraction values. However, the average Bejan and Nusselt numbers decreased as the num‐ ber of heated wings increased. Additionally, the average rate of heat transfer diminished as increased due to the decrease in the temperature gradients. Furthermore, higher values of caused the irreversibility of the heat transfer to be dominant compared to the fluid friction irreversibility. Finally, we observed that the increase in the volume fraction Figures 7 and 8 illustrate the profiles of the average liquid fraction *β*, average Bejan number *Beavg*, and average Nusselt number *Nuavg* under with different numbers of heated wings, time parameters, and volume fraction parameters. The results revealed that case 4, in which eight heated wings were assumed, produced the highest average liquid fraction values. However, the average Bejan and Nusselt numbers decreased as the number of heated wings increased. Additionally, the average rate of heat transfer diminished as *ϕ* increased due to the decrease in the temperature gradients. Furthermore, higher values of *ϕ* caused the irreversibility of the heat transfer to be dominant compared to the fluid friction irreversibility. Finally, we observed that the increase in the volume fraction parameter *ϕ* enhanced the mushy zone and, hence, the average liquid fraction rose. *Nanomaterials* **2022**, *12*, x 10 of 14

30 **Figure 7.** *Cont*.

Nuavg

Nuavg

=0 =0.02 =0.04 =0.08

**Figure 7.** The influences of geometry on the liquid fraction, Nusselt number, and Bejan number.

 case 1 case 2 case 3 case 4

0 100 200 300 400 500 600

Time(s)

0 100 200 300 400 500 600

Time(s)

0 100 200 300 400 500 600

0 100 200 300 400 500 600

Time(s)

Time(s)

*Nanomaterials* **2022**, *12*, x 10 of 14

*Nanomaterials* **2022**, *12*, x 10 of 14

0 100 200 300 400 500 600

0 100 200 300 400 500 600

Time(s)

Time(s)

 case 1 case 2 case 3 case 4

 case 1 case 2 case 3 case 4

case 1

0.0

0.2

0.4

0.2

30

0.4

0.6

0.8

1.0

0.6

Beavg

0.8

Beavg

1.0

0.2

0.4

0.0

0.2

0.4

0.6

0.8

1.0

Liquid fraction

0.6

0.8

1.0

Liquid fraction

 case 1 case 2 case 3 case 4

 case 1 case 2 case 3 case 4

**Figure 7.** The influences of geometry on the liquid fraction, Nusselt number, and Bejan number. **Figure 7.** The influences of geometry on the liquid fraction, Nusselt number, and Bejan number. **Figure 7.** The influences of geometry on the liquid fraction, Nusselt number, and Bejan number.

**Figure 8.** The influences of nanoparticles' concentration on liquid fraction, Nusselt. Number, and Bejan number. **Figure 8.** The influences of nanoparticles' concentration on liquid fraction, Nusselt. Number, and Bejan number.

This paper presented a numerical investigation into the impacts of melting on the convective flow of phase change materials within cylindrical tubes containing cross‐shape

and case 4 (eight heated wings). The unsteady case was considered and completely melted conditions were assumed. The finite element method (FEM) with the Poisson pressure equation was applied to solve the governing system. The following are our majorfindings: Distributions of the temperature, velocity, and Bejan number increase as with an in‐ creasing number of heated wings due to the augmentation in the buoyancy convec‐ tive case. Additionally, the melted area was controlled for the most of the flow do‐

 For small time values, the increases in temperature, velocity, and liquid fraction oc‐ cur around the inner heated shapes, but over time, a good isothermal and melted

Increases in cause an enhancement in the dynamic viscosity of the mixture; hence,

With time, the irreversibility due to fluid friction becomes more dominant compare

**Author Contributions** "Conceptualization, S.E.A. and A.A.; methodology, A.A.; software, A.A.; validation, W.K.H., R.A.A. and A.A.; formal analysis, O.Y., and S.A; investigation, S.E.A.; resources, A.A.; data curation, S.A.; writing—original draft preparation, S.A.; writing—review and editing, W.K.H., R.A.A., and O.Y.; visualization, W.A.; supervision, A.A.; project administration, A.A.; fund‐ ing acquisition, S.A. All authors have read and agreed to the published version of the manuscript."

**4. Conclusions**

main in case 4.

flow domain is obtained.

to heat transfer irreversibility.

the velocity decreases as increases.
