*5.1. Evolution of the Temperature Field*

Isotherm evolution for the three cases of no fins, straight fins, and twisted fins considered in this study over the three-time spans of solidification 600, 1200, and 1800 s are shown in Figure 4. During the initial span (*t* ≤ 600 s), the isotherms of low temperatures around 290 K (the blue zones) seem to be the dominant group over the entire cavity in all examined cases. This is attributed to the fact that a major part of PCM is still liquid during this duration, and so, not much influence of solidification can be seen. Therefore, the isotherms seem similar between and close to the fin ligaments. During the subsequent span (600 < *t* ≤ 1200 s), the isotherms of high temperatures above 305 K (the red zones) slightly start to shrink throughout the entire domain, particularly in the cases with fins. By this time, the cooling effect of the heat-transfer fluid (HTF) on the PCM becomes more effective so that the solidifying layer increases in size to occupy the whole domain. Moreover, the existence of fins further supports the heat communication between the PCM and the HTF, as the existence of fins allows for better heat removal from the PCM domain. Comparing the cases with fins to the base case of no fins suggests a larger shrink in the layer of isotherms with high temperatures (the red zone), particularly when moving downward. This implies that there is little influencing role of convection compared to the role of conduction in the case of fins, particularly in the upper portion of the domain. During the final span (*t* ≥ 1800 s), the isotherms seem to be more uniform and consistent in shape than in previous durations, particularly in the lower portion of the domain. This would be due to the dominating role of heat conduction in this region, which helps earlier completion of the solidification process. Indeed, the existence of twisted fins further supports the role of heat conduction due to its curving structure, which limits the role of natural convection, and the relatively larger heat transfer area, which assists the overall heat transfer process.

**Figure 4.** *Cont*.

**Figure 4.** Evolution of temperature field for the cases of no fins, straight fins, and twisted fins at different time spans of solidification progress.

Figure 5 compares the temperature field of the cases with two, four, and six twisted fins over three different time spans of the solidification evolution. Not a big difference in the distribution of isotherms can be noticed as the number of fins increases during the early duration (*t* ≤ 600 s) as a major part of PCM is not yet solidified. However, as time proceeds to *t* ≥ 1200 s, the size of solidifying layer increases due to the enhanced heat removal from the PCM with the existence of a higher number of fins. The effect of increasing the number of fins seems to be more noticeable in the lower portion of the domain. The movement of liquid PCM in the vertical TES units is typically governed by the dominance of gravity effect over the buoyancy effect, which after all results in a larger temperature gradient throughout the domain. This is why the PCM at the bottom within the case of six fins is early terminated solidification and appeared completely blue clearer than that in other cases, as shown in Figure 5 (*t* = 1800 s).

**Figure 5.** Evolution of temperature field for the three different cases of twisted fins at different time spans of solidification progress.

The effects of adding straight and twisted fins on the average temperature of the PCM over different time spans are shown in Figure 6. Adding fins typically provides better heat removal from the PCM as fins work as direct passageways for heat communication between the PCM and the cooling walls. However, the data from Figure 6 indicate that applying twisted fins serves better for lowering the PCM temperature so that higher solidifying rates can be obtained. Figure 7 compares the effects of using different numbers of twisted fins on the time histories of PCM temperature during solidification mode. It can be seen that increasing the number of fins does not introduce a significant difference at the early stage (i.e., for *t* ≤ 500 s) of solidification. However, as time progresses, the difference in the behavior curves of the average temperature becomes more noticeable as the number of fins is changed. The average temperature declines to its minimum value within a shorter time when the number of fins is doubled from two to four and six fins. Therefore, faster heat discharge rates from the PCM can be produced by increasing the number of installed twisted fins.

**Figure 6.** Time history of average temperature for the cases of no fins, straight fins, and twisted fins.

**Figure 7.** Time history of average temperature for the three different cases of twisted fins.

#### *5.2. Evaluation of Velocity Distribution*

In addition to the significant effect of fins on the melting rate through the conduction heat transfer, twisted fins can be effective on the natural convection effect in the storage system. Figure 8 displays the z-velocity in the middle cross-section of the system for different studied cases. As shown, higher velocities can be seen in the systems with twisted fins showing the higher effect of natural convection in the domain. It should be noted that the direction of the twisted fins is along with the gravity direction which can be helpful in boosting the buoyancy effect in the domain. In other words, the twisted fins do not prevent the circulation of melted PCM in the domain due to the twisted shape of the fins.

**Figure 8.** Evolution of z-velocity field for different studied cases at the time of 1200 s.

#### *5.3. Evolution of the Liquid-Fraction Field*

The liquid-fraction contours, including the solidifying fronts (shown in light green), are illustrated in Figure 9 for the cases of no fins, straight fins, and twisted fins at five different vertical positions within time spans of 600, 1200, and 1800 s of solidification. During the early time span (*t* ≤ 600 s) and where no fins are present, the solidifying fronts

(shown by light green) take almost the shape of uniform circles that are identical along the vertical direction. However, the addition of fins, particularly twisted fins, results in the formation of relatively bigger solidifying layers (blue areas) adjacent to and surrounding the fins, as seen in Figure 9. The fronts typically do not move away from one another because only thin solid layers can be formed during this time range. Moving to the next time span (600 < *t* ≤ 1200 s), the shapes of solidifying fronts get further deformed, particularly in cases when fins are present, due to the higher rates of heat removal at the cooling walls. The size of the solidifying layers (blue zones) appears to be gradually increased towards the bottom as can be seen in Figure 9 (*t* = 1200 s). This is due to the stronger role of convection in the upper portion of the domain compared to that in the lower portion. The solidifying layer better increases in size in the case of twisted fins as the major space turns blue. In the case of twisted fins, a more important role is noticed for natural convection with more deformation in the shape of the solidifying front. The reason is that the flow-resistant forces generated due to the flow-promoting structure of twisted fins are superior compared to that in straight fins. During the last time span (*t* ≥ 1800 s), the liquid-fraction field shows fully solidified zones in the lower portion of the domain due to the minimal convection role in this part, while solidification is slightly delayed in the upper portion due to the stronger local convection. On the effects of fins, the results show that the size of the solid layer increases as the fins (twisted or straight) are added. However, the effect of twisted fins is more noticeable as the size of the solidifying layer is larger, as can be seen by comparing the liquid-fraction field between the three cases considered in Figure 9 (*t* = 1800 s). Thus, applying twisted fins while keeping the total mass of PCM constant leads to faster evolution of solidifying frons than that when applying straight fins.

**Figure 9.** *Cont*.

**Figure 9.** Evolution of liquid-fraction field for the cases of no fins, straight fins, and twisted fins at different time spans of solidification progress.

Figure 10 compares the liquid-fraction field of the cases with two, four, and six twisted fins over three different time spans of the solidification evolution. A noticeable deformation in the shape of solidifying fronts can be observed as the number of fins increases due to the enhanced heat removal from the PCM with increasing the number of fins. The size of the solidifying layers (blue zones) appears to be gradually increased towards the bottom as can be seen in Figure 10 (*t* = 1800 s). The solidification is observed to be earlier completed in the lower parts of the vertical TES unit, indicating a strong conduction involvement in this area. A little delay occurs in all cases of twisted fins in the upper parts of the domain. The existence of fins impacts the buoyant flow of liquid PCMs in these parts so that only minor involvement of the convection is expected in the heat transfer process. Increasing the number of twisted fins from two to six greatly aids the solidification process at the upper parts of the domain. The twisted fins technically improve heat transport so that the solidifying front appears to travel quicker in two ways. First, their enormous surface area aids for superior heat transfer by conduction between the various portions of the PCM. Second, the twisted structure of fins allows for higher flow-resistant forces so that only a minor role can be played by natural convection.

**Figure 10.** Evolution of liquid-fraction field for the three different cases of twisted fins at different time spans of solidification progress.

The temporal evolution of the liquid fraction throughout the PCM solidification mode has been also tracked to better evaluate the potential of solidification enhancement when twisted fins are applied. Figure 11 compares the liquid-fraction evolution history for the three cases of no fins, straight fins, and twisted fins. The TES system in the three cases is designed to carry out the same PCM mass (*m* = 0.335 kg). Data from the figure indicate that the case with twisted fins provides the best potential for solidification enhancement among the cases considered. Table 2 shows that the case with four twisted fins can reduce the solidifying time from 2739 s in the base case of no fins to only 2229 s so that a time saving of about 20% is achieved. In addition, twisted fins can cut solidifying time from 2512 s in the case of straight fins to 2229 s, saving roughly 8% of the entire solidifying time. Regarding the discharge rate, data from Table 2 imply that applying twisted fins can remove heat from PCM at the rate of 34.25 W while applying straight fins would remove 30.45 W of heat compared to only 27.87 W in the base case of no fins. This results in an increase in the heat discharge rate of around 29% and 10%, respectively, when compared to the reference case of applying no fins.

**Figure 11.** Temporal evolution of the PCM liquid fraction with straight and twisted fins.

**Table 2.** The improvement in solidifying time and discharging rate due to the inclusion of straight and twisted fins.


The time histories for liquid-fraction evolution in the cases of two, four, and six twisted fins are compared in Figure 12. As seen in the figure, applying six twisted fins leads to the greatest possible reduction in solidifying time. Based on the data obtained for the number of fins considered, the solidifying rate generally increases as the number of twisted fins increases. The time data from Table 3 reveals that the TES system design with four and six twisted fins do require 2229 and 1922 s, respectively, to reach the status of complete solidification of PCM. This implies that an 11 to 20 % reduction in solidifying time is possible when doubling the number of twisted fins in use from two to four or six fins, respectively. Data from Table 3 also reveal that applying twisted fins can improve the heat discharge rate from 30.45 W to 34.25 W and 38.20 W when the number of twisted fins in use increases from two to four or six fins, respectively. This results in an increase in the heat discharge rate of around 13% to 26%, respectively.

**Figure 12.** Temporal evolution of the PCM liquid fraction with different numbers of twisted fins.

**Table 3.** The improvement in solidifying time and discharging rate due to the inclusion of different twisted fins.


To have a non-dimensional analysis, the dimension less time is defined as follows:

$$
\pi = \frac{t\alpha}{D} \tag{8}
$$

where *D* is the hydraulic diameter of the PCM container equals 0.02 m. It should be noted that the mass of the PCM is considered constant in all the studied cases. Table 4 presents the dimensionless solidification time for different studied cases. The use of six fins results in the lower non-dimensional solidification time which is almost 27.3% less than that for the case without fins.

**Table 4.** Dimensionless solidification time for different studied cases.


#### *5.4. Impact of HTF Reynolds Number on Solidification of a PCM with Twisted Fins*

Different flow rates of the HTF that are corresponding to Reynolds number values of 500, 1000, and 1500 are examined in terms of liquid-fraction profile and average temperature profile in Figure 13a,b, respectively. The data from Figure 13a indicate that a shorter solidifying time is needed when a higher Reynolds number of the HTF are used. This is due to the fact that higher Reynolds numbers inspire a greater convective heat transfer coefficient at the thermally-active walls so that greater heat removal rates from the PCM are achieved during solidification. The corresponding data in Figure 13b show that almost lower values of the average PCM temperature can be recorded when higher Reynolds numbers of the HTF are used. This also implies that a better cooling effect can be done on the PCM side when HTF with a higher Reynolds number is used. The PCM with twisted fins takes around 2100 s to complete the solidification at Re = 500, but only 1980 and 1850 s at Re = 1000 and 1500, respectively. Therefore, when the HTF Reynolds number is increased from 500 to 1000 and 1500, the total solidification time can be saved by around 5% and 12%, respectively.

**Figure 13.** Evolution of the PCM liquid-fraction and average temperature profiles at three different Reynolds numbers of the HTF. (**a**) Liquid-fraction profile, (**b**) Average PCM temperature profile.

## *5.5. Impact of HTF Temperature on Solidification of a PCM with Twisted Fins*

Figure 14 shows the impact of varying the HTF temperature on the time-wise evolution of the PCM liquid fraction and average temperature, respectively for *THTF* = 10, 15, and 20 ◦C. As can be seen in this figure the values of liquid fraction and average PCM temperature decreases as the HTF temperature increases. In other words, lowering the HTF temperature promotes a greater cooling impact on the PCM side. This is basically due to the fact that utilising a cooler HTF allows for a quicker solidification rate of PCM. This trend appears to be increasingly pronounced as the process approaches the point of solidification completion. As explained earlier, the contribution of conduction in the heat removal process from the PCM becomes more effective and controlling within the final period (*t* ≥ 1800 s) of solidification. Data from the figure shows that a PCM with twisted fins takes around 2700 s to complete the solidification at *THTF* = 20, but only 2100 and 1650 s at *THTF* = 15 and 10 ◦C, respectively. Therefore, when the HTF temperature is decreased from 20 to 15 and 10 ◦C, the total solidification time can be saved by around 28% and 40%, respectively.

#### **6. Conclusions**

A combination of twisted fins with a triple-tube thermal energy storage system was explored and assessed in the three-dimensional modeling during the solidification process. This work involved the influence of planting the twisted fins compared with the cases of straight fins and no-fins. The effects of the inserted fins' number, inlet temperature, and the flow rate (represented with the Re) of the heat transfer fluid were evaluated. The PCM was located between the inner and the outer tubes, which include the heat transfer fluid flows in an opposite direction as the best technique for releasing heat from the PCM. The performance of the unit was evaluated by analysing the reduction of the liquid fraction and the thermal profile, as well as the solidification time and discharge rate. The outcomes specify the benefits of combining the twisted fins with the TES. The results reveal that the utilizing of four twisted fins reduced the solidification time by 12.7% and 22.9% compared with four straight fins and the no-fins cases (assuming the same mass of the PCM), respectively. Likewise, applying four twisted fins enhanced the discharging rate compared with four straight fins and the no-fins. Increasing the number of fins from two to four and six, the solidification time reduces by 11.9% and 25.6%, respectively. Adding fins enhances the thermal removal from the PCM as fins work as direct passageways for heat communication between the PCM and the cooling walls. The solidification rate increases with increasing the Reynolds number (Re); When the Re of the HTF is increased from 500 to 1000 and 1500, the solidification time is reduced by 5% and 12%, respectively. Further, the solidification rate increase with decreasing the heat transfer fluid temperature; when the HTF temperature is reduced from 20 to 15 and 10 ◦C, the discharge time decreased by 28% and 40%, respectively. This work offers an innovative design for adding fins to improve the thermal efficiency of the LHTES units.

**Author Contributions:** Conceptualization, X.S., P.T.; methodology, X.S., P.T.; software, P.T.; validation, P.T.; formal analysis, X.S., J.M.M., H.I.M., H.S.M., W.Z. and P.T.; investigation, X.S., J.M.M., H.I.M., H.S.M., W.Z. and P.T.; resources, X.S., P.T.; writing—original draft preparation, X.S., J.M.M., H.I.M., H.S.M., W.Z. and P.T.; writing—review and editing, X.S., J.M.M., H.I.M., H.S.M., W.Z. and P.T.; visualization, X.S., W.Z. and P.T.; supervision, X.S., P.T.; All authors have read and agreed to the published version of the manuscript.

**Funding:** The study was supported by the Jiangsu Provincial Basic Research Program (Natural Science Fund) (Grant no. BK20191050), Natural Science Research Project of Jiangsu Province Colleges and Universities (Grant no. 18KJD560001), Philosophy and Social Science Project of Jiangsu Province Colleges and Universities (Grant no. 2019SJA1659).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data will be available on request.

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