Leaf-vein bionic fin configurations for enhanced thermal energy storage performance of phase change materials in smart heating and cooling systems

In the present study, we investigated the effect of different structures of a novel leaf vein bionic fin and various arrangements in the tube on the complete melting time of phase change materials (PCM) in a triplex-tube thermal energy storage (TES) system. RT82 was adopted as the phase change material. The enthalpy-porosity method was employed for this numerical study. The numerical model was validated against experimental data from a previous reference. The simulation results demonstrate that the novel fins deliver significant reductions in the duration of complete melting. Based on fin-branched vein numbers of 1, 2 and 3, increasing the fin angle from 30 ◦ to 60 ◦ can reduce the complete melting time by up to 14.3%. Additionally, adjusting the fin arrangement can save up to 6.35% of the complete melting time. The proper arrangement of the fins can improve the heat transfer performance of the PCM. The non-dimensional quantities analysis of the calculated results shows that the melting time is negatively correlated with the non-dimensional angle. As the non-dimensional parameter, fin arrangement number decreases from 1, the complete melting time corresponding to the fins of different structures first decreases and then increases for the phase change material.


H I G H L I G H T S
• Integration of solar energy and energy storage for heating and cooling systems.
• Phase change materials as energy storage unit for the sustainable built environment.
• Leaf vein bionic fins to enhance phase change material charging performance.
• Efficient melting performance is achieved in thermal energy storage systems.
• Non-dimensional quantities analysis for melting enhancement evaluation. In the present study, we investigated the effect of different structures of a novel leaf vein bionic fin and various arrangements in the tube on the complete melting time of phase change materials (PCM) in a triplex-tube thermal energy storage (TES) system. RT82 was adopted as the phase change material. The enthalpy-porosity method was employed for this numerical study. The numerical model was validated against experimental data from a previous reference. The simulation results demonstrate that the novel fins deliver significant reductions in the duration of complete melting. Based on fin-branched vein numbers of 1, 2 and 3, increasing the fin angle from 30 • to 60 • can reduce the complete melting time by up to 14.3%. Additionally, adjusting the fin arrangement can save up to 6.35% of the complete melting time. The proper arrangement of the fins can improve the heat transfer performance of the PCM. The non-dimensional quantities analysis of the calculated results shows that the melting time is negatively correlated with the non-dimensional angle. As the non-dimensional parameter, fin arrangement number decreases from 1, the complete melting time corresponding to the fins of different structures first decreases and then increases for the phase change material.

Introduction
It is now widely acknowledged that the consumption of fossil fuels, which has been a long-standing reliance for humanity, generates substantial greenhouse gas emissions and contributes to global warming, with grave consequences for human life [1,2]. In response to this critical issue, the Paris Agreement was signed by 193 contracting parties, and the use of renewable energy has continued to grow globally in recent years [3,4]. Solar energy, which boasts a wide distribution, enormous energy potential, and a safe and pollution-free exploitation process, presents significant promise as a source of renewable energy [5][6][7]. However, the availability of solar energy is heavily influenced by time and seasonal differences, which results in a mismatch between solar energy supply and human demand [8]. To address this issue, a dependable solution has been identified in the form of the thermal energy storage (TES) system, which can correct this mismatch by adjusting the energy output capacity, output location, and output time [9].
One essential element of the TES system is the energy storage material, with phase change materials (PCM) being more favorable compared to other materials [10]. PCM has a high energy storage density and can store at least 5 times more heat compared to the same volume of sensible heat material [11]. Additionally, it has the ability to maintain a nearly constant temperature during its operation, which can contribute to reducing heat loss [12]. Due to its versatility, PCM can be integrated into an extensive range of systems and is used in various applications, including TES systems, as well as thermal management systems such as Lithium-ion capacitors [13], solar stills [14], building equipment [15], electronic equipment [16], air conditioners [17], and unmanned underwater vehicles [18].
However, it is widely recognized that PCM has poor thermal conductivity, and the temperature difference between the heat transfer fluid (HTF) and PCM decreases in the flow direction, thereby severely impacting the heat transfer performance [19]. To reduce the response time of PCM, several effective strategies have emerged, such as adding nanoparticles [20], using diverse types of porous media fillers [21][22][23][24][25][26][27], inserting heat pipes [28], rotating heat exchangers [29], and inserting fins [30]. Among these, inserting fins is the most commonly employed method due to its simplicity, reliability, and ease of use in engineering applications [31]. The insertion of fins enhances the heat transfer surface area and changes the convection of the PCM fluid [32]. An experiment conducted on the melting of a PCM with straight fins inserted in a rectangular cavity revealed that an increase in the number of fins not only led to an increase in total contact area but also reduced performance since the fins impeded fluid flow [33]. Therefore, the contribution of the fins to the melting of the PCM needs to be further investigated.
Modifying the fin pattern alters the amount of contact area between the PCM and HTF, and can cause the PCM to divide within the TES system. Researchers have examined the influence of different fin designs on the melting response time of PCMs, including linear [34], combined fractal [35], helical [36], tee [37], L-shaped [38], Y-shaped fins [39], and irregular structure fins [40]. Additionally, the heat transfer performance of fins with the same structure can vary by altering geometric factors, like adjusting the step ratio of stepped fins [41], modifying the angles of Y-shaped fins [42], or widening triangular fins [43]. These investigations have summarized the attributes of various novel fin structures, and suggest that complex fin shapes outperform simpler ones.
Previous studies have confirmed that thermal convection plays a significant role in melting rates, whereby the PCM solids sink to the bottom during the melting process [44]. Madhi et al. observed improved heat transfer performance when fins were inserted at the bottom of the system, after varying the fin arrangement [45]. Similar conclusions were drawn by Liu et al., who studied the impact of straight fin layout on the melting time of PCM within a vertical cavity structure [46]. In the case of double fin length arrangement, Ji et al. found that heat transfer enhancement mainly occurs during the strong convection phase [47]. Furthermore, it has been demonstrated that dividing the PCM can enhance heat transfer [48,49]. These studies highlight the significant role of natural convection in heat transfer and suggest that investigating fin arrangement is crucial for enhancing heat transfer in PCMs.
The shell-and-tube and double-pipe heat transfer systems are highly recommended for efficient heat transfer. However, the triplex-tube TES system offers a larger contact area compared to the double-pipe TES system. This triplex-tube TES system is composed of three concentric copper tubes, with the HTF flowing through the internal and external tubes while the intervening annular space contains the PCM. This structure has the added benefit of reducing the response time of the TES system, optimizing the use of low-temperature HTF, and minimizing energy waste.
In order to enhance heat transfer in PCM energy storage systems, a novel fin based on a bionic leaf vein structure was employed in this study. The inspiration for this new design was derived from the way plant leaves transport resources to the cells within the leaf. Since the main drawback of PCM materials in industrial applications is their sluggish heat response time, achieving higher heat transfer efficiency without modifying the total heat storage capacity holds considerable significance as compared to existing methods. The novelties include: a) This study proposes a leaf vein bionic fin, discusses its structural characteristics, and analyzes the influence of different morphologies of the fin on the heat transfer performance of PCM. Under the same surface area, this structure shows an improvement in the heat transfer efficiency of fins relative to the structure previously proposed by the research group. b) This study proposes a quantitative model for the arrangement of fins and finds that this approach can improve the shortest complete melting time. c) The non-dimensional analysis of heat transfer results were conducted, and the influence of non-dimensional length, nondimensional angle, and non-dimensional arrangement on the nondimensional heat transfer time were analysed.

PCM in heating and cooling systems
In Fig. 1, a cross-sectional schematic of the triplex-tube TES system with leaf vein bionic fins is shown. These tubes are arranged horizontally. This proprietary system consists of three concentric copper tubes arranged horizontally. The inner and outer tubes of the system are filled with water as HTF, while the left circular zone is filled with RT82 as PCM [50]. RT82 is a type of paraffin that can be obtained online with detailed information [51]. This material was chosen because it is widely available, inexpensive, stable, non-toxic at the temperature at which its phase change cycle is utilised and has a high latent heat of phase change. The characteristics of the materials utilized in this investigation are detailed in Table 1. The fin surface area fraction (φ) remains constant, while the sum of the branch lengths of the fins is equivalent to the length of the main stem. The fins' width measures 0.5 mm. Fig. 1(b) displays fins comprising two branched veins. In each fin, the bifurcation points for the branched veins divide the fin's main stem, with dimensions provided in Table 2. To preserve computational resources, only half of the TES system model serves as the computational domain for this study. At the outset, the PCM exists in solid form at 300 K, and the HTF maintains a temperature of 363 K.

Governing equations
The utilization of the enthalpy-porosity model serves to demonstrate the melting process of PCM, with this particular model being the prevalent method for addressing precarious heat transfer issues [52]. The phase change of PCM constitutes a non-linear process that requires the application of governing equations [53,54].
λ is the liquid fraction of PCM, it is defined as: h and T can be calculated as [55]: The fin percentage is, The non-dimensional time and fin angles are calculated as, The fin arrangement number is defined as, The accuracy of the simulation data relative to the experimental data is calculated as,

Implementation and validation of numerical models
At the onset of the melting process, the PCM exists in a solid state and maintains a temperature of 300 K. Meanwhile, the HTF temperature remains constant at 363 K. To simulate the working conditions for case  3, three separate scenarios are employed with corresponding grid numbers of 55908, 78299, and 101168. A simulation time step of 0.3 s is utilized, as it has been shown to offer a balance between precision and efficiency. The liquid volume fraction's curves within the computational domain are depicted over time in Fig. 2. The difference between the results obtained using grid numbers 78,299 and 101,168 is negligible. Therefore, it is deemed reliable to proceed with the use of grid number 78299. This investigation utilizes the data from reference [56] to authenticate the simulation model, and the authentication is executed under identical operating and boundary conditions as stated in the reference. The findings obtained arecompared with the initial experimental data, as shown in Fig. 3. The accuracy of the simulation data in comparison to the experimental data is demonstrated in Table 3. The simulation outcomes concur with the experimental results, implying that the simulation model utilized in this study is dependable.

Effects of fin branched vein numbers
For this study, three types of fins were selected with fin branched vein numbers of 1, 2 and 3. Numerical simulations were conducted for each of these cases. Fig. 4 illustrated the evolution of solid-liquid distribution contours at various time intervals during the melting process. It was observed that the melting process was similar for all the different fin structures and can be described in three stages.
During the initial stage (t < 10 min), heat was absorbed by the PCM solids through the walls and fins, resulting in the formation of a PCM melt layer along the walls and edges of the fins. However, due to the early stage of the melting process, the thickness of the melt layer remained low and no significant deformation of the PCM solids took    place. Heat conduction was the primary mode of heat transfer during this stage, and the effect of fins on the melting rate was not very prominent. In the second stage (10 min < t < 30 min), as the thickness of the melting layer increased, the shape of the solid-liquid boundary started to change, and the PCM solids became fully separated from the walls and fins. The key mode of heat transfer became thermal convection due to the increasing thickness of the melting layer. As the density of the liquid was lower than that of the solid, buoyancy led to natural convection, causing continuous upward movement of the liquid PCM. The PCM liquid at the top of the tube started to accumulate while the PCM liquid at the bottom of the tube started to impact the PCM solids, resulting in a higher degree of PCM deformation compared to the rest of the PCM solids. During this stage, the fin structure divided the PCM solids, leading to independent melting in different areas of the tube. Fins with a higher fin branched vein number resulted in lower uniformity of partitioning of PCM solids. As the melting process progressed, the crosssectional area of gradually formed individual PCM solids increased as the fin branched vein number increased. In the third stage (t > 30 min), the liquid PCM occupied the main area of the tube. Due to increased natural convection, the PCM at the top of the tube melted completely. The rest of the PCM solids deposited at the bottom of each area gradually melted until they were completely melted. During this stage, the fins mainly helped to divide the area, ensuring faster heat transfer rates by separating the individual sinking solids. The melting time directly reflects the effect of the fin on the resistance, which represents the energy consumption. The definition of thermal resistance is.
ΔT is the temperature difference, and the heat power is represented by P T . Varying the form of the formula.
Considering the time from the beginning of melting of PCM to the time of complete melting Δt, integrating over it yields.
As the total volume of PCM is fixed, the total heat required for complete melting, Q, is constant, and the temperature difference, ΔT, is also constant. Therefore, the thermal resistance is proportional to the complete melting time. The thermal resistance here, R T , refers to the equivalent thermal resistance of the complete melting process, which represents the equivalent thermal resistance of the fin under the actual working conditions. Therefore, the thermal resistance of the fins can be known by obtaining the melting time. Fig. 5 depicted the liquid fraction variation curve during the melting process for the three cases illustrated in Fig. 4. Although all three cases possessed the same fin surface area, differing structures affect the complete melting time. Specifically, the case with l = 1 saved 16.9% of the time compared to the case with l = 3. In the initial stage of melting, fins with a higher fin branched vein number had a slower melting rate because the initial split point of the fins was closer to the wall as the fin branched vein number increases. The surface temperature of the fins decreased as the distance between the fins and the wall increased. Thus, a fin with a small fin branched vein number had a higher temperature in the first branch, leading to a larger surface area in the region of higher fin temperature, which facilitated heat conduction. However, when the melting layer reached a certain thickness, this effect became less significant, and thermal convection dominated the heat exchange process. Fins with fewer branches split the PCM solids more evenly, resulting in a faster melting rate in the middle stage. This explained the difference in the slope of the curves in this region in Fig. 5. In the third stage, remaining solids were deposited at the bottom, and the heat transfer rate slowed down significantly. This behavior was evident in Fig. 5, where the curve's slope decreased rapidly and remained low until it reached zero. At this stage, the fin heat transfer capability had less impact on the PCM melting rate than the uniformity of the PCM solids split by the fins, which can affect individual residual PCM solid sizes.

Effects of fin angles
The preceding section examined the impact of fin construction on heat transfer. However, it is equally important to explore the heat transfer capacity of fins with the same configuration. Altering the fin angle is akin to modifying the number of branched veins on the fin, both of which can divide the solid PCM. Fig. 6 displayed the progression of solid-liquid distributions for the corresponding case by adjusting the fin tension angle under l = 1 at different intervals throughout the melting process. This analysis compared the uniformity of splitting of the same structural fins on melting time. A comparison of the two cases in Fig. 6 with Case 1 in Fig. 4 revealed that the cross-sectional area of individual split PCM solids was larger in the case with a smaller fin angle during the second stage of melting. During this process, the PCM solids between the main stem of the fins and the branch melt more rapidly. Conversely, larger PCM solids between the fins melt slowly due to the ineffective improvement of heat transfer. In Case 1 with a larger fin angle, the shape of the PCM solids was more restricted by the fins compared to the remaining solids in Case 4, which were better divided by the fins. As a result, there was an effective reduction in the size of the PCM solids. After 50 min of heat transfer, the case with a fin angle of 30 • still had small pieces of PCM solids, whereas Case 1 with a fin angle of 60 • had almost entirely melted, leaving only small portions of residue. Fig. 7 depicted the evolution of PCM at various fin angles, and the three graphs exhibit similar melting patterns. Firstly, the curves aligned with the findings in Fig. 6, where the complete melting time of the PCM solids increased as the fin angle decreased. Secondly, altering the same fin angle for fins with the same branched vein number produced a comparable effect on the melting time of the PCM solids. However, the time difference resulting from variations in fin angles was less than one minute. Furthermore, the impact of changing the fin angle on melting time decreased as the number of branched veins on the fin increases. In comparison to Case 4, Case 1 saved 14.3% of the time, while Case 3 only saved 7.1% of the time compared to Case 6. Additionally, modifying the fin angle had a weaker effect on heat transfer than altering the fin branches number. This was because changing the fin angle had a limited impact on the extension distance of the fin branches, whereas variation in the number of fin branches can directly affect the length of the fin branches, leading to a significant change in their extension distance.
Based on the analysis in the previous sections, the novel fin used in this study has demonstrated excellent heat transfer performance. Compared to other types of fins, this new fin also demonstrated an improvement in heat transfer efficiency. Under the same fin arrangement, the minimum required melting time using the proposed fin was found to be 1.9% less than that of Y-shaped fins with the same surface area, indicating a more efficient heat transfer [42].

Effects of fin arrangements
In addition to altering the fin configuration, changing the arrangement of fins can also affect the overall heat transfer of fins. Previous research has shown that during thermal convection, melted PCM collected in the upper part of a tube, while solids were deposited in the lower portion. When fins were added, the liquid PCM still rose through the intervals between the fins, but the flow was impeded and split into multiple parts. Increasing the fin density in the lower half of the tube can improve heat transfer by moderating this effect. To accomplish this, the angular distribution of the four intervals of the fin arrangement was given as an arithmetic progression, setting the first term of the series (φ) to achieve a total of 180 • in the calculation domain. Fig. 8 showed the solid-liquid distribution of different fin arrangements under three fin configurations. Increasing the fin density in the lower part of the tube with the same fin structure led to higher melting efficiency, as seen in cases 10, 13, and 16 after 30 min of melting time. However, overly sparse fins caused PCM solids to accumulate between the inner tube and fins in the upper part of the tube. When fins with a fin branched vein number of 1 were used, the fins in the lower part of the tube may lean together, reducing their efficiency. Therefore, fin arrangement should be adjusted within a certain range to effectively divide PCM solids in the upper part of the tube and avoid excessive fin density in the lower part. Fig. 9 illustrated the development of the liquid component of the PCM under different fin arrangements, which correspond to the cases presented in Fig. 8. It was evident that altering the fin arrangement had a minor impact on the first and second stages of the PCM melting process, with the curves for all scenarios nearly coinciding during these initial stages. The primary disparity lay in the third stage of the melting process. The difference was particularly notable when the fin branched vein number was one, whereas the variance was negligible when it was three. This can be attributed to the fact that the distribution of fins primarily affected the partitioning of solid PCM, which, in turn, directly impacted the size of residual solid PCM when deposition took place in the third stage. Conversely, the effects of fin distribution on PCM melting did not vary significantly during the heat conduction process in the first stage and the thermal convection process in the second stage. Therefore, the difference in melting rate between these two stages was minimal. Furthermore, it was discovered that the impact of fin arrangements on improving the melting time increased with a higher fin branched vein number. Compared to case 1, case 12 only saved 1.85% of melting time, while case 17 saved 6.35% of melting time compared to case 3. Additionally, it was determined that a reduction in φ resulted in an increased effect on the complete melting time when the number of fin branches rose. In addition, the fastest melting time obtained by changing the arrangement of fins increased by 1.85% compared to the situation where the arrangement was not changed.

Non-dimensional quantities analysis
To succinctly summarize the data analysis results, a dimensionless method was employed to process the data. The complete melting time for the reference case without enhanced heat transfer methods was utilized to derive the complete melting dimensionless time for each case. The parameters investigated in this study, namely fin angles and fin arrangements, were independently dimensionlessized. The nondimensional fin angle for each operating condition was calculated using equation (11) with the reference angle set at 90 • . Similarly, the fin arrangement number was defined as the ratio of the first term of the equal difference series, which represented the minimum interval angle, to the reference angle of 45 • in the case of uniform fin arrangement.
The utilization of novel fins effectively enhanced the efficiency of heat transfer in TES systems, resulting in significant reductions in nondimensional melting times across all cases. Fig. 10(a) illustrated the impact of non-dimensional fin angle on non-dimensional melting time. At a 30 • , 45 • , and 60 • fin angle, the corresponding non-dimensional fin angles were 0.33, 0.5, and 0.67, respectively. It is demonstrated that for each fin structure, the non-dimensional melting time exhibited a negative correlation to the non-dimensional fin angle. Furthermore, the melting time decreased more significantly for cases with a smaller fin branched vein number as the fin angle increases, as shown by the comparison of the three curves. In Fig. 10(b), the effect of fin arrangement on melting time was depicted as the fin arrangement number decreases from 1. The complete melting time corresponding to different fin structures decreased and then increased. The trend of the curves revealed that the optimal fin arrangement number to improve melting time decreased as the fin branched vein number increased.

Conclusions
In this study, a novel leaf vein bionic fin was introduced into the triplex-tube TES storage system, and the performance enhancement of PCMs in this system was numerically investigated. This study aims to positively use the novel fin as a compound enhancement technique to achieve better charging (melting) process in TES systems. and the following conclusions were obtained.
PCM melting involves three stages, each controlled by a different melting mode. Changing the fin branched vein number can save up to 16.9% of the total melting time, while increasing the fin angle can reduce the total melting time by up to 14.3%. Modifying the fin arrangement angle according to an arithmetic progression can save up to an additional 6.35% of the total melting time for different fin structure. The melting time curve decreased and then increased when the minimum fin interval angle decreased. The fastest melting time increases by 1.85% by changing the fin arrangement compared to not changing the arrangement. Non-dimensional analysis can help integrate the above conclusions.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
Data will be made available on request.