Heat Transfer Augmentation Using Duplex and Triplex Tube Phase Change Material (PCM) Heat Exchanger Configurations

Abstract

The significance of latent heat thermal energy storage is more substantial when compared to sensible energy storage due to its higher energy storage capability. In this paper, heat transfer enhancement techniques for melting (charging) and solidification (discharging) by using external fins and internal–external fins for a phase change material (PCM) in duplex and triplex tube heat exchangers (DTHX and TTHX) are investigated numerically. A two-dimensional analysis is carried out using ANSYS Fluent for various configurations. Moreover, the effect of different critical parameters, number of fins, fin length, fin thickness, and the heat exchanger tube material are evaluated in terms of the total time of complete phase change of the PCM. Four cases are investigated; cases 1 and 2 are based upon a DTHX while cases 3 and 4 are TTHXs. By considering case 1 as a reference case, it is found that case 2 and case 3 reduce the total melting time by 48.76% and 90.12%, respectively. Case 4 achieves the shortest time for complete melting of the PCM, and the total melting time is decreased by 92%. Solidification behaviour for all four cases is also investigated. The novel configurations increase (doubled) the supply of heat transfer fluid (HTF) while at the same time significantly enhance the melting/solidification characteristics for all the cases without disrupting the convectional currents during phase change of the PCM. Tube materials with different thermophysical properties are also investigated with the heat transfer rate and melting time significantly improved with a high thermal diffusivity material. Moreover, the heat transfer is found to increase with fin length and fin thickness.

1. Introduction

A latent thermal energy storage unit (LTESU) works on the principle of the heat of fusion, which is the required energy for the change of phase from solid to liquid. Throughout the phase change process, the temperature of the phase change material (PCM) remains almost constant while heat transfer occurs with the heat transfer fluid (HTF) [1]. The main advantage of an LTESU over a sensible heat storage unit is the ability to charge a large quantity of energy in a comparatively low volume using a PCM, the material which stores heat [2]. Ever-increasing energy demands due to intensive industrialization have resulted in rapid fuel consumption and massive growth in emissions of greenhouse gases. A drastic increase in the depletion of fossil fuels in recent years and global warming have paved the way for the integration of latent thermal storage with renewable energy sources for a sustainable future [3]. Solar thermal energy is available in abundance and highly suitable for non-industrial applications. PCMs have many applications: thermal storage for heating and cooling structures, electric gadgets, automobile engines, spacecraft, the food industry, the medical industry, heat pump systems, waste heat recovery, cool suits, and cold storage [4]. Nonetheless, storing solar thermal energy for subsequent efficient use is a major challenge [4]. Khan et al. [5] numerically and experimentally studied the effect of fin orientation in a horizontal LTES unit by using stearic acid as a PCM located in an isothermal copper annular tube with an adiabatic steel shell. Five angular geometries were studied by systematically varying the fins’ relative angle ${(0}^0\le \theta \le {90}^0)$.The geometries of the fin arrangement mimicked the shape of a 𝜆 and Y, with the former at ${90}^0$ and the latter at ${30}^0$. The Y-shaped fin arrangement was found to be more effective than the 𝜆-shaped fin arrangement, resulting in a significant increase in the rate of melting. Furthermore, we noticed that a higher length-to-thickness ratio of the fins improved the melting rate of the phase change material (PCM) due to a greater shell diffusivity. Latent thermal energy storage (LTES) involving PCMs is an efficient heat transfer technique to address this specific issue. Nevertheless, the thermal conductivity of PCMs is relatively low, thereby limiting the heat transfer rate and resulting in long phase change times. Many methods, including the diffusion of nanoparticles [6,7] and metallic foam [8,9] in PCMs, along with the employment of fins [10,11], have been proposed to enhance the heat transfer rate of PCMs. Amongst these methods, fin usage is the most promising choice to improve the efficiency of shell and tube PCM heat exchangers [12]. This technique is easier to implement, effective, and economical [12,13]. A PCM used in a finned heat exchanger is an efficient way of charging and discharging a large quantity of heat throughout the melting and solidification processes, respectively. It has been found that the concentric-tube form of a heat exchanger is capable of a latent heat transfer that results in the highest effectiveness for low-volume flow rates. In this type of heat exchanger, the PCM surrounds the tube’s concentric shell space with attached fins while the HTF flows through the tube [14].

Nie et al. [15] explored the enhancement of repeated charging and discharging of a PCM in an axially concentric tubes heat exchanger using fins. The effect of different configurations, number, and length of fins on heat storage was investigated. It was found that the fins of a conventional design had a more complex heat transfer augmentation than other configurations that have fewer fins. Moreover, this arrangement resulted in an improvement of the melting/solidification rate due to higher rates of thermal energy release and storage. Yu et al. [16] numerically analyzed a transient melting concentric-tubes heat exchanger having tree-like fins. The fins, like fractal trees, significantly increased the melting of the PCM. The spreading of the fins in a multi-level three-dimensional fashion carried a point-to-area network for larger heat flows. The conduction heat transfer enhancement is more significant than the dissipation of natural convection. It was found that the overall melting time of a heat exchanger having such fins decreased by 26.7%, and the equivalent heat storage rate increased by 45.4% relative to those of heat exchangers having plate fins. Nóbrega et al. [17] numerically and experimentally investigated horizontally finned tubes for cold applications using water as the PCM. The results showed that the increase in width and number of fins and the reduction of the tube’s wall temperature reduced the total time for solidification. The results showed that improving the heat transfer in the PCM required maximum spreading of the fins. Improvement of melting was proportional to fin height, number, and thickness. It was concluded in the research that internal fins are the most efficient technique to improve the thermal conductivity of PCMs. Cheng et al. [18] examined the melting behaviour for an LTES by the incorporation of gradient fins consisting of axial fins with a gradient central angle and inclined thickness. The melting time of the LTESU with such fins was found to decrease by 12.5%. Free convection was observed to perform a significant part in the melting enhancement of the LTESU, with melting-rate increments of up to 180% as without free convection.

Chen et al. [19] performed a novel experimental study on the performance of an LHTS with the PCM filled inside the tube. The experimental setup recorded the phase boundary as well as the PCM’s temperature as a function of time. The PCM filled both the internal and external sides of the tube at different filling ratios (40–90%). It was shown from the results that the PCM at the inner side of tube had the greatest enhancement at a filling ratio of more than 62%. Mehta et al. [20] analyzed the effect of spiral fins on the melting and solidification rate of stearic acid as the PCM and water as the HTF in an experimental setup. The results concluded that the spiral fins increased the charging and discharging rates due to the redistribution of heat at the mid and axial planes. Both the charging and discharging time was found to decrease by 31.82% due to these spiral fins. Wu et al. [21] numerically examined the transient heat transfer models during phase change processes in LTESUs with natural convection. The fractal tree-shaped fins employed in this model were found to improve the heat transfer effectiveness of the LTESU with a multi-level spatial structure primarily due to increased heat transfer area. Kalapala et al. [22] and Mehta et al. [23] experimentally examined the various inclinational positions of concentric-tubes LTESUs. Experimental results showed that there are major effects on charging and discharging behaviour due to the orientation of the LTESU. In horizontal configurations, the melting rate was testified to be greater at the upper portion till half the PCM melted. In contrast, in the vertical configuration, it was found to decrease for the complete melting time. Lamrani et al. [24] studied the viability of employing a coupled solar parabolic trough collector in an LTESU for water heating in large buildings. Both melting and solidification were investigated for the LTESU for three types of PCMs. These results demonstrated that for a mass flow rate of $1800\raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$h$}\right.$, utilizing RT-55 as a PCM was more favorable than RT-42 and RT-65. Rezaei et al. [25] performed a parametric study to estimate the sensitivity of a Eutectic Si-Mg PCM in vertical SiC tubes for high-temperature LTESUs. The results showed that high-temperature LTESUs having 95% efficiency and a maximum storage energy of 810 $\mathrm{MJ}/{\mathrm{m}}^3$ can be achieved. Zhang et al. [26] investigated the solidification rate for an LTESU. An advanced fractal-tree-shaped configuration using metallic fins for the concentric-tubes LTESU was used. It was shown that the tree-shaped fin significantly increased the solidification performance of concentric-tubes LTESUs. Al-Abidi et al. [12,27] experimentally and numerically studied using a TTHX with internal–external fins in an LTESU. Several methods using multiple families of PCMs and different configurations were analyzed. It was shown that the inlet temperature of the HTF has a higher influence on the charging process than the mass flow rates. Moreover, it was also found [28] that LTESU usage in air-conditioning systems was much more sensitive to PCMs in different aspects of the air-spreading network. Y. Wang et al. [29] studied a new method for enhancing heat transfer in a drag-reducing surfactant solution using a photo-rheological counterion. They found that a concentration of 4 mM/5 mM was most efficient for improving heat transfer, and rheological tests showed that the counterion altered the solution’s viscoelastic properties to enhance heat transfer. Bo Yu et al. [30] performed direct numerical simulations (DNSs) to investigate turbulent channel flows with heat transfer for both Newtonian and drag-reducing fluids. They used the roper orthogonal decomposition (POD) method to extract coherent structures and analyze the mechanisms of turbulent drag and heat transfer reduction. Their study provides insights into the energy distribution and coherent structures of drag-reducing flows with heat transfer.

From the literature survey presented, several parameters regarding the use of single and multiple PCMs, spirals and gradient fins, longitudinal fins, different flow rates of HTFs, metallic foam usage in PCM to enhance the conductivity, fin length-to-thickness ratio, different configurations of fins, the study of LTESUs at different positions, duplex tube LTESUs, and triplex-tube LTESUs have been studied, but there is a clear research gap in the analysis of critical design parameters in different fin and shell configurations of duplex and triplex concentric tubes using different PCMs. There are few research studies on individual analyses of duplex and triplex tubes, but there is no comparative study between the two, especially with external and internal–external fin arrangements. In view of this potential gap, a numerical model is validated in this paper as a basis for subsequent investigation, formulating the basis of comparison of duplex- and triplex-tube LTESUs for four different cases. In this present study, the optimization of the performance of a duplex and triplex concentric-tubes heat exchanger using stearic acid as a PCM is primarily performed.

2. Numerical Model

2.1. Computational Domain

Figure 1a,b show the two-dimensional computational domain for the concentric tube cross section known as a duplex and triplex tube heat exchanger (DTHX and TTHX) along with the fin configuration and the passage for the HTF and PCM. Figure 1c shows the three-dimensional cross section of this heat exchanger. Four cases are studied based on their configurations. Cases 1 and 2 are based on a DTHX while case 3 and 4 are based on a TTHX. Cases 1 and 2 have $\lambda$- and Y-shaped external fin configurations, respectively, while cases 3 and 4 have $\lambda$- and Y-shaped with both external and internal fin configurations, respectively, as shown in Figure 2. For these cases, the inner tube is made of copper, having an inner diameter of $D_t=32.1 \mathrm{mm}$ with a thickness of $t_t=3 \mathrm{mm}$. This inner copper tube is inserted into an outer tube made of steel, also called a steel shell, with an inner diameter of $D_s=121 \mathrm{mm}$. The steel shell’s thickness is kept the same as the thickness of the copper tube. The steel shell is used as a duplex tube for cases 1 and 2, while in cases 3 and 4, it is used as a triplex tube with an external diameter of $D_s=143 \mathrm{mm}$. Diameter and thickness of the intermediate tube is same as the diameter and thickness of the steel shell in the DTHX. The fin length is $l_f=36 \mathrm{mm}$, and the fin thickness is $t_f=3 \mathrm{mm}$. The inner gap between the copper tube and steel shell is filled with stearic acid as the PCM, and the copper tube is filled with water as the HTF. The choice of copper for the inner tube and fins is due to its excellent thermal conductivity. Steel is chosen for the shell due to its lower thermal conductivity, as it helps in insulation of the heat exchanger while minimizing corrosion and providing mechanical strength.

2.2. Thermophysical Properties

Stearic acid was selected as the PCM for the present study, which is from an extended family of organic PCMs generated from fatty acids. Stearic acid is used due to its nontoxic and noncorrosive characteristics. Additionally, it has a comparatively low melting temperature, and phase change occurs at an approximately constant temperature. It is thermally and chemically stable throughout the phase change process. Relevant thermophysical properties of stearic acid, steel, and copper are summarized in

Table 1. Thermophysical parameters of the geometry and PCM [5].

Properties	Copper	Steel	Stearic Acid
Thermal expansion coefficient, $\beta \left(1/\mathrm{K}\right)$	-	-	0.00081
Thermal conductivity, $k\left(\mathrm{W}/\mathrm{m}·\mathrm{K}\right)$	387.6	16.27	0.18
Specific heat of PCM, solid, ${cp}_s\left(\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·K\right)$	0.381	0.5025	2.83
Specific heat of PCM, liquid, ${cp}_l\left(\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}\right)$	-	-	2.38
The density of PCM, solid, $\rho s\left(\mathrm{kg}/{\mathrm{m}}^3\right)$	8798	8030	1150
The density of PCM, liquid, $\rho l\left(\mathrm{kg}/{\mathrm{m}}^3\right)$	-	-	1008
Melting temperature, TM $\left(k\right)$	-	-	327–337
Latent heat of fusion, Lf $(\mathrm{kJ}/\mathrm{kg})$	-	-	186.5
Dynamic viscosity, μ $\left(\mathrm{kg}/\mathrm{m}·s\right)$	-	-	0.0078

.

2.3. Governing Equations

The governing equations for the thermal analysis are the continuity, momentum, and energy equations shown as Equations (1)–(3) respectively:

$$\frac{\partial \rho }{\partial t}+\frac{\partial {\rho u}_i}{{\partial x}_i}=0$$

(1)

$$\frac{\partial \rho u_i}{\partial t}+\frac{\partial \rho u_i{u}_j}{{\partial x}_j}=-\frac{\partial p}{{\partial x}_i}+\mu \frac{{\partial }^2{u}_i}{{{\partial x}_j}^2}+{\mathrm{S}}_{{\mathrm{B}}_{\mathrm{i}}}+S_{Mi}$$

(2)

$$\frac{\partial \rho h}{\partial t}+\frac{\partial \rho u_{i}h}{{\partial x}_i}=\frac{\partial }{{\partial x}_i}k\frac{\partial T}{{\partial x}_i}$$

(3)

where ‘$u$’ is taken as fluid velocity, ‘μ’ represents dynamic viscosity, ‘p’ denotes the pressure, ‘g’ represents the gravitational constant, ‘Si’ is taken as source term and ‘ρ’ signifies the density. The buoyancy forced flow of the PCM liquid is generated due to the melting process that is an unsteady, incompressible laminar flow. The natural convection is due to a density difference for the duration of the melting process because of the gravitational effects. The density discrepancy is estimated by applying Boussinesq’s correlation, which includes the thermal expansion coefficient ‘β’, temperature difference ‘$\mathrm{T}-{\mathrm{T}}_{\mathrm{l}}$’, density changes ‘$\mathsf{\Delta }\mathsf{\rho }$’, and the buoyancy source term ‘$S_{B_i}=\mathsf{\rho }\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{l}}\right)\mathrm{g}$.’

Furthermore, ‘k’ denotes the thermal conductivity, ‘T’ represents the temperature, and ‘h’ is the enthalpy in Equations (1)–(3). The enthalpy of the PCM can have three possibilities:

(a)

PCM = totally solid,

(b)

PCM = liquid and solid,

(c)

PCM = totally liquid.

It is computed though the enthalpy difference depicted mathematically in Equation (4), known as the enthalpy method.

$$h=\left\{\begin{array}{cc}{{\displaystyle \int }}_{T_R}^T{C}_{ps}dT,\hfill & \hfill T<T_s\\ {{\displaystyle \int }}_{T_R}^{T_s}{C}_{ps}dT+\alpha h_{∆},\hfill & \hfill T_s<T<T_l\\ {{\displaystyle \int }}_{T_R}^{T_s}{C}_{ps}dT+h_{∆}+{{\displaystyle \int }}_{T_l}^T{C}_{pl}dT,\hfill & \hfill T>T_l\end{array}\right.$$

(4)

where ‘$T_M$’ represents the reference temperature of 300 K, ‘Cp’ signifies specific heat, ‘$h_{∆}$’ is the latent heat of fusion and its value for solid PCM is zero, while ‘$L_f$’ represents the latent heat of fusion for the liquid PCM. The fraction for PCM melting is represented by ‘α’, which appears during the phase change process $\left(T_s<T<T_l\right)$. The melting fraction can be computed through the correlation given in Equation (5).

$$\alpha =\frac{h_{∆}}{L_f}=\left\{\begin{array}{cc}0\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T<T_s\\ \frac{T-T_s}{{T_l-T}_s}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_s<T<T_l\\ 1\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T>T_l\end{array}\right.$$

(5)

The source term for momentum ‘$S_{Mi}$’ in Equation (2) for the convection heat transfer impact of liquid PCM is a damping term established by Darcy’s law [31] in Equation (6).

$$S_{Mi}=C_{Mushy}\frac{{(1-\alpha )}^2}{{\alpha }^3+\epsilon }{u}_i$$

(6)

‘$C_{Mushy}$’ is accountable for velocity damping that represents the mushy zone constant. Its value usually varies from ${10}^4$ to ${10}^7$. A high value of the mushy zone constant indicates a higher velocity damping rate. As for greater values, more significant fluctuations will be generated in the mushy zone due to lack of velocity damping. The recommended value of the mushy zone constant, as used in the literature, is ${10}^5$ [2]. While ‘$\alpha$’ is also visible in the denominator of the source term equation, so, as the PCM is solid, the source term is infinite. A small number $\epsilon =0.001$ is included in the denominator of $S_{Mi}$ to avoid fluctuations.

2.4. Solver Settings

Laminar two-dimensional buoyant incompressible transient flow simulations are carried out. In this module, the double precision model is used for the simulations to improve the precision and range of magnitudes. The energy, solidification, and melting modes are employed to simulate the physics. Third-order MUSCL (monotone upstream-centered schemes for conservation laws) is used for the energy equation and convective part of the momentum equation that balances the upwind scheme and central differencing. It improves numerical diffusion and spatial precision resulting in higher numerical stability for large time steps. The second-order central differencing method is used for the diffusive momentum equation. Temporal discretization is performed by employing an inherently stable second-order implicit scheme, and the residuals convergence criteria is set at ${10}^{-6}$.

2.5. Initial and Boundary Conditions

Figure 3 presents the boundary conditions implemented for the simulations. HTF with a constant temperature is made to flow through the inner tube where there is a negligible temperature drop across the tube. Consequently, the temperature for the inner surface of the HTF tubes is set as $T_t=358 K$ and considered as an isothermal wall. The boundary condition for the outer surface of the HTF tube is of a coupled wall because it has a boundary contact with the PCM. The temperature difference of the outer surface of the HTF tube with the PCM is linked with a coupled boundary condition. At the start of the simulation, the temperature of the outer surface of the HTF tube and solid PCM is 300 K. The inner surface of the outer tube is known as the shell. For cases 1 and 2, shell conduction is switched on, which allows the shell to conduct heat from the inner side, but an adiabatic boundary condition is implemented to avoid heat loss from the outer surface of the shell. For cases 3 and 4, the intermediate tube has outer side contact with the HTF and inner-side contact with the PCM. Coupled wall boundary condition is provided for these intermediate tubes. The outer tube is considered an adiabatic shell.

3. Verification and Validation of the Model

3.1. Verification

A five-step verification scheme [19] is employed to verify the CFD simulation results. The first step is a grid convergence study. It is conducted to confirm spatial discretization independence. The melting fraction of the PCM is examined using five dissimilar mesh sizes. The computational domain consists of M1 = 6000, M2 = 10,000, M3 = 12,000, M4 = 15,000, and M5 = 24,000 cells having complete melting in 79.8 min, 82 min, 83.3 min, 84.65 min, and 88.21 min, respectively, plotted against melt fractions as presented in Figure 4. The intermediate cell size M3 consisting of 12,000 cells with complete melting time of 83.3 min is selected as it is the most appropriate mesh having an appropriate computational time with efficient results. The second verification step involves temporal discretization convergence, certified for a transient solution by changing the initial time step. The time step sizes of 100 ms, 400 ms, and 80 ms are investigated. The solution remains stable with the current methodology for all time steps, but the computational time increases by decreasing the time step. However, the results largely remain unaffected by the choice of time step. Therefore, a time step of 400 ms is chosen for current numerical simulations as it converges the solution with a reasonable real-time computational period with acceptable accuracy. Thirdly, all seven residuals according to the governing equations of Section 2.4 converge at 10⁻⁶. Due to the simple physics of 2D laminar flow in concentric pipes with the phase change process of the PCM, this value is reached after 80 iterations. Fourthly, to check the consistency of the results, 10 exactly similar simulations are performed with the exact boundary conditions, and the achieved precision of the results lies in the range of 99%. Finally, to check for any abnormality in any thermal parameter, a sensitivity analysis is performed to verify the results theoretically. Variation in heat transfer rate by the temperature of the HTF is directly proportional to PCM phase change, as expected from the literature [31,32].

3.2. Validation

To validate the present numerical methodology, it is imperative to compare the simulation results with those in the literature. To that end, the melting fraction of a Y-fin configuration LTESU having stearic acid as the PCM calculated from the present study is compared to the results of Khan et al. [5]. The PCM is set in the spherical concentric tube with an inner tube containing the HTF at a constant temperature. Initially, PCM is maintained at 300 K, and the heated tube with the HTF has a temperature of 358 K. The simulations result from the present study are found to be in close agreement with the published results [5], as shown in Figure 5a. The fins’ presence significantly accelerates the melting process since complete melting is achieved after 84 min. Moreover, the temperature change of the spherical concentric LTESU is also compared with the results of Khan [5]. Stearic acid is utilized as the PCM with a copper inner tube and steel shell. The temperature profile is similarly extracted from the simulation results and is shown in Figure 5b. The temperature firstly increases linearly but after this initial rapid increase, it shows a slow rise and becomes asymptotic stable by the end of the simulation. If the PCM is heated up, the density difference makes the liquid PCM rise in the annular tube. After the liquid PCM moves toward the bottom of the cell because of convection, the average PCM temperature increases rapidly. Nevertheless, the temperature grows constantly once the liquid PCM moves toward the bottom of the unit.

4. Results and Discussion

4.1. Melting Fraction and Temperature Profile

The melting fraction and temperature profile for cases 1–4 are presented in Figure 6. The PCM melting rate remains constant initially for all the cases. The melting process occurs rapidly in the first five minutes, and it appears to slow down as the time progresses. It is worth mentioning that the convection process is negligible at the start due to PCM being present in solid form. In cases 1 and 2 based on the DTHX, a significant variation is observed after 20 min when the melting fraction reaches 40% and the convection process becomes dominant thereby improving the heat transfer rate. At the time of 20 min, the approximate percentage of the melt fraction is 38% and 43% for cases 1 and 2, respectively. As expected, it is clear from the results that fin employment plays a substantial role in improving the heat transfer rate. The melting rate increases for case 2 as compared to case 1. At the time of 83 min, case 2 (Y-fin configuration) results in complete melting due to the fin configuration in Y-shape, while case 1 ($\lambda$-fin configuration) records 86% melt fraction at the same time. It is because the solid PCM moves to the bottom due to gravitational effects providing greater access for heat transfer for the Y-fin configuration. On the other hand, the $\lambda$-fin configuration does not have extended fins, limiting its access to the bottom side of the container. This results in a higher heat transfer for the Y-fin configuration as compared to the $\lambda$ configuration.

In cases 3 and 4 based on the TTHX, rapid melting is observed in the system partly due to the double supply of HTF and partly due to the employment of internal and external fins increasing the surface area available for heat transfer as reported in the literature [11]. Cases 3 and 4 take 16 and 13 min, respectively, to complete melting. It is clear from the trends in Figure 6 that there is a great potential for heat transfer enhancement by the proposed modification. In the DTHX LTESU, case 1 requires a maximum time for complete melting of the PCM while case 2 takes approximately half of that time to melt completely. As far as the TTHX LTESU configuration is concerned, case 3 and case 4 take 90% and 92.12%, respectively, less time than case 1. It is worth mentioning that the difference of total melting time between case 1 and case 2 (DTHX configuration) is 50% while it reduces to 18.75% between case 3 and case 4 (TTHX configuration). This comparison clearly demonstrates that the employment of TTHX, large number of fins, and double supply of HTF accelerates the melting process significantly.

In order to elaborate these heat transfer characteristics for all four cases, melt-fraction contours and temperature contours are shown in Figure 7. Natural convection effects are noticeable from the start in both the temperature and melt-fraction contours. The rate of melting of the PCM at the top region is enhanced by a rise in temperature because of the convection effect. At $t=25 \mathrm{m}\mathrm{i}\mathrm{n}$ the temperature distribution around the fin is significant, resulting in melting of the PCM to be significant for case 2 (300 or Y). Contrary to this, there is a high-temperature region below the HTF tube between the two lower fins and at the top tip of $\lambda$-shaped fins in case 1 (900 or $\lambda$) at the same time. The convectional effects let the charged PCM move upwards due to its low density, but the PCM from the lower section gets stuck between the lower fins of the $\lambda$-shaped configuration of case 1 (900 or $\lambda$). This makes a high-temperature zone because there is an increase in the temperature continuously. The upper fin of the $\lambda$-shaped configuration of case 1 (900 or $\lambda$) gives way to melted PCM, which attains the temperature from the HTF tube and rises from the bottom due to buoyancy effects. The melted PCM gathers at the top of the upper fin, forming a high-temperature zone, whereas, in case 2 (300 or Y) there are no high-temperature zones. Thus, the average PCM temperature remains the same even if the melting rate varies. Nevertheless, after 33 min the variation of average temperature becomes significant as shown in the contour plots where the upper section completely melts while the bottom section only partially melts. After the melting of the upper region of the PCM, the role of free convection becomes more important. In case 2 (300 or Y) the effect of convection is improved when the melted PCM attains higher temperatures rapidly. The temperature of case 2 (300 or Y) is considerably greater than case 1 (900 or $\lambda$) and this trend continues with time. At $t=84 \mathrm{m}\mathrm{i}\mathrm{n}$, case 2 (300 or Y) melts the PCM completely and has the PCM average temperature of 356.1K while in case 1 (900 or $\lambda$) there is solid PCM left over at the bottom section and the average temperature and melting fraction is 351.3 K and 88.2%, respectively. Therefore, case 2 (300 or Y) has less meting time while case 1 (900 or $\lambda$) has more melting time as is evident in Figure 7.

To elaborate the difference of the heat transfer mechanism for TTHX configurations, the melting and temperature contours for case 3 and case 4 are shown in Figure 8. At $t=100 \mathrm{s}$, there is a constant heat transfer in case 3 and case 4 from both the inner tube and outer shell. At the start, conduction occurs between the PCM and tubes due to the solid state of the PCM. At $t=200 \mathrm{s}$, the high-temperature zones stat to develop at the top side for both case 3 and case 4. The intensity of the high-temperature strength increases at the top side with the passage of time. It is remarkable to see that the temperature of case 4 is higher than case 3 at $t=500 \mathrm{s}$. Afterwards, the temperature for case 3 starts increasing more rapidly than case 4. It may be attributed to the PCM being trapped between the fins where there are high-temperature zones present due to a constant supply of heat coming from the HTF through conduction between the tubes and fins. The increase in temperature for case 3 becomes less than that of case 4 after $t=700 \mathrm{s}$. This decrease in temperature is due to the internal–external Y-fin arrangement employed in case 4, which is found to be more highly effective than the internal–external $\lambda$-fin arrangement used in case 3. The internal–external Y-fin arrangement is found to melt at the same amount of PCM in $t=760 \mathrm{s}$, while the internal–external $\lambda$-fin arrangement takes $t=960 \mathrm{s}$ for complete melting of PCM, as shown in Figure 7.

4.2. Comparison between Melting and Solidification

The PCM melting process is known as charging while the solidification process is discharging.

Table 2. Melting and solidification time with their comparison.

Cases	Melting Time (min)	Solidification Time (min)	Solidification vs. Melting Ratio
Case 1	162	190	1.18
Case 2	83	240	2.89
Case 3	16	44	2.75
Case 4	13	47	3.61

shows the charging and discharging times while the discharging contours for all four cases at different time instances are shown in Figure 9.

Cases 1 and 2 have a single supply of HTF from the inner-side tube, and the total number of external fins are three and are DTHX based; therefore it takes more time for the melting and solidification processes. However, the fin configuration becomes the most important factor in all four cases. The Y-fin configuration of case 2 is more effective than the $\lambda$-fin configuration of case 1. This is because the Y-fin arrangement reduces the buoyancy effect by providing a fin to the bottom so that the heat transfer rate increases as buoyancy ensures the melted PCM moves upward while holding down the solid PCM during charging, as elaborated in Figure 8. Now the bottom fin of the Y-fin configuration provides heat transfer access to the solid PCM that holds itself to the bottom, resulting in the melting and solidification rate to increase compared to the $\lambda$-fin configuration of case 1. The $\lambda$-fin configuration of case 1 does not have access to the bottom-side PCM, causing the heat transfer rate at the bottom of the duplex tube of case 1 to be less than case 2.

On the other hand, cases 3 and 4 consist of TTHX with a double supply of HTF at constant temperature provided to the PCM from both sides with double fins as shown in Figure 8. Cases 1 and 2 have only external fins with a single supply of HTF while cases 3 and 4 have both internal and external fins with a double supply of HTF. Therefore, the number of internal and external fins and double supply of HTF become the most important factors in cases 3 and 4. Consequently, the source of providing constant heat and the surface area of the fins available for heat transfer doubles. Due to this double supply of HTF and fins in cases 3 and 4, the melting time and solidification time improve significantly.

At the upper side of the concentric tubes, fins are useful, but not as useful as compared to the bottom. This is because the flow of heat transfer rate is more towards the upward direction due to the low density of PCM shown in the Figure 8. The charged PCM is less dense than the uncharged PCM, hence this uncharged PCM holds itself to the bottom. Although a constant supply of heat is coming from HTF in the tube, the fin arrangement increases the rate of heat transfer throughout the concentric tubes. Hence the fin configuration becomes a more important factor for the total melting and solidification compared to the double effect.

For the melting process, case 1 takes t = 162 min to fully melt, which is marked by the completion of the melting process, whereas cases 2, 3, and 4 have a charging time of 83, 16, and 13 min, respectively. On the other hand, the discharging process (marked by the completion of the solidification process) is completed in 190, 240, 44, and 47 min, for cases 1–4, respectively. In all the configurations used, case 4 is found to have the maximum value of solidification-to-melting-time ratio, which is 3.61, while cases 1–3 have time ratios of 1.18, 2.89, and 2.75, respectively.

4.3. Effect of Tube Materials

To elaborate the effects of different materials for the concentric-tubes heat exchanger based on TTHX, a material analysis has been performed. Aluminum, gold, copper, and steel are selected to check the melting fraction for the optimum case 4 configuration. The thermophysical properties of these materials are given in the

Table 3. Thermophysical properties of different tube materials.

Property	Copper	Aluminum	Gold	Steel
Density of PCM, solid, $\mathsf{\rho }$s (kg/m3)	8978	2719	19,320	8030
Specific heat of PCM, solid Cps (J/kg K)	381	871	129.81	502.48
Thermal conductivity, solid, k (W/m K)	387.6	202.4	297.73	16.27

. It is clear from Figure 10 and Figure 11 that aluminum, gold, and copper exhibit almost similar melting times for the PCM while steel has the longest time to completely melt the same amount of PCM due to its lowest thermal conductivity. The time taken for complete melting for aluminum, gold, copper, and steel is 13, 12.6, 14, and 18.28 min, respectively. Gold achieves the shortest complete melting time, so it is taken as a reference material. The difference of complete melting time between aluminum-gold, copper–gold, and steel–gold is 0.4, 1.4, and 5.56 min, respectively. It is to be noted that all other materials except steel are rather expensive and undergo complex tube manufacturing techniques. On the other hand, steel is more suitable due to its low cost and ease of fabrication. Moreover, the melting time using steel as the tube material is 18.28 min and still feasible for the internal–external fins used in the TTHX configuration.

4.4. Effect of Fin Length on Melting

To investigate the effect of fin length, by taking a constant fin thickness, three different lengths of 18 mm, 26 mm, and 36 mm are used for the optimal case 4. It is clear from Figure 12 that the melting time decreases with the increase in the fin length.

Table 4. Melting percentage for case 4 for different fin lengths and thicknesses at t = 12 min.

Case 4 (Internal–External Y-Fin)							Time (Min)
Fin thickness (mm)	3	2	1	3	3	3	12
Fin length (mm)	36	36	36	36	26	18
Melting fraction (%)	94.02	73.14	62.38	94.02	61.85	58.19

shows the melting fraction at $t=12 \mathrm{m}\mathrm{i}\mathrm{n}$ for different fin lengths and thicknesses. Melting fraction for fin lengths of 18 mm, 26 mm, and 36 mm are 58.19%, 61.85%, and 94.02%, respectively. Hence, this implies that the longer the fin length, the more the melting due to larger heat transfer areas as reported in the literature [27].

4.5. Effect of Fin Thickness on Melting

The fin thickness effect is investigated by taking constant fin length for three different fin thicknesses of 1mm, 2 mm, and 3 mm tested for the optimal case 4 TTHX configuration as summarized in Figure 13. It is clearly observed that an increase in fin thickness results in the reduction of the melting time. The melting time of 1 mm, 2 mm, and 3 mm, thickness at 12 min was 62.38%, 73.14%, and 94.02%, respectively. As the fin thickness increases, the surface area available for heat transfer also increases, although the surface-area-to-volume ratio decreases. Consequently, fin thickness improves the heat transfer rate and melting time but not as much as fin length does, as is found in the literature [27].

5. Conclusions

Heat transfer enhancement for DTHX and TTHX by using internal and internal–external fins to speed up the melting rate of stearic acid as a PCM were studied numerically. Different operational and design parameters including the DTHX and TTHX configurations, tube material, fin length, and fin thickness were analyzed. Based on the results from the simulation, these parameters have a substantial effect on the time for both complete charging and discharging of the PCM. A double supply of the HTF has a significant effect on the melting rate (charging). The different orientational effects of fins, i.e., $\lambda$-fin and Y-fin configurations, on the performance of melting augmentation for the LTESU were also studied. The temperature-stimulated buoyancy effect of PCM movement alongside its convective heat transfer features were examined in detail. It was found that the large number of fins and double supply of HTF accelerates the melting process significantly. Although this strategy enhances the heat transfer, complex morphology and design also increased the cost.

Different storage configurations were studied; case 1 ($\lambda$-fins) and case 2 (Y-fins) were based upon duplex tubes while case 3 ($\lambda$-fins) and case 4 (Y-fins) were based on triplex tubes. In the duplex tube arrangement, case 2 was found to be more efficient than case 1 because the melting time taken in case 2 is around 85 min while case 1 takes 160 min. Initial convectional currents enable complete melting early on in case 2. The heat transfer enhancement in the Y-fins (case 2), as compared to the $\lambda$-fin configuration (case 1) decreased the melting time by 48.76%. In triplex tubes configurations, case 4 had a better thermal performance (13 min total melting time) as compared to the melting time for case 3 of 16 min. Cases 2, 3, and 4 had 51.87%, 90%, and 91.87% improvement in melting rate, respectively, in comparison to the melting time of case 1. These four cases were also investigated for the total time taken for PCM solidification for (discharging). Case 4 had the highest solidification-to-melting time ratio of 3.61. Cases 1, 2, and 3 had solidification-to-melting-time ratios of 1.19, 2.82, and 2.75, respectively.

Due to the high melting rate, the optimal case 4 configuration was further investigated for various tube materials. It was found that gold and copper have the better thermal performance as compared to aluminum and steel. However, steel was more appropriate for usage considering its low cost and ease of manufacturing. Moreover, the effects of fin thickness and fin length were studied for this optimal case 4. Heat transfer characteristics are found to improve with the increase in fin thickness and fin length due to the availability of a higher surface area for heat transfer.