Impact of Tube Bundle Placement on the Thermal Charging of a Latent Heat Storage Unit

: The melting process of a multi-tube’s thermal energy storage system in the existence of free convection effects is a non-linear and important problem. The placement of heated tubes could change the convective thermal circulation. In the present study, the impact of the position of seven heat exchanger tubes was systematically investigated. The energy charging process was numerically studied utilizing liquid fraction and stored energy with exhaustive temperature outlines. The tubes of heat transfer ﬂuid were presumed in the unit with different locations. The unit’s heat transfer behavior was assessed by studying the liquid fraction graphs, streamlines, and isotherm contours. Each of the design factors was divided into four levels. To better investigate the design space for the accounted ﬁve variables and four levels, an L16 orthogonal table was considered. Changing the location of tubes could change the melting rate by 28%. The best melting rate was 94% after four hours of charging. It was found that the tubes with close distance could overheat each other and reduce the total heat transfer. The study of isotherms and streamlines showed the general circulation of natural convection ﬂows at the ﬁnal stage of melting was the most crucial factor in the melting of top regions of the unit and reduces the charging time. Thus, particular attention to the tubes’ placement should be made so that the phase change material could be quickly melted at both ends of a unit.


Introduction
The effective methods to correct the irregular global energy consumption are the major point to increase the save and production of power, which fill the gap between the power generation and the demands. Accordingly, thermal energy storage (TES) was considered an encouraging solution for comprehensive renewable systems [1]. Sensible, latent heat, and thermochemical heat are the available configurations for storing energy in TES units. The latent form using the phase change materials (PCMs) is the best conductive type due to the the case of a faster discharging rate with the HTF tube length of 0.51 m with a single air channel, the discharging rate dropped by 57%, the exit temperature improved by 38%, and the temperature variance between both ends of the discharging rate improved by 3 times compared with the unit with the straight air tube.
The literature review shows that the natural convection regime could be an important mechanism in the melting of PCM at the final charging stages. The location of tubes (heat sources) could notably impact the circulation flows and the heat transfer and studying the optimization of multi-tubes locations, giving wider knowledge in this field. Thus, the present study, for the first time, aimed to systematically optimize the performance of a shell-and-tube LHTES to reduce the charging time and enhance the heat transfer rate from the HTF to the PCM. The distance of the heating tubes from the bottom wall was optimized to gain the lowest melting time. The parameters of the liquid fraction (LF), and the time of the charging process, and maps of phase change were investigated for different locations of the seven heat transfer tubes using the Taguchi method for specifying the locations. The goal was to shift the HTF tubes' position across the PCM domain to reach the best position and take full advantage of natural convection flows to minimize the charging time.

Model Description
The latent heat thermal energy storage (LHTES) investigated in the current work is a shell-and-tube thermal exchanger incorporating several tubes in a rectangular PCM container. Figure 1 illustrates the diagram of the proposed unit, including seven tubes that could be extended by repeating the pattern in the horizontal direction. This assumption can be taken into account considering a long width for the heat exchanger. Due to the repetition of the inner tubes in both sides of the heat exchanger, the symmetrical boundary condition was applied for the left and right boundaries, denoted by dash lines. The heat exchanger's upper and lower walls were insulated to eliminate the impact of environmental conditions with no-slip boundary conditions. The tubes' wall temperature and original temperature of the domain were presumed to be 50 and 15 • C, respectively. The gravity was considered in the vertical direction toward the bottom of the heat exchanger.  The nominal diameter of the tube (D) was 12.7 mm (0.5 in) with an outer diameter of 15.875 mm. The height of the shell was assumed 11D (314.325 mm), and the width of each repeated section was considered 3 √ 3D/2 (74.24 mm). Four variables of HL0, HL1, HL2, and HL3 were defined independently, locating the tubes on the left-hand side of the domain, which was going to be changed to produce a maximum melting rate. The optimization process is discussed comprehensively in the results and discussion. The locations of the tubes in the right-hand side of the domain were determined based on the tubes in the opposite side, as illustrated in the figure. The thermophysical conditions of paraffin RT35 (type of PCM) is presented in Table 1.

Governing Equations
The governing equation in this case for laminar, incompressible, transient, and Newtonian fluid flow is provided in details (as follows) built on the enthalpy-porosity technique defined by Brent et al. [29]: Mass conservation: ∂u ∂x where ρ is the mass density and → V is the velocity vector. x-momentum: y-momentum: where the x-velocity (u) and y-velocity (v) are the velocity components of vector → V, and P is the pressure. In the above equations, the dynamic viscosity (µ), volume expansion coefficient (β), gravity acceleration (g), and reference temperature (T ref ) are used. The symbol λ shows the liquid volume fraction (melted fraction of PCM in an element), and A m is the mushy constant, which is selected as 10 5 following [30]. Energy: where the heat capacity (C p ), thermal conductivity (k), and latent heat of phase change (L f ), and temperature field (T) are used to define the equation. The following assumptions were employed for driving the governing equations: • Employing Boussinesq approximation for the natural convection effect due to lowtemperature variation in the domain [29]; • Neglecting viscous dissipation because of the nonappearance of high velocities [29].
The melt volume fraction (λ) was defined using the fusion temperature as follows [31]: where the total enthalpy of an element is the combination of latent heat (∆H) and sensitive enthalpy (h) as H = ∆H + h. Thus, the total enthalpy could be computed by integration of H over the computational domain.

Numerical Method and Validation
The FLUENT package was employed to simulate the proposed system using computational fluid dynamics (CFD). The SIMPLE algorithm was applied to solve the governing equations numerically using an iterative algorithm after setting the boundary and initial conditions. For natural-convective flow in this problem, the pressure staggering option (PRESTO) was employed to calculate the pressure on the center of control volume accurately using staggered grids rather than interpolating the pressure on the faces of a control volume. The quadratic upstream interpolation for convective kinematics (QUICK) scheme as a high-order differencing scheme was also utilized for solving the convection-diffusion equation as a more accurate interpolation compared with second-order schemes. The convergence values (an acceptable differences of the parameter values between two iterations) were set to 10 −4 for continuity and 10 −6 for the momentum and energy formulae.
The phase change simulations could be impacted by grid size and time-steps significantly. This was since the phase change appeared in a limited area in a space that could be captured by just a few elements. Thus, a coarse grid may not have been capable of capturing the phase change interface and caused numerical errors and solver instability. The time-step was another crucial factor since the phase change was controlled by a temperature-dependent source term in the heat formula. Thus, a solver with large time-steps may fail to capture the temperature and phase-field variations (liquid fraction) adequately. Thus, here a grid study was performed following a time-step investigation. For the grid study, a case with HL 0 /D = 2.4, HL 1 /D = 1.8, HL 2 /D = 3.0, HL 3 /D = 2.4 was simulated with four different grid sizes (28,000, 50,500, 75,000, and 100,000 cells) and a time-step of 0.2 s. Figure 2 shows the impact of the mesh size on the liquid fraction. The case of 75,000 cells has not been plotted here to avoid congestion. The results show that a fine grid made of 50,500 cells could provide accurate results, and increasing the cell numbers by two-fold could not add a notable accuracy to the computations. The phase change simulations could be impacted by grid size and time-steps significantly. This was since the phase change appeared in a limited area in a space that could be captured by just a few elements. Thus, a coarse grid may not have been capable of capturing the phase change interface and caused numerical errors and solver instability. The time-step was another crucial factor since the phase change was controlled by a temperature-dependent source term in the heat formula. Thus, a solver with large time-steps may fail to capture the temperature and phase-field variations (liquid fraction) adequately. Thus, here a grid study was performed following a time-step investigation. For the grid study, a case with HL0/D=2.4, HL1/D=1.8, HL2/D=3.0, HL3/D=2.4 was simulated with four different grid sizes (28,000, 50,500, 75,000, and 100,000 cells) and a time-step of 0.2 s. Figure 2 shows the impact of the mesh size on the liquid fraction. The case of 75,000 cells has not been plotted here to avoid congestion. The results show that a fine grid made of 50,500 cells could provide accurate results, and increasing the cell numbers by two-fold could not add a notable accuracy to the computations.
Various time-steps (0.1, 0.2, and 0.4 s) were tested for the selected grid with 50,500 cells. It was found that the computed data for a time-step of 0.1 and 0.2 s were almost identical. Thus, the time-step of 0.2 s was selected to provide sufficient accuracy with a feasible computational cost. Figure 3 displays the computational grid after grid-independence analysis.   Various time-steps (0.1, 0.2, and 0.4 s) were tested for the selected grid with 50,500 cells. It was found that the computed data for a time-step of 0.1 and 0.2 s were almost identical. Thus, the time-step of 0.2 s was selected to provide sufficient accuracy with a feasible computational cost. Figure 3 displays the computational grid after grid-independence analysis. The investigation of Mat et al. [32] was regenerated to verify the employed numerical simulation in the present study. Mat et al. [32] examined a finned type dual-pipe LHS numerically and experimentally using organic PCM. Eight fins were implemented in the longitude direction connected into both internal and external tubes to improve the heat transfer exchanged between the working fluid in the inner tube and PCM in the outer tube. RT82 was implemented to store energy during the phase change. Fifteen thermocouples were used to measure the temperature inside the PCM domain. The comparison is illustrated in Figure 4 between the numerical and practical results of the mean temperature, and numerical data of liquid fraction belongs to the study of Mat et al. with the present numerical study, which showed an excellent agreement.
As another comparison, we examined the melting interface for melting PCM in a rectangular enclosure with a height of 6.35 and widths of 8.89 cm. The PCM was at an initial super-cold temperature of 28.3 °C, while the phase change temperature was 29.8 °C. All walls were insulated except the left wall, which was exposed to an isothermal temperature of 38 °C. The melting interface was measured during the melting process [33]. The model was solved in the present study. After 2, 6, and 10 min, the melting interfaces are plotted in Figure 5 and compared with the experimental work of Gau and Viskanta [33] and the literature of [34][35][36][37]. The differences between some of the literature The investigation of Mat et al. [32] was regenerated to verify the employed numerical simulation in the present study. Mat et al. [32] examined a finned type dual-pipe LHS numerically and experimentally using organic PCM. Eight fins were implemented in the longitude direction connected into both internal and external tubes to improve the heat transfer exchanged between the working fluid in the inner tube and PCM in the outer tube. RT82 was implemented to store energy during the phase change. Fifteen thermocouples were used to measure the temperature inside the PCM domain. The comparison is illustrated in Figure 4 between the numerical and practical results of the mean temperature, and numerical data of liquid fraction belongs to the study of Mat et al. with the present numerical study, which showed an excellent agreement.    [33][34][35][36][37]; L is the enclosure's characteristic length, L = 6.35 cm.

Results and Discussion
The purpose of the current work was to adjust the placement of thermal tubes in an LHTES unit. The present model consisted of seven tubes whose locations in the unit were controlled by four geometrical variables. The details of these variables were summarized in Table 2. As seen in this table, each design parameter was divided into four levels. In order to better explore the design space for these five variables and four levels, an L16 orthogonal table was considered. The orthogonal table ensured an orthogonal combination of variables with maximum coverages of design space. The L16 table with 16 design cases is depicted in Table 2. The simulations were executed for the cases of this table. The MVF (liquid fraction in the LHTES unit after four hours of heating) was computed and brought in Table 3. The base case of the L16 table was case 4 with 94% melting, while the worst case was case 1 with a 68% melting fraction. The difference between these two cases was 28%. Thus, just changing the tubes' location without changing any other design As another comparison, we examined the melting interface for melting PCM in a rectangular enclosure with a height of 6.35 and widths of 8.89 cm. The PCM was at an initial super-cold temperature of 28.3 • C, while the phase change temperature was 29.8 • C. All walls were insulated except the left wall, which was exposed to an isothermal temperature of 38 • C. The melting interface was measured during the melting process [33]. The model was solved in the present study. After 2, 6, and 10 min, the melting interfaces are plotted in Figure 5 and compared with the experimental work of Gau and Viskanta [33] and the literature of [34][35][36][37]. The differences between some of the literature simulations and the experimental data were significant, which could be due to the selection of a large phase change range or not adequately large mushy constant (A m ). As seen, the results of the present study were in good agreement with the experimental observations.    [33][34][35][36][37]; L is the enclosure's characteristic length, L = 6.35 cm.

Results and Discussion
The purpose of the current work was to adjust the placement of thermal tubes in an LHTES unit. The present model consisted of seven tubes whose locations in the unit were controlled by four geometrical variables. The details of these variables were summarized in Table 2. As seen in this table, each design parameter was divided into four levels. In order to better explore the design space for these five variables and four levels, an L16 orthogonal

Results and Discussion
The purpose of the current work was to adjust the placement of thermal tubes in an LHTES unit. The present model consisted of seven tubes whose locations in the unit were controlled by four geometrical variables. The details of these variables were summarized in Table 2. As seen in this table, each design parameter was divided into four levels. In  space for these five variables and four levels, an L16  orthogonal table was considered. The orthogonal table ensured an orthogonal combination  of variables with maximum coverages of design space. The L16 table with 16 design cases  is depicted in Table 2. The simulations were executed for the cases of this table. The MVF (liquid fraction in the LHTES unit after four hours of heating) was computed and brought in Table 3. The base case of the L16 table was case 4 with 94% melting, while the worst case was case 1 with a 68% melting fraction. The difference between these two cases was 28%. Thus, just changing the tubes' location without changing any other design variable could add a notable enhancement to the charging performance of an LHTES unit.  Figure 6 shows the liquid fraction during the heating for six selected cases of the L16 table. Case 1 and case 4 were the worst and best cases, respectively. The other cases led to the liquid fraction (LF) in between these two cases. In the first four cases, the first tube was at the bottom of the unit (HL 0 /D = 1.2), while for cases 11 and 14, the first tube was at HL 0 /D = 2.4 and HL 0 /D = 3.0, respectively. Figure 6 depicted that all the cases had a semi-linear similar behavior at the beginning of melting, and then the differences became significant. The streamlines and melting maps, as well as isotherms, were plotted  Table 3 Figure 7 shows small melted regions around the HTF tubes just one hour start of charging. When the tubes were close to each other (cases 1, 11, and 14), th region was thick but short. When the tubes were well spread in the unit (cases the molten region was narrow but tall. All cases showed comparatively weak a circulations around tubes. Since the molten region was quite small, the natura tion flows were confined by the solid PCM, and hence, the natural convection flo weak. When tubes were close to each other, their circulation flows merged to flow. The tubes with distance, 3D, produced almost independent circulation flow Figure 8a displays the molten regions after two hours. The molten reg grown, and the melted regions between the first and second column of tubes had and merged at this stage. Case 1 (1.2D distance between tubes) produced two circulation flows, each for a column of tubes. Cases 2 and 3 with 1.8D and 2.8D d between tubes produced a dominant circulation for the left column of tubes showed the same pattern as cases 2 and 3, but it pushed the melting interface to unit's top solid regions.
The streamlines for cases 11 and 14 followed the same pattern as that of ca since the first tube was placed high for cases 11 and 14, there was a notable solid the bottom. The isotherms showed a warm region at the center of the unit betw umns and cold regions at the top and bottom. The liquid around the tubes wa and moved upward. A hot plume of warm liquid moved toward the top regio unit, absorbing the heat of PCM. From Figure 6, the amount of molten PCM highest to lowest could be ranked as cases 4, 3, 14, 2, 11, and 1.  Table 3.  From Figure 9, it can be seen that the solid PCM at the bottom for cases 1-3 had been melted adequately. Thus, regardless of the top tubes' placement, the case with the first tube at the bottom had melted the bottom solid PCM. The cases with the first tube at high locations had not melted the PCM at the bottom.
After about 1080 s (three hours) of melting, a short slow-down in the trend of the liquid fraction could be seen. The reason was attributed to the shape of the melting interface, depicted in the streamlines and melting fields. The comparison between the shape of melting interfaces between Figure 8a (two hours of melting) and Figure 9a (three hours of melting) showed that there was a wedge-shaped solid PCM at the early time. This solid PCM region was placed above the first heated tube, and it could be effectively heated by the tube. Thus, this region could be easily melted and boost the overall melting rate. However, at about the three-hour margin, this region was fully melted to a level below/equal to the location of the first heating tube. Thus, the melting rate slowed down temporarily until the convection flow became stronger at the top region and again accelerated the melting rate.
(a) (b) Figure 9. The melting behavior of the unit after three hours: (a) The streamlines and melting field; (b) the isotherms. Figure 10 shows almost similar results to Figure 9, but in this case, tubes competed to push the melting interface toward the top regions. After one hour (compared to Figure  9), the cases with the first tube not at the lowest location (cases 11 and 14) could not show any significant progress in charging of the PCM at the base of the unit. After four hours, all of the tubes were surrounded by liquid, and there was sufficient space for the flow circulation in the liquid region. The streamlines could mainly dictate the difference be- Figure 9. The melting behavior of the unit after three hours: (a) The streamlines and melting field; (b) the isotherms.
perature gradients were small, and thus, the tubes could not effectively contribute to heat transfer. This was since the strong general convection flows had been formed above the tube section, and they could partially reach the tubes. Almost the same behavior could be seen in case 2. The circulation for the right column of tubes could not reach the top cold regions. However, since the top tubes' distance was slightly larger than in case 1, the temperature gradients around the tubes were slightly larger (a thin red region), and the heat transfer was also slightly stronger.
Cases 3 and 4 not only reached the top of the enclosure but also melted the PCM at the bottom. In case 4, two strong and symmetric general circulation flows could be seen that did not weaken each other. Case 3 showed almost the same behavior as case 4; however, the convection flows were not as symmetric as case 4, and thus, the melting rate for case 3 was lower than case 4.

Conclusions
A geometrical design of an LHTES unit heated by HTF tubes was systematically investigated. The model of LHTES consisted of seven HTF tubes where five variables could control the location of these tubes. Four levels for each variable were considered, and the design space was explored using an L16 orthogonal table. The impact of tubes' location on the charging heat transfer actions of the LHTES unit was addressed by using the time history of the liquid fraction, melting maps, streamlines, and isotherms. The findings indicated that the solid PCM at the bottom of a unit was hard to melt, and thus, the first tube should be located at the base of the unit. However, the heated plume of the first tube could help with the melting of PCM around the top tubes. Thus, the top tubes should be placed with their maximum design distance.
Moreover, tubes with close distances could overheat each other and reduce the total heat transfer. The study of isotherms and streamlines showed the general circulation of natural convection flows at the final stage of melting was the most crucial factor in the melting of top regions of the unit and reducing the charging time. A comparison between the best and worst design revealed that the variation of tubes' location changed the charging rate by 28%. It should be noted that such an improvement was almost at no cost since no extra material should be used in the construction of an LHTES unit.
In the current work, the impact of distance between the tube columns was not investigated. The distance between the columns could be another essential geometrical parameter since it effectively could control the merging of initial molten regions and the interaction of general circulation flows. The investigation of the distance between the columns of tubes could be subject to future investigations.   At the beginning of melting, the PCM was super cold and solid. Thus, the heat slowly diffused around the heated tubes. Thus, each tube behaved like a tube embedded in a semi-infinity domain. After a while, a large area around a tube felt the heat, and the temperature fields for each tube collided with another tube. During this time, the tubes starting the temperature of PCM reached the fusion temperature, and melting began.
Attention to Figure 6 shows that there was a slight difference between LF for all cases. However, cases 3 and 4 with narrow and tall melted regions led to a slightly higher liquid fraction (LF) compared to other cases. The isotherms showed a completely cold region at the top of the unit. When the first tube was at the bottom (Cases 1-4), the cold region at the bottom of the unit had been melted. However, for cases where the first tube had been shifted upward, there was a cold unmelted region at the bottom. Figure 7 shows small melted regions around the HTF tubes just one hour after the start of charging. When the tubes were close to each other (cases 1, 11, and 14), the molten region was thick but short. When the tubes were well spread in the unit (cases 3 and 4), the molten region was narrow but tall. All cases showed comparatively weak and small circulations around tubes. Since the molten region was quite small, the natural convection flows were confined by the solid PCM, and hence, the natural convection flows were weak. When tubes were close to each other, their circulation flows merged to a larger flow. The tubes with distance, 3D, produced almost independent circulation flows. Figure 8a displays the molten regions after two hours. The molten region had grown, and the melted regions between the first and second column of tubes had reached and merged at this stage. Case 1 (1.2D distance between tubes) produced two general circulation flows, each for a column of tubes. Cases 2 and 3 with 1.8D and 2.8D distances between tubes produced a dominant circulation for the left column of tubes. Case 4 showed the same pattern as cases 2 and 3, but it pushed the melting interface toward the unit's top solid regions.
The streamlines for cases 11 and 14 followed the same pattern as that of case 1, but since the first tube was placed high for cases 11 and 14, there was a notable solid region at the bottom. The isotherms showed a warm region at the center of the unit between columns and cold regions at the top and bottom. The liquid around the tubes was hottest and moved upward. A hot plume of warm liquid moved toward the top regions of the unit, absorbing the heat of PCM. From Figure 6, the amount of molten PCM from the highest to lowest could be ranked as cases 4, 3, 14, 2, 11, and 1.
From Figure 9, it can be seen that the solid PCM at the bottom for cases 1-3 had been melted adequately. Thus, regardless of the top tubes' placement, the case with the first tube at the bottom had melted the bottom solid PCM. The cases with the first tube at high locations had not melted the PCM at the bottom.
After about 1080 s (three hours) of melting, a short slow-down in the trend of the liquid fraction could be seen. The reason was attributed to the shape of the melting interface, depicted in the streamlines and melting fields. The comparison between the shape of melting interfaces between Figure 8a (two hours of melting) and Figure 9a (three hours of melting) showed that there was a wedge-shaped solid PCM at the early time. This solid PCM region was placed above the first heated tube, and it could be effectively heated by the tube. Thus, this region could be easily melted and boost the overall melting rate. However, at about the three-hour margin, this region was fully melted to a level below/equal to the location of the first heating tube. Thus, the melting rate slowed down temporarily until the convection flow became stronger at the top region and again accelerated the melting rate. Figure 10 shows almost similar results to Figure 9, but in this case, tubes competed to push the melting interface toward the top regions. After one hour (compared to Figure 9), the cases with the first tube not at the lowest location (cases 11 and 14) could not show any significant progress in charging of the PCM at the base of the unit. After four hours, all of the tubes were surrounded by liquid, and there was sufficient space for the flow circulation in the liquid region. The streamlines could mainly dictate the difference between the rate of heat transfer. Tubes for case 1 were quite close to each other, and they produced two general circulations in the enclosure. However, the circulation flow was weakened by the thermal interaction between the right and left columns. There was a thick red region (Figure 10b) around the tubes for this case, which shows that the temperature gradients were small, and thus, the tubes could not effectively contribute to heat transfer. This was since the strong general convection flows had been formed above the tube section, and they could partially reach the tubes. Almost the same behavior could be seen in case 2. The circulation for the right column of tubes could not reach the top cold regions. However, since the top tubes' distance was slightly larger than in case 1, the temperature gradients around the tubes were slightly larger (a thin red region), and the heat transfer was also slightly stronger.
Cases 3 and 4 not only reached the top of the enclosure but also melted the PCM at the bottom. In case 4, two strong and symmetric general circulation flows could be seen that did not weaken each other. Case 3 showed almost the same behavior as case 4; however, the convection flows were not as symmetric as case 4, and thus, the melting rate for case 3 was lower than case 4.

Conclusions
A geometrical design of an LHTES unit heated by HTF tubes was systematically investigated. The model of LHTES consisted of seven HTF tubes where five variables could control the location of these tubes. Four levels for each variable were considered, and the design space was explored using an L16 orthogonal table. The impact of tubes' location on the charging heat transfer actions of the LHTES unit was addressed by using the time history of the liquid fraction, melting maps, streamlines, and isotherms. The findings indicated that the solid PCM at the bottom of a unit was hard to melt, and thus, the first tube should be located at the base of the unit. However, the heated plume of the first tube Moreover, tubes with close distances could overheat each other and reduce the total heat transfer. The study of isotherms and streamlines showed the general circulation of natural convection flows at the final stage of melting was the most crucial factor in the melting of top regions of the unit and reducing the charging time. A comparison between the best and worst design revealed that the variation of tubes' location changed the charging rate by 28%. It should be noted that such an improvement was almost at no cost since no extra material should be used in the construction of an LHTES unit.
In the current work, the impact of distance between the tube columns was not investigated. The distance between the columns could be another essential geometrical parameter since it effectively could control the merging of initial molten regions and the interaction of general circulation flows. The investigation of the distance between the columns of tubes could be subject to future investigations.