Thermal performance evaluation of non-uniform fin array in a finned double-pipe latent heat storage system

Journal Pre-proof Thermal performance evaluation of non-uniform fin array in a finned double-pipe latent heat storage system Amin Shahsavar, Abbas Goodarzi, Hayder I. Mohammed, Alireza Shirneshan, Pouyan Talebizadehsardari PII:

S0360-5442(19)32495-8

DOI:

https://doi.org/10.1016/j.energy.2019.116800

Reference:

EGY 116800

To appear in:

Energy

Received Date: 2 August 2019 Revised Date:

11 December 2019

Accepted Date: 17 December 2019

Please cite this article as: Shahsavar A, Goodarzi A, Mohammed HI, Shirneshan A, Talebizadehsardari P, Thermal performance evaluation of non-uniform fin array in a finned double-pipe latent heat storage system, Energy (2020), doi: https://doi.org/10.1016/j.energy.2019.116800. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Thermal performance evaluation of non-uniform fin array in a finned double-pipe latent heat storage system Amin Shahsavar1, Abbas Goodarzi1, Hayder I. Mohammed2, Alireza Shirneshan3,4, Pouyan Talebizadehsardari5,6,* 1

Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, Iran. 2

3

Department of Physics, College of Education, University of Garmian, Kurdistan, Iraq.

Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

4

Modern Manufacturing Technologies Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran

5

Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam. 6

Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

Abstract This paper aims to evaluate the effects of fin arrangement in a vertical finned double-pipe latent heat storage system based on the locations, thickness and diameter of the fins in melting and solidification mechanisms. The PCM is placed in the outer tube while water is passed through the inner tube. The fins are circularly placed around the inner pipe in the PCM zone considering a constant fins number. The results show significant advantages of fins addition in reducing the melting time along with a lower advantage on the solidification time. For the uniform fin array compared with the non-fined case, the melting and solidification times reduces by 41.4% and 9.7%, respectively. For the best fin array compared with the uniform case, the melting time reduces by 23.9%. For the solidification process, the best case is the uniform distribution of the fins when the rate of heat recovery increases by 11.4% compared with the non-finned case. Furthermore, the melting time reduces by decreasing the thickness of the fins. The evaluation of fins’ diameter shows that after a *

Corresponding author E-mail address: [email protected] (Pouyan Talebizadehsardari)

1

certain diameter, increasing the diameter of the fins shows an adverse effect due to suppressing the natural convection effect.

Keywords: Latent heat storage; Finned double-pipe; Heat storage; Heat recovery; Phase Change Material.

Nomenclature
  



Mushy zone constant -1

Specific heat, J kg K



-1



Gravity vector, m s-2 -1



Thermal conductivity, W m K



PCM Mass, kg



Rate of heat storage, W



melting/solidification time, s





-1

Latent heat of fusion, J kg-1



Heat storage capacity, kJ



Pressure, Pa

Melting temperature, K Initial temperature, K Temperature, K Velocity vector, m/s

Greek symbols

 

Thermal expansion coefficient, K-1



Dynamic viscosity, kg m-1 s-1



Liquid fraction

∆

PCM Density, kg m-3 Latent heat (J kg-1)

End temperature, K

1. Introduction To modify heat transfer in heat exchangers, fins or extended metals have been utilized widely. Different fin configurations are employed to increase the heat transfer area and also to transfer the heat from the source heat to all the domain more rapidly by conduction which can be spread out by convection [1-6]. Thermal energy storage has been widely employed in different applications for different purposes including heating, cooling, energy consumption reduction, waste energy harvesting, and peak shaving [7-9]. Due to the high capacity of latent heat storage (LHS) compared with the sensible one by utilizing phase change materials (PCMs), there has been lots of attention to the LHS systems recently [9-11]. However, the low thermal conductivity of PCMs limits

2

their utilization in different applications [12, 13]. Geometry modification, fin addition, use of nanoparticles, metal foams, multiple-PCMs and encapsulation are among various methods to make the use of PCMs possible to have an acceptable rate of heat storage/recovery [14-23]. However, the use of metal foams and encapsulation cause a reduction in the PCM volume and the use of nanoparticles reduces the effective latent heat of fusion [24-26]. Geometry modification may be difficult in some applications and also there is a need to design a complex geometry which is expensive, difficult to fabricate and unendurable [27]. Among all methods, fin addition is desirable regarding the price, PCM volume reduction, PCM properties and system design. However, a proper design of the fins addition is required to modify the rate of charging/discharging process to an acceptable range [28]. Therefore, different studies in the literature have been worked on the application of fin addition in PCM based heat storage system [29-32]. Lamberg and Siren [33] developed an analytical model for temperature distribution in the finned LHS system during the melting process. Wang and Yang [34] studied the effect of a number of fins in a finned heat sink filled with PCM. They showed that the duration of melting time can be controlled based on the number of fins in the system. Abduljalil et al. [35] added longitudinal fins to a triple-tube LHS system experimentally to reduce the melting time. They examined different geometrical parameters of the fins numerically and showed a 34.7% reduction in the melting time for the best case. Andrew et al. [36] studied on longitudinal and circular fins addition to a multi-tube LHS system to examine the effects of geometry modification and fin addition simultaneously. Celador et al. [37] studied an innovative fined type heat storage unit for domestic application compared with a conventional hot water storage tank. They showed that by employing a compact LHS system with almost half of the volume of the hot water storage tank, the required thermal energy can be achieved. Wang et al. [38] simulated the phase-change process of the PCM in a sleeve-pipes compared with a fined tube. They found that by

3

considering the natural convection, the effective angle of the tubes was between 60o – 90o. Zhu et al. [39] studied the performance of a heat storage system with CH3COONa PCM using various porosity of the metal foam accompanied by metal fins. It was found that the combination accelerates the melting rate and improve PCM storage performance. Mahdi and Nsofor [40] investigated the effect of nanoparticles and fins addition on the solidification rate of the PCMs and found that using the fins improves the phase-change rate better than alone nanoparticles or the combination of both. Jia et al. [41] studied on the effect of fin addition in a sphere cold storage system and optimized the length of the fins. Using six fins reduces the melting time by more than 50%. One of the simplest heat exchangers used in different heating applications is a double-pipe heat exchanger and therefore, for the energy storage applications, there are a lot of studies working on the heat transfer enhancement with the aid of fins using the effects of so-called close contact melting/solidification [42, 43]. Kozak et al. [44] analysed experimentally and numerically on a vertical double-pipe energy storage unit with circumferential fins for the inner tube where the PCM was placed in the outer tube. They experimentally showed a lower melting time by a factor of 2.5 and presented a good agreement between the experimental and numerical data. In another study on a horizontal double-pipe heat exchanger with longitudinal plate fins [45], they showed that the melting time reduces by a factor of 2.5 experimentally. Tao et al. [46] investigated the performance of the energy storage system under the effect of natural convection. They found that adding the fins to the tube accelerates the melting rates and makes the melting more uniform on the boundary. Rozenfeld et al. [47] performed both experimental and numerical investigations on a novel vertical double-pipe LHS unit with helical fins. They showed a higher advantage of fin addition on the melting process compared with the solidification process. They showed that the melting time reduces by a factor of 3 using a close contact melting enhancement technique. Mahdi et al. [48] studied numerically

4

on the location of PCM in a horizontal double-pipe LHS system and showed that by placing the PCM in the outer tube, the melting time is shortened by 50% while the solidification time is longer by 43.4%. Based on the literature review, all previous publications have studied on a uniform fin distribution inside an LHS system and the study on a fined LHS system with a variable distance between the fins cannot be seen. The aim of this study is to analyse the locations of the fins as well as the diameter and thickness in a vertical non-uniform finned double-pipe LHS unit. The fins are placed as circular disks around the inner tube inside the PCM zone located in the outer pipe. The performance of the system is analysed on both melting and solidification processes regarding the melting/solidification time and rate of heat storage/recovery. The number of fins is considered constant. The significant advantages of non-uniform fins arrangement are shown compared with the uniform arrangement which is done for the first time in this study. This paper can shed a light on a novel design of LHS systems using fins as a heat transfer enhancement technique.

2. Mathematical modelling Standard mathematic model is applied as outlined in the following for a phase change problem including the continuity, momentum and energy equations as [2]:

  = 0 + ∇. 

 

(1)

 

 . !"

 = −! +  ! $

 " − %   − % " − & +





  " = !' !( − &) + ! 



(2)

(3)

The assumptions for the above equations can be found in the authors' previous work [49]. It should be noted that as shown by Vogel et al. [50], volume expansion plays a minor role in the performance of PCM based energy storage systems and Boussinesq approximation is 5

applicable. Therefore, for simplicity, the PCM without volume expansion is simulated in this study. The vector & is defined as [51]: & =


'1 − ($ 

+ + 0.001

(4)

It should be noted that the constant parameter in the Kozeny–Carman equation (
 ) for modelling natural convection-driven phase change depends on different geometrical and operational parameters of the problem and should be determined accurately comparing with the similar experimental study of the computational domain [52]. The high values of


result in slower melting rates and low values of
 result in unphysical predictions of the melt front development [53]. Based on the verification report done in this study and previous studies of the authors [49, 54-57],
 is considered 105 as also recommended in various

studies in the literature [58-60].  is the liquid fraction which is defined as [35]: 0 12  < 456789 > 1 12  > );8789 ∆ = = .  − 456789 = -);8789 − 456789 12 456789 <  < );8789 , < / -

(5)

∆ varies between 0 (PCM is solid) and (PCM is liquid). &) is defined as [61]: &) =

   " + ! 



(6)

The rate of heat storage/recovery rate is calculated as [49]:  ?@9567 A B + + @6;87 A BC   = =  

≈



'A ' −  ( + + A ' −  (( 

(7)

Note that for heat transfer and fluid flow in the inner tube, the source terms are eliminated from the momentum and energy equations.

3. Problem description 6

The studied LHS unit, as well as the boundary conditions, are shown in Fig. 1 for a nonfinned vertical double-pipe heat storage with a length of 25 cm and the inner and outer diameters of 4 cm and 8 cm, respectively. The finned case with uniform fin array is shown in Fig. 1-b as Case 1 in the axisymmetric condition. The axisymmetric model is employed for the double-pipe heat exchanger to model the conveying heat transfer fluid (HTF) through the inner tube in the opposite direction of the gravity and the PCM in the annular space [62, 63]. Different fins’ diameters of 5, 6 and 7 mm and various thicknesses of 1, 2 and 3 mm are also studied. The materials of the inner pipe with a thickness of 2 mm and the fins are considered copper. The HTF flow is considered laminar with the Reynolds number of 1800 in both melting and solidification. Note that the properties of water are considered at the inlet temperature of the water and therefore the inlet velocity is calculated based on the defined Reynolds number and HTF properties in both melting and solidification. The HTF inlet temperature is considered 285 K and 323 K, respectively, in the solidification and melting processes. The initial temperature of the PCM is considered 323 K and 285 K, respectively, in the solidification and melting processes. No-slip boundary condition is also considered for the walls. It should be noted that based on the considered length, five fins is selected for the fined case and so for the uniform finned case, the fins spacing is considered 4 cm. The number of fins is considered constant as a constraint in this study. In other words, the locations of the non-uniformed fins as well as the dimensions of the fins are determined considering a constant number of the fins.

7





a)

b)

Fig. 1. Schematic of the studied double-pipe LHS system.

RT-35 is employed as the PCM and its physical properties are listed in Table 1.

Table 1. RT 35 properties [27]. Property

Values

 (kg/m3)

L (kJ/kg)

 (kJ/kgK)

(W/mK)

 (Pas)

 (1/K)

Liquidus temperature (K)

Solidus temperature (K)

815

170

2.0

0.2

0.023

0.0006

309

302

Note that for all the cases including non-finned and finned cases with different lengths and thicknesses, the length of the heat storage unit is changed to have a constant PCM mass and volume for a reasonable comparison.

4. Fins evaluation process To find the best fin’s performance, first, the location of the fins are evaluated by the following procedure: 8

1- Changing the location of the first fin from the pipe’s bottom considering the constant distance of 40 mm between other fins as shown in Fig. 2. For this purpose, the location of the first fin is changed from y=40 mm to y=10 mm considering four different locations of 10, 20, 30 and 40 mm.



 Case 1

Case 2



 Case 2

Case 4

Fig. 2. Schematic of the fin arrangement evaluation process to find the best location for the first fin.

2- After finding the best location for the first fin from the bottom, change the location of the second fin considering constant distance for the other remaining fins. 3- Do the same procedure for the third, fourth and fifth fins. Then, for the best performance of the fin arrangement, the effects of fins’ diameter and thickness are studied. It should be noted that, in vertical double-pipe LHS systems when the HTF enter the unit from the bottom, after the initial moments when the heat is transferred by conduction through 9

the inner wall, natural convection is generated inside the melted PCM which helps the melting process [27]. Due to the direction of the gravity, PCM melts from the top and the melting process is terminated from the bottom [64]. Therefore, the use of fins is more critical at the bottom of the system to provide a uniform melting inside the system. Therefore, in this study, the selection process is started from the bottom of the unit.

5. Numerical modelling and validation ANSYS-FLUENT code is employed using the finite volume method employing SIMPLE algorithm. PRESTO pressure interpolation scheme is used due to buoyancy, while the quadratic upwind discretisation, QUICK, scheme is employed for the momentum and energy equations. The grid independence analysis is done for the uniform distribution of the fins. Fig. 3 illustrates the computational grid for the uniform fin distribution considering 22,360 cells. A higher number of grids were analysed results in less than 0.5% in the melting time and therefore this mesh was selected. Note that in the near fin region, a higher number of nodes is generated. The size of time-step was considered 0.2 s and no change was indicated employing a lower time-step size.

Fig. 3. The computational grid for uniform fin distribution.

10

For the code verification, a finned LHS system studied experimentally and numerically by Mat et al. [58] is employed to validate the code. Mat et al. [58] performed experimental and numerical investigations on a double-pipe LHS system using longitudinal plate fins. The fins were added in both inner and outer tubes in the annular space filled with RT-58 as the PCM. The temperature of the inner and outer tubes are kept constant as the boundary conditions of the problem. Figs. 4-a and 4-b illustrate the mean PCM temperature and liquid fraction in comparison with the results of Mat et al. [58], respectively, showing an excellent agreement.

Mean temperature (°C)

100 80 60 40

Present results [58] (experimental) Mat et al. [35] [58] (numerical) Mat et al. [35]

20 0 0

10

20

30

40

50

Time (min) a) 1

Liquid Fraction

0.8 0.6 0.4

Present results [58] Mat et al. [32]

0.2 0 0

10

20

30

40

50

Time (min) b) Fig. 4. Validation study presenting PCM (a) average temperature and (b) liquid fraction compared with Mat et al. [58].

11

6. Results and discussion In this section, first, the effect of uniform fin distribution are discussed compared with a nonfinned case in both melting and solidification processes. Then, the processes to find the locations, as well as the diameter and thickness of the fins, are discussed.

6.1. Effect of uniform fins addition Adding fins to the inner pipe results in heat transfer enhancement by two mechanisms inside the PCM. First, a higher heat transfer area is provided. Furthermore, due to the low thermal conductivity of PCM, the presence of fins causes transferring source heat to the middle part of the PCM domain results in a better distribution of heat source and therefore enhance the heat transfer mechanism. After the generation of liquid PCM, the effect of natural convection can be more pronounced due to the presence of the fins which will be discussed in details. Fig. 5 illustrates the contour plots of temperature distribution in 15 minutes time intervals for the finned double-pipe heat storage system compared with the non-finned one. In the HTF zone, the temperature is almost constant due to the short length of the pipe. The temperature of the fins is almost constant equal to the HTF temperature due to the small length and thickness of the fins. As shown, due to the presence of fins, the heat is penetrated inside the PCM tube which causes a higher temperature in the area around the fins. Therefore, through time, higher PCM temperature can be seen in the finned case compared with the non-finned case. The effect of natural convection is more pronounced after the initial time when the heat is just transferred by the pure conduction. Natural convection causes the circulation of melted PCM in the domain. The melted PCM moves toward the top wall of the pipe and therefore higher temperature of PCM can be seen in the top layers. For the finned case, in addition to the higher PCM temperature near the inner pipe, a higher temperature is seen near the fins.

12

Then, when natural convection starts to circulate the melted PCM, the heat is more penetrated from the fins to the PCM which can be shown perfectly at the time of 45 and 60 minutes. In other words, natural convection intensifies the effect of fins since the heat can spread more by natural convection from the middle layers of the PCM when the heat is transferred by the fins.

15 min

30 min

45 min

a)

60 min

75 min

90 min

Uniform finned double-pipe heat storage system

13

105 min

120 min

b) Non-finned double-pipe heat storage system Fig. 5. Temperature distribution in the (a) uniform finned and (b) non-finned double-pipe heat storage units at 15 minutes time intervals during the melting process.

Fig. 6 displays the contour plots of PCM liquid fraction for the uniform finned heat storage unit in comparison with the non-fin case. The effect of natural convection causes a higher liquid fraction at the top of the PCM zone compared with the bottom layer. The presence of fins results in a higher liquid fraction at an identical time for the finned case compared with the non-finned case. The fins can separate the PCM zone into several parts and therefore melts the PCM in a shorter time. Higher penetration of heat due to the presence of fins can be seen in the finned case results in converting more solid PCM to the liquid PCM around the fins. Then, natural convection helps the fins to spread the heat in the entire domain.

15 min

30 min

45 min

60 min

75 min

14

90 min

105 min

120 min

a)

Uniform finned double-pipe heat storage system

b) Non-finned double-pipe heat storage system Fig. 6. Liquid fraction distribution in the (a) uniform finned and (b) non-finned double-pipe heat storage units at 15 minutes time intervals during the melting process.

The variation of PCM mean temperature and liquid fraction versus time for the uniform finned unit are illustrated in Figs. 7-a and 7-b, respectively, in comparison with the nonfinned case. After a sharp increase in the temperature from the initial temperature (285 K) to the solidus temperature (302 K), the rate of temperature enhancement reduces due to the phase change effect until the liquidus temperature. After that, temperature increases toward 15

the temperature of HTF. In the finned case, higher mean temperature can be seen during the melting process. However, after the liquidus temperature, the difference between the mean temperature of the finned and non-finned cases enhances. The reason is that the huge amount of PCM melts in the finned case especially near the inner pipe and the fins and therefore natural convection helps to increase the rate of heat transfer and therefore, a higher temperature is achieved. As shown in Fig. 7-b, after almost 75 minutes, the rate of liquid fraction enhancement reduces. This can also be seen in the temperature profile (Fig. 7-a). The reason is that, as also shown in Fig. 6, at the time of 75 minutes, all the PCM placed above the fins almost melts. However, almost 10% of the PCM is still in the solid zone placed below the first fin. The reason in that the first fin at the bottom cannot be very helpful for the amount of PCM between the first fin and bottom of the unit and due to the effect of natural convection, the melted PCM moves toward the top layers of the system which is not effective for the PCM at the bottom. Therefore, the location of the fins should be optimized so that they can be more effective on the performance of the system. Furthermore, by using the finned unit, the melting time reduces by 43.5%. Note that the rate of heat storage based on Eq. (9) is 21.1 W and 12.37 W, respectively, for the uniform finned and non-finned cases which means that the use of uniform fins can enhance the heat transfer rate by almost 70.6 %.

16

325

Temperature (K)

320 315 310 305 300 295

No-fin

290

Uniform fins

285 280 0

50

100

150

200

250

Time (min) a)

1

225

0.6 137

Liquid fraction

0.8

0.4

No-fin

0.2

Uniform fins 0 0

50

100

150

200

250

Time (min) b) Fig. 7. The variation of PCM a) mean temperature and b) liquid fraction in terms of time for the finned double-pipe heat storage system compared with the non-finned case during the melting process

Fig. 8 illustrates the contour plots of temperature distribution during the solidification process for the finned double-pipe heat storage system in comparison with the non-finned case at 30 minutes time intervals. Similar to the melting process, the presence of fins results in a higher area for heat transfer from the PCM to the HTF. However, in the near area of fins, when the heat is transferred to the HTF, the PCM solidifies and therefore natural convention cannot help noticeably to improve the solidification process in contrast to the melting process.

17

Natural convection generates a clockwise circulation in the liquid PCM due to PCM lower temperature near the inner pipe and therefore the PCM solidifies more from the bottom rather the top of the system.

30 min

60 min

90 min

a)

120 min

150 min

180 min

210 min

240 min

Uniform finned double-pipe heat storage system

b) Non-finned double-pipe heat storage system Fig. 8. Temperature distribution in the (a) uniform finned and (b) non-finned double-pipe heat storage units at

18

30 minutes time intervals during the solidification process.

Fig. 9 displays the contour plots of liquid fraction for the uniform finned heat storage unit in comparison with the non-finned case at 30 minutes time interval. As shown, the area near the fins solidifies due to a higher conduction heat transfer in the presence of fins; however, the effect of natural conduction is almost similar for both cases of finned and non-finned units.

30 min

60 min

90 min

120 min

150 min

19

180 min

210 min

240 min

a)

Uniform finned double-pipe heat storage system

b) Non-finned double-pipe heat storage unit Fig. 9. Liquid fraction distribution in the a) uniform finned and b) non-finned double-pipe heat storage unit at 30 minutes time intervals during the solidification process

The variation of PCM mean temperature and liquid fraction versus time is illustrated in Figs. 10-a and 10-b, respectively, for the heat storage system with uniform fin distribution in comparison with the non-finned unit. Similar to the melting process, after a sharp reduction in the mean temperature, the temperature reduction rate reduces due to the phase change phenomena between the solidus and liquidus temperature and then the temperature reduces with a higher rate. According to the temperatures of initial, HTF, solidus and liquidus and also the properties of HTF, the solidification and melting times vary. As shown in Fig. 10-b, the solidification time for the non-finned unit is 40% higher than the melting time. Furthermore, the difference between the temperatures of finned and non-finned cases are smaller than that for the melting process results in a lower time-saving during the solidification process. The solidification time reduces by almost 10% using the uniform distribution of the fins compared with the non-finned case. Note that the rate of heat recovery based on Eq. (9) is 9.2 W and 10.1 W, respectively, for the uniform finned and non-finned

20

cases which means that the use of uniform fins can enhance the heat transfer rate by almost 10.5 %.

325

Temperature (K)

No-fin 315

Uniform fins

305

295

285 0

50

100

150

200

250

Time (min) a)

1

No-fin Uniform fins 317

0.6 286

Liquid fraction

0.8

0.4 0.2 0 0

50

100

150

200

250

300

350

Time (min) b) Fig. 10. The variation of PCM (a) mean temperature and (b) liquid fraction versus time for the finned doublepipe heat storage system compared with the non-finned case during the solidification process.

To better understand the difference between the melting and solidification mechanisms for the uniform finned array, Fig. 11 displays the velocity vectors at the time of 20 and 60 minutes in the PCM zone. Note that the scale of the arrows is similar in both mechanisms and

21

the black arrows in the solidification process are added to the picture to better show the direction of velocity vectors in the domain. A higher velocity magnitude can be seen in the melting mode than that for the solidification mode at the same time. Three vortices are generated around the fin in the melting mode due to the effect of natural convection and are grown through the time. However, in the solidification process, the general circulation of the liquid PCM can be seen and natural convection is not affected by the fins significantly.

20 min Solidification

60 min Melting

Solidification

22

Melting

Fig. 11. The velocity vectors of the PCM at different times for the melting process compared with the solidification process

6.2.Fins arrangement during the melting process In this section, the locations of the fins are obtained considering 6 mm as the diameter of the fins and 2 mm as the thickness of the fins as follows:

6.2.1. Finding the location of the first fin As mentioned, four different locations are evaluated to find the best location of the first fin considering the constant distance of 40 mm for the distance between the other fins. Table 2 presents the melting time, the percentage of time-saving compared with the uniform finned case and also the rate of heat storage. Note that based on Eq. (9), the total heat capacity of the unit is 173.8 kJ. By decreasing the distance of the first fin to the bottom of the unit, the melting time reduces and the heat storage rate increases. For the best case with a distance of 10 mm from the bottom, the melting time reduces by almost 24% and the rate of heat storage enhances by 31.5%.

Table 2. The melting time, time-saving percentage and heat storage rate for different locations of the first fin. The distance between the first fin and the bottom (mm)

Melting time

Time-saving compared with the uniform fin distribution

Heat storage rate

Case 1 (uniform)

40

137.33

-

21.10

Case 2 Case 3 Case 4

30 20 10

128.67 116.92 104.50

6.31 14.87 23.91

22.52 24.78 27.73

Case number

As mentioned, the problem of uniform distribution of the fins is that the amount of PCM between the first fin and the bottom wall cannot be affected considerably by the fin heat transfer enhancement due to natural convection and moving the warm liquid PCM toward the top wall. Therefore, it is expected that decreasing the distance between the first fin and the 23

bottom wall of the pipe can be effective. Fig. 12 illustrates the contour plots of PCM mean temperature (on the left) and the liquid fraction (on the right) for different non-uniform cases. Decreasing the distance of the first fin to the bottom of the pipe results in a faster spreading of heat in the domain by natural convection results in a lower melting time. It should be noted that by decreasing the distance between the first fin and the pipe’s bottom, the distance between the last fin and the top wall increases which causes an unmelted PCM until the time of 75 minutes for Case 3 and 4 compared with Case 2 and Case 1 (shown in Fig. 12-a); however, due to the natural convection effect and circulation of the melted PCM in the domain, this amount of PCM melts quickly during the time that the remaining PCM between the first fin and bottom wall is still going to melt. Therefore, there is a challenge between the distance of the first fin from the bottom and distance of the last fin from the top which should be evaluated in proper design of finned double-pipe LHS systems.

Liquid fraction

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 2

Time

24

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 3 Case 4 Fig. 12. PCM mean temperature and liquid fraction distributions for different non-uniform finned heat storage units at 15 minutes time intervals during the solidification process.

6.2.2. Finding the location of the second fin After locating the first fin which is placed at the distance of 10 mm from the bottom, the distance between the first and second fin is obtained considering a constant distance of 40 mm for the other fins. Table 3 presents the melting time and the rate of heat storage to examine the effect of the second fin location. The melting times for different positions of the second fin are close. As shown, the minimum melting time happens for a distance of 40 mm from the first fin. Note that Case 4-1 is the same as Case 4 in Table 2 and the second number is used to indicate the location of the second fin.

25

Table 3. The melting time and heat storage rate for the effect of the second fin location. Case number Case 4-1 Case 4-2 Case 4-3 Case 4-4

The distance between the first and second fins (mm) 40 30 20 10

Melting time (min)

Heat storage rate (W)

104.50 106.25 105.83 105.75

27.73 27.27 27.38 27.40

As mentioned, for the finned case compared with the non-fined case, the PCM at the bottom of the pipe melts with the same rate. By changing the location of the first fin, the problem of heat transfer enhancement to the bottom part of the pipe can be solved. Therefore, as shown in Table 3, the location of the second fin has negligible effects on the melting time. To better understand the behaviour of the second fin location, Fig. 13 illustrates the contour plots of PCM temperature and liquid fraction at 30 minutes time interval for Cases 4-2 to 4-3. The distribution of temperature and liquid fraction for different Cases are almost similar. As shown, especially obvious for Case 4-4, a part of PCM in the distance between the last fin and the top wall is still solid similar to the bottom of the pipe; however, the huge effect of natural convection causes that the PCM melts completely in the upper part of the pipe while a part of PCM is still not melted in the bottom. Liquid fraction

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-2

Time

26

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-3 Case 4-4 Fig. 13. PCM mean temperature and liquid fraction distribution for different non-uniform finned heat storage units at 30 minutes time intervals during the melting process.

6.2.3. Finding the location of the third fin After finding the best location for the first fin (10 cm from the bottom) and the distance between the first and second fins (40 cm), Table 4 presents the melting time and heat storage rate for the location of the third fin related to the second fin. Instead of Case 4-1-4 which the second and third fins are very close, the melting times for the other cases are almost similar. The best performance happens for the Case 4-1-2 which has the melting time of 104.17 minutes. Note that Case 4-1-1 is the same as Case 4-1 in Table 3 or Case 4 in Table 2.

Table 4. The melting time and heat storage rate for the effect of the third fin location.

27

Case number Case 4-1-1 Case 4-1-2 Case 4-1-3 Case 4-1-4

The distance between the second and third fins (mm) 40 30 20 10

Melting time (min)

Heat storage rate (W)

104.50 104.17 105.83 109.75

27.73 27.81 27.38 26.40

Fig. 14 illustrates the contour plots of PCM temperature and liquid fraction for cases 4-1-2 to 4-1-3. At the time of 105 minutes when all the PCM melts for the Case 4-1-2, a small amount of PCM still remains solid for the other cases.

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-2

Time

Liquid fraction

28

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-3 Case 4-1-4 Fig. 14. PCM mean temperature and liquid fraction distribution for different non-uniform finned heat storage units at 15 minutes time intervals during the solidification process.

6.2.4. Finding the location of the fourth fin For the fourth fin, Table 5 presents the melting time and heat storage rate for the effect of the fourth fin location related to the third fin. The results of the Cases 4-1-2-1 and 4-1-2-2 are almost the same where the distance between the third and fourth fins is 40 mm and 30 mm, respectively. However, decreasing to a lower distance results in increasing the melting time which is not desirable. Note that Case 4-1-2-1 is the same as Case 4-1-2 in Table 4.

Table 5. The melting time and heat storage rate for the effect of the fourth fin location. Case number

The distance between the second and third fins (mm)

29

Melting time (min)

Heat storage rate (W)

Case 4-1-2-1 Case 4-1-2-2 Case 4-1-2-3 Case 4-1-2-4

40 30 20 10

104.17 104.83 108.00 111.58

27.81 27.64 26.83 25.97

Fig. 15 illustrates the contour plots of PCM temperature and liquid fraction for cases 4-1-2-2 to 4-1-2-4. The advantages of case 4-1-2-2 are shown compared with the other cases on having a higher temperature in larger area results in a higher liquid fraction at the same time.

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-2-3

Case 4-1-2-2

Time

Liquid fraction

30

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-2-4 Fig. 15. PCM mean temperature and liquid fraction distribution for different non-uniform finned heat storage units at 30 minutes time intervals during the melting process.

6.2.5. Finding the location of the fifth fin For the last fin, Table 6 presents the melting time and heat storage rate for the effect of the last fin location related to the fourth fin. The first case in Table 6 which has the largest distance from the fourth fin has the minimum melting time and the maximum heat storage rate. Reducing the distance between the fourth and fifth fins from 40 to 10 mm results in increasing the melting time by 7.5% and reducing the heat storage rate by almost 2 W showing the critical role of fin location during the working procedure of latent heat storage systems. Note that Case 4-1-2-1-1 is the same as Case 4-1-2-1 in Table 5.

Table 6. The melting time and heat storage rate for the effect of the fifth fin location. Case number Case 4-1-2-1-1 Case 4-1-2-1-2 Case 4-1-2-1-3 Case 4-1-2-1-4

The distance between the second and third fins (mm) 40 30 20 10

31

Melting time (min)

Heat storage rate (W)

104.17 107.17 110.00 111.83

27.81 27.04 26.34 25.91

Fig. 16 illustrates the contour plots of PCM temperature and liquid fraction for cases 4-1-2-12 to 4-1-2-1-4. As shown, the advantages of case 4-1-2-1-2 are shown compared with the other cases on having a higher temperature and as a result a higher liquid fraction.

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-2-1-3

Case 4-1-2-1-2

Time

Liquid fraction

32

15 min

30 min

45 min

60 min

75 min

90 min

105 min

Case 4-1-2-1-4 Fig. 16. PCM mean temperature and liquid fraction distribution for different non-uniform finned heat storage units at 30 minutes time intervals during the melting process.

As a summary, Case 4-1-2-1-1 has the shortest melting time and is considered for further evaluation of fin’s length and thickness.

6.3.Effect of fins’ diameter Three different diameters of 5, 6 and 7 mm are studied for the best case achieved in the fins location study which is Case 4-1-2-1-1. The contour plots of liquid fraction and temperature for the fins’ diameter of 6 mm was shown in Fig. 14. Fig. 17 displays the contour plots of liquid fraction and temperature for the fins’ diameters of 5 mm at the top and 7 mm at the bottom. By increasing the diameter of fins and as a result higher penetration of solid fins with a higher temperature inside the PCM domain, an almost higher temperature is achieved for the PCM which results to a higher liquid fraction in the domain. However, by increasing the diameter of the fins more than a certain value, the effect of natural convection decreases due to the flow blockage made by the fins since the distance between the fins and the outer wall is small.

33

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

15 min

30 min

45 min

60 min

75 min

90 min

105 min

7 mm

5 mm

Time

Liquid fraction

Fig. 17. PCM mean temperature and liquid fraction distribution for different fins’ diameters for Case 4-1-2-11 at 15 minutes time intervals during the melting process.

Table 7 presents the melting time and heat storage rate for different diameters of the fins for the Case of 4-1-2-1-1. The difference between the melting time and heat storage rate for different length of the fins are small. The melting time reduces by 4.5% by increasing the fins’ diameter from 5 mm to 6mm. Furthermore, by increasing the diameter of the fins from 6 mm to 7 mm, no changes can be seen in the melting time since it starts blocking the fluid flow movement and suppressing the natural convection effect in addition to the positive effect of the fin addition. Therefore, the diameter of 6 mm is selected for the fins.

34

Table 7. The melting time and heat storage rate for different diameters of the fins. Case number Case 4-1-2-1-1 Case 4-1-2-1-1 Case 4-1-2-1-1

Fins’ diameter (mm) 5 6 7

Melting time (min) 109.08 104.17 104.17

Heat storage rate (W) 26.56 27.81 27.81

6.4.Effect of fin’s thickness Three different thickness of 1, 2 and 3 mm are studied for Case 4-1-2-1-1. The contour plots of liquid fraction and temperature for the case with the thickness of 2 mm was shown in Fig. 14. Fig. 18 displays the contour plots of liquid fraction and temperature for the fin’s thickness of 1 mm at the top and 3 mm at the bottom. By increasing the thickness of fins inside the PCM domain, more heat is transferred by conduction through the fins; however, on the other hand, to have a similar PCM mass in different cases, the volume of the heat exchanger should be increased for the case with the higher thickness of the fins. As shown, after 105 min, all the PCM melts for the case of 1 mm; however, for the case of 3 mm, there is still an amount of unmelted PCM at the bottom of the domain.

Liquid fraction

Temperature

15 min

30 min

45 min

60 min

75 min

90 min

105 min

1 mm

Time

35

15 min

30 min

45 min

60 min

75 min

90 min

105 min

3 mm Fig. 18. PCM mean temperature and liquid fraction distribution for different fins’ thicknesses for Case 4-1-21-1 at 15 minutes time intervals during the melting process.

Table 8 presents the melting time and heat storage rate for different thicknesses of the fins for the Case of 4-1-2-1-1. The difference between the melting time and heat storage rate for the thickness of 1 mm and 2 mm are very similar. The melting time for the unit with 1 mm fin’s thickness is 5.6 % less than that for the case of 3 mm fin’s thickness as a result of a 6% higher heat storage rate.

Table 8. The melting time and heat storage rate for different thicknesses of the fins. Case number Case 4-1-2-1-1 Case 4-1-2-1-1 Case 4-1-2-1-1

Fins’ thickness (mm) 1 2 3

Melting time (min) 103.75 104.17 110.0

Heat storage rate (W) 27.93 27.81 26.34

6.5. Solidification process As mentioned, the effect of fin addition to the inner pipe when the first fin starts from the bottom of the heat exchanger is more significant in the melting mechanism rather than solidification. For the solidification, the fin affects the heat transfer performance by increasing the heat transfer area and therefore varying the locations of the fins is expected to be negligible in the solidification time. Furthermore, as mentioned, due to natural convection, in addition to the near-wall region, the PCM solidifies from the bottom and the process is 36

terminated from the top layers of the pipe. Therefore, the process of finding the location of the fins for the melting process is different from the solidification which is expected to employ the first fin near the top wall. Therefore, since the same system is used in both melting and solidification process, this study focuses on the melting process due to the lower effect of fin addition in the solidification when the PCM is placed in the annular space in double-pipe LHS systems [48]. Table 9 presents the solidification time as well as heat recovery rate for Cases 1-4 compared with the non-fined unit. As discussed, in the solidification process, the case with a uniform distribution of the fins has the best performance among all the other configurations which reduces the solidification time by 9.7% and enhance the heat recovery rate by 11.4% compared with the non-finned case. As shown, for the first fin, increasing the distance from the bottom wall results in a lower solidification time. In general, as presented, for the solidification mechanism, the effect of different fins distribution has a lower effect compared with the melting process.

Table 9. The solidification time, time saving percentage compared with the non-finned case and heat recovery rate for different studied cases during the solidification. Case number Case 0-non-finned Case 1-uniform fins Case 2 Case 3 Case 4

Solidification (min) 317.33 286.42 293.00 297.50 300.75

time

Time-saving compared with the non-finned unit (%) 9.74 7.67 6.25 5.22

Heat recovery rate (W) 9.09 10.12 9.89 9.74 9.63

It should be noted that for Case 4-1-2-1-1 with the fin thickness of 1 mm and diameter of 6 mm which has the minimum melting time, the solidification time is 305.83 min which is almost 3.6% higher than that for the non-finned case. Therefore, for the applications required high rates of melting and low rates of solidification, Case 4-1-2-1-1 is the best case.

37

7. Conclusion The effects of adding fins non-uniformly to a double-pipe latent heat storage system were studied numerically in both melting and solidification processes. After presenting the advantages of a uniform array of the fins, the locations of the fins, as well as the diameter and thickness of the fins, were obtained considering a constant number of the fins. The proposed system was a vertical double-pipe with the PCM in the outer while the water is passed through the inner tube opposite to the gravity direction. The fins are placed around the inner pipe in the PCM side. The results indicated the significant advantages of fin addition in the melting mechanism. For the uniform fin distribution, the melting time decreases by 41.4% and the heat storage rate increases by 70.6%, for the fins’ diameter of 6 mm and the fins’ thickness of 2 mm. By evaluating the arrangement of the fins, compared with the uniform fins distribution, the melting time reduces by 23.9% and the heat storage rate increases by 31.4%, for the fins’ diameter of 6 mm and fins’ thickness of 2 mm. For the solidification mechanism, the effect of fins addition with the same configuration as the melting process is low since the PCM is placed in the outer tube and the flow direction is opposite to the gravity direction. For the best case which is the uniform fin distribution, the solidification time reduces by 9.7% and the heat recovery rate enhances by 11.4%, compared with the nonfinned case, for the fins’ diameter of 6 mm and fins’ thickness of 2 mm. Furthermore, the melting time reduces by decreasing the thickness of the fins. The evaluation of fins’ diameter shows that after a certain value, increasing the diameter of the fins shows an adverse effect due to suppressing the natural convection effect. The results of this paper indicate the important role of fins array in the fined latent heat storage systems.

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Highlights •

Evaluate the thermal performance of a vertical finned double-pipe LHS system.

•

Evaluate the fins array to have a higher heat storage/recovery rate.

•

41.4/9.7% reduction in melting/solidification time using a uniform fins array.

•

54% reduction in melting time in the best distribution of the fins.

•

Higher effect of fins array on natural convection for higher fins’ diameters.

Declaration of interests ☒ 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: