Thermal performance optimization and evaluation of a radial finned shell-and-tube latent heat thermal energy storage unit

Journal Pre-proofs Thermal performance optimization and evaluation of a radial finned shelland-tube latent heat thermal energy storage unit Liang Pu, Shengqi Zhang, Lingling Xu, Yanzhong Li PII: DOI: Reference:

S1359-4311(19)34976-2 https://doi.org/10.1016/j.applthermaleng.2019.114753 ATE 114753

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Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

19 July 2019 8 November 2019 30 November 2019

Please cite this article as: L. Pu, S. Zhang, L. Xu, Y. Li, Thermal performance optimization and evaluation of a radial finned shell-and-tube latent heat thermal energy storage unit, Applied Thermal Engineering (2019), doi: https://doi.org/10.1016/j.applthermaleng.2019.114753

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Thermal performance optimization and evaluation of a radial finned shell-and-tube latent heat thermal energy storage unit Liang Pu*, Shengqi Zhang, Lingling Xu ,Yanzhong Li * Corresponding author: [email protected] Department of Refrigeration and Cryogenic Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, P. R. China. 710049 Abstract: Add of fin is a simple and effective way to enhance the thermal behavior of latent heat thermal energy storage (LHTES). The target of this research is finding the most appropriate radial fins arrangement to accelerate phase change material (PCM) melting process in vertical LHTES unit. The effect of adding aluminum fins which possess the much higher thermal conductivity compared to PCM on melting process was investigated by numerical simulation firstly. The full melting time of radial finned LHTES unit decreases 44.0% compared with LHTES unit without fin. Then, the effects of fin height and pitch on thermal performance are simultaneously considered. The dimensionless fin height of 0.642 is recommended in this study. The thermal performances of vertical finned LHTES with four different fin arrangements (lower fins, upper fins, middle fins and arithmetic fins) were explored. Numerical results show that the complete melting time could be reduced 49.9% by arithmetic fins. The root mean square (RMS) of liquid fraction is proposed to describe the uniformity of the melting process of PCM. In this study, the root mean square of the LHTES unit using arithmetic fins is the lowest in the four fins arrangements, following by lower fins, middle fins and upper fins. For maximizing thermal performance, adding arithmetic fins to LHTES is recommended. Keywords: latent heat thermal energy storage; root mean square; thermal performance; 1

phase change material

NOMENCLATURE C

-3 -1 mushy zone constant, kg  m  s

Cp

-1 -1 specific heat, J  kg  K

E

root mean square

g

Gravitational acceleration, m  s -2

h

average heat transfer coefficient, W  m 2  K 1

H

total length of the tube, mm

l

dimensionless fin height

L

fin height, mm

N

fin number

p

fin pitch, mm

r

radius of the inner surface of steel tube, mm

R

radius of the inner surface of Perspex tube, mm

t

fin thickness, mm

T

temperature, K

2

Tm

PCM melting temperature, K

u

velocity in x direction, m  s -1

v

velocity in y direction, m  s -1

GREEK SYMBOLS 

thermal expansion coefficient



liquid fraction



thickness of the tube， Pa  s



thermal conductivity, W  m 1  K 1



3 density, kg  m

Subscripts

1.

full

complete melting

i

initial state

in

inlet

k

the number of small domain

l

liquid phase

ref

reference

s

solid phase

Introduction

The increasing energy demand and deteriorating environment are two major problems 3

all over the world. Developing new energy sources such as wind energy, solar energy to cope with energy crisis, reducing environmental pollution, and promoting sustainable development of energy is extremely necessary. However, new energy sources such as solar energy, wind energy are intermittency and affected by the weather. Thermal energy storage (TES) system can efficiently solve energy mismatching in space and time. There are three types of TES: sensible heat thermal energy storage (SHTES), thermochemical heat thermal energy storage (THTES) and latent heat thermal energy storage (LHTES). LHTES with the advantages of low temperature fluctuation and high energy density is regarded as the most promising technique[1]. The low thermal conductivity of energy storage materials (such as paraffin) limits the further development of LHTES. In order to enhance the thermal performance of LHTES, much work has been carried out on LHTES heat transfer enhancement techniques. According to the heat transfer equation, the heat transfer performance of LHTES can be enhanced from three aspects: using porous media[2, 3] and nanomaterial additive[4] to improve the conductivity of PCM, adding fins[5, 6] and encapsulated PCM[7, 8] to extend heat transfer surface, setting multiple PCMs[9] to improve melting process uniformity. Among the three kinds of methods, adding fins is a simple and effective way to improve the thermal behavior of LHTES [10, 11]. Some studies were performed to research the effects of fin parameters on thermal performance of LHTES [12-14]. Erek et al.[15] researched the melting front evolution of a radially finned LHTES which is full of PCM in the annular shell space. The effects of fin parameters and flow parameters on solidification behavior are performed and 4

compare with experimental results. Parsazadeh and Duan[14] studied the effects of fin angle and nanoparticles on liquid-solid interface in an annular finned LHTES unit. They found a strong interaction between fin angle and nanoparticles. With fin angle variation from -45° to 45° and fin pitch increasing from 45 mm to 65 mm, fin angle of 35° contributes to the shortest melting time. Jmal and Baccar [16]numerically researched the heat transfer in a coaxial tubes thermal storage unit with external and internal radial fins. The effect of natural convection on the melting front and the solidification front evolution had been investigated. They also discussed the influence of fin number on performance enhancement. Yang et al.[17] numerically studied the melting process of PCM in a vertical LHTES with radial fins. Fin structure parameters are chosen as optimization parameters under the premise of the fin volume ratio of 2%. The results show that the complete melting time has a maximum reduction of 65% by inserting radial fins into PCM, they recommended an optimal group fin parameter in their study. Zhao and Tan [18]proposed combining a new-style of TES unit to conventional airconditioner to enhance cooling performance. Iemn and Mounir[19] considered the discharging process in a rectangular LHTES equipped with fins. The effect of fins number on performance enhancement of the heat exchanger and the temporal evolution of the solidification front has been investigated. Ismail and Lino[20] reported the effects of turbulence promoters and radial fins on the thermal performance enhancement of a LHTES. The tests were carried out on tubes without fins, tubes with fins and finned tubes equipped with turbulence promoter. They found that using turbulence promoter results in less time for full solidification. Besides radial fins, several other types of fin 5

pattern have been adopted to improve thermal performance of LHS, such as longitude fins[21-23], helical fins[24], triplex fins[25]. Moreover, Tao [26] and Kazemi[27] studied installing all fins at lower half and upper half respectively. However, the above studies mainly focus on single parameter optimization, such as increasing fin number and height, decreasing fin pitch, little research have been devoted to multi-parameter optimization. Increasing fins number or height would occupy the volume of annular space and decrease the heat storage capacity of LHTES. The combined effect of fin height and pitch were not investigated under the premise of a certain fin volume. Otherwise, most fins arrangements are evenly embedded on the tube, large melting non-uniformity formed in the PCM region due to local natural convection. Only a few researches focus on the non-uniform arrangement of fins. Little attention has been paid to compare the thermal behavior of different arrangements of fins for vertical LHTES and solve the non-uniformity of melting process. Based on the problems of single parameter and non-uniformity of melting process, heat transfer enhancement with fins is studied in a shell-and-tube LHTES unit in this paper. Comparison studies of different fins arrangement (middle fins, upper fins, lower fins and arithmetic fins) were carried out. The combined effects of fin height and pitch at a fixed fin volume ration on the thermal performance of LHTES were investigated. Furthermore, the uniformity of melting process for different radial finned LHTES is firstly quantified using root mean square of liquid fraction in this study. The new findings in this paper can provide guidelines for the design of radial finned LHTES units. 6

2.

Computation model and numerical method

2.1 Physical model The structure of a vertical radial-finned LHTES unit is shown as Fig. 1. Two coaxial cylinder and radial fins are the key components of the unit. The geometry parameters of the unit are designed as follows: the length of the unit is 400 mm. The outer cylinder which is made of Perspex, with thickness of 2 mm, has an internal diameter of 44 mm. The inner cylinder, with thickness of 2.5mm, which is made of steel, has an internal diameter of 15 mm. Fins are distributed on the outer surface of the steel tube. The fin thickness (t) is a fixed value of 1 mm. In order to make fin numbers an integer, the dimensionless fin height l  L / ( R  r  2.5) is selected as 0.417, 0.642, 0.833 and 0.983. The corresponding fin height L is 5 mm, 7.7 mm, 10 mm and 11.8 mm while the corresponding fin number is 24, 14, 10 and 8. Paraffin RT35 with a melting temperature of 308 K provided by Rubitherm was chosen as the PCM. The HTF is water, which flows from top to bottom at a temperature of 325 K and a velocity of 0.01 m  s -1 . The corresponding Reynold number is lower than 2300, which can be considered to laminar. Table 1 shows the thermal physical properties of water, RT35 and Aluminum. Because of the rotational symmetry nature of the physical model, a two-dimensional computational domain is considered in the following cases.

Table 1 Thermal physical properties of the HTF, PCM[28] and Fin

Property

PCM(RT35)

HTF

Fin(Aluminum)

Density ( kg  m )

880(s)/760((l)

998.2

2719

-1 -1 Specific heat ( J  kg  K )

2400(s)/1800(l)

4182

871

-3

7

Thermal conductivity ( W  m 1  K 1 )

0.2

0.6

202.4

Dynamic viscosity ( Pa  s )

0.0029

0.001003

-

Thermal expansion coefficient ( K1 )

0.001

-

-

Melting temperature range(K)

308

-

-

Latent heat ( kJ  kg )

157

-

-

-1

Fig. 1 Schematic diagram of the radial-finned shell-and-tube LHTES unit and computation domain

2.2 Governing equation The heat transfer equation for fins is as follows: c

T t



 x

(

T x

)

 y

(

T y

)

(1)

Here T denotes the temperature of fin; c is specific heat of fin. λ is thermal conductivity. Enthalpy-porosity model is used to simulate the phase change process of PCM, instead of tracing the liquid-solid interface explicitly in the PCM region. The thermal physical 8

properties of both HTF and PCM are constant or independent on temperature, except density of liquid PCM. Natural convection is taken into consideration during PCM melting process by Boussinesq approximation. The governing equations to describe flow and heat transfer of PCM and HTF are as follows [17, 29]: Continuity equation:   (  u )  (  v)   0 t x y

(2)

 (  u )  (  uu )  (  uv) P  u  u     (  )  (  )  Su t x y x x x y y

(3)

 (  v)  (  uv)  (  vv) P  v  u     (  )  (  )  Sv t x y x x x y y

(4)

Momentum equation:

Here T denotes the temperature, cp is specific heat. ρ represents the density and k is thermal conductivity. Where Su and Sv are the momentum source items. For HTF region, Su=0, Sv=0. For PCM region, Su=Au, Sv=Av+ρgα(T-Tm). g represents the gravitational acceleration. A is “porosity function”, which is defined as: AC

(1   ) 2 2 

(5)

Where C is the mushy zone constant, which equals to 106 in this study[30].  is a small number, which is set as 0.001 to prevent division by zero.  is the liquid volume fraction, which is defined as: 0   T  Ts    Tl  Ts 1 

T  Ts Ts  T  Tl

(6)

Tl  T

Energy equation:  (  h)  (  uh)  (  vh)  T  T    (k )  (k )S t x y x x y y

9

(7)

Where S is the source item. For HTF region, S=0. For PCM region, S    L  TT cP dT   ref  h    TTrefs cP dT  H  T T   Trefs cP dT  H   Tl cP dT

 t

.

T  Ts Ts  T  Tl

(8)

Tl  T

Where Tref is the reference temperature, which equals to 295 K. 2.3 Initial and boundary conditions In this study, the initial temperature of all computation domains is 295 K. The initial velocity of the computation domain is zero:

t  0, T =T0 , u  0, v  0

(9)

The velocity and temperature of the HTF at the inlet of the tube can be expressed as: Tin  325 K, u  0.01 m  s -1 , v  0

(10)

The left side of the computation domain is symmetrical: x  0, v  0,

u T  0, 0 y y

(11)

The bottom and top wall are regarded as thermal insulation: T 0 y

(12)

There has a temperature difference between the right wall and ambient, thus taking the heat loss into consideration in the numerical simulations. A natural convection empirical correlation appropriate for infinite space is chosen in this study. The heat transfer coefficient of natural convection can be calculated by: h

Nu D

(13)

Nu  0.59  Gr Pr m

1/ 4

(14)

Where  represents the thermal conductivity of air, D represents the characteristic length of LHTES, and subscript m represents the qualitative temperature equals to the 10

mean temperature of the boundary layer. The boundary condition of the right wall is: 

T  h T  T0  n

(15)

Where T0 denotes ambient temperature. 2.4 Numerical methodology 2D axisymmetric numerical simulation is carried out by FLUENT 16.1. The structured tetrahedral grids are used to discrete the computation domain. The SIMPLE algorithms are employed for pressure and velocity coupling, and the QUICK scheme is used to spatial discretization of energy and momentum equation. PRESTO! Scheme is adopted for pressure correction. The under-relaxation factors for energy, liquid fraction, pressure correction and momentum are set as 1, 0.9, 0.7 and 0.3 respectively. To ensure the residual of continuity term less than 10-5 and the residual of energy term less than 10-12, the max iteration of every time step is set as 30, which is enough for the convergence criterion.

3.

Validation of numerical simulation

3.1 Model validation The proposed simulation model is verified through Martin’s experimental study[28]. All the conditions, including the geometric parameters, the operation conditions and the materials are set as the same with their experiment. Fig. 2(a) shows the scheme of the experimental LHTHS unit without fin and thermocouple position. It can be observed from Fig. 2(b) that the results of simulation agree well with the experiment and the maximum deviation is less than 0.64%. Moreover, another validation is performed by comparing the numerical result with the numerical prediction by Yang who also 11

operated the same validation using their numerical model. Fig. 3 depicts that the liquid fraction is in good agreement with Yang’s[17]. experiment in literature[38]

325

present simulation

320

T/K

315 310 305 300 295 0

1000 2000 3000 4000 5000 6000 7000 8000 t/s

(a)

(b)

Fig. 2 Comparison of the experimental results and numerical solution

Yang's simulation[17] Present simulation

1.0

Liquid fraction

0.8

0.6

0.4

0.2

0.0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time(s)

Fig. 3 Comparison of liquid fraction in the present simulation versus simulation results of Yang’s

3.2 Grid validation and time step validation Independence studies of time step and grid size are also verified, which are shown in Fig. 4 and Fig. 5. The distributions of liquid fraction with time were examined by four 12

time steps (0.02 s, 0.05 s, 0.1 s and 0.5 s) and grid numbers ( 2.0 105 , 2.7 105 , 3.9 105 and 6.1 105 ). It can be observed from Fig. 4, the relative deviation between

time steps equal to 0.02 s, 0.05 s and 0.1 s is very small. In order to save computing resources, 0.1 s is chosen to use in present study. As shown in Fig. 5, a grid number of 2.7 105 is enough to ensure the calculation accuracy.

time step=0.02 time step=0.05 time step=0.1 time step=0.5

1.0

Liquid fraction

0.8

0.6

0.4

0.2

0.0

0

2000

6000

4000

8000

Time(s)

Fig. 4 Independence study of time step

13

10000

1.0

9.7104 2.0105

Liquid fraction

0.8

2.7105 3.9105

0.6

0.4

0.2

0.0

0

2000

4000

6000

8000

Time(s)

Fig. 5 Independence study of grid number

4.

Results and discussion

4.1 Effect of fins on melting process On the one hand, adding fins to LHTES unit may contribute to increasing heat transfer area and promoting heat transfer coefficient; on the other hand, local flow rate of melting PCM would be weakened due to the presence of fins. Therefore, the effect of adding fins on melting front evolution is investigated firstly. The fin number N, fin height L and fin thickness t are set as 10 mm, 10 mm and 1 mm, respectively. Fig. 6 shows the melting front evolution in two vertical LHTES unit: LHTES without fin and LHTES with fins. The HTF injects to the tube with a speed of 0.01 m  s -1 from top to bottom, and the corresponding Reynolds number is lower than 2300, which meets a laminar flow. Such a low velocity makes more heat carried by HTF transfer to PCM. In the initial stage of melting, there is a thin melt PCM layer which is parallel to the tube and fins surface, because of the dominant of heat conduction mechanism. The 14

thickness of melt PCM layer at top is thicker than that at bottom owing to the entrance effect of the HTF temperature, which gradually decreases along the flow direction. As time goes on, it can be seen that PCM adjacent to the tube and fin surface melt completely, because the temperature of liquid PCM at top is higher than that at bottom, the shape of the solid-liquid interface changes due to the effect of buoyancy, and in this stage, the heat transfer mechanism is dominated by natural convection. Moreover, the melting rate of PCM at lower part in finned LHTES is much high than that in LHTES without fin because of the presence of fins. Fig. 7 depicts the variations of liquid fraction versus time for the two LHTES unit. It can be observed that the complete melting time is 4258 s for finned LHTES unit and 7605 s for LHTES unit without fin. The full melting time can be reduced 44.0% by finned LHTES. However, the temperature at top is much higher than that at bottom, which causes larger nonuniformity for the temperature distribution and melting front during the PCM melting process. Making the temperature distribution and melting process uniformity is the main content of our study, this will be discussed in section 4.4.

15

Fig. 6

The contours of melting front in vertical LHTES units at different times: 300 s, 1000 s, 2000 s, 2800 s. (a) LHTES without fin; (b) LHTES with fins

To illustrate the influence of fins on heat transfer performance, an average heat transfer coefficient representing the thermal behavior of finned LHTES over the total heat transfer process is proposed in literature[17], which is defined as: h



tfull

0

qw dt

tfull (Tw  Ti )

(16)

Where h is the average heat transfer coefficient, qw denotes the heat flux on the tube and fin surface, tfull is the entire melting time, Ti is the temperature of initial state, Tw is the area-weighted temperature of heat transfer wall surface.

The average heat transfer coefficient of LHTES without fin and finned LHTES is 11.73 and 23.71 respectively. Obviously, adding fins can dramatically improve the thermal behavior of LHTES unit. 16

1.2

LHTES finned LHTES

Liquid fraction

1.0

0.8

0.6

0.4

0.2

0.0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time(s)

Fig. 7 Effect of fins on PCM liquid fraction versus time for vertical LHTES units

Table 2 Structure parameters of different fin heights under a fixed fin volume ratio

Fin volume ratio

l

N

p/mm

t/mm

2%

0.417

24

16.7

1

2%

0.642

14

28.6

1

2%

0.833

10

40.0

1

2%

0.983

8

50.0

1

4.2 Combined effect of fin pitch and height In this section, by fixing the fin volume ratio at 2% and the fin height at 1 mm , four dimensionless

fin

heights

are

proposed,

the

dimensionless

fin

height

l  L / ( R  r  2.5) is selected as 0.417, 0.642, 0.833 and 0.983 (the corresponding fin

height equals to 5 mm, 7.7 mm, 10 mm and 11.8 mm), respectively. With dimensionless 17

fin height increase from 0.417 to 0.983, the fin number decrease from 24 to 8, and the fin pitch increase from 16.7 mm to 50.0 mm. Table 2 presents the structure parameters of different fin heights under a fixed fin volume ratio. The effect of dimensionless fin height on thermal behavior is studied. Fig. 8 shows the combined effect of fin height and pitch on the melting front evolution at different time: 300 s, 1000 s, 2000 s and 2800 s. It can be seen when the dimensionless fin height is 0.417, 0.642, 0.833, not only does local natural convection form in the PCM between two adjacent fins, but also heat exchange occurs in the PCM between the adjacent spaces through the gap between fins and the inner surface of Perspex. As for the dimensionless height equals to 0.983, local natural convection only occurs in every annular space between two adjacent fins and adjacent spaces don’t exist heat exchange because of the entire obstruction of fins. Therefore, as seen in Fig. 9, the liquid fraction of the dimensionless height l equals to 0.983 is always lower than other cases. It can also be seen from Fig. 9 that at first stage (t<2400 s), the liquid fraction of the dimensionless height l equals to 0.417 is the highest, followed by l equals to 0.642, 0.833 and 0.983. The reason is that the closer the fin is near to the tube wall, the higher the fin temperature is, and more heat can transfer to PCM from HTF. As time goes on, more fins of l equals to 0.417 are out of action, so at second stage (t>2400 s), the liquid fraction of l equals to 0.417 is lower than that of l equals to 0.642 and 0.833, which also can be seen in Fig. 8. The liquid fraction increases from 0.95 to 1.0 spend much time and storage little heat energy, so when the liquid fraction reaches 0.95, it can be considered that all the PCM have been melted completely. When dimensionless fin height increases from 0.417 to 18

0.983, the full melting time is 3050 s, 2891 s, 3000 s and 3115 s, respectively, saving 46.0%, 48.8%, 46.9% and 44.9% in comparison with no fin case. Furthermore, as shown in Fig. 10, the average heat transfer coefficient of l equals to 0.417, 0.642, 0.833 and 0.983 is 22.5, 24.5, 23.7 and 22.6, respectively. Although short fin height has a good early effect, the temperature impact range is small in later stage. In addition, the case of dimensionless fin height l equals to 0.983 almost obstructs heat transfer between adjacent spaces between fins thoroughly. Under the combined effect of these two factors, medium fin height shows the best thermal performance. The result indicated that the dimensionless height of 0.642 is recommended. Therefore, in the design of LHTES unit, choosing the appropriate fin height is important under finite fin volume ratio.

Fig. 8 The contours of melting front for different dimensionless lengths at different time: t equals to 300 s, 1000 s, 2000 s and 2800 s 19

1.0

l=0.417 l=0.642 l=0.833 l=0.983

liquid fraction

0.8

0.6

0.96 0.92 0.88

0.4

0.84 0.80

0.2

0.0

0

500

1000

1500

2400

2600

2000

2500

2800

3000

3200

3000

3500

Time(s) Fig. 9 Effect of dimensionless fin height on liquid fraction

24.5

Average heat transfer coefficient

25

23.7

22.5

22.6

20

15

10

5

0

l=0.417

l=0.833

l=0.642

l=0.983

Fig. 10 Averange heat transfer coefficient of finned LHTES with different fin height

20

Fig. 11 Different fin arrangements of vertical finned LHTES: (a) middle fins, (b) upper fins, (c) lower fins, (d) arithmetic fins

4.3 Comparison of different fin arrangements The thermal performance of LHTES can be significantly promoted by adding fins from section 4.1. For the purpose of optimizing thermal behavior of radial finned LHTES unit, identifying the optimal fin arrangement needs further discussion. As Fig. 11 shows, four different fin arrangements, named as upper fins, lower fins, middle fins and arithmetic fins, is proposed. The upper/lower fins means that all the fins are evenly distributed on the upper/lower half of the tube. The arithmetic fins means that fins appear an arithmetic progression distribution on the tube from bottom to top. The middle fins means that all the fins are evenly distributed on the entire tube. In order to illustrate the heat transfer phenomena of different arrangements, Fig.12 reveals the effects of fin arrangements on melting front evolution and temperature distribution 21

during the charging process. β represents the area-weighted average melting fraction in Fig. 12. It can be observed that upper fins accelerate the melting rate of PCM at upper half owing to the uniformity arrangement of fins at upper half. In the same way, the role of middle fins is to improve the melting rate of the entire area. With fins adding to the LHTES unit, the space is divided into eleven small domains. It can be observed that the temperature in the bottom domain is much lower than that in the top domain for the cases of upper fins and middle fins, because high temperature HTF injects from top to bottom. The temperature of HTF decreases along the axis of the tube and the PCM at the top acquire more heat than that at the bottom. Moreover, lots of heat which is used to melt PCM at bottom is transferred to melting PCM at top because of buoyancy. The combined effects cause large temperature and melting process non-uniform of LHTES. As shown in Fig. 12(a) and (b), at 2800 s, the PCM in the upside has a completing melt for the cases of upper fins and middle fins. Simultaneously, a large amount of PCM has not melt in the downside. Therefore, a new fin arrangement is proposed, arranging the fins in the lower half of the LHTES unit. As Fig. 12(c) shows, the lower fins just promote the melting rate of the lower part. The PCM melting rate at lower half is higher than that at upper half. The reason is that the effect of adding fins is better than the temperature difference between PCM and HTF. According to the above results and discussion, another fin arrangement is further proposed, arranging the fins as an arithmetic progression distribution on the tube from downside to upside. As seen in Fig. 12(d), compared to the other three fin arrangements, the temperature distribution and melting process of this structure is more uniform. In 22

order to evaluate the melting process uniformity of different arrangements, root mean square (RSM) is selected to quantify the uniformity of melting process, which is defined as: 11

E   (  k   ) 2 Pk

(17)

k 1

Where  denotes the average liquid fraction at every small domains,  denotes the average liquid fraction of eleven small domains; k represents the number of every small domains; P represents volume ratio of every small domains.

23

Fig. 12 The contours of temperature(left) and melting front (right) for different fin arrangements at different time: t=300 s, 1000 s, 2000 s and 2800 s

24

0.25

arithmetic fins upper fins middle fins lower fins

Root mean square

0.20

0.15

0.10

0.05

0.00

0

500

1000

1500

2000

2500

3000

Time(s)

Fig. 13 The root mean square (RSM) of liquid fraction of the eleven domains for different fin arrangements.

1.0

no fin middle fins lower fins upper fins arithmetic fins

Liquid fraction

0.8

0.6

inflection point

0.4

0.2

0.0

0

1000

2000

3000

4000

5000

6000

Time(s)

Fig. 14 The liquid fraction evolution of different fin arrangements and no fin case

25

Fig. 13 shows the root mean square of liquid fraction of the eleven domains for the four fin arrangements. Generally speaking, the root mean square of the LHTES with arithmetic fins is the lowest in the four fins arrangements, following by lower fins, middle fins and upper fins. Moreover, the RSM of liquid fraction of lower fins and arithmetic fins is much lower than that of upper fins and middle fins, which can also be seen in Fig.12. Fig. 14 depicts the variations of liquid fraction with time for different fin arrangements. It can be observed from the simulation results that the full melting time is 3000s, 2827s, 3450s and 4862s for the case of middle fins, arithmetic fins, lower fins and upper fins, saving 46.9%, 49.9%, 38.9% and 13.9% in comparison with no fin case. The complete melting time has a huge difference with different fin arrangements. Thus, it is vital to arrange fins in LHTES unit for the sake of maximizing the thermal behavior of LHTES unit. As Fig 14 shows, for the case of upper fins, the slope of the curve is sharply decreasing at 2000s, after that moment, the slope of the curve is parallel with the curve of no fin case. This represents that the fins have lost their effect after that moment, which is named as inflection point in literature[31]. The inflection point reached earlier for the case of lower fins and middle fins. The inflection point for arithmetic fins case is not evident, the slope of liquid fraction curve of arithmetic fins case is approximate constant, which also can explain that the melting process uniformity of arithmetic fins is better than other cases. Fig. 15 reveals the full melting time, the heat storage capacity at the end of melting process and the average temperature of PCM at the end of melting process. The heat storage capacity includes three parts: the absorbed sensible heat of 26

PCM from 295 K-301 K (sensible heat of solid PCM), the absorbed latent heat of PCM from 301 K-313 K (latent heat), and the absorbed sensible heat of PCM more than 313 K (sensible heat of liquid PCM). Sensible heat of solid PCM and latent heat are the same for all cases, but sensible heat of liquid PCM of upper fins is larger than other cases, following by lower fins, middle fins and arithmetic fins. This is caused by the different melting time of all the cases. The melting time is longer, the temperature of PCM at the end is higher, and the more sensible heat of liquid PCM is required. The difference in the heat storage capacity of all finned LHTES is small, but there is a considerable difference in full melting time. 100

322

6000

Sensible heat II Latent heat

320 318

80

316 60 4000 40

312 310 308

20

0

314

Temperature/K

5000

Time/s

Heat storage capacity/Kj

Sensible heat I

3000

0

1

2

3

4

5

6

306 304

case number

Fig. 15 The full melting time, average temperature at full melting moment and the heat storage capacity at full melting moment of all finned LHTES; the LHTES without fin is set as contrast

27

Average heat transfer cofficient

35 30 25.9 25

23.7 19.9

20 15

13.3

11.7 10 5 0

no fin

middle fins arithmetic fins lower fins

upper fins

Fig. 16 Averange heat transfer coefficient of all finned LHTES; the LHTES without fin is set as contrast

no fin middle fins lower fins upper fins arithmetic fins

80

Heat storage capacity

70 60 50 40 30 20 10 0

0

500

1000

1500

2000

2500

3000

3500

4000

Time(s)

Fig. 17 Variations of heat storage capacity versus time of different fin arrangements; LHTES without fin is set as contrast

28

As Fig.16 shows, the average heat transfer coefficient of arithmetic fins is higher than that of the other fin arrangements, following by middle fins, lower fins, upper fins and no fin case. The average heat transfer coefficient has obviously positive correlation with the full melting time. The result is consistent with the results of Fig. 14. This finding supports a design strategy towards heat transfer performance maximum: fins should be equipped with proper design. Fig. 17 shows time-wise variation of heat storage capacity of all fin arrangements. The heat storage capacity of LHTES with arithmetic fins is higher than that of the other cases. Due to the storage of latent heat of PCM, the result of Fig. 17 is also consistent with the result of Fig. 14. In the first 1000 s, the liquid fraction curve of arithmetic fins, middle fins, lower fins and upper fins are almost coincident. Thus, the heat storage capacity of these cases is overlapped. As time goes by, the upper fin is out of effect, and the heat storage capacity of upper fins case is lower than other cases. That means adding arithmetic fins to vertical LHTES is the most appropriate structure.

5.

Conclusion

Adding fins to LHTES unit is a simple and effective way to improve the heat transfer performance of the heat exchanger. Numerical studies on vertical finned LHTES unit were carried out and the design parameters including fin height and pitch are chosen to optimize the thermal behavior of the LHTES unit. The thermal behavior of four different fin arrangements was also investigated. The main conclusions are shown bellow： (1). Embedded fins to LHTES can significantly improve the thermal behavior of 29

LHTES unit. The full melting time is 4258 s for finned case and 7605 s for no fin case, the full melting time can be saved by 44.0% for finned LHTES. (2). Dimensionless fin height should be appropriate selected. With dimensionless fin height increasing from 0.417 to 0.983, the full melting time is 3050 s, 2891 s, 3000 s and 3115 s, saving 46.0%, 48.8%, 46.9% and 44.9% in comparison with no fin case. And the average heat transfer coefficient of l equals to 0.417, 0.642, 0.833 and 0.983 is 22.5, 24.5, 23.7 and 22.6 respectively. The dimensionless fin height recommended is 64.2% for the present study. (3). The arithmetic fin arrangement is recommended in this study. The complete melting time is 3000 s, 2827 s, 3450 s and 4862 s for the case of middle fins, arithmetic fins, lower fins and upper fins, saving 46.9%, 49.9%, 38.9% and 13.9% compared with no fin case. And the average heat transfer coefficient of arithmetic fins is the highest, following by middle fins, lower fins, and upper fins. The average heat transfer coefficient of no fin case is the lowest. (4). The root mean square of the LHTES unit using arithmetic fins is the lowest in the four fins arrangements, following by middle fins and lower fins, the RSM of the LHTES unit using upper fins is the highest. Moreover, the RSM of liquid fraction for upper fins and middle fins is much higher than that of lower fins and arithmetic fins. These indicate that arithmetic fins have a uniformity melting process compared with other fin arrangements, contributed to a good thermal performance.

Acknowledgements This research work was jointly supported by the National Natural Science Foundation 30

of China (No.51641608) and the Fundamental Research Funds for the Central Universities of China (No. 022019058).

References [1] Y.B. Tao, Y.-L. He, A review of phase change material and performance enhancement method for latent heat storage system, Renewable and Sustainable Energy Reviews, 93 (2018) 245-259. [2] Z. Liu, Y. Yao, H. Wu, Numerical modeling for solid–liquid phase change phenomena in porous media: Shell-and-tube type latent heat thermal energy storage, Applied Energy, 112 (2013) 1222-1232. [3] Y. Zhao, C.Y. Zhao, Z.G. Xu, H.J. Xu, Modeling metal foam enhanced phase change heat transfer in thermal energy storage by using phase field method, International Journal of Heat and Mass Transfer, 99 (2016) 170-181. [4] Y.B. Tao, C.H. Lin, Y.L. He, Effect of surface active agent on thermal properties of carbonate salt/carbon nanomaterial composite phase change material, Applied Energy, 156 (2015) 478-489. [5] C. Zauner, F. Hengstberger, M. Etzel, D. Lager, R. Hofmann, H. Walter, Experimental characterization and simulation of a fin-tube latent heat storage using high density polyethylene as PCM, Applied Energy, 179 (2016) 237-246. [6] J. Shon, H. Kim, K. Lee, Improved heat storage rate for an automobile coolant waste heat recovery system using phase-change material in a fin–tube heat exchanger, Applied Energy, 113 (2014) 680-689. [7] L. Pu, L. Xu, S. Zhang, Y. Li, Optimization of ground heat exchanger using 31

microencapsulated phase change material slurry based on tree-shaped structure, Applied Energy, 240 (2019) 860-869. [8] S. Bellan, J. Gonzalez-Aguilar, M. Romero, M.M. Rahman, D.Y. Goswami, E.K. Stefanakos, D. Couling, Numerical analysis of charging and discharging performance of a thermal energy storage system with encapsulated phase change material, Applied Thermal Engineering, 71 (2014) 481-500. [9] J. Guo, X. Huai, The heat transfer mechanism study of three-tank latent heat storage system based on entransy theory, International Journal of Heat and Mass Transfer, 97 (2016) 191-200. [10] A.M. Abdulateef, S. Mat, J. Abdulateef, K. Sopian, A.A. Al-Abidi, Geometric and design parameters of fins employed for enhancing thermal energy storage systems: a review, Renewable and Sustainable Energy Reviews, 82 (2018) 1620-1635. [11] C. Ji, Z. Qin, Z. Low, S. Dubey, F.H. Choo, F. Duan, Non-uniform heat transfer suppression to enhance PCM melting by angled fins, Applied Thermal Engineering, 129 (2018) 269-279. [12] X. Cao, Y. Yuan, B. Xiang, L. Sun, Z. Xingxing, Numerical investigation on optimal number of longitudinal fins in horizontal annular phase change unit at different wall temperatures, Energy and Buildings, 158 (2018) 384-392. [13] A.A. Rabienataj Darzi, M. Jourabian, M. Farhadi, Melting and solidification of PCM enhanced by radial conductive fins and nanoparticles in cylindrical annulus, Energy Conversion and Management, 118 (2016) 253-263. [14] M. Parsazadeh, X. Duan, Numerical study on the effects of fins and nanoparticles 32

in a shell and tube phase change thermal energy storage unit, Applied Energy, 216 (2018) 142-156. [15] A. Erek, Z. lken, M.A. Acar, Experimental and numerical investigation of thermal energy storage with a finned tube, International Journal of Energy Research, 29 (2005) 283-301. [16] I. Jmal, M. Baccar, Numerical study of PCM solidification in a finned tube thermal storage including natural convection, Applied Thermal Engineering, 84 (2015) 320-330. [17] X. Yang, Z. Lu, Q. Bai, Q. Zhang, L. Jin, J. Yan, Thermal performance of a shelland-tube latent heat thermal energy storage unit: Role of annular fins, Applied Energy, 202 (2017) 558-570. [18] D. Zhao, G. Tan, Numerical analysis of a shell-and-tube latent heat storage unit with fins for air-conditioning application, Applied Energy, 138 (2015) 381-392. [19] I. Jmal, M. Baccar, Numerical investigation of PCM solidification in a finned rectangular heat exchanger including natural convection, International Journal of Heat and Mass Transfer, 127 (2018) 714-727. [20] K.A.R. Ismail, F.A.M. Lino, Fins and turbulence promoters for heat transfer enhancement in latent heat storage systems, Experimental Thermal and Fluid Science, 35 (2011) 1010-1018. [21] Z. Khan, Z.A. Khan, An experimental investigation of discharge/solidification cycle of paraffin in novel shell and tube with longitudinal fins based latent heat storage system, Energy Conversion and Management, 154 (2017) 157-167. [22] M.K. Rathod, J. Banerjee, Thermal performance enhancement of shell and tube 33

Latent Heat Storage Unit using longitudinal fins, Applied Thermal Engineering, 75 (2015) 1084-1092. [23] M.J. Hosseini, A.A. Ranjbar, M. Rahimi, R. Bahrampoury, Experimental and numerical evaluation of longitudinally finned latent heat thermal storage systems, Energy and Buildings, 99 (2015) 263-272. [24] R. Amini, M. Amini, A. Jafarinia, M. Kashfi, Numerical investigation on effects of using segmented and helical tube fins on thermal performance and efficiency of a shell and tube heat exchanger, Applied Thermal Engineering, 138 (2018) 750-760. [25] A. Sciacovelli, F. Gagliardi, V. Verda, Maximization of performance of a PCM latent heat storage system with innovative fins, Applied Energy, 137 (2015) 707-715. [26] Y.B. Tao, Y.L. He, Effects of natural convection on latent heat storage performance of salt in a horizontal concentric tube, Applied Energy, 143 (2015) 38-46. [27] M. Kazemi, M.J. Hosseini, A.A. Ranjbar, R. Bahrampoury, Improvement of longitudinal fins configuration in latent heat storage systems, Renewable Energy, 116 (2018) 447-457. [28] M. Longeon, A. Soupart, J.-F. Fourmigué, A. Bruch, P. Marty, Experimental and numerical study of annular PCM storage in the presence of natural convection, Applied Energy, 112 (2013) 175-184. [29] S. Deng, C. Nie, G. Wei, W.-B. Ye, Improving the melting performance of a horizontal shell-tube latent-heat thermal energy storage unit using local enhanced finned tube, Energy and Buildings, 183 (2019) 161-173. [30] M. Fadl, P.C. Eames, Numerical investigation of the influence of mushy zone 34

parameter Amush on heat transfer characteristics in vertically and horizontally oriented thermal energy storage systems, Applied Thermal Engineering, 151 (2019) 90-99. [31] P. Wang, H. Yao, Z. Lan, Z. Peng, Y. Huang, Y. Ding, Numerical investigation of PCM melting process in sleeve tube with internal fins, Energy Conversion and Management, 110 (2016) 428-435.

35

Highlights 

Effect of dimensionless fin height on thermal behavior of LHTES was investigated.



Arithmetic fin arrangement showed a superior heat transfer performance.



Root mean square of liquid fraction was proposed to describe melting uniformity.



A uniform PCM melting process was observed in arithmetic fin arrangement.

36

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my coauthors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

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