Numerical investigation on optimal number of longitudinal fins in horizontal annular phase change unit at different wall temperatures

Accepted Manuscript Title: Numerical investigation on optimal number of longitudinal fins in horizontal annular phase change unit at different wall temperatures Authors: Xiaoling Cao, Yanping Yuan, Bo Xiang, Liangliang Sun, Zhang Xingxing PII: DOI: Reference:

S0378-7788(17)30939-8 https://doi.org/10.1016/j.enbuild.2017.10.029 ENB 8048

To appear in:

ENB

Received date: Revised date: Accepted date:

21-3-2017 22-9-2017 5-10-2017

Please cite this article as: Xiaoling Cao, Yanping Yuan, Bo Xiang, Liangliang Sun, Zhang Xingxing, Numerical investigation on optimal number of longitudinal fins in horizontal annular phase change unit at different wall temperatures, Energy and Buildings https://doi.org/10.1016/j.enbuild.2017.10.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

Numerical investigation on optimal number of longitudinal fins in horizontal annular phase change unit at different wall temperatures Xiaoling Caoa, Yanping Yuana,, Bo Xianga, Liangliang Suna, Zhang Xingxingb a

School of Mechanical Engineering, Southwest Jiaotong University, 610031, Chengdu, P. R. China

b

School of Industrial Technology and Business Studies, Dalarna University, SE-79188, Falun,

Sweden 

Corresponding author, Phone: +86 13880871068, Fax: +86 2887634937,

E-mail: [email protected].

Highlights 

The fin number has an optimal value for a specific boundary temperature.



The heat transfer enhancement is more effective under a lower wall temperature.



Number of fins beyond a value result in the decrease of heat transfer rate.



The optimal number of fins is 10 in this study.

Abstract: The number, shape, and size of the fins utilized in a phase change unit form the key parameters affecting the heat transfer process in the unit. To the best of our knowledge, there are no relevant literature reports on the optimal number of fins in a horizontal annular phase change unit for different wall temperatures. To investigate the correlation between the number of fins and the wall temperature, in this study, based on a numerical simulation using the enthalpy-porosity model, we examine the phase-change material (PCM) melt with different numbers of installed fins (n = 4, 6, 8, 10, 12) at five different constant wall temperatures. The

1

calculated results are analyzed and compared with those of the case without fins. The study results indicate that the addition of fins can effectively accelerate the PCM melting rate and shorten the melting time; further, the wall temperature also significantly affects the melting rate. ̅̅̅̅ We define the time-average Nusselt number (𝑁 𝑢 ) as an indicator of heat transfer efficiency to analyze the heat transfer mechanism. The influence of the fin number and wall temperature on ̅̅̅̅ 𝑁𝑢 and the variation in ̅̅̅̅ 𝑁𝑢 at different stages of the melting process are investigated. According to the variation in the average wall heat flux, it can be concluded that the fin number has an optimal value for a specific wall temperature. The optimal number of fins according to this study is 10. Nomenclature n

number of the fins

cp

specific heat capacity at constant pressure (J/kg·K)

𝐿

latent heat(J/g)

T

temperature (K)

Ts

temperature of solid region of PCM (K)

Tl

temperature of liquid region of PCM (K)

Tm

phase change temperature (K)

p

pressure(Pa)

→

velocity(m/s)

t

time(s)

g

gravitational acceleration (m/s2)

→

momentum source term

H

enthalpy(J/kg)

𝑉

𝑆

H’ sensible enthalpy(J/kg) h

heat transfer coefficient (w·m-2·K)

Q

heat transfer capacity (J)

q

wall heat flux (W)

D

equivalent diameter (m) 2

ℎ𝑓

height of fin (m)

Nu

Nusselt number

Fo

Fourier number, Fo = α𝑡/𝐷2

Ste Stefan number for constant cylinder surface temperature, Ste = Ra

Rayleigh number,Ra =

𝑐𝑝 (𝑇𝑤 −𝑇𝑚 ) 𝐿

𝑔𝛽𝐷 3 𝛼𝜈

Greek symbols 𝛽

expansion coefficient(1/K)

𝜇

dynamic viscosity(Pa·s)

𝜈

kinematics viscosity(m2/s)

𝛼

thermal diffusivity(m2/s)

λ

thermal conductivity (W/m·K)

ε

small constant value

𝜌

density (kg/m3)

Subscripts s

solid phase of PCM

l

liquid phase of PCM

f

fin

i

inner annulus

o

outer annulus

Keywords: Latent heat storage; melting; heat transfer enhancement; fins 1. Introduction When compared with other heat storage methods, latent heat storage as a new energy storage technology offers several advantages such as high energy storage density, system stability, and a near-isothermal heat storage/release process [1]. Thus, latent heat storage offers unique advantages when utilized for building enclosure structures [2], heating and air conditioning [3], solar thermal electric power [4], construction waste heat recovery [5] and 3

thermal management [6]. Phase-change materials (PCMs), phase-change containers, and phase-change heat transfer devices are the key components of a phase-change energy storage system [7]. At present, in the field of building energy conservation, organic materials such as fatty acids, paraffin, and inorganic compounds such as hydrated salts [8] are commonly used as PCMs. Unlike salt hydrates, fatty acids and paraffin are limited by their small thermal transfer coefficient [9, 10], thereby resulting in a mismatch between the heat exchange rate and the heat exchange demand in a building energy application system. Consequently, researchers have conducted extensive studies on phase-change heat transfer enhancement from various aspects including the development of new composite PCMs [11], thermal conductivity enhancement of PCMs [12], PCM microencapsulation [13], phase-change container heat transfer enhancement, and system structure optimization [9]. The addition of fins is a simple and economical method for the heat transfer enhancement of a phase-change container, and the fins can effectively improve the melting and solidification rate of PCMs [14, 15]. In this regard, Li and Wu [16] found that the melting and solidification time of horizontal shell and tube phase change units with fins was reduced by at least 14% with respect to the case without fins. Annular phase change enclosures with longitudinal and annular fins have been studied both experimentally and theoretically. In this regard, Agyenim et al. [17] conducted experiments to investigate the effect of mass flow rates and inlet heat transfer fluid (HTF) temperatures on the thermal behavior of a finned PCM storage system. Liu and Groulx [18] studied and compared the heat transfer enhancement performances of both straight and angled longitudinal fins inside a 4

horizontal cylindrical latent heat energy storage system. It was observed that the complete melting time is strongly affected by the HTF inlet temperature but very slightly by the HTF flow rates. The influences of the fin number, height, and thickness as well as aspect ratio of the annular spacing in a vertical circular cylinder on solidification process have been studied by Ismail et al. [19]. Their study indicates that the annular space size, radial length of the fin, and number of fins strongly influence the solidified mass fraction. In addition, their results confirmed the undesirable effects of natural convection during the phase change processes. Further, Hosseini et al. [20] focused on the thermal performance of a double pipe heat exchanger with longitudinal fins of different fins heights and Stefan numbers. Kozak and Rozenfeld [21, 22] investigated the effects of close-contact melting in a horizontal and vertical double-pipe concentric enclosure. They proposed that the fins should not be considered merely as extended heat transfer surfaces; when properly designed and oriented, they cause the material to melt in their proximity throughout the melting process, thus significantly accelerating melting. Agyenim et al. [23] performed a comparative analysis on the effect of annular and longitudinal fins on tubes and concluded that systems with longitudinal fins afford the best performance with increased thermal response during charging and reduced subcooling during discharging. Natural convection plays an important role in the melting process of PCMs, and it can effectively improve the heat transfer coefficient. In this context, Lamberg et al. [24] suggested that the heat transfer rate could be doubled considering natural convection based on numerical simulations. Here, we note that while fins can increase the heat transfer surface area and therefore enhance heat transfer, the addition of fins blocks the natural convection of liquid 5

[25], which is a problem that has attracted the attention of many researchers. Lacroix and Benmadda [26] reported that reducing the distance between vertically oriented fins weakens the natural convection strength along a horizontal heated wall. The melting and solidification of a PCM within three different horizontal annulus configurations were investigated numerically by Darzi et al. [27], and their results indicated that natural convection plays important roles in the melting process, where the melting rate at the bottom section of the annulus is lower than that at the top section. While fins significantly enhance the melting and solidification rate, they also suppress natural convection during the melting process, Dhaidan and Khodadadi[28] indicated that this problem should be considered by the designer through selecting the optimum positions and orientation of the fins. Here, we note that the heat transfer efficiency is a result of two conflicting measures: increased heat exchange area and reduced natural convection. Therefore, for a specific wall temperature, there is an optimum value for the fin number [29]. From the above literature review, we can conclude that fins can significantly improve the heat transfer performance of latent heat storage enclosures and systems. Many analyses have previously been performed on horizontal annular phase change enclosures, and it has been observed that fins attached to the inner tube increase the heat transfer area and accelerate the diffusion of heat. During the melting process, the natural convection is weakened by fins. In addition, increasing the number of fins will reduce the actual PCM volume. Despite these studies, to our knowledge, no studies have addressed the optimum fin number during the melting process for different wall temperatures. To investigate this problem, taking the horizontal annular phase change enclosure as a research object, we conducted 6

numerical simulations to research the effect of the fin number on the melting rate, Nusselt number, and wall heat flux. On the basis of heat transfer mechanism analysis, we formulated the quantitative relationship between the Nusselt number, fin number, wall temperature, melting time, and other factors. The average wall heat flux was determined as a metric corresponding to the optimal number of fins. We believe that the results of this study can further contribute toward enhancing the phase change heat transfer process. 2. Mathematical model and numerical solution method 2.1. Physical models In our study, we considered a horizontal annular phase change unit with longitudinal fins of a fixed height. As the experimental results of Agyenim et al., for temperature gradients in the axial direction, there is essentially a two-dimensional (2D) heat transfer in the unit [30]. Therefore, we treat the melting process as a 2D process, and the boundary condition of the inner tube is considered isothermal. The 2D physical model of a horizontal tube-in-shell storage unit is shown in Fig. 1, where n represents the number of fins. The diameters of the inner tube ri and outer tube ro are 40 mm and 80 mm, respectively. The fins are evenly distributed in the inner tube. The fin height is 10 mm, and the fin thickness is 1 mm. In our study, we chose lauric acid (LA) as the PCM owing to its many advantages such as a low supercooling degree, stable chemical properties, noncorrosive nature, and low cost. The LA used in this study (purity of 98%) was purchased from Shanghai Aladdin Industries. Fig. 2 shows the differential scanning calorimetry (DSC) testing curve of LA. Its main physical properties are listed in Table 1.

7

outer tube

inner tube

outer tube

PCM

PCM PCM

inner tube

Fig. 1 2D illustration of phase change units with various fin numbers o 44.2 C

6

173.8 J/g

4

Heat flow (W)

2 0 -2 -4 -6

o 43 C

173.4J/g

-8 20

30

40

50

60

o

Temperature ( C)

Fig. 2 Differential scanning calorimetry (DSC) testing curve of lauric acid (LA) Table 1 Major physical properties of lauric acid 8

Tm (°C)

44.22

λ (W/m·K)

0.147

cp (J/kg·K)

2300

𝛽 (1/K)

0.000615

L (J/g)

173.8

𝜇(Pa·s)

0.005336(60 °C) / 0.004269(70 °C) / 0.003469(80 °C)

𝜌(kg/m3)

862.9(60 °C) / 856(70 °C) / 848.3(80 °C)

To simplify the heat transfer model, we made the following assumptions in this study: (1) The Boussinesq approximation is adopted in which the material properties of each phase are considered constants; only the influence of the change in the fluid density on buoyancy lift was considered in the study. (2) The mushy zone is regarded as a porous medium, ignoring convective acceleration and diffusion, and therefore pressure drop within the porous region can be computed by Darcy's Law. (3) The coefficient of thermal expansion of LA is small, and the liquid phase flow induced by density difference is negligible compared with that by natural convection, and therefore, the density difference between the solid and the liquid phase is ignored. 2.2. Mathematical model and solution （1）PCM We adopt the enthalpy method to describe the energy equation of PCM in the unit. The main idea is that the enthalpy and temperature are both functions to be determined, and a unified energy equation is established for the entire area (including the liquid phase, solid phase, and two-phase interface). We use numerical simulations to calculate the enthalpy distribution and determine the two-phase interface. The governing equation can be expressed as follows: 9

Continuity equation: 𝜕𝜌 𝜕𝑡

+

1 𝜕(𝑟𝜌𝑉𝑟 ) 𝑟

𝜕𝑟

+

1 𝜕(𝜌𝑉𝜃 ) 𝑟

𝜕𝜃

=0

(1)

Momentum equation: 𝜕𝑉𝜃 𝑉𝑉 1 𝜕𝑝 2 𝜕𝑉𝑟 𝑉𝜃 ⃗ . ∇)𝑉𝜃 − 𝑟 𝜃 = 𝑔𝛽(𝑇 − 𝑇𝑚 ) sin 𝜃 − + (𝑉 + 𝜈 (∇2 𝑉𝜃 + 2 − )+𝑆 𝜕𝑡 𝑟 𝜌𝑟 𝜕𝜃 𝑟 𝜕𝜃 𝑟 2 (2) 𝜕𝑉𝑟 𝑉2 1 𝜕𝑝 2 𝜕𝑉𝜃 𝑉𝑟 ⃗ . ∇)𝑉𝑟 − 𝜃 = −𝑔𝛽(𝑇 − 𝑇𝑚 ) cos 𝜃 − + (𝑉 + 𝜈 (∇2 𝑉𝑟 − 2 − )+𝑆 𝜕𝑡 𝑟 𝜌 𝜕𝑟 𝑟 𝜕𝜃 𝑟 2 Here, parameter S is introduced as the source term, and it is defined as

1  f  S

f 3 

2

Amush V ,

(3)

where Amush represents the constant reflecting the mushy zone morphology, which describes how steeply the velocity is reduced to zero when the material solidifies, with a value in the range of 104–107. Further,  is a small constant introduced to avoid division by zero. In this study, we set Amush = 105 and  = 0.001 [31]. Energy conservation equation: 𝜕 𝜕𝑡

⃗ 𝐻) = 𝜆 ( (𝜌𝐻) + ∇ ∙ (𝜌𝑉

1 𝜕2 𝑇

𝑟 2 𝜕𝜃2

1 𝜕

𝜕𝑇

+ 𝑟 𝜕𝑟 (𝑟 𝜕𝑟 ))

(4)

Here, H is the total volumetric enthalpy (J), which can be expressed as the sum of sensible and latent heats: 𝐻 = 𝐻 ′ + ∆𝐻

(5)

Here, 𝑇

𝐻 ′ = ℎ𝑟𝑒𝑓 + ∫𝑇 H  fL

𝑟𝑒𝑓

𝑐𝑝 𝑑𝑇

(6) (7)

Parameter L represents the phase-change latent heat (J/g) and f the melting fraction in the 10

range of 0 (solid) to 1(liquid). The melting fraction f can be expressed as 0，𝑖𝑓 𝑇 < 𝑇𝑠 𝑓={

𝑇−𝑇𝑠 𝑇𝑙 −𝑇𝑠

， 𝑖𝑓 𝑇𝑠 < 𝑇 < 𝑇𝑙

(8)

1, 𝑖𝑓 𝑇 > 𝑇𝑙

For the liquid phase region, continuity equation, momentum equation and energy equation should be solved to obtain the distribution of temperature and velocity, but for the solid phase region, only the energy conservation equation need to be solved. Once obtaining the distribution of enthalpy through solving above control equations in each time-step, the temperature distribution can be calculated by equation (6), and then the liquid fraction is obtained. （2）fins The heat transfer of fins is two-dimensional unsteady thermal conduction, and the governing equation is：

ρ𝑓 𝑐𝑓

𝜕𝑇𝑓 𝜕𝑡

1 𝜕2 𝑇𝑓

= 𝜆𝑓 (𝑟 2

𝜕𝜃2

1 𝜕

+ 𝑟 𝜕𝑟 (𝑟

𝜕𝑇𝑓 𝜕𝑟

))

(9)

Boundary conditions: For the momentum equation, the liquid-solid interface between the liquid and solidified material, a zero-gradient velocity is used. And the boundary condition is noslip velocity boundary condition for the wall. The boundary conditions corresponding to the inner and outer tubes are expressed as (T|𝑟 = 𝑟𝑖 ) = 𝑇𝑤 11

(10)

∂T

(

∂r

|𝑟 = 𝑟𝑜 ) = 0

(11)

T=Tw

(12)

AB:

BC, CD and DA: The boundary conditions on fin surface are coupled boundary with PCM. In this study, we considered five different wall temperatures: Tw = 60 °C, 65 °C, 70 °C, 75 °C, and 80 °C. The initial conditions: The initial temperature of all calculated regions is consistent, T|𝑡=0 = T𝑓 |

𝑡=0

=20 oC

(13)

The numerical solution was obtained by use of the solidification & melting model in the Fluent 6.3.26 software package. Structured grid is applied in this study, as shown in Fig 3, the cells are refined in the vicinity of the inner tube and fins to obtain the velocity and temperature gradients accurately. Next, we conducted a grid-independence study by taking n = 0 as an example. Fig. 4 shows the calculation results for grid numbers of 63000, 125730 (double), and 252000 (quadruple). From the result, we can conclude that increasing the number of grids to 125730 yields an acceptable accuracy while the computational time is also economical. For configuration with fins, the same method was also applied to confirm the appropriate grid number is about 132400.

12

Fig. 3 Computational mesh of the simulation domain 0.60

0.55

0.50

10min 20min

f

0.45

0.40

0.35

0.30

0.25 50000

100000

150000

200000

250000

grid number

Fig. 4 Results of grid number independence study The momentum and energy equations utilized a second-order upwind difference grid for discretization. The under-relaxation factors were 0.3, 1, 0.7, 0.9, and 1, corresponding to pressure, density, body forces, and momentum, respectively, and the calculation time step was taken as 0.02 s. The maximum number of iterations for each time step was set to 20, which satisfies the convergence criteria of the energy equation (10-6) and the speed (10-3). To verify the numerical simulation, we compared the experimental results for the phasechange unit without fins under the condition of Tw = 80 °C. Fig. 5 shows the photograph of the experimental system. The inner annulus is heating surface and electrical heating wire is placed inside it, and the temperature was controlled by a PID controller. A glass observation window was installed in front of the heat transfer unit to observe changes in the phase interface. The other parts were constructed of stainless steel. Based on interface tracking method, the location 13

of the solid/liquid phase interface was photographed every 5 min and obtained the pattern of the change in liquid fraction with time. The ambient temperature was 20 °C.

(a) Photograph of experimental system used in our study

(b) Front view of the unit Fig. 5 Experimental system utilized in the study The aforementioned numerical calculation methods are validated via comparing liquid fraction during melting process between simulation and experimental results. Fig. 6 shows the comparison of f obtained from the experimental and numerical simulation results. Based on Fig. 6(a), we know that the melting rate obtained from the numerical simulation is higher than that obtained from the experiment, which is due to the following reason: (1) At the beginning of the experiment, there is a period that the temperature of heating surface reaches the preset temperature (80 oC), (2) and there is thermal contact resistance between solid PCM and heating surface. The numerical simulation ignores these effects that lead to a fixed difference between the results of the numerical simulation and that of experiment, if the difference is 14

subtracted, namely, the Fo of Fig. 6(a) for experiment results is shifted to the left, resulting in Fig. 6(b). The simulation and experimental curves coincide with each other as shown in Fig. 6(b) (error≤6%). So we concluded the aforementioned numerical calculation models can be used to study the melting characteristics of the PCM in an annular unit.

0.6

0.6

f

0.8

f

0.8

0.4

0.4

Experimental results Numerical simulation

Experimental results Numerical simulation

0.2 0.02

0.04

0.06

0.08

0.10

0.12

0.2 0.02

0.04

0.06

Fo

Fo

(a)

(b)

0.08

0.10

Fig. 6 Comparison of f obtained from the experiment and the numerical simulation 3. Analysis of results 3.1. Melting rate Fig. 7 reflects the variation in the liquid fraction when the heated wall temperature is 80°C. We note that the liquid fraction increases over time but the rate gradually reduces. With increase in the fin number, the liquid fraction of the PCM increases, but the incensements are less obvious. We remark here that adding fins also increases the cost and reduces the effective volume of the unit. As a result, the melting rate is the combinational result of the heat transfer temperature gap and the effective heat transfer area.

15

1.0

melting fraction

0.8

0.6

n=0 n=4 n=6 n=8 n=10 n=12

0.4

0.2

0.0 0.00

0.05

0.10

0.15

0.20

Fo

Fig. 7 Variation in liquid fraction over time (tw = 80 °C) Next, we defined the melting enhancement ratio r as the melting fraction ratio before and after the fins were added. This ratio is utilized to study the heat transfer enhancement characteristics of fins at different stages of the melting process. Parameter R is expressed as

𝑅=

𝑓0 𝑓

(14)

In this equation, 𝑓0 and 𝑓 represent the liquid fractions before and after the addition of fins at the same wall temperature. The change in 𝑅 over time is shown in Fig. 8 for different heated wall temperatures. Lower wall temperatures correlate to larger R values; for a given wall temperature, a larger number of fins is correlated with a larger value of 𝑅. The variation in R indicates that the presence of fins is more significant in the early period of the melting process. In particular, in the early melting stage corresponding to the Fourier number interval Fo < 0.0085 in Fig. 8, the same number of fins results in the same R value although the wall temperatures are different. The melting fraction is only affected by the number of fins, and a larger number of fins results in a larger value of R. The calculation results show that the temperatures of the fins reach the wall temperature rapidly and increase the heat transfer area, and therefore, the melting rate is mainly affected by the fin number in this period that is dominated by heat 16

conduction. As melting progresses, R under each condition increases and reaches its highest value and gradually decreases. In the late stage of the melting process corresponding to Fo > 0.15, R depends mainly on the wall temperature, and the effect of the number of fins becomes very small. In general, a larger number of fins can result in a faster melting rate based on the heat transfer efficiency enhancement ratio shown in Fig. 8, and when the fin number reaches a certain value (n = 10 and 12), the enhancement in the melting rate is limited. Simultaneously, at a lower heated wall temperature, the fins exhibit a larger influence on the melting rate. As the rise of inner wall temperature, the differences of R due to fins number decreases, therefore, the optimal number of fins should be determined by considering the heated wall temperature. Fo=0.0085

Fo=0.15

5.0

melting ehancement ratio

4.5 4.0 3.5

n=10 n=12 n=8 n=6 n=4

3.0 Tw=60 oC

2.5 Tw=70 oC

2.0 1.5 Tw=80 oC 1.0 0.00

0.05

0.10

0.15

0.20

0.25

Fo

Fig. 8 Variation in melting enhancement ratio as function of number of fins Fig. 9 shows the temperature and flow field distributions for different numbers of fins. The results indicate that because of the increased heat transfer area, in the initial stage, fins facilitate conduction by diffusing heat from the heated wall to the PCM, and the temperature of the PCM around the fin is relatively high. The PCM melts at both the upper part of the tube and the area around the fins. Due to the density difference in the fluid, the hot fluid flows upward while cold fluid flows down, and this process immediately generates vortexes. 17

Because of the increased flow, the melting rates are greatly improved in the upper parts. Since the flow is significantly weaker in the lower part, the phase interface moves slowly. With the melting of the PCM, smaller vortexes gradually merge and form larger ones. The fins slightly suppress the expansion of the vortexes. However, because of the existing local temperature difference, the addition of fins still enhances the melting rate. As the liquid fraction increases (e.g., at 30 min in Fig. 9), the temperature distribution is uniform, and the effect of the fin in enhancing heat transfer is very weak at this point. At a later stage in the melting process, because the fins are located in the liquid phase region with a homogeneous temperature distribution, the melting rate is essentially unaffected. Therefore, an appropriate increase in the fin height and area at the bottom of the ring can strengthen the heat conduction in this late stage.

18

n=0

n=4

n=6

n=8

n=10

3min

15min

30min

Fig. 9 Temperature and flow field distributions in melting process

19

n=12

3.2. Heat transfer characteristics The temporal heat transfer coefficient can be obtained as ℎ = 𝐴(𝑇

𝑄

, where Q denotes

𝑤− 𝑇𝑚 )

the instantaneous heat transfer capacity and A the area of the heating wall. In order to investigate the variation in the heat transfer performance during the melting process, we define the average heat transfer coefficient as follows: 𝑄(∆𝑡) ℎ̅ = 𝐴(𝑇 𝑇 )∆𝑡

(15)

𝑤− 𝑚

Here, 𝑄(∆𝑡) denotes total heat transfer to the PCM from the start of heating in every elapsed time ∆𝑡 (in this study,∆𝑡 = 1 min). It includes three parts: the latent heat that is absorbed by the PCM during the phase transformation, the temperature rise in the solid PCM to its melting point and in superheating of the liquid phase, the sensible heating of the solid phase in additional. We ignore the sensible heat absorbed by the solid and liquid material because it is much smaller than latent heat. Further, A denotes the total heat transfer area. Without fins: A = 2π𝑟𝑖 With fins: A = 2π𝑟𝑖 + 2𝑛ℎ𝑓 , The temporal Nusselt number can be expressed as 𝑁𝑢 =

ℎ𝐷

(16)

𝑘𝑙

Next, the time-averaged Nusselt number is defined as ̅̅̅̅ 𝑁𝑢 = ℎ̅𝐷/𝑘𝑙

(17) n=0 n=4 n=6 n=8 n=10 n=12

50

40

Nu

30

20

10

0 0

20

40

60

80

Time(min)

̅̅̅̅ Fig. 10 Variation in time-averaged Nusselt number (𝑁 𝑢 ) with time (Tw = 80 °C) 20

Fig. 10 depicts the variation in ̅̅̅̅ 𝑁𝑢 before and after the addition of fins at the boundary temperature of 80 °C. Parameter ̅̅̅̅ 𝑁𝑢 decreases rapidly in the initial stage, which indicates that convection in the liquid region is relatively weak in the beginning of the melting period, and an increase in the liquid phase thickness increases the heat transfer thermal resistance and leads to decrease in the thermal exchange efficiency. At this point, ̅̅̅̅ 𝑁𝑢 is essentially the same for all fin numbers. Subsequently, natural convection phenomena begin to occur in the liquid phase, without the addition of fins in the phase-change unit, the value of ̅̅̅̅ 𝑁𝑢 remains stable or even increases because the effect of natural convection is enhanced in this liquid region, which overcomes the thermal resistance between the thermal boundary and phase interface. This result indicates that the heat transfer mechanism is dominated by natural convection. Parameter ̅̅̅̅ 𝑁𝑢 is relatively small after adding fins, and it decreases with increasing number of fins. Here, we speculate that the fins reduce heat convection from the vertical direction to the wall surface and weaken the intensity of natural convection. With increase in the liquid fraction and a more uniform internal temperature distribution, the natural convection intensity is weakened. During this period, the heat transfer mechanism is dominated by heat conduction. With gradual increase in thermal resistance, ̅̅̅̅ 𝑁𝑢 continues to decline and finally remains at a small value until the end of the melting process. The time-average Nusselt number is correlated with dimensionless parameters including the Fo number, Ste number, and Ra number as well as the number of fins. Fig. 11 shows the curve of ̅̅̅̅ 𝑁𝑢 after normalization on the basis of simulation data.

21

50

40

Nu

30

20

10

0 0

2

4

6

8 0.2

10

12

14

16

0.8

FoSteRa (1+N)

Fig. 11 ̅̅̅̅ 𝑁𝑢 as a function of FoSteRa0.2(1 + n)0.8 (1.33×107≤ Ra≤4.48×107, 0.21≤ Ste≤ 0.47) Fig. 12 shows the change in wall heat flux at the boundary temperature of 80 °C. In the initial stage, the fins increase the area of heat conduction, so a larger number of fins results in a larger wall heat flux. As dominant heat transfer mechanism converts from heat conduction to natural convection, PCM containers respond differently to the heat transfer efficiency, with and without fins (as shown in Fig. 10). So the heat transfer enhancement effect of fins reduces and heat fluxes decreases rapidly, while the wall heat flux in PCM container without fins keeps in stably even rises slightly in this stage. With the progression of the melting process, n = 0 yields the maximum wall heat flux, and the difference in heat flux for different n values significantly narrows in comparison with the previous period. In the late melting stage, the heat flux is very small for any condition because the heat resistance is the maximum and heat transfer enhancement measure has no effect.

22

n=0 n=4 n=6 n=8 n=10 n=12

2500

2000

q(W)

1500

1000

500

0 0

10

20

30

40

50

60

Time(min)

Fig. 12 Change in wall heat flux over time during heat transfer process (Tw = 80 °C) To further understand the influence of fins on the whole melting process, we depict the average Nusselt number (ave Nu) and the average heat flux (ave q) during the entire melting process in Fig. 13. Because an increase in the wall temperature results in an increase in natural convection in the unit, the two parameters increase accordingly. With an increase in the fin number, the average Nusselt number decreases gradually, whereas the average wall heat flux increases at first, and its maximum value is reached when n = 10 under this condition. Next, the average wall heat flux starts to decrease with an increase in the number of fins, thereby suggesting that the positive effect on the heat exchange efficiency induced by an increased heat transfer area is not sufficient to overcome the negative effect induced by the weakened natural convection intensity. In addition, an excessive number of fins will reduce the effective volume of the PCM. Therefore, it can be concluded that the strategy of adding fins to enhance the phasechange heat transfer should be determined based on the specific thermal boundary conditions; that is, each specific boundary temperature has an optimal number of fins. For the boundary temperature examined in this study, the optimum fin number is 10. In addition, according to 23

the trend shown in the curve of the graph, the optimal number of fins will increase with increasing boundary temperature.

600 Ave Q Ave Nu

60oC 80oC

70oC 70oC

80oC 60oC

65oC 75oC

14

75oC 65oC

500

Ave Nu

Ave q (W)

12 400

300 10 200

100

8 4

6

8

10

12

fin numbers

Fig. 13 Variation in average wall heat flux and average Nu with fin numbers 4. Conclusion In this study, we numerically simulated the melting process in a horizontal annular phasechange unit with longitudinal fins. We analyzed the effect of the fin number on the melting ̅̅̅̅ rate, time-average Nusselt number (𝑁 𝑢 ), and wall heat flux for different wall temperatures. We examined the heat transfer enhancement characteristics due to fin addition and determined the optimal fin number. Fins can effectively shorten the melting time of PCMs; the larger the fin number, the lesser is the melting time. However, under the same wall temperature, with an increase in the fin number, the increase in melting rate declines gradually. The heat transfer enhancement is more effective for a lower wall temperature. Based on the temperature distribution, flow field distribution, and variation in ̅̅̅̅ 𝑁𝑢 , we can conclude that in the initial stage of the melting process, fins spread the heat quickly to the 24

surrounding area owing to high thermal conductivity of fin. The heat transfer rate increases with increase in the number of fins. With increase in the liquid fraction, natural convection is intensified, and fins weaken the intensity of natural convection to some extent. In the late stage of melting, fins in the liquid regions with uniform temperature distributions provide little heat transfer enhancement. The dominant heat transfer mechanism is heat conduction, and the heat transfer rate is mainly affected by the boundary temperature. Fins can increase the heat transfer area but can also lead to a decline in the heat transfer coefficient. For a specific wall temperature, the fin number can be optimized. If the number of fins exceeds the optimal value, the heat flow will decrease. In this study, when the fin number was more than 10, the enlarged heat transfer area was found insufficient to overcome the effect of the decreased transfer coefficient; this resulted in a decrease in the average heat flux, and therefore, we determined that the optimal fin number was 10. Acknowledgements The work was supported by the National Natural Science Foundation of China (Grant Nos. 51378426 and 51678488) and the Sichuan Province Youth Science and Technology Innovation Team of Building Environment and Energy Efficiency (Grant No. 2015TD0015).

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