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Performance evaluation and analysis of a vertical heat pipe latent thermal energy storage system with fins-copper foam combination
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Performance evaluation and analysis of a vertical heat pipe latent thermal energy storage system with fins-copper foam combination ⁎
Chunwei Zhang, Yubin Fan, Meng Yu, Xuejun Zhang , Yang Zhao Key Laboratory of Refrigeration and Cryogenic Technology of Zhejiang Province, Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China
H I GH L IG H T S
fins with copper foam in a vertical HP LTES system to enhance heat transfer. • Combine influence of three key parameters on heat transfer performance is investigated. • The exergy analysis is used to assess the conduction and convection. • The • The evolution of solid-liquid interfaces and isotherms is analyzed.
A R T I C LE I N FO
A B S T R A C T
Keywords: Latent thermal energy storage Phase change material Combination enhancement Heat pipe Fin Copper foam
As one of the most promising thermal storage techniques, latent thermal energy storage (LTES) using phase change materials (PCMs) is gaining importance. However, as is well known, most PCMs still suffer from the poor thermal conductivity for both the solid and liquid phases. To overcome this drawback, a combination of fins and copper foam based on the heat pipe (HP) heat exchanger is proposed and applied. The thermal performances, including the liquid fraction, the exergy transfer rate, the evolution of solid-liquid interfaces, and temperature distributions are evaluated by a two-dimensional numerical model considering natural convection. The results indicate that the combination shows a greater enhancement performance than using either copper foam alone or fins alone, and the best enhancement can be achieved with a specific fin volume ratio (γfin = 0.5) when the total volume fraction is fixed. Besides, the exergy analysis further suggests that increasing the total volume fraction accelerates the phase change rate by improving the effective thermal conductivity, but meanwhile, the impact of natural convection is gradually reduced. This study illustrates an effective method for enhancing the heat transfer of LTES by combining different techniques.
1. Introduction Thermal energy storage (TES) plays a critical role in optimizing thermal processes and overcoming the mismatch between energy supply and demand. Generally, there are three kinds of TES methods, including sensible thermal energy storage, latent thermal energy storage, and thermochemical energy storage. The LTES with PCMs as the storage media has the advantages of high energy storage density, small storage volume, and nearly isothermal storage [1–3] and has received increasing attention in previous decades. However, most PCMs are characterized by the low thermal conductivity [4], substantially limiting the large-scale application of LTES. To improve the thermal performance of LTES systems, numerous investigations on the heat transfer enhancement techniques have been conducted. Most of them can be classified into three ways: extending the heat transfer area, improving ⁎
thermal conductivity, and using the multi-stage or cascaded LTES technique [5]. Among the techniques above, the researches about fins [6–8] and the HP [4,9] are quite common, which can augment the heat exchange area between the phase change material (PCM) and the heat transfer fluid (HTF) significantly, leading to the increase of heat flux. Kamkari and Shokouhmand [10] reported that, by immersing one and three fins into PCMs, the total melting time were reduced by 18% and 37%, respectively, relative to the enclosure without the fin. Influences of the geometry of fins have also been fully explored. Sharifi et al [11] simulated the melting of the PCM in a HP LTES system and performed a parametric study. Results showed that melting rates were significantly higher than those associated with the rod or tube. Whereas using metal foams [12,13] or high thermal conductivity particles [14,15] can be resorted to increasing the thermal conductivity, improving heat transfer
Corresponding author. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.applthermaleng.2019.114541 Received 20 May 2019; Received in revised form 23 September 2019; Accepted 15 October 2019 1359-4311/ © 2019 Published by Elsevier Ltd.
Please cite this article as: Chunwei Zhang, et al., Applied Thermal Engineering, https://doi.org/10.1016/j.applthermaleng.2019.114541
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Nomenclature
ts tc T Tm Tlower Tupper ΔT
Abbreviations HP HTF PCM LTES TES
heat pipe heat transfer fluid phase change material latent thermal energy storage thermal energy storage
u v → U
Symbols asp A CF c df dp E E¯ fl g hsp H k K L M P q Pr rHPi rHPo Re Rtotal t tm
specific coefficient surface area of metallic foams (m2) inner surface area of the thermal energy storage section of the HP inertia coefficient (m2) specific heat (J/kg K) ligament diameter (m) pore diameter of metal cells (m) exergy (J) average exergy transfer rate (J/min) liquid fraction gravity acceleration (m/s2) interfacial heat transfer coefficient (W/m2 K) height (m) thermal conductivity (W/m K) permeability of porous foam (m2) latent heat of fusion (J/kg K) mushy zone constant pressure (Pa) boundary heat flux (W/m2) Prandtl number inner radius of the HP (m) outer radius of the HP (m) Reynolds number total thermal resistance of the HP (W/K) time (s) melting time (s)
solidification time (s) complete cycle time (s) temperature (K) melting temperature lower PCM melting temperature upper PCM melting temperature difference between the HTF temperature and the melting point of PCM velocity in r-direction (m/s) velocity in z-direction (m/s) velocity vector (m/s)
Greek symbols ρ φt γfin η µ ε ω ξ
density (kg/m3) total volume fraction fin volume ratio thermal expansion coefficient (1/K) dynamic viscosity (kg/m s) copper foam porosity pore density (PPI) a small number (0.0001)
Subscripts c cf d eff envir i int l n pcm ref s t
charging copper foam discharging effective value environment thermal element counter initial liquid phase of PCM total number of discretized cells phase change material reference solid phase of PCM total
However, as could be found, the combination might not always lead to a better enhancement effect. Therefore, it is still worth to propose a novel combination method considering the merits of each technique reasonably and explore its thermal performances, which are the aim of this study. In consequence, the combination of fins and copper foam based on the HP is proposed and investigated during the complete cycle that consists of melting and solidification processes. Furthermore, the exergy analysis is also carried out to assess the heat transfer performance.
in the LTES system effectively. Esapour et al. [16] found that, in a multi-tube heat exchanger, by inserting the metallic foam with porosities of 0.9 and 0.7, the melting time were reduced by 14% and 55%, respectively. Li et al. [17] adopted multi-wall carbon nanotubes (MWCNTs) to improve the heat transfer performance of palmitic acid. It was found that the thermal conductivity of composite phase change material was increased by approximately 30% by adding 1 wt% MWCNTs. The enhancement effect by increasing the heat exchange area alone or improving the thermal conductivity alone is relatively limited in achieving a rapid thermal response of LTES system. Therefore, the combination of different techniques, aiming to make a further enhancement, has attracted increasing attention recently. Xie et al. [18] conducted an experimental study and reported that the effective thermal conductivity of the PCM enhanced with the fins-copper foam combination is 2.7 times as that of the PCM enhanced with copper foam only. Sharifi et al. [19] indicated that melting and solidification rates of HP-foil configurations were increased by about 3 and 9 times compared to configurations involving the rod with no foil, respectively. Feng et al. [20] found that the enhancement performance of finned metal foam is superior to plate fin and metal foam structures by numerical simulation. Mahdi and Nsofor [21,22] examined the effects of three techniques, including fins, nanoparticles, and the combination of both numerically. Results indicated that the application of fins alone is more effective than using other techniques during both melting and solidification.
2. Physical model Fig. 1 shows the schematic diagram of the vertical HP LTES unit with the combination. It can be observed that the HP, which is concentrically positioned in a cylindrical enclosure, is subdivided into three axial regions according to different functions. The middle section of the HP is encased in a PCM enclosure, as the thermal energy storage section. The bottom and top sections of the HP are exposed into the HTF with a specified temperature in different operation modes. During the melting process, the bottom section of the HP absorbs thermal energy as the evaporator, and the top section is assumed to be adiabatic. During the solidification process, the top section of the HP releases thermal energy as the condenser, and the bottom section is assumed to be adiabatic. The details of the two fundamental modes used can be found in Ref. [23]. In addition, the PCM enclosure walls, as well as the top and 2
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Fig. 1. Schematic illustration of the vertical HP LTES system with the combination: (a) physical model, (b) computational domain.
temperature is set to 638 K (578 K) which is about 30 K above (below) Tm. As a result, the temperature difference (ΔT) is equal to 30 K, and this value is dependent on the temperature of the HTF since the melting point (Tm) is constant. Moreover, the mass flow rate of the HTF is also constant. The volume fraction [21] is used to make volume usage comparable. Specifically, φt represents the total volume fraction of fins and copper foam. Additionally, to investigate the effect of proportions of fins and copper foam in the combination, a fin volume ratio, γfin, is defined. These parameters can be calculated as follows:
bottom end caps of the HP, are considered adiabatic. The initial geometric dimensions of the HP and fins are given as follows. The heights of three axial regions, HC, HTES, and HD are set to 60 × 10−3 m, 75 × 10−3 m, and 60 × 10−3 m, respectively. The inner radius of the HP, rHPi, and wall thickness, bHP, are set to 2 × 10−3 m and 2 × 10−3 m, respectively. The inner radius, outer radius, and thickness of each fin are set to 4 × 10−3 m, 21 × 10−3 m, and 3 × 10−3 m, respectively. The pore density of the copper foam is set to 20 PPI. The inner radius of the PCM enclosure is equal to the outer radius of the HP. The PCM volume is constant, 2.14 × 10−4 m3, for all cases. The physical properties of the PCM [24], fins, and copper foam [21] are displayed in Table 1. Besides, in the melting (solidification) process, the initial PCM temperature is set to 603 K (613 K), the HTF
φt =
3
Vfin + Vcf Vpcm + Vfin + Vcf
(1)
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e. Energy equation for copper foam:
Table 1 Physical parameters used in the numerical modeling. Properties
PCM [24]
HP wall, fin and metal foam [21]
Material Density (kg/m3) Thermal conductivity (W/m K)
KNO3 2109 0.425 (liquid) 0.5 (solid) 953 2.59 × 10−3 608 200 × 10−6
copper 8920 400
95 × 103
N/A
(1 − ε )
∂ (ρcf Tcf ) ∂ k cf,eff ∂Tcf 1 − ε ∂ k cf,eff ∂Tcf = (1 − ε ) ( )+ (r ) − ST ∂t ∂z ccf ∂z r ∂r ccf ∂r (7)
Specific heat (J/kg K) Dynamic viscosity (Pa s) Melting point (K) Thermal expansion coefficient (K−1) Latent heat of fusion (J/kg)
The three source terms above are given as follows:
→η (T − T ) − ( μ + ρ C |→ 2 3 Su = ρg ref l F U |) u − M [(1 − fl ) /(fl + ξ )] u K
380 N/A N/A N/A
Sv = −(
→ μ + ρCF |U |) v − M [(1 − fl )2 /(fl3 + ξ )] v K
ST = hsp asp (Tcf − Tpcm )
(8) (9) (10)
→ + ρ l CF |U |) u
γfin =
Vfin Vfin + Vcf
μ → where ρg η (T − Tref ) is the Boussinesq approximation, ( K → μ and ( K + ρCF |U |) v are the Forchheimer-Darcy terms, and are employed to describe the additional flow resistance, M [(1 − fl )2 /(fl3 + ξ )] is the source term for damping the velocity in the solid or mushy zone. The effective thermal conductivity of the copper foam, kcf,eff, and the effective thermal conductivity of the PCM, kpcm,eff, are calculated as same as the work of Tian and Zhao [27]. fl is the liquid fraction and can be described as follows:
(2)
where Vfin, Vcf, and Vpcm are the volume of fins, copper foam, and PCM, respectively. 3. Mathematical description In the mathematical model, the effective heat capacity method [25,26] is utilized to simulate the melting and solidification of the PCM. The two-temperature energy equation model is used to express thermal non-equilibrium between the PCM and the copper foam. The natural convection flow in the liquid PCM is modeled according to the Boussinesq approximation. Besides, some assumptions are made and listed as follows:
⎨ ⎩ (T − Tlower )/(Tupper − Tlower ), Tlower ⩽ T ⩽ Tupper
3.1. Parameters of copper foam The characteristics of the copper foam can be described by some pore morphological parameters [16], which can be represented as follows:
df dp
Based on these assumptions, the continuity, momentum, and energy equations [16] can be expressed as follows:
= 1.18
dp =
1 ∂ (ρ l v ) 1 ∂ (ρ l uv ) 1 1 ∂ (rρ l v 2) + 2 + 2 × ε ∂t ε ∂z ε r ∂r μ ∂P ∂ μl ∂v 1 ∂ μl ∂v =− + ( )+ (r ) − 2l v + Sv ∂r ∂z ε ∂z r ∂r ε ∂r r
(4)
asp =
∂t ∂ k pcm,eff =ε ( ∂z c pcm
(15)
3πdf (1 − e−(1 − ε )/0.04 ) (0.59dp)2
0.4 Pr 0.37 k / d (1 ⩽ Re ⩽ 40) 1 f ⎧ 0.76Re ⎪ (40 ⩽ Re ⩽ 1000) hsp = 0.52Re 0.5 Pr 0.37 k1/ df ⎨ ⎪ 0.26Re 0.6 Pr 0.37 k1/ df (1000 ⩽ Re ⩽ 20, 000) ⎩
(5)
d. Energy equation for PCM:
1 ∂ (rρpcm vTpcm ) r ∂r ∂z ∂Tpcm ε ∂ k pcm,eff ∂Tpcm )+ (r ) + ST c pcm ∂r ∂z r ∂r
0.0254 ω
(14)
In Eq. (10), asp is the specific coefficient surface area of the copper foam, hsp is the interfacial heat-transfer coefficient between the liquid PCM and the copper foam, and can be calculated by using the empirical correlation [28]:
c. Momentum equation in z-direction:
∂ (ρpcm uTpcm )
(13)
−1.63
df CF = 0.00212(1 − ε )−0.132 ⎛⎜ ⎞⎟ ⎝ dp ⎠
(3)
1 ∂ (ρ l u) 1 ∂ (ρ l u2) 1 1 ∂ (rρ l uv ) + 2 + 2 × ε ∂t ε ∂z ε r ∂r ∂P ∂ μl ∂u 1 ∂ μl ∂u =− + ( )+ (r ) + Su ∂z ∂z ε ∂z r ∂r ε ∂r
+
(12)
−1.11
b. Momentum equation in r-direction:
∂ (ρpcm Tpcm )
1−ε ⎛ 1 ⎞ ε ⎝ 1 − e−(1 − ε )/0.04 ⎠
df K = 0.00073(1 − ε )−0.224 ⎛⎜ ⎞⎟ d dp2 ⎝ p⎠
a. Continuity equation:
ε
(11)
The melting and solidification processes of the PCM are assumed to take place over a given temperature range (Tlower ~ Tupper), where Tlower and Tupper are set to be 0.5 K lower and higher than the melting point Tm, respectively.
• The contact thermal resistances are negligible. • The structure of the copper foam is considered to be homogeneous and isotropic. • The volume variation of the PCM during the phase change process is neglected. • The flow of the liquid PCM is incompressible and laminar. • No-slip boundary conditions for velocities at walls are applied.
∂u 1 ∂ (rv ) + =0 ∂z r ∂r
T < Tlower T > Tupper
0, 1,
⎧ fl =
(16)
(17)
It is noted that only conduction heat transfer exists in the solid PCM. As a result, the interfacial heat transfer coefficient is not applicable at that time. To solve this problem, hsp in Eq. (10) is replaced with kpcm-cf /rp when Re < 1, where rp is the pore radius of metal cells, kpcm-cf is the effective thermal conductivity between the PCM and the copper foam.
+
(6) 4
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3.2. Thermos-physical properties of PCM
Ec n
=
The thermos-physical properties of the PCM (liquid or solid) used in the simulation can be calculated by using the formulations provided by Alipanah and Li [29], as follows:
⎧ ρpcm =
c pcm
⎨ + − ⩽ ⩽ ⎩ ρs (ρ l ρs ) fl , Tlower T Tupper
c s, T < Tlower ⎧ ⎪ c l, T > Tupper = ⎨ c l + cs L ⎪ 2 + (Tupper − T lower ) , Tlower ⩽ T ⩽ Tupper ⎩ ⎧
k pcm =
i=1
⎨ ⎩ k s + (kl − k s) fl , Tlower ⩽ T ⩽ Tupper
n
= (18)
) + L (fi − fint )(1 −
Tenvir ⎤ ) Tm ⎥ ⎦
⎣
Tint T ) + L (fint − fi )(1 − envir )⎤ Ti,pcm Tm ⎥ ⎦ (26)
(19)
where L is the latent heat, Tenvir is the environment temperature, and n denotes the total number of discretized cell volumes of the PCM in the computational domain. Moreover, an average exergy transfer rate over the melting or solidification process is defined to compare the thermal performances quantitatively. It can be calculated as follows.
(20)
E¯ =
Ec
⎧ tm
Melting
⎨ Ed ⎩ ts
Solidification
(27)
where tm and ts are the time to reach complete melting and solidification, respectively.
The heat transfer through the HP is simplified with the thermal resistance network model and is incorporated into the boundary conditions. The mathematical details of the thermal resistance network model can be found in Ref. [30]. As a consequence, the boundary conditions [31] can be described as follows:
∂T [rHPi, z , t ] T − T [rHPi, z , t ] = HTF ∂r Rtotal
∑ mi ⎡⎢cpcm (Tint − Ti,pcm − Tenvir ln i=1
3.3. Boundary conditions
− kA
Tint
Ed
T < Tlower T > Tupper
k s, kl,
⎣
Ti,pcm
(25)
T < Tlower T > Tupper
ρs , ρl ,
∑ mi ⎡⎢cpcm (Ti,pcm − Tint − Tenvir ln
4. Numerical procedure A MATLAB program is developed in this study. The governing equations are discretized with the finite volume approach, and a fullyimplicit scheme [33] is adopted for the time term. The corresponding algebraic equations are solved numerically by the Gauss-Seidal Iteration in combination with a super-relaxation technique. The SIMPLEC algorithm, combined with the collocated grid arrangement, is employed for the pressure correction equation. The convergence criteria for the continuity, momentum, and energy equations are set to 10−5, 10−5, and 10−6, respectively. All simulations are conducted on a computer, which features 3.40 GHz quad-core Intel Xeon processors. The numerical model is validated against the experimental work of Allen et al. [9] when the volume of fins is set to zero. They investigated the melting and solidification of the PCM in a HP configuration with metal foam. Fig. 2 shows the comparison of the temperature profiles of the melting and solidification processes between the present work and the experimental data under the same conditions. As can be observed, the simulated results are in good agreement with the experimental data, which indicates the proposed model can be used for further analysis. The sensitivity studies of grid size and time step based on the liquid fraction have been carried out to obtain an independent solution. The geometric parameters of the physical model (φt = 0.10) are used as an input to the sensitivity studies. Three grid sizes of 7500 cells, 11,250
(21)
∂T [rHPo, z , t ] =0 ∂r
(22)
∂T [r , 0, t ] =0 ∂z
(23)
∂T [r , HTES, t ] =0 ∂z
(24)
where A is the inner surface area of the thermal energy storage section of the HP, Rtotal is the total thermal resistance related to the HP. 3.4. Exergy analysis To further analyze the thermal performances of the combination, an exergy analysis [32] on the complete cycle is performed is applied. The net exergy charged or discharged at any instant is calculated based on the second law of thermodynamics as follows:
Fig. 2. Comparison of the temperature profiles between the present work and the experimental data by Allen et al. [9]. 5
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become shorter. This is attributable to the increase of height augments the heat transfer area between the HP walls and the PCM. Meanwhile, the flow resistance is also reduced due to the increase of the gap between the fins, consequently boosting the natural convection. As a result, the heat can be transferred to the PCM more quickly. Moreover, the initial variations of the liquid fraction do not show a significant difference, which means that the natural convection has not enough impact at that time. In summary, a larger number of fins, a higher PCM enclosure, or a lower porosity could induce a higher phase change rate for both melting and solidification when the PCM volume is fixed. The same advantage of them is that the heat transfer area can be increased substantially. Thus, these three parameters are significant and need to be considered in the design of LTES systems.
cells, and 15,000 cells and three time steps of 0.02 s, 0.05 s, and 0.1 s are selected and tested. Comparison results are shown in Fig. 3. As a result, a grid size of 11,250 cells and a time step of 0.05 s are finally chosen because more grids or smaller time steps do not produce appreciable improvement in accuracy.
5. Results and discussion 5.1. Effects of parameters in the combination
5.2. Characteristics of the combination The problem of determining what proportions of fins and copper foam to use in the combination is crucial because the heat transfer enhancement performance varies with the volume proportion. Therefore, the effect of the fin volume ratio (γfin) is investigated to obtain an optimal combination design. The fin volume ratio can be adjusted by changing the number of fins or the copper foam porosity. In addition, the HTF temperature is also considered, because a greater temperature gradient can accelerate the melting and solidification processes. The PCM enclosure height (HTES) is set to 0.075 m. In consequence, as shown in Fig. 7, the complete cycle time of cases with different fin volume ratios are calculated under the conditions of two total volume fractions (φt = 0.07 and 0.10) and two temperature differences (ΔT = 30 K and 50 K). It can be found that the best enhancement performance can be obtained with a specific fin volume ratio (γfin = 0.5), and this ratio is not subject to temperature differences. Also, the use of the combination is better than using either copper foam alone (γfin = 0) or the fins alone (γfin = 1) under the same total volume fraction limit. 5.3. Performance comparison of cases with different total volume fractions The duration of the phase change process can estimate the charge or discharge rate of thermal energy, while the exergy can evaluate the quantity and quality of thermal energy simultaneously. They are two useful parameters for investigating the characteristics of the HP LTTS system with the combination. Three different total volume fractions (φt = 0, 0.07 and 0.10) are selected for testing, and higher total volume
1.0
1.0
0.8
0.8 Liquid fraction
Liquid fraction
To investigate the influence of fins, copper foam, and the HP on the heat transfer performance, the number of fins (Nfin), copper foam porosity (ε), and PCM enclosure height (HTES), are selected as three key parameters. Other parameters, such as fin length and pore density, have a relatively little effect on heat transfer, which has been proved by numerous researchers. That is the reason why they are considered to be constant in this study. The histories of the liquid fraction are tracked and analyzed to evaluate the heat transfer performance in both melting and solidification processes, and the results are shown in Figs. 4–6, respectively. Fig. 4 displays the liquid fraction evolution of the cases with three different fins numbers (Nfin = 1, 3, and 5). The copper foam porosity (ε) and the PCM enclosure height (HTES) are set to 0.93 and 0.075 m, respectively. It can be seen that, as the number of fins increases, the phase change time required decreases sharply. The melting and solidification time in the case (Nfin = 5) are 3.2 min and 4.0 min, respectively, while their values are almost 8.3 min and 9.8 min in the case (Nfin = 1). It is because increasing the number of fins expands the heat transfer area and intensifies the thermal penetration. Also, melting and solidification rates gradually abate as time progresses, owing to the decrease of the temperature difference and the weakening of natural convection. Fig. 5 compares the liquid fraction evolution of the cases with three different porosities (ε = 0.96, 0.93, and 0.90). The number of fins (Nfin) and the PCM enclosure height (HTES) are set to 3 and 0.075 m, respectively. As shown in the figure, as the porosity decreases, melting and solidification rates also augment obviously, reducing the phase change time. The primary reason is that the heat transfer area and the effective thermal conductivity are enhanced. Fig. 6 presents the liquid fraction evolution of the cases with three different PCM enclosure heights (HTES = 0.055 m, 0.075 m, and 0.095 m). The number of fins (Nfin) and the copper foam porosity (ε) are set to 3 and 0.93, respectively. Because the PCM volume is fixed, the enclosure outer radius changes accordingly in different cases. It is found that, as the height increases, both the melting and solidification time
0.6
7500 Cells 11250 Cells 15000 Cells
0.4
0.02 s 0.05 s 0.1 s
0.4
0.2
0.2
0.0
0.6
0
1
2
3
4
0.0
5
0
1
2
3
Time (minutes)
Time (minutes)
(a)
(b)
Fig. 3. (a) Mesh independency. (b) Time step independency. 6
4
5
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1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
C. Zhang, et al.
0.6 İ= 0.93 , HTES= 0.075 m 0.4 Nfin = 1 Nfin = 3
0.2
0.0
2
4
6
Nfin = 1
0.6
Nfin = 3 Nfin = 5
0.4
0.2
Nfin = 5
0
İ= 0.93 , HTES= 0.075 m
0.0
8
0
2
4
6
8
10
Time (minutes)
Time (minutes)
(a)
(b)
1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
Fig. 4. Effects of the number of fins on liquid fraction during (a) melting and (b) solidification.
0.6 Nfin = 3 , HTES= 0.075 m 0.4 İ= 0.90 İ= 0.93 İ= 0.97
0.2
0.0
0
2
Nfin = 3 , HTES= 0.075 m İ= 0.90 İ= 0.93 İ= 0.97
0.6
0.4
0.2
4
0.0
6
0
2
4
6
Time (minutes)
Time (minutes)
(a)
(b)
1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
Fig. 5. Effects of the porosity on liquid fraction during (a) melting and (b) solidification.
0.6 Nfin = 3 , İ= 0.93 0.4
Nfin = 3 , İ= 0.93 HTES=0.055 m
0.6
HTES=0.075 m HTES=0.095 m
0.4
HTES=0.055 m HTES=0.075 m
0.2
0.0
0.2
HTES=0.095 m
0
2
4
0.0
6
0
2
4
Time (minutes)
Time (minutes)
(a)
(b)
6
Fig. 6. Effects of the enclosure height on liquid fraction during (a) melting and (b) solidification.
exergy are displayed in Tables 2 and 3, respectively. Note that all the melting time considering natural convection are shorter than those under the pure conduction condition, which suggests that the contribution of natural convection in the melting process is not negligible. The comparisons of the phase change time and the average exergy transfer rates are shown in Figs. 8 and 9, respectively. Fig. 8 illustrates that, as the total volume fraction increases, the melting and
fractions are not considered because the dominant heat transfer mechanism varies with the total volume fraction. The cases are calculated under two conditions of convection and conduction, which can analyze the improvement of the effective thermal conductivity and separate the impact of natural convection comparably. The initial temperature difference and the fin volume ratio (γfin) are set to 30 K and 0.5, respectively. The time to complete the phase change and the corresponding 7
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Fig. 9. Comparison of the average exergy transfer rates. Fig. 7. Complete cycle time in cases with different fin volume ratios.
that the solidification has a bigger enhancement potential. Fig. 9 shows that the average exergy transfer rates increase with the total volume fraction. Based on the definition above, the average exergy transfer rate under the pure conduction condition can indicate the degree of effective thermal conductivity. And the corresponding values in the cases (φt = 0.07 and 0.1) are much higher than that in the base case (φt = 0). Thus, the effective thermal conductivity is increased with the total volume fraction. Furthermore, it is also found that natural convection can significantly accelerate the melting process. For example, if the natural convection is considered, the average exergy transfer rate can be improved by 566% for the case (φt = 0), 41.90% for the case (φt = 0.07), and 21.59% for the case (φt = 0.10), respectively. Therefore, the effect of natural convection in the melting process gradually decreases with the increase of the total volume fraction, whereas it is almost negligible in the solidification process.
Table 2 Melting and solidification time of the cases under convection and conduction. Total volume fraction
φt = 0 φt = 0.07 φt = 0.10
Melting time (min)
Solidification time (min)
Convection
Conduction
Convection
Conduction
21.15 8.35 5.15
121.95 11.80 6.15
147 12 6.05
121.95 11.80 6.15
Table 3 Exergy changed and discharged of the cases under convection and conduction. Total volume fraction
φt = 0 φt = 0.07 φt = 0.10
Exergy charged (kJ)
Exergy discharged (kJ)
Convection
Conduction
Convection
Conduction
28.78 26.73 26.56
24.82 26.61 26.09
26.85 26.54 25.99
24.80 26.45 25.97
5.4. Evolution of solid–liquid interfaces The solid–liquid distributions of the three cases at different times (t = 1 min, 2 min, 3 min, 4 min, and 5 min) during melting and solidification are displayed in Figs. 10 and 11, respectively. During the melting process, the solid-liquid interface in the case (φt = 0) deforms remarkably due to the impact of natural convection. The liquid PCM accumulates in the upper part of the enclosure, and the melting front propagates downward. As a result, the liquid layer in the upper part is thicker than that in the lower part. After (t = 3 min), the liquid PCM has already occupied the entire upper part, but a large amount of solid PCM still exists in the lower part. The solid-liquid interface becomes shorter as time progresses; the impact range of natural convection decreases accordingly. Besides, the solid-liquid interface is quite clear, reflecting the thermal stratification caused by the high thermal resistance of pure PCM. In the other two cases (φt = 0.07 and 0.10), the solid-liquid interfaces show a different evolution. In the beginning, a thin liquid PCM layer occurs, and it is closely parallel to the fins and HP surfaces. As time progresses, the melting front propagates smoothly by absorbing more heat. Due to less deformation in the shape of the solid-liquid interfaces, the conduction is dominant in the heat transfer, and the convection may exert only a little effect. Meanwhile, the solid-liquid interface becomes wider. That is because the fins and copper foam exist simultaneously, higher thermal conductivity and stronger flow resistance are produced, leading to a better overall thermal diffusion and severely suppressing the natural convection. During the solidification process, the solid-liquid interfaces in all cases are closely parallel to the cold walls and change their positions evenly. It implies that the effect of natural convection is much weaker than that in the melting process. There are two reasons for this. On the
Fig. 8. Melting and solidification time in cases with different total volume fractions.
solidification time of the PCM decrease obviously. When φt increases from 0.07 to 0.1, the complete cycle time are reduced by 87.90% and 93.34% compared to the base case (φt = 0). Therefore, using the combination can make a remarkable improvement in heat transfer performance. Also, it is found that the solidification time in the case (φt = 0) is higher significantly than the melting time, which implies 8
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
fl = 0.070
fl = 0.228
fl = 0.372
fl = 0.494
fl = 0.606
fl = 0.178
fl = 0.349
fl = 0.512
fl = 0.662
fl = 0.801
fl = 0.265
fl = 0.516
fl = 0.738
fl = 0.919
fl = 1
Fig. 10. Solid-liquid distributions of the three cases at different times during melting.
one hand, the solid layer of the PCM around the HP or fins becomes thicker, decreasing the temperature gradient of the liquid PCM, which is the driving force of natural convection. On the other hand, the solid region also propagates continuously, reducing the space of natural convection flow. However, a slight delay of solidification in the lower part of the PCM enclosure is observed in the case (φt = 0), suggesting that the natural convection still has a little influence. In the other two cases (φt = 0.07 and 0.10), most solidification phenomena are similar to those occurring in the melting process. The use of the combination does not show a significant impact on the shape of solid-liquid interfaces but causes a visible influence on the total amount of the solid
PCM. For example, at (t = 4 min), the liquid fractions are 0.468 for the case (φt = 0.07) and 0.196 for the case (φt = 0.10), respectively. This indicates that the solidification rate improves proportionally as the total volume fraction increases. 5.5. Analysis of temperature distributions The temperature distributions of the three cases at different times (t = 1 min, 2 min, 3 min, 4 min, and 5 min) during melting and solidification are displayed in Figs. 12 and 13, respectively. During the melting process, in the case (φt = 0), a large temperature 9
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
fl = 0.975
fl = 0.965
fl = 0.959
fl = 0.954
fl = 0.950
fl = 0.835
fl = 0.697
fl = 0.576
fl = 0.468
fl = 0.371
fl = 0.751
fl = 0.531
fl = 0.348
fl = 0.196
fl = 0.067
Fig. 11. Solid-liquid distributions of the three cases at different times during solidification.
During the solidification process, the liquid PCM at the lower part solidifies faster than that at the upper part in the case (φt = 0). Consequently, the impact of natural convection still exists. However, since the liquid PCM temperature at the lower part gradually increases, the effect of natural convection is relatively limited. In the other two cases (φt = 0.07 and 0.10), the isotherms are approximately parallel to the clod walls, and their distributions are also relatively uniform. As time passes, the temperature gradient gradually decreases.
gradient occurs near the hot walls due to the low thermal conductivity, consequently boosting the natural convection of the liquid PCM. Thus, the natural convection at the upper part of the PCM enclosure is relatively strong. The hot liquid PCM moves upward under the buoyancy effect and accumulates at the top, and the position is replaced by the cold liquid PCM accordingly. As a result, a great deal of thermal energy used to heat the PCM in the lower part is transferred to the upper part. Therefore, the melting rate in the upper part is higher, and the corresponding isotherms also look more deformed in shape. In the other two cases (φt = 0.07 and 0.10), the temperature contours look relatively uniform and unified in color relative to those in earlier melting stages. 10
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
Fig. 12. Temperature distributions of the three cases at different times during melting.
• The exergy analysis indicates that, with the increase of the total
6. Conclusions
volume fraction, the effective thermal conductivity improves, but the impact of the natural convection weakens. In the melting process, if the natural convection is considered, the average exergy transfer rate can be improved by 566% for the case (φt = 0), 41.90% for the case (φt = 0.07), and 21.59% for the case (φt = 0.10), respectively.
In this study, a series of numerical calculations have been done to investigate the thermal performances of a vertical HP LTES system with the fins-copper foam combination. The effective heat capacity method and the thermal resistance network model are used to simulate the phase change process and the heat transfer through the HP, respectively. The main conclusions can be depicted as follows.
Overall, this study shows the superior thermal performances of the combination and demonstrates its great potential in heat transfer enhancement.
• The parametric study shows that a larger number of fins, a higher • •
PCM enclosure, or a lower porosity could induce a higher heat transfer rate for both melting and solidification as the PCM volume is fixed. The use of the combination shows a greater enhancement performance than using either copper foam alone or fins alone, and the best enhancement can be achieved with a specific fin volume ratio (γfin = 0.5). The phase change rate increases substantially with the increase of the total volume fraction. Compared to the base case (φt = 0), the complete cycle time are reduced by 87.90% for the case (φt = 0.07) and 93.34% for the case (φt = 0.10), respectively.
Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. Acknowledgements The present research project was supported by the National Key R& D Program of China (2017YFB0603702). 11
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
Fig. 13. Temperature distributions of the three cases at different times during solidification.
Appendix A. Supplementary material
Heat Mass Transf. 50 (2007) 3163–3175. [7] C.X. Guo, W.J. Zhang, Numerical simulation and parametric study on new type of high temperature latent heat thermal energy storage system, Energy Convers. Manage. 49 (2008) 919–927. [8] G. Ravi, J.L. Alvarado, C. Marsh, D.A. Kessler, Laminar flow forced convection heat transfer behavior of a phase change material fluid in finned tubes, Num. Heat Transfer A: Appl. 55 (2009) 721–738. [9] M.J. Allen, T.L. Bergman, A. Faghri, N. Sharifi, Robust heat transfer enhancement during melting and solidification of a phase change material using a combined heat pipe-metal foam or foil configuration, J. Heat Transfer 137 (2015) 102301. [10] B. Kamkari, H. Shokouhmand, Experimental investigation of phase change material melting in rectangular enclosures with horizontal partial fins, Int. J. Heat Mass Transf. 78 (2014) 839–851. [11] N. Sharifi, S.M. Wang, T.L. Bergman, A. Faghri, Heat pipe-assisted melting of a phase change material, Int. J. Heat Mass Transf. 55 (2012) 3458–3469. [12] S. Krishnan, J.Y. Murthy, S.V. Garimella, A two-temperature model for solid-liquid phase change in metal foams, J. Heat Transfer-Trans. ASME 127 (2005) 995–1004. [13] A. Siahpush, J. O'Brien, J. Crepeau, Phase change heat transfer enhancement using copper porous foam, J. Heat Transfer-Trans. ASME 130 (2008) 318–323. [14] J. Fukai, M. Kanou, Y. Kodama, O. Miyatake, Thermal conductivity enhancement of energy storage media using carbon fibers, Energy Convers. Manage. 41 (2000) 1543–1556. [15] J. Fukai, Y. Hamada, Y. Morozumi, O. Miyatake, Effect of carbon-fiber brushes on conductive heat transfer in phase change materials, Int. J. Heat Mass Transf. 45 (2002) 4781–4792. [16] M. Esapour, A. Hamzehnezhad, A.A.R. Darzi, M. Jourabian, Melting and solidification of PCM embedded in porous metal foam in horizontal multi-tube heat
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114541. References [1] B. Zalba, J.M. Marin, L.F. Cabeza, H. Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, Appl. Therm. Eng. 23 (2003) 251–283. [2] M.M. Farid, A.M. Khudhair, S.A.K. Razack, S. Al-Hallaj, A review on phase change energy storage: materials and applications, Energy Convers. Manage. 45 (2004) 1597–1615. [3] F. Agyenim, N. Hewitt, P. Eames, M. Smyth, A review of materials, heat transfer and phase change problem formulation for latent heat thermal energy storage systems (LHTESS), Renew. Sustain. Energy Rev. 14 (2010) 615–628. [4] S. Lohrasbi, M. Gorji-Bandpy, D.D. Ganji, Thermal penetration depth enhancement in latent heat thermal energy storage system in the presence of heat pipe based on both charging and discharging processes, Energy Convers. Manage. 148 (2017) 646–667. [5] Y.B. Tao, Y.L. He, A review of phase change material and performance enhancement method for latent heat storage system, Renew. Sustain. Energy Rev. 93 (2018) 245–259. [6] K. Ermis, A. Erek, I. Dincer, Heat transfer analysis of phase change process in a finned-tube thermal energy storage system using artificial neural network, Int. J.
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2979–2988. [25] K.A.R. Ismail, C.L.F. Alves, M.S. Modesto, Numerical and experimental study on the solidification of PCM around a vertical axially finned isothermal cylinder, Appl. Therm. Eng. 21 (2001) 53–77. [26] K.A.R. Ismail, J.R. Henriquez, L.F.M. Moura, M.M. Ganzarolli, Ice formation around isothermal radial finned tubes, Energy Convers. Manage. 41 (2000) 585–605. [27] Y. Tian, C.Y. Zhao, A numerical investigation of heat transfer in phase change materials (PCMs) embedded in porous metals, Energy 36 (2011) 5539–5546. [28] A. Žkauskas, Heat transfer from tubes in crossflow, Advances in Heat Transfer, Vol. 18, Elsevier, 1987, pp. 87–159. [29] M. Alipanah, X.L. Li, Numerical studies of lithium-ion battery thermal management systems using phase change materials and metal foams, Int. J. Heat Mass Transf. 102 (2016) 1159–1168. [30] K. Nithyanandam, R. Pitchumani, Analysis and optimization of a latent thermal energy storage system with embedded heat pipes, Int. J. Heat Mass Transf. 54 (2011) 4596–4610. [31] A. Khalifa, L. Tan, A. Date, A. Akbarzadeh, A numerical and experimental study of solidification around axially finned heat pipes for high temperature latent heat thermal energy storage units, Appl. Therm. Eng. 70 (2014) 609–619. [32] T. Watanabe, A. Kanzawa, Second law optimization of a latent heat storage system with PCMs having different melting points, Heat Recovery Syst. CHP 15 (1995) 641–653. [33] S. Patankar, Numerical Heat Transfer and Fluid Flow, CRC Press, 1980.
storage system, Energy Convers. Manage. 171 (2018) 398–410. [17] T.X. Li, J.H. Lee, R.Z. Wang, Y.T. Kang, Enhancement of heat transfer for thermal energy storage application using stearic acid nanocomposite with multi-walled carbon nanotubes, Energy 55 (2013) 752–761. [18] Y.Q. Xie, J. Song, P.T. Chi, J.Z. Yu, Performance enhancement of phase change thermal energy storage unit using fin and copper foam, Applied Mechanics and Materials, Vol. 260, Trans Tech Publ, 2013, pp. 137–141. [19] N. Sharifi, T.L. Bergman, M.J. Allen, A. Faghri, Melting and solidification enhancement using a combined heat pipe, foil approach, Int. J. Heat Mass Transf. 78 (2014) 930–941. [20] S.S. Feng, M. Shi, Y.F. Li, T.J. Lu, Pore-scale and volume-averaged numerical simulations of melting phase change heat transfer in finned metal foam, Int. J. Heat Mass Transf. 90 (2015) 838–847. [21] J.M. Mahdi, E.C. Nsofor, Melting enhancement in triplex-tube latent thermal energy storage system using nanoparticles-fins combination, Int. J. Heat Mass Transf. 109 (2017) 417–427. [22] J.M. Mahdi, E.C. Nsofor, Solidification enhancement of PCM in a triplex-tube thermal energy storage system with nanoparticles and fins, Appl. Energy 211 (2018) 975–986. [23] N. Sharifi, A. Faghri, T.L. Bergman, C.E. Andraka, Simulation of heat pipe-assisted latent heat thermal energy storage with simultaneous charging and discharging, Int. J. Heat Mass Transf. 80 (2015) 170–179. [24] H. Shabgard, T.L. Bergman, N. Sharifi, A. Faghri, High temperature latent heat thermal energy storage using heat pipes, Int. J. Heat Mass Transf. 53 (2010)
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Performance evaluation and analysis of a vertical heat pipe latent thermal energy storage system with fins-copper foam combination ⁎
Chunwei Zhang, Yubin Fan, Meng Yu, Xuejun Zhang , Yang Zhao Key Laboratory of Refrigeration and Cryogenic Technology of Zhejiang Province, Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China
H I GH L IG H T S
fins with copper foam in a vertical HP LTES system to enhance heat transfer. • Combine influence of three key parameters on heat transfer performance is investigated. • The exergy analysis is used to assess the conduction and convection. • The • The evolution of solid-liquid interfaces and isotherms is analyzed.
A R T I C LE I N FO
A B S T R A C T
Keywords: Latent thermal energy storage Phase change material Combination enhancement Heat pipe Fin Copper foam
As one of the most promising thermal storage techniques, latent thermal energy storage (LTES) using phase change materials (PCMs) is gaining importance. However, as is well known, most PCMs still suffer from the poor thermal conductivity for both the solid and liquid phases. To overcome this drawback, a combination of fins and copper foam based on the heat pipe (HP) heat exchanger is proposed and applied. The thermal performances, including the liquid fraction, the exergy transfer rate, the evolution of solid-liquid interfaces, and temperature distributions are evaluated by a two-dimensional numerical model considering natural convection. The results indicate that the combination shows a greater enhancement performance than using either copper foam alone or fins alone, and the best enhancement can be achieved with a specific fin volume ratio (γfin = 0.5) when the total volume fraction is fixed. Besides, the exergy analysis further suggests that increasing the total volume fraction accelerates the phase change rate by improving the effective thermal conductivity, but meanwhile, the impact of natural convection is gradually reduced. This study illustrates an effective method for enhancing the heat transfer of LTES by combining different techniques.
1. Introduction Thermal energy storage (TES) plays a critical role in optimizing thermal processes and overcoming the mismatch between energy supply and demand. Generally, there are three kinds of TES methods, including sensible thermal energy storage, latent thermal energy storage, and thermochemical energy storage. The LTES with PCMs as the storage media has the advantages of high energy storage density, small storage volume, and nearly isothermal storage [1–3] and has received increasing attention in previous decades. However, most PCMs are characterized by the low thermal conductivity [4], substantially limiting the large-scale application of LTES. To improve the thermal performance of LTES systems, numerous investigations on the heat transfer enhancement techniques have been conducted. Most of them can be classified into three ways: extending the heat transfer area, improving ⁎
thermal conductivity, and using the multi-stage or cascaded LTES technique [5]. Among the techniques above, the researches about fins [6–8] and the HP [4,9] are quite common, which can augment the heat exchange area between the phase change material (PCM) and the heat transfer fluid (HTF) significantly, leading to the increase of heat flux. Kamkari and Shokouhmand [10] reported that, by immersing one and three fins into PCMs, the total melting time were reduced by 18% and 37%, respectively, relative to the enclosure without the fin. Influences of the geometry of fins have also been fully explored. Sharifi et al [11] simulated the melting of the PCM in a HP LTES system and performed a parametric study. Results showed that melting rates were significantly higher than those associated with the rod or tube. Whereas using metal foams [12,13] or high thermal conductivity particles [14,15] can be resorted to increasing the thermal conductivity, improving heat transfer
Corresponding author. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.applthermaleng.2019.114541 Received 20 May 2019; Received in revised form 23 September 2019; Accepted 15 October 2019 1359-4311/ © 2019 Published by Elsevier Ltd.
Please cite this article as: Chunwei Zhang, et al., Applied Thermal Engineering, https://doi.org/10.1016/j.applthermaleng.2019.114541
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Nomenclature
ts tc T Tm Tlower Tupper ΔT
Abbreviations HP HTF PCM LTES TES
heat pipe heat transfer fluid phase change material latent thermal energy storage thermal energy storage
u v → U
Symbols asp A CF c df dp E E¯ fl g hsp H k K L M P q Pr rHPi rHPo Re Rtotal t tm
specific coefficient surface area of metallic foams (m2) inner surface area of the thermal energy storage section of the HP inertia coefficient (m2) specific heat (J/kg K) ligament diameter (m) pore diameter of metal cells (m) exergy (J) average exergy transfer rate (J/min) liquid fraction gravity acceleration (m/s2) interfacial heat transfer coefficient (W/m2 K) height (m) thermal conductivity (W/m K) permeability of porous foam (m2) latent heat of fusion (J/kg K) mushy zone constant pressure (Pa) boundary heat flux (W/m2) Prandtl number inner radius of the HP (m) outer radius of the HP (m) Reynolds number total thermal resistance of the HP (W/K) time (s) melting time (s)
solidification time (s) complete cycle time (s) temperature (K) melting temperature lower PCM melting temperature upper PCM melting temperature difference between the HTF temperature and the melting point of PCM velocity in r-direction (m/s) velocity in z-direction (m/s) velocity vector (m/s)
Greek symbols ρ φt γfin η µ ε ω ξ
density (kg/m3) total volume fraction fin volume ratio thermal expansion coefficient (1/K) dynamic viscosity (kg/m s) copper foam porosity pore density (PPI) a small number (0.0001)
Subscripts c cf d eff envir i int l n pcm ref s t
charging copper foam discharging effective value environment thermal element counter initial liquid phase of PCM total number of discretized cells phase change material reference solid phase of PCM total
However, as could be found, the combination might not always lead to a better enhancement effect. Therefore, it is still worth to propose a novel combination method considering the merits of each technique reasonably and explore its thermal performances, which are the aim of this study. In consequence, the combination of fins and copper foam based on the HP is proposed and investigated during the complete cycle that consists of melting and solidification processes. Furthermore, the exergy analysis is also carried out to assess the heat transfer performance.
in the LTES system effectively. Esapour et al. [16] found that, in a multi-tube heat exchanger, by inserting the metallic foam with porosities of 0.9 and 0.7, the melting time were reduced by 14% and 55%, respectively. Li et al. [17] adopted multi-wall carbon nanotubes (MWCNTs) to improve the heat transfer performance of palmitic acid. It was found that the thermal conductivity of composite phase change material was increased by approximately 30% by adding 1 wt% MWCNTs. The enhancement effect by increasing the heat exchange area alone or improving the thermal conductivity alone is relatively limited in achieving a rapid thermal response of LTES system. Therefore, the combination of different techniques, aiming to make a further enhancement, has attracted increasing attention recently. Xie et al. [18] conducted an experimental study and reported that the effective thermal conductivity of the PCM enhanced with the fins-copper foam combination is 2.7 times as that of the PCM enhanced with copper foam only. Sharifi et al. [19] indicated that melting and solidification rates of HP-foil configurations were increased by about 3 and 9 times compared to configurations involving the rod with no foil, respectively. Feng et al. [20] found that the enhancement performance of finned metal foam is superior to plate fin and metal foam structures by numerical simulation. Mahdi and Nsofor [21,22] examined the effects of three techniques, including fins, nanoparticles, and the combination of both numerically. Results indicated that the application of fins alone is more effective than using other techniques during both melting and solidification.
2. Physical model Fig. 1 shows the schematic diagram of the vertical HP LTES unit with the combination. It can be observed that the HP, which is concentrically positioned in a cylindrical enclosure, is subdivided into three axial regions according to different functions. The middle section of the HP is encased in a PCM enclosure, as the thermal energy storage section. The bottom and top sections of the HP are exposed into the HTF with a specified temperature in different operation modes. During the melting process, the bottom section of the HP absorbs thermal energy as the evaporator, and the top section is assumed to be adiabatic. During the solidification process, the top section of the HP releases thermal energy as the condenser, and the bottom section is assumed to be adiabatic. The details of the two fundamental modes used can be found in Ref. [23]. In addition, the PCM enclosure walls, as well as the top and 2
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Fig. 1. Schematic illustration of the vertical HP LTES system with the combination: (a) physical model, (b) computational domain.
temperature is set to 638 K (578 K) which is about 30 K above (below) Tm. As a result, the temperature difference (ΔT) is equal to 30 K, and this value is dependent on the temperature of the HTF since the melting point (Tm) is constant. Moreover, the mass flow rate of the HTF is also constant. The volume fraction [21] is used to make volume usage comparable. Specifically, φt represents the total volume fraction of fins and copper foam. Additionally, to investigate the effect of proportions of fins and copper foam in the combination, a fin volume ratio, γfin, is defined. These parameters can be calculated as follows:
bottom end caps of the HP, are considered adiabatic. The initial geometric dimensions of the HP and fins are given as follows. The heights of three axial regions, HC, HTES, and HD are set to 60 × 10−3 m, 75 × 10−3 m, and 60 × 10−3 m, respectively. The inner radius of the HP, rHPi, and wall thickness, bHP, are set to 2 × 10−3 m and 2 × 10−3 m, respectively. The inner radius, outer radius, and thickness of each fin are set to 4 × 10−3 m, 21 × 10−3 m, and 3 × 10−3 m, respectively. The pore density of the copper foam is set to 20 PPI. The inner radius of the PCM enclosure is equal to the outer radius of the HP. The PCM volume is constant, 2.14 × 10−4 m3, for all cases. The physical properties of the PCM [24], fins, and copper foam [21] are displayed in Table 1. Besides, in the melting (solidification) process, the initial PCM temperature is set to 603 K (613 K), the HTF
φt =
3
Vfin + Vcf Vpcm + Vfin + Vcf
(1)
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e. Energy equation for copper foam:
Table 1 Physical parameters used in the numerical modeling. Properties
PCM [24]
HP wall, fin and metal foam [21]
Material Density (kg/m3) Thermal conductivity (W/m K)
KNO3 2109 0.425 (liquid) 0.5 (solid) 953 2.59 × 10−3 608 200 × 10−6
copper 8920 400
95 × 103
N/A
(1 − ε )
∂ (ρcf Tcf ) ∂ k cf,eff ∂Tcf 1 − ε ∂ k cf,eff ∂Tcf = (1 − ε ) ( )+ (r ) − ST ∂t ∂z ccf ∂z r ∂r ccf ∂r (7)
Specific heat (J/kg K) Dynamic viscosity (Pa s) Melting point (K) Thermal expansion coefficient (K−1) Latent heat of fusion (J/kg)
The three source terms above are given as follows:
→η (T − T ) − ( μ + ρ C |→ 2 3 Su = ρg ref l F U |) u − M [(1 − fl ) /(fl + ξ )] u K
380 N/A N/A N/A
Sv = −(
→ μ + ρCF |U |) v − M [(1 − fl )2 /(fl3 + ξ )] v K
ST = hsp asp (Tcf − Tpcm )
(8) (9) (10)
→ + ρ l CF |U |) u
γfin =
Vfin Vfin + Vcf
μ → where ρg η (T − Tref ) is the Boussinesq approximation, ( K → μ and ( K + ρCF |U |) v are the Forchheimer-Darcy terms, and are employed to describe the additional flow resistance, M [(1 − fl )2 /(fl3 + ξ )] is the source term for damping the velocity in the solid or mushy zone. The effective thermal conductivity of the copper foam, kcf,eff, and the effective thermal conductivity of the PCM, kpcm,eff, are calculated as same as the work of Tian and Zhao [27]. fl is the liquid fraction and can be described as follows:
(2)
where Vfin, Vcf, and Vpcm are the volume of fins, copper foam, and PCM, respectively. 3. Mathematical description In the mathematical model, the effective heat capacity method [25,26] is utilized to simulate the melting and solidification of the PCM. The two-temperature energy equation model is used to express thermal non-equilibrium between the PCM and the copper foam. The natural convection flow in the liquid PCM is modeled according to the Boussinesq approximation. Besides, some assumptions are made and listed as follows:
⎨ ⎩ (T − Tlower )/(Tupper − Tlower ), Tlower ⩽ T ⩽ Tupper
3.1. Parameters of copper foam The characteristics of the copper foam can be described by some pore morphological parameters [16], which can be represented as follows:
df dp
Based on these assumptions, the continuity, momentum, and energy equations [16] can be expressed as follows:
= 1.18
dp =
1 ∂ (ρ l v ) 1 ∂ (ρ l uv ) 1 1 ∂ (rρ l v 2) + 2 + 2 × ε ∂t ε ∂z ε r ∂r μ ∂P ∂ μl ∂v 1 ∂ μl ∂v =− + ( )+ (r ) − 2l v + Sv ∂r ∂z ε ∂z r ∂r ε ∂r r
(4)
asp =
∂t ∂ k pcm,eff =ε ( ∂z c pcm
(15)
3πdf (1 − e−(1 − ε )/0.04 ) (0.59dp)2
0.4 Pr 0.37 k / d (1 ⩽ Re ⩽ 40) 1 f ⎧ 0.76Re ⎪ (40 ⩽ Re ⩽ 1000) hsp = 0.52Re 0.5 Pr 0.37 k1/ df ⎨ ⎪ 0.26Re 0.6 Pr 0.37 k1/ df (1000 ⩽ Re ⩽ 20, 000) ⎩
(5)
d. Energy equation for PCM:
1 ∂ (rρpcm vTpcm ) r ∂r ∂z ∂Tpcm ε ∂ k pcm,eff ∂Tpcm )+ (r ) + ST c pcm ∂r ∂z r ∂r
0.0254 ω
(14)
In Eq. (10), asp is the specific coefficient surface area of the copper foam, hsp is the interfacial heat-transfer coefficient between the liquid PCM and the copper foam, and can be calculated by using the empirical correlation [28]:
c. Momentum equation in z-direction:
∂ (ρpcm uTpcm )
(13)
−1.63
df CF = 0.00212(1 − ε )−0.132 ⎛⎜ ⎞⎟ ⎝ dp ⎠
(3)
1 ∂ (ρ l u) 1 ∂ (ρ l u2) 1 1 ∂ (rρ l uv ) + 2 + 2 × ε ∂t ε ∂z ε r ∂r ∂P ∂ μl ∂u 1 ∂ μl ∂u =− + ( )+ (r ) + Su ∂z ∂z ε ∂z r ∂r ε ∂r
+
(12)
−1.11
b. Momentum equation in r-direction:
∂ (ρpcm Tpcm )
1−ε ⎛ 1 ⎞ ε ⎝ 1 − e−(1 − ε )/0.04 ⎠
df K = 0.00073(1 − ε )−0.224 ⎛⎜ ⎞⎟ d dp2 ⎝ p⎠
a. Continuity equation:
ε
(11)
The melting and solidification processes of the PCM are assumed to take place over a given temperature range (Tlower ~ Tupper), where Tlower and Tupper are set to be 0.5 K lower and higher than the melting point Tm, respectively.
• The contact thermal resistances are negligible. • The structure of the copper foam is considered to be homogeneous and isotropic. • The volume variation of the PCM during the phase change process is neglected. • The flow of the liquid PCM is incompressible and laminar. • No-slip boundary conditions for velocities at walls are applied.
∂u 1 ∂ (rv ) + =0 ∂z r ∂r
T < Tlower T > Tupper
0, 1,
⎧ fl =
(16)
(17)
It is noted that only conduction heat transfer exists in the solid PCM. As a result, the interfacial heat transfer coefficient is not applicable at that time. To solve this problem, hsp in Eq. (10) is replaced with kpcm-cf /rp when Re < 1, where rp is the pore radius of metal cells, kpcm-cf is the effective thermal conductivity between the PCM and the copper foam.
+
(6) 4
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3.2. Thermos-physical properties of PCM
Ec n
=
The thermos-physical properties of the PCM (liquid or solid) used in the simulation can be calculated by using the formulations provided by Alipanah and Li [29], as follows:
⎧ ρpcm =
c pcm
⎨ + − ⩽ ⩽ ⎩ ρs (ρ l ρs ) fl , Tlower T Tupper
c s, T < Tlower ⎧ ⎪ c l, T > Tupper = ⎨ c l + cs L ⎪ 2 + (Tupper − T lower ) , Tlower ⩽ T ⩽ Tupper ⎩ ⎧
k pcm =
i=1
⎨ ⎩ k s + (kl − k s) fl , Tlower ⩽ T ⩽ Tupper
n
= (18)
) + L (fi − fint )(1 −
Tenvir ⎤ ) Tm ⎥ ⎦
⎣
Tint T ) + L (fint − fi )(1 − envir )⎤ Ti,pcm Tm ⎥ ⎦ (26)
(19)
where L is the latent heat, Tenvir is the environment temperature, and n denotes the total number of discretized cell volumes of the PCM in the computational domain. Moreover, an average exergy transfer rate over the melting or solidification process is defined to compare the thermal performances quantitatively. It can be calculated as follows.
(20)
E¯ =
Ec
⎧ tm
Melting
⎨ Ed ⎩ ts
Solidification
(27)
where tm and ts are the time to reach complete melting and solidification, respectively.
The heat transfer through the HP is simplified with the thermal resistance network model and is incorporated into the boundary conditions. The mathematical details of the thermal resistance network model can be found in Ref. [30]. As a consequence, the boundary conditions [31] can be described as follows:
∂T [rHPi, z , t ] T − T [rHPi, z , t ] = HTF ∂r Rtotal
∑ mi ⎡⎢cpcm (Tint − Ti,pcm − Tenvir ln i=1
3.3. Boundary conditions
− kA
Tint
Ed
T < Tlower T > Tupper
k s, kl,
⎣
Ti,pcm
(25)
T < Tlower T > Tupper
ρs , ρl ,
∑ mi ⎡⎢cpcm (Ti,pcm − Tint − Tenvir ln
4. Numerical procedure A MATLAB program is developed in this study. The governing equations are discretized with the finite volume approach, and a fullyimplicit scheme [33] is adopted for the time term. The corresponding algebraic equations are solved numerically by the Gauss-Seidal Iteration in combination with a super-relaxation technique. The SIMPLEC algorithm, combined with the collocated grid arrangement, is employed for the pressure correction equation. The convergence criteria for the continuity, momentum, and energy equations are set to 10−5, 10−5, and 10−6, respectively. All simulations are conducted on a computer, which features 3.40 GHz quad-core Intel Xeon processors. The numerical model is validated against the experimental work of Allen et al. [9] when the volume of fins is set to zero. They investigated the melting and solidification of the PCM in a HP configuration with metal foam. Fig. 2 shows the comparison of the temperature profiles of the melting and solidification processes between the present work and the experimental data under the same conditions. As can be observed, the simulated results are in good agreement with the experimental data, which indicates the proposed model can be used for further analysis. The sensitivity studies of grid size and time step based on the liquid fraction have been carried out to obtain an independent solution. The geometric parameters of the physical model (φt = 0.10) are used as an input to the sensitivity studies. Three grid sizes of 7500 cells, 11,250
(21)
∂T [rHPo, z , t ] =0 ∂r
(22)
∂T [r , 0, t ] =0 ∂z
(23)
∂T [r , HTES, t ] =0 ∂z
(24)
where A is the inner surface area of the thermal energy storage section of the HP, Rtotal is the total thermal resistance related to the HP. 3.4. Exergy analysis To further analyze the thermal performances of the combination, an exergy analysis [32] on the complete cycle is performed is applied. The net exergy charged or discharged at any instant is calculated based on the second law of thermodynamics as follows:
Fig. 2. Comparison of the temperature profiles between the present work and the experimental data by Allen et al. [9]. 5
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become shorter. This is attributable to the increase of height augments the heat transfer area between the HP walls and the PCM. Meanwhile, the flow resistance is also reduced due to the increase of the gap between the fins, consequently boosting the natural convection. As a result, the heat can be transferred to the PCM more quickly. Moreover, the initial variations of the liquid fraction do not show a significant difference, which means that the natural convection has not enough impact at that time. In summary, a larger number of fins, a higher PCM enclosure, or a lower porosity could induce a higher phase change rate for both melting and solidification when the PCM volume is fixed. The same advantage of them is that the heat transfer area can be increased substantially. Thus, these three parameters are significant and need to be considered in the design of LTES systems.
cells, and 15,000 cells and three time steps of 0.02 s, 0.05 s, and 0.1 s are selected and tested. Comparison results are shown in Fig. 3. As a result, a grid size of 11,250 cells and a time step of 0.05 s are finally chosen because more grids or smaller time steps do not produce appreciable improvement in accuracy.
5. Results and discussion 5.1. Effects of parameters in the combination
5.2. Characteristics of the combination The problem of determining what proportions of fins and copper foam to use in the combination is crucial because the heat transfer enhancement performance varies with the volume proportion. Therefore, the effect of the fin volume ratio (γfin) is investigated to obtain an optimal combination design. The fin volume ratio can be adjusted by changing the number of fins or the copper foam porosity. In addition, the HTF temperature is also considered, because a greater temperature gradient can accelerate the melting and solidification processes. The PCM enclosure height (HTES) is set to 0.075 m. In consequence, as shown in Fig. 7, the complete cycle time of cases with different fin volume ratios are calculated under the conditions of two total volume fractions (φt = 0.07 and 0.10) and two temperature differences (ΔT = 30 K and 50 K). It can be found that the best enhancement performance can be obtained with a specific fin volume ratio (γfin = 0.5), and this ratio is not subject to temperature differences. Also, the use of the combination is better than using either copper foam alone (γfin = 0) or the fins alone (γfin = 1) under the same total volume fraction limit. 5.3. Performance comparison of cases with different total volume fractions The duration of the phase change process can estimate the charge or discharge rate of thermal energy, while the exergy can evaluate the quantity and quality of thermal energy simultaneously. They are two useful parameters for investigating the characteristics of the HP LTTS system with the combination. Three different total volume fractions (φt = 0, 0.07 and 0.10) are selected for testing, and higher total volume
1.0
1.0
0.8
0.8 Liquid fraction
Liquid fraction
To investigate the influence of fins, copper foam, and the HP on the heat transfer performance, the number of fins (Nfin), copper foam porosity (ε), and PCM enclosure height (HTES), are selected as three key parameters. Other parameters, such as fin length and pore density, have a relatively little effect on heat transfer, which has been proved by numerous researchers. That is the reason why they are considered to be constant in this study. The histories of the liquid fraction are tracked and analyzed to evaluate the heat transfer performance in both melting and solidification processes, and the results are shown in Figs. 4–6, respectively. Fig. 4 displays the liquid fraction evolution of the cases with three different fins numbers (Nfin = 1, 3, and 5). The copper foam porosity (ε) and the PCM enclosure height (HTES) are set to 0.93 and 0.075 m, respectively. It can be seen that, as the number of fins increases, the phase change time required decreases sharply. The melting and solidification time in the case (Nfin = 5) are 3.2 min and 4.0 min, respectively, while their values are almost 8.3 min and 9.8 min in the case (Nfin = 1). It is because increasing the number of fins expands the heat transfer area and intensifies the thermal penetration. Also, melting and solidification rates gradually abate as time progresses, owing to the decrease of the temperature difference and the weakening of natural convection. Fig. 5 compares the liquid fraction evolution of the cases with three different porosities (ε = 0.96, 0.93, and 0.90). The number of fins (Nfin) and the PCM enclosure height (HTES) are set to 3 and 0.075 m, respectively. As shown in the figure, as the porosity decreases, melting and solidification rates also augment obviously, reducing the phase change time. The primary reason is that the heat transfer area and the effective thermal conductivity are enhanced. Fig. 6 presents the liquid fraction evolution of the cases with three different PCM enclosure heights (HTES = 0.055 m, 0.075 m, and 0.095 m). The number of fins (Nfin) and the copper foam porosity (ε) are set to 3 and 0.93, respectively. Because the PCM volume is fixed, the enclosure outer radius changes accordingly in different cases. It is found that, as the height increases, both the melting and solidification time
0.6
7500 Cells 11250 Cells 15000 Cells
0.4
0.02 s 0.05 s 0.1 s
0.4
0.2
0.2
0.0
0.6
0
1
2
3
4
0.0
5
0
1
2
3
Time (minutes)
Time (minutes)
(a)
(b)
Fig. 3. (a) Mesh independency. (b) Time step independency. 6
4
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1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
C. Zhang, et al.
0.6 İ= 0.93 , HTES= 0.075 m 0.4 Nfin = 1 Nfin = 3
0.2
0.0
2
4
6
Nfin = 1
0.6
Nfin = 3 Nfin = 5
0.4
0.2
Nfin = 5
0
İ= 0.93 , HTES= 0.075 m
0.0
8
0
2
4
6
8
10
Time (minutes)
Time (minutes)
(a)
(b)
1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
Fig. 4. Effects of the number of fins on liquid fraction during (a) melting and (b) solidification.
0.6 Nfin = 3 , HTES= 0.075 m 0.4 İ= 0.90 İ= 0.93 İ= 0.97
0.2
0.0
0
2
Nfin = 3 , HTES= 0.075 m İ= 0.90 İ= 0.93 İ= 0.97
0.6
0.4
0.2
4
0.0
6
0
2
4
6
Time (minutes)
Time (minutes)
(a)
(b)
1.0
1.0
0.8
0.8
Liquid fraction
Liquid fraction
Fig. 5. Effects of the porosity on liquid fraction during (a) melting and (b) solidification.
0.6 Nfin = 3 , İ= 0.93 0.4
Nfin = 3 , İ= 0.93 HTES=0.055 m
0.6
HTES=0.075 m HTES=0.095 m
0.4
HTES=0.055 m HTES=0.075 m
0.2
0.0
0.2
HTES=0.095 m
0
2
4
0.0
6
0
2
4
Time (minutes)
Time (minutes)
(a)
(b)
6
Fig. 6. Effects of the enclosure height on liquid fraction during (a) melting and (b) solidification.
exergy are displayed in Tables 2 and 3, respectively. Note that all the melting time considering natural convection are shorter than those under the pure conduction condition, which suggests that the contribution of natural convection in the melting process is not negligible. The comparisons of the phase change time and the average exergy transfer rates are shown in Figs. 8 and 9, respectively. Fig. 8 illustrates that, as the total volume fraction increases, the melting and
fractions are not considered because the dominant heat transfer mechanism varies with the total volume fraction. The cases are calculated under two conditions of convection and conduction, which can analyze the improvement of the effective thermal conductivity and separate the impact of natural convection comparably. The initial temperature difference and the fin volume ratio (γfin) are set to 30 K and 0.5, respectively. The time to complete the phase change and the corresponding 7
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Fig. 9. Comparison of the average exergy transfer rates. Fig. 7. Complete cycle time in cases with different fin volume ratios.
that the solidification has a bigger enhancement potential. Fig. 9 shows that the average exergy transfer rates increase with the total volume fraction. Based on the definition above, the average exergy transfer rate under the pure conduction condition can indicate the degree of effective thermal conductivity. And the corresponding values in the cases (φt = 0.07 and 0.1) are much higher than that in the base case (φt = 0). Thus, the effective thermal conductivity is increased with the total volume fraction. Furthermore, it is also found that natural convection can significantly accelerate the melting process. For example, if the natural convection is considered, the average exergy transfer rate can be improved by 566% for the case (φt = 0), 41.90% for the case (φt = 0.07), and 21.59% for the case (φt = 0.10), respectively. Therefore, the effect of natural convection in the melting process gradually decreases with the increase of the total volume fraction, whereas it is almost negligible in the solidification process.
Table 2 Melting and solidification time of the cases under convection and conduction. Total volume fraction
φt = 0 φt = 0.07 φt = 0.10
Melting time (min)
Solidification time (min)
Convection
Conduction
Convection
Conduction
21.15 8.35 5.15
121.95 11.80 6.15
147 12 6.05
121.95 11.80 6.15
Table 3 Exergy changed and discharged of the cases under convection and conduction. Total volume fraction
φt = 0 φt = 0.07 φt = 0.10
Exergy charged (kJ)
Exergy discharged (kJ)
Convection
Conduction
Convection
Conduction
28.78 26.73 26.56
24.82 26.61 26.09
26.85 26.54 25.99
24.80 26.45 25.97
5.4. Evolution of solid–liquid interfaces The solid–liquid distributions of the three cases at different times (t = 1 min, 2 min, 3 min, 4 min, and 5 min) during melting and solidification are displayed in Figs. 10 and 11, respectively. During the melting process, the solid-liquid interface in the case (φt = 0) deforms remarkably due to the impact of natural convection. The liquid PCM accumulates in the upper part of the enclosure, and the melting front propagates downward. As a result, the liquid layer in the upper part is thicker than that in the lower part. After (t = 3 min), the liquid PCM has already occupied the entire upper part, but a large amount of solid PCM still exists in the lower part. The solid-liquid interface becomes shorter as time progresses; the impact range of natural convection decreases accordingly. Besides, the solid-liquid interface is quite clear, reflecting the thermal stratification caused by the high thermal resistance of pure PCM. In the other two cases (φt = 0.07 and 0.10), the solid-liquid interfaces show a different evolution. In the beginning, a thin liquid PCM layer occurs, and it is closely parallel to the fins and HP surfaces. As time progresses, the melting front propagates smoothly by absorbing more heat. Due to less deformation in the shape of the solid-liquid interfaces, the conduction is dominant in the heat transfer, and the convection may exert only a little effect. Meanwhile, the solid-liquid interface becomes wider. That is because the fins and copper foam exist simultaneously, higher thermal conductivity and stronger flow resistance are produced, leading to a better overall thermal diffusion and severely suppressing the natural convection. During the solidification process, the solid-liquid interfaces in all cases are closely parallel to the cold walls and change their positions evenly. It implies that the effect of natural convection is much weaker than that in the melting process. There are two reasons for this. On the
Fig. 8. Melting and solidification time in cases with different total volume fractions.
solidification time of the PCM decrease obviously. When φt increases from 0.07 to 0.1, the complete cycle time are reduced by 87.90% and 93.34% compared to the base case (φt = 0). Therefore, using the combination can make a remarkable improvement in heat transfer performance. Also, it is found that the solidification time in the case (φt = 0) is higher significantly than the melting time, which implies 8
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
fl = 0.070
fl = 0.228
fl = 0.372
fl = 0.494
fl = 0.606
fl = 0.178
fl = 0.349
fl = 0.512
fl = 0.662
fl = 0.801
fl = 0.265
fl = 0.516
fl = 0.738
fl = 0.919
fl = 1
Fig. 10. Solid-liquid distributions of the three cases at different times during melting.
one hand, the solid layer of the PCM around the HP or fins becomes thicker, decreasing the temperature gradient of the liquid PCM, which is the driving force of natural convection. On the other hand, the solid region also propagates continuously, reducing the space of natural convection flow. However, a slight delay of solidification in the lower part of the PCM enclosure is observed in the case (φt = 0), suggesting that the natural convection still has a little influence. In the other two cases (φt = 0.07 and 0.10), most solidification phenomena are similar to those occurring in the melting process. The use of the combination does not show a significant impact on the shape of solid-liquid interfaces but causes a visible influence on the total amount of the solid
PCM. For example, at (t = 4 min), the liquid fractions are 0.468 for the case (φt = 0.07) and 0.196 for the case (φt = 0.10), respectively. This indicates that the solidification rate improves proportionally as the total volume fraction increases. 5.5. Analysis of temperature distributions The temperature distributions of the three cases at different times (t = 1 min, 2 min, 3 min, 4 min, and 5 min) during melting and solidification are displayed in Figs. 12 and 13, respectively. During the melting process, in the case (φt = 0), a large temperature 9
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
fl = 0.975
fl = 0.965
fl = 0.959
fl = 0.954
fl = 0.950
fl = 0.835
fl = 0.697
fl = 0.576
fl = 0.468
fl = 0.371
fl = 0.751
fl = 0.531
fl = 0.348
fl = 0.196
fl = 0.067
Fig. 11. Solid-liquid distributions of the three cases at different times during solidification.
During the solidification process, the liquid PCM at the lower part solidifies faster than that at the upper part in the case (φt = 0). Consequently, the impact of natural convection still exists. However, since the liquid PCM temperature at the lower part gradually increases, the effect of natural convection is relatively limited. In the other two cases (φt = 0.07 and 0.10), the isotherms are approximately parallel to the clod walls, and their distributions are also relatively uniform. As time passes, the temperature gradient gradually decreases.
gradient occurs near the hot walls due to the low thermal conductivity, consequently boosting the natural convection of the liquid PCM. Thus, the natural convection at the upper part of the PCM enclosure is relatively strong. The hot liquid PCM moves upward under the buoyancy effect and accumulates at the top, and the position is replaced by the cold liquid PCM accordingly. As a result, a great deal of thermal energy used to heat the PCM in the lower part is transferred to the upper part. Therefore, the melting rate in the upper part is higher, and the corresponding isotherms also look more deformed in shape. In the other two cases (φt = 0.07 and 0.10), the temperature contours look relatively uniform and unified in color relative to those in earlier melting stages. 10
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
Fig. 12. Temperature distributions of the three cases at different times during melting.
• The exergy analysis indicates that, with the increase of the total
6. Conclusions
volume fraction, the effective thermal conductivity improves, but the impact of the natural convection weakens. In the melting process, if the natural convection is considered, the average exergy transfer rate can be improved by 566% for the case (φt = 0), 41.90% for the case (φt = 0.07), and 21.59% for the case (φt = 0.10), respectively.
In this study, a series of numerical calculations have been done to investigate the thermal performances of a vertical HP LTES system with the fins-copper foam combination. The effective heat capacity method and the thermal resistance network model are used to simulate the phase change process and the heat transfer through the HP, respectively. The main conclusions can be depicted as follows.
Overall, this study shows the superior thermal performances of the combination and demonstrates its great potential in heat transfer enhancement.
• The parametric study shows that a larger number of fins, a higher • •
PCM enclosure, or a lower porosity could induce a higher heat transfer rate for both melting and solidification as the PCM volume is fixed. The use of the combination shows a greater enhancement performance than using either copper foam alone or fins alone, and the best enhancement can be achieved with a specific fin volume ratio (γfin = 0.5). The phase change rate increases substantially with the increase of the total volume fraction. Compared to the base case (φt = 0), the complete cycle time are reduced by 87.90% for the case (φt = 0.07) and 93.34% for the case (φt = 0.10), respectively.
Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. Acknowledgements The present research project was supported by the National Key R& D Program of China (2017YFB0603702). 11
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t = 1 min
t = 2 min
t = 3 min
t = 4 min
t = 5 min
Fig. 13. Temperature distributions of the three cases at different times during solidification.
Appendix A. Supplementary material
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