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A new heat transfer model of phase change material based on energy asymmetry
Applied Energy 212 (2018) 1409–1416
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A new heat transfer model of phase change material based on energy asymmetry
T
⁎
Xing Jina,b, , Huoyan Huc, Xing Shia,b, Xin Zhoua,b, Liu Yangc, Yonggao Yinc, Xiaosong Zhangc a
School of Architecture, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China Key Laboratory of Urban and Architectural Heritage Conservation (Southeast University), Ministry of Education, No. 2 Sipailou, Nanjing 210096, PR China c School of Energy and Environment, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China b
H I G H L I G H T S heat transfer model of PCM based on energy asymmetry is built. • AAnnew facility is built to validate the model. • Theexperimental • asymmetrical model has higher accuracy than the symmetrical model.
A R T I C L E I N F O
A B S T R A C T
Keywords: Phase change material (PCM) Heat transfer Numerical model Energy asymmetry
The melting process and the solidifying process of phase change material (PCM) are usually considered symmetrical in the traditional PCM heat transfer models, but there are inevitable calculation errors using these models. In this paper, a new heat transfer model of PCM based on the energy asymmetry was built, and it was validated by experimental data. It was found that the two main reasons for the energy asymmetry of the PCM were the melting temperature range and the solidifying temperature range were not the same and the supercooling problem during the cooling process. No matter for only one thermal cycle or for the multiple thermal cycles, the results from the asymmetrical model and the symmetrical model were different. Apparently, compared with the symmetrical model, the asymmetrical model had higher accuracy, and the heating/cooling process in the asymmetrical model was more consistent with the real heating/cooling process of the PCM.
1. Introduction Energy storage can reduce the time, space or rate mismatch between energy supply and demand, thereby it is playing an important role in energy conservation. Energy storage could save energy and make the system more effective by reducing the wastage of energy [1]. Among all the energy storage technologies, the thermal energy storage is recognized as one of the most effective technique to improve the performances of the energy systems [2,3]. Especially, it possesses a great adaption to the renewable energy [4]. Thermal energy storage is generally classified as sensible heat storage and latent heat storage. While, the latent heat storage system with phase change materials (PCMs) has attracted more and more researches in the last two decades [5,6]. PCM absorbs or releases the latent heat when the temperature of the material undergoes its phase change process. The advantages of PCM system include higher energy storage density, smaller temperature variation during the phase change process
⁎
while requiring smaller masses and volumes of material [7,8]. Therefore, PCMs have been used in many fields, not only in traditional areas such as solar energy storage and building heating/cooling, but also in new areas such as battery thermal management, photovoltaic cells and food preservation, etc. Phase change of a material is described by a particular kind of boundary value problem for partial differential equations, where phase boundary can move with time [9]. The heat balance equation in three dimensions at the solid-liquid interface of this phase change condition (i.e., Stefan condition) is:
ρH
ds (τ ) ∂t ∂t = λs s −λl l dτ ∂n ∂n
(1)
where ρ is the density, H is the heat of fusion of PCM, s(τ) is the solidliquid interface position, λ is the thermal conductivity, t is the temperature, τ is the time, n is the normal to the interface, subscript s and l refers to solid and liquid phases.
Corresponding author at: School of Architecture, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China. E-mail address: [email protected] (X. Jin).
https://doi.org/10.1016/j.apenergy.2017.12.103 Received 24 August 2017; Received in revised form 30 November 2017; Accepted 30 December 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.
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Eq. (1) is suitable for both melting process and solidifying process. Usually, the solidifying process of PCM is recognized as the exact inverse process of its melting process, which means that there is only one PCM enthalpy-temperature curve which is used for both cooling process and heating process. The enthalpy of PCM is only dependent on its temperature. In other words, for each small temperature range, the amount of absorbed heat when the PCM is heated is the same as the amount of released heat when it is cooled. Therefore, the melting process and the solidifying process of PCM could be considered symmetrical [10]. Accordingly, in most of the PCM models, the melting temperature and the solidifying temperature are usually considered to be the same or the melting temperature range and the solidifying temperature are considered to be the same. Saffari et al. [11] proposed an optimization method coupled with an innovative PCM enthalpytemperature function to find out the optimum PCM melting temperature, the melting temperature range and the solidifying temperature range in their model were set to be the same. Tay et al. [12] built a new simplified two dimensional mathematical model to characterize phase change material systems, the melting temperature and the solidifying temperature of the chosen PCM were both 35 °C. Zhao and Tan [13] developed a numerical model for the PCM thermal storage unit, the phase change temperature range (i.e., both melting and solidifying) in their model was 19–23 °C. Jin et al. [14,15] built the PCM wall models and tried to find the optimal PCM location in the wall, the melting temperature range and the solidifying temperature range in the models were also the same. Other PCM models, such as natural cooling energy storage model [16], tube-in-tank latent thermal energy storage model [17], PV/PCM cell heat transfer model [18], PCM floor model [19,20], and PCM board model [21,22], also showed that the melting temperature ranges and the solidifying temperature ranges were all the same. However, because the melting temperature range and the solidifying temperature range of PCM are unlikely to be exactly the same, and the PCM may have supercooling problem [23,24], our previous research has showed that the melting process and the solidifying process of PCM are asymmetrical [10]. Therefore, the PCM models are divided into two types in this research, which are symmetrical model and asymmetrical model respectively. The PCM will absorb or release an amount of latent heat during its phase change process. Usually, the latent heat of PCM is represented by the effective heat capacity indirectly, which considers the phase change process as a sensible process with an increased (effective) heat capacity. The effective heat capacities of PCM during heating/cooling process in symmetrical model and asymmetrical model are shown in Fig. 1. The effective heat capacity of PCM in its phase change temperature range was assumed to be uniform. As shown in this figure, the melting temperature range of the PCM is [tm1, tm2], the solidifying temperature range is [ts1, ts2]. If [tm1, tm2] and [ts1, ts2] are the same, and for a certain temperature value, the effective heat capacity in the heating process is always equal to that in the cooling process (only considering the absolute values), just as shown in Fig. 1(a). It means the melting process and the solidifying process are symmetrical. Here, the model is referred to as the symmetrical model. As shown in Fig. 1(b), [tm1, tm2] and [ts1, ts2] are not the same, and the PCM could not release the latent heat until it is cooled down to tc, then the PCM is heated up to ts2 from tc by a part of latent heat released by itself. The solidifying temperature range of PCM is still [ts1, ts2], so it will release the rest of latent heat when it is cooled to ts1 from ts2. The PCM temperature variations with time during the heating process and the cooling process are shown in Fig. 2, which also shows the differences between the symmetrical model and asymmetrical model. It will be discussed in detail later in Section 2. In Fig. 1(a), if the PCM temperature is known, it is very easy to find that the value of the effective heat capacity in the heating process is always equal to that in the cooling process. However, in Fig. 1(b), they are not the same. For example, when the PCM temperature is between ts2 and tm2, the effective
Fig. 1. Effective heat capacity of PCM in the heating process and the cooling process (a) Symmetrical model; (b) Asymmetrical model.
heat capacity in the heating process is much higher than that in the cooling process. Therefore, the melting process and the solidifying process of PCM are asymmetrical. Accordingly, the model is referred to as the asymmetrical model. Because the heating/cooling process of the PCM in the symmetrical model does not fully accord with its real heating/cooling process, the calculation errors of the model are inevitable, and the errors may be significant, especially when there is a serious supercooling problem. According to the published papers in the databases, the enthalpy method [25–28], the effective heat capacity method [15,21,29–31], and the heat source method [29,32–34] are the most common methods to build the numerical heat transfer PCM models. However, the melting processes and the solidifying processes in the models with these methods are all symmetrical. The most popular kinds of commercial software such as EnergyPlus, ANSYS Fluent, and ESP-r also have the phase change modules, but which are still based on the energy symmetry. In only a few PCM models, the supercooling in the solidifying process is taken into account. Gunther et al. [35] introduced an algorithm for a simple but expedient modeling of supercooling and solidification in the PCM, the method was based on the explicit finite volume method and the enthalpy method. Uzan et al. [36] developed a novel mathematical model of multidimensional PCM solidification with supercooling. It was shown that the model could reflect the experimental results fairly well, especially when predicting temperatures at various locations inside the material. Bédécarrats et al. [37] presented a numerical study on the behavior of a test plant which was a tank filled with spherical nodules containing a phase change material. In the numerical model, the supercooling of PCM was considered during its crystallization process. Bony and Citherlet [38] built a numeric model 1410
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2. Asymmetrical energy model of PCM 2.1. Symmetrical model and asymmetrical model Here in this research, to simplify the model, as shown in Fig. 1, the effective heat capacity of the PCM in its melting temperature range or solidifying temperature range was assumed to be constant. In addition, the magnitude of the melting temperature range and the magnitude of the solidifying temperature range were assumed to be the same. The models based on Fig. 1(a) and (b) were referred to as the symmetrical model and the asymmetrical model, respectively. The differences between the symmetrical model and asymmetrical model of the PCM could also be presented based on the PCM temperature variations with time during the heating and cooling processes. As shown in Fig. 2, the PCM was heated up and then cooled down. The heating temperature during heating process and the cooling temperature during cooling process were both constant, and it was assumed that there was no temperature gradient inside the PCM. As shown in Fig. 2(a), the PCM was in its melting process from τ1 to τ2, and it was in its solidifying process from τ4 to τ5. The melting temperature range and the solidifying temperature range of the PCM were the same. The heating process of Fig. 2(b) was the same with Fig. 2(a), but their cooling processes were very different. As shown in Fig. 2(b), the PCM could not release its latent heat until it was cooled down to tc, and then it was heated to ts2 from tc by the latent heat released by itself. Then, it would release the rest of the latent heat from τ5 to τ6. Therefore, the heating process and the cooling process in Fig. 2(a) were considered symmetrical, while they were asymmetrical in Fig. 2(b).
2.2. Case study of PCM asymmetrical energy model In order to compare the symmetrical model and the asymmetrical model, a simple one-dimensional phase change heat transfer case for a thin PCM layer was analyzed. As shown in Fig. 3, the physical model was consisted of an insulting layer, an electric heater, two stainless steel plates and a PCM layer. The PCM was heated by the electric heater and then cooled by the air whose temperature was assumed to be constant. The PCM used here is CT26, which is a kind of inorganic PCM and has the supercooling problem. The thermophysical properties of the materials are shown in Table 1. To simplify the heat transfer model, the following assumptions are applied:
Fig. 2. Temperature variations with time during the heating process and the cooling process (a) Symmetrical model; (b) Asymmetrical model.
to simulate heat transfer process between PCM and water in a tank. It was assumed that as soon as a PCM part reached the point of crystallization, the whole PCM would turn into solid. However, only the solidifying process of PCM was analyzed in these models, the melting process was not included. Accordingly, there was no thermal cycle in these models. There are at least two different PCM effective heat capacity curves in thermal cycles if the melting process and the solidifying process of PCM are not symmetrical, one is for the melting process and the other is for the solidifying process. While only one PCM effective heat capacity curve is needed if only the solidifying process is simulated. Therefore, the model for thermal cycles is very different from that for only solidifying process. In this research, a new PCM model will be built, and it is not only used for solidifying process of PCM but also for thermal cycles. In addition, a new method is proposed to simulate the supercooling process of PCM accurately. It is assumed that there is an internal heat source in the PCM during the steep temperature increase process, and the energy of the heat source in this process is from the latent heat of PCM. It should be noticed that this method is totally different from the heat source method mentioned before, which is still like the enthalpy method or the effective heat capacity method, but could not simulate the supercooling process, either. This new asymmetrical model is expected to show the real thermal cycles of PCM and have higher accuracy than the traditional models.
x=0 x=į
x
6 5
41 2 3
1. Stainless steel plate #1; 2. PCM; 3 Stainless steel plate #2; 4. Electric heater; 5. Insulating layer; 6. Air-conditioned environment. Fig. 3. Physical model.
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Table 1 Thermophysical properties of materials. Material
Density, kg/ m3
Thermal conductivity, W/(m K)
Specific heat capacity (Solid), kJ/(kg K)
Specific heat capacity (Liquid), kJ/(kg K)
Latent heat, kJ/kg
Melting temperature range, °C
Thickness, mm
PCM Stainless steel plate Insulation (XPS)
1500 7850
0.3 15.0
2.5 0.5
3.2 /
165 /
23–28 /
10 2
29
0.03
1470
/
/
/
50
(1) Because the thickness of all the layers is much less than their width and height, the heat transfer through all the layers is one-dimensional. (2) The convection effect in the molten PCM is neglected. (3) As shown in Fig. 2(b), the temperature increase stage during the supercooling process could not be simulated well with traditional methods. In this model, when the PCM has supercooling, it is assumed that there is an internal heat source in the PCM from τ4 toτ5. (4) All the materials are assumed to be isotropic media. The governing equation of the model is
ρn cpn
∂t ∂ 2t = λn 2 + S ∂τ ∂x
(2)
where cp is the specific heat capacity, S is the internal heat source. n = 1 for the stainless steel plate #1, n = 2 for the PCM, n = 3 for the stainless steel plate #2. aHρ
S=
⎧ τ5 − τ4 when τ4 < τ < τ5 ⎨ 0 others ⎩
(3)
where a is the released rate of the latent heat from τ4 toτ5. The boundary conditions of the governing equation are
t (0,τ ) = theat
(x = 0, and heating)
(4)
λ1
∂t = 0 (x = 0, and cooling) ∂x
(5)
λ3
∂t = h (te−ta) (x = δ ) ∂x
(6)
where theat is the surface temperature of the electric heater, h is the heat transfer coefficient between the exterior surface of stainless steel plate #2 and the air, te is the exterior surface temperature of the stainless steel plate #2, ta is the air temperature, δ is the total thickness of plate #1, PCM and plate #2. The governing equation along with the boundary conditions was discretized using the finite difference method (FDM). Central difference was applied in space and fully implicit method (backwards difference) was applied in time. The space grid size was 0.2 mm and the time step was 60 s. The equations system was solved by tridiagonal matrix algorithm (TDMA). 3. Results
Fig. 4. Experimental facility (a) Front view (b) Side view.
3.1. Model validation
central interface of PCM layer, the temperature of the interface between PCM layer and steel plate #2, the exterior surface temperature of steel plate #2, and the air temperature were all measured. In addition, the surface heat flux of the steel plate #2 was also measured. The numbers, installation locations and accuracies of the thermocouple and heat flux meter are shown in Table 2. The whole experimental facility was located in an air-conditioned room, the indoor air temperature of which was constant. Two experiments were conducted to validate the model using this facility. The first experiment included a heating process and a cooling process. The electric heater temperature was set to be 65 °C, and the
To validate the numerical model, an experimental facility was built, as shown in Fig. 4. This facility was designed based on Fig. 3. It was consisted of two 300 mm × 300 mm × 2 mm stainless steel plates, an electric heater, and an insulating layer. The PCM was put in the 10 mm empty space enclosed by the two stainless steel plates and four pieces of plexiglass. The electric heater was used to heat the PCM, and a temperature controller was used to control the surface temperature of the electric heater. During the experiments, the temperature of the interface between electric heater and steel plate #1, the temperature of the interface between steel plate #1 and PCM layer, the temperature of the 1412
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Table 2 Numbers, installation locations and accuracies of sensors. Sensor
Number
Thermocouple Heat flux meter
Interface between heater and plate #1
Interface between plate #1 and PCM
Central face of PCM
Interface between PCM and plate #2
Exterior surface of plate #2
Air
4 /
4 /
4 /
4 /
4 4
2 /
o
Temperature ( C)
40
30
20
10
0
5
10
15
20
Time (hours) Fig. 5. Comparisions of the numerical temperatures in asymmetrical model and the measured temperatures (single thermal cycle).
400 350
Measured exterior surface heat flux of stainless steel plate #2 Numerical exterior surface heat flux in symmetrical model Numerical exterior surface heat flux in asymmetrical model
2
Heat flux (W/m )
300 250 200
50 5
10
15
0.4 °C 3%
As mentioned in the Introduction, the two main reasons for the PCM energy asymmetry were the melting temperature range and the solidifying temperature range were not the same and the PCM had supercooling. Therefore, the effects of these two aspects were analyzed in this section. Firstly, it was assumed that the PCM had no supercooling, only the melting temperature range and the solidifying temperature range were not the same. In the symmetrical model, the melting temperature range and the solidifying temperature range were both 23–28 °C. While in the asymmetrical model, the melting temperature range was 23–28 °C, the
100
0
−200 to 350 °C 0–200 KW/m2
3.2. Model analyses
150
0
Error
In this experiment, the electric heater temperature was also set to be 65 °C. Firstly, the PCM was heated for 3 h, then the heater was turned off and the PCM was cooled down for 1.5 h, then the heater was turned on and the PCM was heated for 18 min, then the heater was turned off and the PCM was cooled down for 1.3 h, then the heater was turned on and the PCM was heated for 25 min, then the heater was turned off and the PCM was cooled down for about 12 h. The heating times during the second and third thermal cycles were realatively short was because the highest PCM temperatures during these two cycles were controlled, then the temperature variations here were very similar to the temperature variations of the exterior surface of a wall during a day in summer. The wall temperature goes up gradually in a sunny day, then the wall is cooled down because of a short heavy rain at noon, and then the wall temperature goes up again because of the strong solar radiation. Finally, the wall temperature goes dwon gradually from the afternoon. Fig. 7 shows the comparisions of the numerical temperatures in the asymmetrical model and the measured temperatures. Fig. 8 shows the comparisions of the numerical surface heat fluxes and the measured surface heat fluxes. It was also found that the numerical results from the asymmetrical model had very good agreement with the experimental results, and the asymmetrical model had higher accuracy than the symmetrical model.
Measured central temperature of PCM layer Numerical central temperature of PCM layer in asymmetrical model Measured exterior surface temperature of plate #2 Numerical exterior surface temperature of plate #2 in asymmetrical model
50
Range
20
Measured central temperature of PCM layer Numerical central temperature of PCM layer in asymmetrical model Measured exterior surface temperature of plate #2 Numerical exterior surface temperature of plate #2 in asymmetrical model
Time (hours) 50
Temperature ( C)
Fig. 6. Comparisions of the numerical surface heat fluxes and the measured surface heat fluxes (single thermal cycle).
o
PCM was heated for nearly five hours, then the heater was turned off and the PCM was cooled down for nearly 20 h. Fig. 5 shows the comparisions of the numerical temperatures in the asymmetrical model and the measured temperatures. As shown in the figure, the numerical data had very good agreement with the measured data. Fig. 6 shows the comparisions of the numerical surface heat fluxes and the measured surface heat fluxes. As shown in the figure, the results from the asymmetrical model also had very good agreement with the measured results. It was very obvious to observe that the differences between the symmetrical model and the measured data were large, especially during the supercooling process. The previous experiment showed that the asymmetrical model had high accuracy in a single thermal cycle. To validate the accuacy of the model in multiple thermal cycles, another experiment was conducted.
40
30
20
10
0
5
10
15
Time (hours) Fig. 7. Comparisions of the numerical temperatures in asymmetrical model and the measured temperatures (multiple thermal cycles).
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150
400 350
2
Heat flux (W/m )
300 250
Heat flux (W/m 2 )
Measured exterior surface heat flux of stainless steel plate #2 Numerical exterior surface heat flux in symmetrical Model Numerical exterior surface heat flux in asymmetrical Model
200 150 100
Symmetrical model o Asymmetrical model (3 C) o Asymmetrical model (5 C) o Asymmetrical model (7 C)
100
50
50 0
0
5
10
0
15
Time (hours) Fig. 8. Comparisions of the numerical surface heat fluxes and the measured surface heat fluxes (multiple thermal cycles).
10
Time (hours)
150 Symmetrical model o Asymmetrical model (3 C) o Asymmetrical model (5 C) o Asymmetrical model (7 C) 2
Heat flux (W/m )
100
50
0
5
10
15
15
20
respectively. The cooling times of the three cooling processes were 2 h, 2 h and 15 h, respectively. It was also found that the heating processes and the cooling processes of the asymmetrical model were all different from the symmetrical model expect for the first heating process. Then, the combined effects of supercooling and ΔT on the heating/ cooling process of the PCM were analyzed. In the symmetrical model, the melting temperature range and the solidifying temperature range were both 23–28 °C. While in the asymmetrical model, the solidifying temperature range was 20–25 °C. In addition, the degrees of supercooling (DSC) in the asymmetrical model were 0 °C, 2 °C, 4 °C and 6 °C, respectively. The system was heated for 3 h with the heating temperature of 35 °C, and then it was cooled down in the air with the temperature of 10 °C. Fig. 11 shows the exterior surface heat fluxes of the stainless steel plate #2. Actually, the black line and the red line in this figure were the same with the black line and the red line in Fig. 9. The effects of DSC could be observed based on the four heat flux lines of the asymmetrical model. It was very easy to find that the cooling processes of PCM with supercooling were very different from the cooling process of PCM with no supercooling. Because of the supercooling, the starting solidifying time was delayed, and the period for releasing latent heat was short. Compared with the asymmetrical model, when DSC were 0 °C, 2 °C, 4 °C and 6 °C, the maximum absolute errors of the symmetrical model were 27.7 W/m2, 24.9 W/m2, 36.0 W/m2 and 45.6 W/m2, and the maximum relative errors were 55.7%, 40.5%, 55.9% and
150
Heat flux (W/m 2)
5
Fig. 10. Surface heat fluxes in the asymmetrical model with different ΔT (multiple thermal cycles).
solidifying temperature ranges were 20–25 °C, 18–23 °C and 16–21 °C, respectively. Here in this research, the temperature difference (ΔT, i.e., hysteresis) between the melting temperature range and the solidifying temperature range was used to present the energy asymmetry degree of PCM. For example, when the solidifying temperature range was 20–25 °C, ΔT was 3 °C. The physical model here was the same with Fig. 3. The PCM was heated for 3 h with the heating temperature of 35 °C, and then it was cooled down in the air with the temperature of 10 °C. Fig. 9 shows the exterior surface heat fluxes of the stainless steel plate #2. As shown in the figure, the cooling processes of the asymmetrical model and the symmetrical model had great differences, which were increased with the increase of ΔT. Compared with the asymmetrical model, when ΔT were 3 °C, 5 °C, and 7 °C, the maximum absolute errors of the symmetrical model were 27.7 W/m2, 31.4 W/m2 and 37.0 W/m2, and the maximum relative errors were up to 55.7%, 78.5% and 92.5%, respectively. It was also found that the starting solidifying time of the asymmetrical model was delayed. In addition, because of the temperature difference between the PCM and the air in the asymmetrical model was smaller than that in the symmetrical model, more time was needed for releasing the latent heat during the solidifying process. Fig. 10 shows the multiple thermal cycles of the PCM. The heating temperature was 35 °C, the air temperature was 10 °C. The heating times of the three heating processes were 3 h, 0.5 h and 0.5 h,
0
0
100
50
0
20
Time (hours)
Symmetrical model o Asymmetrical model (DSC=0 C) o Asymmetrical model (DSC=2 C) o Asymmetrical model (DSC=4 C) o Asymmetrical model (DSC=6 C)
0
5
10
15
20
Time (hours)
Fig. 9. Surface heat fluxes in the asymmetrical model with different ΔT (single thermal cycle).
Fig. 11. Surface heat fluxes in the asymmetrical model with different DSC (single thermal cycle).
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Heat flux (W/m 2 )
150
energy asymmetry was proposed, and an experimental facility was built to validate the model. The numerical results had good agreement with the experimental results. Based on the numerical results, it was found that no matter for a single thermal cycle or the multiple thermal cycles, the results from the asymmetrical model and the symmetrical model were different. The maximum relative error of the symmetrical model could be more than 90% in the simulated conditions. Compared with the symmetrical model, the asymmetrical model had higher accuracy, and the heating/cooling process in the asymmetrical model was more consistent with the real heating/cooling process of the PCM. This asymmetrical model had great potential for real applications.
Symmetrical model o Asymmetrical model (DSC=0 C) o Asymmetrical model (DSC=2 C) o Asymmetrical model (DSC=4 C) o Asymmetrical model (DSC=6 C)
100
50
Acknowledgments 0
0
5
10
15
This research was supported by the Natural Science Foundation of China under Grant No. 51308104, the Ministry of Science and Technology of China under Grant No. 2016YFC0700102, and the Fundamental Research Funds for the Central Universities.
20
Time (hours) Fig. 12. Surface heat fluxes in the asymmetrical model with different DSC (multiple thermal cycles).
References
88.6%, respectively. Heat flux differences between the symmetrical model and the asymmetrical model in multiple thermal cycles could be observed in Fig. 12. The heating temperature, the cooling temperature, the heating time, and the cooling time were all the same with Fig. 10. It was found that the heat flux variations of the asymmetrical model and the symmetrical model were very different, and because the solidifying process of the PCM with supercooling was delayed, the fluctuations of the heat flux were increased with the increase of DSC.
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4. Discussions As mentioned in the Introcution, the solidifying process of PCM was usually recognized as the exact inverse process of its melting process, and the traditional heat transfer model of PCM was usually symmetrical. However, our analyses had shown that the melting process and the solidifying process were asymmetrical, and the numerical results from Section 3 had proved that the asymmetrical model had higher accuracy than the symmetrical model. This asymmetrical model proposed in this research could be used for the real application. Here, the PCM wall in buildings is taken as an example to show the problems of the symmetrical model and the advantages and the actual application of the asymmetrical model. The thermal performance evaluations of the PCM wall with the symmetrical model are right when the outdoor temperature is increasing from morning to noon, but the evaluations are not accurate when the outdoor temperature is decreasing from afternoon to night, just like the results in Figs. 9 and 11. In addition, if the solar radiation or the outdoor air temperature has large fluctuations (e.g., a short rainstorm in a hot sunny day during the daytime), the PCM could be heated and cooled alternately for several times, the heat fluxes through the PCM wall will also have large fluctuations. Accordingly, the thermal performance evaluations of the PCM wall with the symmetrical model will have large errors, just like the results in Figs. 10 and 12. Conversely, the asymmetrical model could demonstrate the real PCM temperature variations and the real surface heat flux variations of a PCM wall, and the thermal performance analyses of the PCM wall with the asymmetrical model will be more consistent with the real conditions, just like the results in Figs. 9–12. Therefore, it is concluded that the asymmetrical model has not only high calculation accuracy but also great potential for real applications. 5. Conclusions In this paper, a new heat transfer model for the PCM based on the 1415
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partially-melted sodiumacetate trihydrate. Energy Build 2014;76:654–60. [24] Zhao B, Cheng M, Liu C, Dai Z. Thermal performance and cost analysis of a multilayered solid-PCM thermocline thermal energy storage for CSP tower plants. Appl Energy 2016;178:784–99. [25] Morales-Ruiz S, Rigola J, Oliet C, Oliva A. Analysis and design of a drain water heat recovery storage unit based on PCM plates. Appl Energy 2016;179:1006–19. [26] Merlin K, Delaunay D, Soto J, Traonvouez L. Heat transfer enhancement in latent heat thermal storage systems: Comparative study of different solutions and thermal contact investigation between the exchanger and the PCM. Appl Energy 2016;166:107–16. [27] Biswas K, Lu J, Soroushian P, Shrestha S. Combined experimental and numerical evaluation of a prototype nano-PCM enhanced wallboard. Appl Energy 2014;131:517–29. [28] Borreguero AM, Sánchez ML, Valverde JL, Carmona M, Rodríguez JF. Thermal testing and numerical simulation of gypsum wallboards incorporated with different PCMs content. Appl Energy 2011;88:930–7. [29] Heim D. Isothermal storage of solar energy in building construction. Renew Energy 2010;35:788–96. [30] Borderon J, Virgone J, Cantin R. Modeling and simulation of a phase change material system for improving summer comfort in domestic residence. Appl Energy 2015;140:288–96. [31] Copertaro B, Principi P, Fioretti R. Thermal performance analysis of PCM in
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refrigerated container envelopes in the Italian context-Numerical modeling and validation. Appl Therm Eng 2016;102:873–81. Mathieu-Potvin F, Gosselin L. Thermal shielding of multilayer walls with phase change materials under different transient boundary conditions. Int J Therm Sci 2009;48:1707–17. Sun X, Zhang Q, Medina MA, Lee KO, Liao S. Parameter design for a phase change material board installed on the inner surface of building exterior envelopes for cooling in China. Energy Conver Manage 2016;120:100–8. Joulin A, Younsi Z, Zalewski L, Lassue S, Rousse DR, Cavrot JP. Experimental and numerical investigation of a phase change material: Thermal-energy storage and release. Appl Energy 2011;88:2454–62. Gunther E, Mehling H, Hiebler S. Modeling of subcooling and solidification of phase change materials. Modelling Simul Mater Sci Eng 2007;15:879–92. Uzan AY, Kozak Y, Korin Y, Harary I, Mehling H, Ziskind G. A novel multi-dimensional model for solidification process with supercooling. Int J Heat Mass Transfer 2017;106:91–102. Bédécarrats JP, Castaing-Lasvignottes J, Strub F, Dumas JP. Study of a phase change energy storage using spherical capsules. Part II: Numerical modelling. Energy Convers. Manage 2009;50:2537–46. Bony J, Citherlet S. Numerical model and experimental validation of heat storage with phase change materials. Energy Build 2007;39:1065–72.
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
A new heat transfer model of phase change material based on energy asymmetry
T
⁎
Xing Jina,b, , Huoyan Huc, Xing Shia,b, Xin Zhoua,b, Liu Yangc, Yonggao Yinc, Xiaosong Zhangc a
School of Architecture, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China Key Laboratory of Urban and Architectural Heritage Conservation (Southeast University), Ministry of Education, No. 2 Sipailou, Nanjing 210096, PR China c School of Energy and Environment, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China b
H I G H L I G H T S heat transfer model of PCM based on energy asymmetry is built. • AAnnew facility is built to validate the model. • Theexperimental • asymmetrical model has higher accuracy than the symmetrical model.
A R T I C L E I N F O
A B S T R A C T
Keywords: Phase change material (PCM) Heat transfer Numerical model Energy asymmetry
The melting process and the solidifying process of phase change material (PCM) are usually considered symmetrical in the traditional PCM heat transfer models, but there are inevitable calculation errors using these models. In this paper, a new heat transfer model of PCM based on the energy asymmetry was built, and it was validated by experimental data. It was found that the two main reasons for the energy asymmetry of the PCM were the melting temperature range and the solidifying temperature range were not the same and the supercooling problem during the cooling process. No matter for only one thermal cycle or for the multiple thermal cycles, the results from the asymmetrical model and the symmetrical model were different. Apparently, compared with the symmetrical model, the asymmetrical model had higher accuracy, and the heating/cooling process in the asymmetrical model was more consistent with the real heating/cooling process of the PCM.
1. Introduction Energy storage can reduce the time, space or rate mismatch between energy supply and demand, thereby it is playing an important role in energy conservation. Energy storage could save energy and make the system more effective by reducing the wastage of energy [1]. Among all the energy storage technologies, the thermal energy storage is recognized as one of the most effective technique to improve the performances of the energy systems [2,3]. Especially, it possesses a great adaption to the renewable energy [4]. Thermal energy storage is generally classified as sensible heat storage and latent heat storage. While, the latent heat storage system with phase change materials (PCMs) has attracted more and more researches in the last two decades [5,6]. PCM absorbs or releases the latent heat when the temperature of the material undergoes its phase change process. The advantages of PCM system include higher energy storage density, smaller temperature variation during the phase change process
⁎
while requiring smaller masses and volumes of material [7,8]. Therefore, PCMs have been used in many fields, not only in traditional areas such as solar energy storage and building heating/cooling, but also in new areas such as battery thermal management, photovoltaic cells and food preservation, etc. Phase change of a material is described by a particular kind of boundary value problem for partial differential equations, where phase boundary can move with time [9]. The heat balance equation in three dimensions at the solid-liquid interface of this phase change condition (i.e., Stefan condition) is:
ρH
ds (τ ) ∂t ∂t = λs s −λl l dτ ∂n ∂n
(1)
where ρ is the density, H is the heat of fusion of PCM, s(τ) is the solidliquid interface position, λ is the thermal conductivity, t is the temperature, τ is the time, n is the normal to the interface, subscript s and l refers to solid and liquid phases.
Corresponding author at: School of Architecture, Southeast University, No. 2 Sipailou, Nanjing 210096, PR China. E-mail address: [email protected] (X. Jin).
https://doi.org/10.1016/j.apenergy.2017.12.103 Received 24 August 2017; Received in revised form 30 November 2017; Accepted 30 December 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.
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Eq. (1) is suitable for both melting process and solidifying process. Usually, the solidifying process of PCM is recognized as the exact inverse process of its melting process, which means that there is only one PCM enthalpy-temperature curve which is used for both cooling process and heating process. The enthalpy of PCM is only dependent on its temperature. In other words, for each small temperature range, the amount of absorbed heat when the PCM is heated is the same as the amount of released heat when it is cooled. Therefore, the melting process and the solidifying process of PCM could be considered symmetrical [10]. Accordingly, in most of the PCM models, the melting temperature and the solidifying temperature are usually considered to be the same or the melting temperature range and the solidifying temperature are considered to be the same. Saffari et al. [11] proposed an optimization method coupled with an innovative PCM enthalpytemperature function to find out the optimum PCM melting temperature, the melting temperature range and the solidifying temperature range in their model were set to be the same. Tay et al. [12] built a new simplified two dimensional mathematical model to characterize phase change material systems, the melting temperature and the solidifying temperature of the chosen PCM were both 35 °C. Zhao and Tan [13] developed a numerical model for the PCM thermal storage unit, the phase change temperature range (i.e., both melting and solidifying) in their model was 19–23 °C. Jin et al. [14,15] built the PCM wall models and tried to find the optimal PCM location in the wall, the melting temperature range and the solidifying temperature range in the models were also the same. Other PCM models, such as natural cooling energy storage model [16], tube-in-tank latent thermal energy storage model [17], PV/PCM cell heat transfer model [18], PCM floor model [19,20], and PCM board model [21,22], also showed that the melting temperature ranges and the solidifying temperature ranges were all the same. However, because the melting temperature range and the solidifying temperature range of PCM are unlikely to be exactly the same, and the PCM may have supercooling problem [23,24], our previous research has showed that the melting process and the solidifying process of PCM are asymmetrical [10]. Therefore, the PCM models are divided into two types in this research, which are symmetrical model and asymmetrical model respectively. The PCM will absorb or release an amount of latent heat during its phase change process. Usually, the latent heat of PCM is represented by the effective heat capacity indirectly, which considers the phase change process as a sensible process with an increased (effective) heat capacity. The effective heat capacities of PCM during heating/cooling process in symmetrical model and asymmetrical model are shown in Fig. 1. The effective heat capacity of PCM in its phase change temperature range was assumed to be uniform. As shown in this figure, the melting temperature range of the PCM is [tm1, tm2], the solidifying temperature range is [ts1, ts2]. If [tm1, tm2] and [ts1, ts2] are the same, and for a certain temperature value, the effective heat capacity in the heating process is always equal to that in the cooling process (only considering the absolute values), just as shown in Fig. 1(a). It means the melting process and the solidifying process are symmetrical. Here, the model is referred to as the symmetrical model. As shown in Fig. 1(b), [tm1, tm2] and [ts1, ts2] are not the same, and the PCM could not release the latent heat until it is cooled down to tc, then the PCM is heated up to ts2 from tc by a part of latent heat released by itself. The solidifying temperature range of PCM is still [ts1, ts2], so it will release the rest of latent heat when it is cooled to ts1 from ts2. The PCM temperature variations with time during the heating process and the cooling process are shown in Fig. 2, which also shows the differences between the symmetrical model and asymmetrical model. It will be discussed in detail later in Section 2. In Fig. 1(a), if the PCM temperature is known, it is very easy to find that the value of the effective heat capacity in the heating process is always equal to that in the cooling process. However, in Fig. 1(b), they are not the same. For example, when the PCM temperature is between ts2 and tm2, the effective
Fig. 1. Effective heat capacity of PCM in the heating process and the cooling process (a) Symmetrical model; (b) Asymmetrical model.
heat capacity in the heating process is much higher than that in the cooling process. Therefore, the melting process and the solidifying process of PCM are asymmetrical. Accordingly, the model is referred to as the asymmetrical model. Because the heating/cooling process of the PCM in the symmetrical model does not fully accord with its real heating/cooling process, the calculation errors of the model are inevitable, and the errors may be significant, especially when there is a serious supercooling problem. According to the published papers in the databases, the enthalpy method [25–28], the effective heat capacity method [15,21,29–31], and the heat source method [29,32–34] are the most common methods to build the numerical heat transfer PCM models. However, the melting processes and the solidifying processes in the models with these methods are all symmetrical. The most popular kinds of commercial software such as EnergyPlus, ANSYS Fluent, and ESP-r also have the phase change modules, but which are still based on the energy symmetry. In only a few PCM models, the supercooling in the solidifying process is taken into account. Gunther et al. [35] introduced an algorithm for a simple but expedient modeling of supercooling and solidification in the PCM, the method was based on the explicit finite volume method and the enthalpy method. Uzan et al. [36] developed a novel mathematical model of multidimensional PCM solidification with supercooling. It was shown that the model could reflect the experimental results fairly well, especially when predicting temperatures at various locations inside the material. Bédécarrats et al. [37] presented a numerical study on the behavior of a test plant which was a tank filled with spherical nodules containing a phase change material. In the numerical model, the supercooling of PCM was considered during its crystallization process. Bony and Citherlet [38] built a numeric model 1410
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2. Asymmetrical energy model of PCM 2.1. Symmetrical model and asymmetrical model Here in this research, to simplify the model, as shown in Fig. 1, the effective heat capacity of the PCM in its melting temperature range or solidifying temperature range was assumed to be constant. In addition, the magnitude of the melting temperature range and the magnitude of the solidifying temperature range were assumed to be the same. The models based on Fig. 1(a) and (b) were referred to as the symmetrical model and the asymmetrical model, respectively. The differences between the symmetrical model and asymmetrical model of the PCM could also be presented based on the PCM temperature variations with time during the heating and cooling processes. As shown in Fig. 2, the PCM was heated up and then cooled down. The heating temperature during heating process and the cooling temperature during cooling process were both constant, and it was assumed that there was no temperature gradient inside the PCM. As shown in Fig. 2(a), the PCM was in its melting process from τ1 to τ2, and it was in its solidifying process from τ4 to τ5. The melting temperature range and the solidifying temperature range of the PCM were the same. The heating process of Fig. 2(b) was the same with Fig. 2(a), but their cooling processes were very different. As shown in Fig. 2(b), the PCM could not release its latent heat until it was cooled down to tc, and then it was heated to ts2 from tc by the latent heat released by itself. Then, it would release the rest of the latent heat from τ5 to τ6. Therefore, the heating process and the cooling process in Fig. 2(a) were considered symmetrical, while they were asymmetrical in Fig. 2(b).
2.2. Case study of PCM asymmetrical energy model In order to compare the symmetrical model and the asymmetrical model, a simple one-dimensional phase change heat transfer case for a thin PCM layer was analyzed. As shown in Fig. 3, the physical model was consisted of an insulting layer, an electric heater, two stainless steel plates and a PCM layer. The PCM was heated by the electric heater and then cooled by the air whose temperature was assumed to be constant. The PCM used here is CT26, which is a kind of inorganic PCM and has the supercooling problem. The thermophysical properties of the materials are shown in Table 1. To simplify the heat transfer model, the following assumptions are applied:
Fig. 2. Temperature variations with time during the heating process and the cooling process (a) Symmetrical model; (b) Asymmetrical model.
to simulate heat transfer process between PCM and water in a tank. It was assumed that as soon as a PCM part reached the point of crystallization, the whole PCM would turn into solid. However, only the solidifying process of PCM was analyzed in these models, the melting process was not included. Accordingly, there was no thermal cycle in these models. There are at least two different PCM effective heat capacity curves in thermal cycles if the melting process and the solidifying process of PCM are not symmetrical, one is for the melting process and the other is for the solidifying process. While only one PCM effective heat capacity curve is needed if only the solidifying process is simulated. Therefore, the model for thermal cycles is very different from that for only solidifying process. In this research, a new PCM model will be built, and it is not only used for solidifying process of PCM but also for thermal cycles. In addition, a new method is proposed to simulate the supercooling process of PCM accurately. It is assumed that there is an internal heat source in the PCM during the steep temperature increase process, and the energy of the heat source in this process is from the latent heat of PCM. It should be noticed that this method is totally different from the heat source method mentioned before, which is still like the enthalpy method or the effective heat capacity method, but could not simulate the supercooling process, either. This new asymmetrical model is expected to show the real thermal cycles of PCM and have higher accuracy than the traditional models.
x=0 x=į
x
6 5
41 2 3
1. Stainless steel plate #1; 2. PCM; 3 Stainless steel plate #2; 4. Electric heater; 5. Insulating layer; 6. Air-conditioned environment. Fig. 3. Physical model.
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Table 1 Thermophysical properties of materials. Material
Density, kg/ m3
Thermal conductivity, W/(m K)
Specific heat capacity (Solid), kJ/(kg K)
Specific heat capacity (Liquid), kJ/(kg K)
Latent heat, kJ/kg
Melting temperature range, °C
Thickness, mm
PCM Stainless steel plate Insulation (XPS)
1500 7850
0.3 15.0
2.5 0.5
3.2 /
165 /
23–28 /
10 2
29
0.03
1470
/
/
/
50
(1) Because the thickness of all the layers is much less than their width and height, the heat transfer through all the layers is one-dimensional. (2) The convection effect in the molten PCM is neglected. (3) As shown in Fig. 2(b), the temperature increase stage during the supercooling process could not be simulated well with traditional methods. In this model, when the PCM has supercooling, it is assumed that there is an internal heat source in the PCM from τ4 toτ5. (4) All the materials are assumed to be isotropic media. The governing equation of the model is
ρn cpn
∂t ∂ 2t = λn 2 + S ∂τ ∂x
(2)
where cp is the specific heat capacity, S is the internal heat source. n = 1 for the stainless steel plate #1, n = 2 for the PCM, n = 3 for the stainless steel plate #2. aHρ
S=
⎧ τ5 − τ4 when τ4 < τ < τ5 ⎨ 0 others ⎩
(3)
where a is the released rate of the latent heat from τ4 toτ5. The boundary conditions of the governing equation are
t (0,τ ) = theat
(x = 0, and heating)
(4)
λ1
∂t = 0 (x = 0, and cooling) ∂x
(5)
λ3
∂t = h (te−ta) (x = δ ) ∂x
(6)
where theat is the surface temperature of the electric heater, h is the heat transfer coefficient between the exterior surface of stainless steel plate #2 and the air, te is the exterior surface temperature of the stainless steel plate #2, ta is the air temperature, δ is the total thickness of plate #1, PCM and plate #2. The governing equation along with the boundary conditions was discretized using the finite difference method (FDM). Central difference was applied in space and fully implicit method (backwards difference) was applied in time. The space grid size was 0.2 mm and the time step was 60 s. The equations system was solved by tridiagonal matrix algorithm (TDMA). 3. Results
Fig. 4. Experimental facility (a) Front view (b) Side view.
3.1. Model validation
central interface of PCM layer, the temperature of the interface between PCM layer and steel plate #2, the exterior surface temperature of steel plate #2, and the air temperature were all measured. In addition, the surface heat flux of the steel plate #2 was also measured. The numbers, installation locations and accuracies of the thermocouple and heat flux meter are shown in Table 2. The whole experimental facility was located in an air-conditioned room, the indoor air temperature of which was constant. Two experiments were conducted to validate the model using this facility. The first experiment included a heating process and a cooling process. The electric heater temperature was set to be 65 °C, and the
To validate the numerical model, an experimental facility was built, as shown in Fig. 4. This facility was designed based on Fig. 3. It was consisted of two 300 mm × 300 mm × 2 mm stainless steel plates, an electric heater, and an insulating layer. The PCM was put in the 10 mm empty space enclosed by the two stainless steel plates and four pieces of plexiglass. The electric heater was used to heat the PCM, and a temperature controller was used to control the surface temperature of the electric heater. During the experiments, the temperature of the interface between electric heater and steel plate #1, the temperature of the interface between steel plate #1 and PCM layer, the temperature of the 1412
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Table 2 Numbers, installation locations and accuracies of sensors. Sensor
Number
Thermocouple Heat flux meter
Interface between heater and plate #1
Interface between plate #1 and PCM
Central face of PCM
Interface between PCM and plate #2
Exterior surface of plate #2
Air
4 /
4 /
4 /
4 /
4 4
2 /
o
Temperature ( C)
40
30
20
10
0
5
10
15
20
Time (hours) Fig. 5. Comparisions of the numerical temperatures in asymmetrical model and the measured temperatures (single thermal cycle).
400 350
Measured exterior surface heat flux of stainless steel plate #2 Numerical exterior surface heat flux in symmetrical model Numerical exterior surface heat flux in asymmetrical model
2
Heat flux (W/m )
300 250 200
50 5
10
15
0.4 °C 3%
As mentioned in the Introduction, the two main reasons for the PCM energy asymmetry were the melting temperature range and the solidifying temperature range were not the same and the PCM had supercooling. Therefore, the effects of these two aspects were analyzed in this section. Firstly, it was assumed that the PCM had no supercooling, only the melting temperature range and the solidifying temperature range were not the same. In the symmetrical model, the melting temperature range and the solidifying temperature range were both 23–28 °C. While in the asymmetrical model, the melting temperature range was 23–28 °C, the
100
0
−200 to 350 °C 0–200 KW/m2
3.2. Model analyses
150
0
Error
In this experiment, the electric heater temperature was also set to be 65 °C. Firstly, the PCM was heated for 3 h, then the heater was turned off and the PCM was cooled down for 1.5 h, then the heater was turned on and the PCM was heated for 18 min, then the heater was turned off and the PCM was cooled down for 1.3 h, then the heater was turned on and the PCM was heated for 25 min, then the heater was turned off and the PCM was cooled down for about 12 h. The heating times during the second and third thermal cycles were realatively short was because the highest PCM temperatures during these two cycles were controlled, then the temperature variations here were very similar to the temperature variations of the exterior surface of a wall during a day in summer. The wall temperature goes up gradually in a sunny day, then the wall is cooled down because of a short heavy rain at noon, and then the wall temperature goes up again because of the strong solar radiation. Finally, the wall temperature goes dwon gradually from the afternoon. Fig. 7 shows the comparisions of the numerical temperatures in the asymmetrical model and the measured temperatures. Fig. 8 shows the comparisions of the numerical surface heat fluxes and the measured surface heat fluxes. It was also found that the numerical results from the asymmetrical model had very good agreement with the experimental results, and the asymmetrical model had higher accuracy than the symmetrical model.
Measured central temperature of PCM layer Numerical central temperature of PCM layer in asymmetrical model Measured exterior surface temperature of plate #2 Numerical exterior surface temperature of plate #2 in asymmetrical model
50
Range
20
Measured central temperature of PCM layer Numerical central temperature of PCM layer in asymmetrical model Measured exterior surface temperature of plate #2 Numerical exterior surface temperature of plate #2 in asymmetrical model
Time (hours) 50
Temperature ( C)
Fig. 6. Comparisions of the numerical surface heat fluxes and the measured surface heat fluxes (single thermal cycle).
o
PCM was heated for nearly five hours, then the heater was turned off and the PCM was cooled down for nearly 20 h. Fig. 5 shows the comparisions of the numerical temperatures in the asymmetrical model and the measured temperatures. As shown in the figure, the numerical data had very good agreement with the measured data. Fig. 6 shows the comparisions of the numerical surface heat fluxes and the measured surface heat fluxes. As shown in the figure, the results from the asymmetrical model also had very good agreement with the measured results. It was very obvious to observe that the differences between the symmetrical model and the measured data were large, especially during the supercooling process. The previous experiment showed that the asymmetrical model had high accuracy in a single thermal cycle. To validate the accuacy of the model in multiple thermal cycles, another experiment was conducted.
40
30
20
10
0
5
10
15
Time (hours) Fig. 7. Comparisions of the numerical temperatures in asymmetrical model and the measured temperatures (multiple thermal cycles).
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150
400 350
2
Heat flux (W/m )
300 250
Heat flux (W/m 2 )
Measured exterior surface heat flux of stainless steel plate #2 Numerical exterior surface heat flux in symmetrical Model Numerical exterior surface heat flux in asymmetrical Model
200 150 100
Symmetrical model o Asymmetrical model (3 C) o Asymmetrical model (5 C) o Asymmetrical model (7 C)
100
50
50 0
0
5
10
0
15
Time (hours) Fig. 8. Comparisions of the numerical surface heat fluxes and the measured surface heat fluxes (multiple thermal cycles).
10
Time (hours)
150 Symmetrical model o Asymmetrical model (3 C) o Asymmetrical model (5 C) o Asymmetrical model (7 C) 2
Heat flux (W/m )
100
50
0
5
10
15
15
20
respectively. The cooling times of the three cooling processes were 2 h, 2 h and 15 h, respectively. It was also found that the heating processes and the cooling processes of the asymmetrical model were all different from the symmetrical model expect for the first heating process. Then, the combined effects of supercooling and ΔT on the heating/ cooling process of the PCM were analyzed. In the symmetrical model, the melting temperature range and the solidifying temperature range were both 23–28 °C. While in the asymmetrical model, the solidifying temperature range was 20–25 °C. In addition, the degrees of supercooling (DSC) in the asymmetrical model were 0 °C, 2 °C, 4 °C and 6 °C, respectively. The system was heated for 3 h with the heating temperature of 35 °C, and then it was cooled down in the air with the temperature of 10 °C. Fig. 11 shows the exterior surface heat fluxes of the stainless steel plate #2. Actually, the black line and the red line in this figure were the same with the black line and the red line in Fig. 9. The effects of DSC could be observed based on the four heat flux lines of the asymmetrical model. It was very easy to find that the cooling processes of PCM with supercooling were very different from the cooling process of PCM with no supercooling. Because of the supercooling, the starting solidifying time was delayed, and the period for releasing latent heat was short. Compared with the asymmetrical model, when DSC were 0 °C, 2 °C, 4 °C and 6 °C, the maximum absolute errors of the symmetrical model were 27.7 W/m2, 24.9 W/m2, 36.0 W/m2 and 45.6 W/m2, and the maximum relative errors were 55.7%, 40.5%, 55.9% and
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Fig. 10. Surface heat fluxes in the asymmetrical model with different ΔT (multiple thermal cycles).
solidifying temperature ranges were 20–25 °C, 18–23 °C and 16–21 °C, respectively. Here in this research, the temperature difference (ΔT, i.e., hysteresis) between the melting temperature range and the solidifying temperature range was used to present the energy asymmetry degree of PCM. For example, when the solidifying temperature range was 20–25 °C, ΔT was 3 °C. The physical model here was the same with Fig. 3. The PCM was heated for 3 h with the heating temperature of 35 °C, and then it was cooled down in the air with the temperature of 10 °C. Fig. 9 shows the exterior surface heat fluxes of the stainless steel plate #2. As shown in the figure, the cooling processes of the asymmetrical model and the symmetrical model had great differences, which were increased with the increase of ΔT. Compared with the asymmetrical model, when ΔT were 3 °C, 5 °C, and 7 °C, the maximum absolute errors of the symmetrical model were 27.7 W/m2, 31.4 W/m2 and 37.0 W/m2, and the maximum relative errors were up to 55.7%, 78.5% and 92.5%, respectively. It was also found that the starting solidifying time of the asymmetrical model was delayed. In addition, because of the temperature difference between the PCM and the air in the asymmetrical model was smaller than that in the symmetrical model, more time was needed for releasing the latent heat during the solidifying process. Fig. 10 shows the multiple thermal cycles of the PCM. The heating temperature was 35 °C, the air temperature was 10 °C. The heating times of the three heating processes were 3 h, 0.5 h and 0.5 h,
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Fig. 9. Surface heat fluxes in the asymmetrical model with different ΔT (single thermal cycle).
Fig. 11. Surface heat fluxes in the asymmetrical model with different DSC (single thermal cycle).
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energy asymmetry was proposed, and an experimental facility was built to validate the model. The numerical results had good agreement with the experimental results. Based on the numerical results, it was found that no matter for a single thermal cycle or the multiple thermal cycles, the results from the asymmetrical model and the symmetrical model were different. The maximum relative error of the symmetrical model could be more than 90% in the simulated conditions. Compared with the symmetrical model, the asymmetrical model had higher accuracy, and the heating/cooling process in the asymmetrical model was more consistent with the real heating/cooling process of the PCM. This asymmetrical model had great potential for real applications.
Symmetrical model o Asymmetrical model (DSC=0 C) o Asymmetrical model (DSC=2 C) o Asymmetrical model (DSC=4 C) o Asymmetrical model (DSC=6 C)
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This research was supported by the Natural Science Foundation of China under Grant No. 51308104, the Ministry of Science and Technology of China under Grant No. 2016YFC0700102, and the Fundamental Research Funds for the Central Universities.
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Time (hours) Fig. 12. Surface heat fluxes in the asymmetrical model with different DSC (multiple thermal cycles).
References
88.6%, respectively. Heat flux differences between the symmetrical model and the asymmetrical model in multiple thermal cycles could be observed in Fig. 12. The heating temperature, the cooling temperature, the heating time, and the cooling time were all the same with Fig. 10. It was found that the heat flux variations of the asymmetrical model and the symmetrical model were very different, and because the solidifying process of the PCM with supercooling was delayed, the fluctuations of the heat flux were increased with the increase of DSC.
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4. Discussions As mentioned in the Introcution, the solidifying process of PCM was usually recognized as the exact inverse process of its melting process, and the traditional heat transfer model of PCM was usually symmetrical. However, our analyses had shown that the melting process and the solidifying process were asymmetrical, and the numerical results from Section 3 had proved that the asymmetrical model had higher accuracy than the symmetrical model. This asymmetrical model proposed in this research could be used for the real application. Here, the PCM wall in buildings is taken as an example to show the problems of the symmetrical model and the advantages and the actual application of the asymmetrical model. The thermal performance evaluations of the PCM wall with the symmetrical model are right when the outdoor temperature is increasing from morning to noon, but the evaluations are not accurate when the outdoor temperature is decreasing from afternoon to night, just like the results in Figs. 9 and 11. In addition, if the solar radiation or the outdoor air temperature has large fluctuations (e.g., a short rainstorm in a hot sunny day during the daytime), the PCM could be heated and cooled alternately for several times, the heat fluxes through the PCM wall will also have large fluctuations. Accordingly, the thermal performance evaluations of the PCM wall with the symmetrical model will have large errors, just like the results in Figs. 10 and 12. Conversely, the asymmetrical model could demonstrate the real PCM temperature variations and the real surface heat flux variations of a PCM wall, and the thermal performance analyses of the PCM wall with the asymmetrical model will be more consistent with the real conditions, just like the results in Figs. 9–12. Therefore, it is concluded that the asymmetrical model has not only high calculation accuracy but also great potential for real applications. 5. Conclusions In this paper, a new heat transfer model for the PCM based on the 1415
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