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Modified computational methods using effective heat capacity model for the thermal evaluation of PCM outfitted walls
International Communications in Heat and Mass Transfer 108 (2019) 104278
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Modified computational methods using effective heat capacity model for the thermal evaluation of PCM outfitted walls
T
Yuan Zhang School of Energy and Power Engineering, Jiangsu University, 301, Xuefu Road, Zhenjiang, Jiangsu 212013, China
ARTICLE INFO
ABSTRACT
Keywords: Phase change material (PCM) Effective heat capacity model Modified computational methods Building envelope
Some modified computational methods when using two-dimensional effective heat capacity model for predicting the transient heat transfer process of the building envelopes outfitted with phase change materials (PCMs) were presented. In the methods, some important improvements were performed, such as considering different melting and solidification temperature ranges, multiple temperature turning points and PCM's liquid fraction during phase change process. The experiments were conducted to verify the accuracy of the proposed methods. Based on the experimental data, the comparative analysis of the presented methods together with the original methods using enthalpy model and effective heat capacity model were conducted. The deviations produced by the original methods were found to be markedly larger than the presented ones. The standard and the maximum deviations of the modified methods were less than 0.25 °C and 0.55 °C, and the corresponding relative errors were under 2.5% and 4.0%.
1. Introduction As one of the important approaches to improve the energy conservation level and indoor thermal comfort of building envelope, more and more studies have been conducted on the thermal performance of the building envelope equipped with phase change materials (PCMs). Effective heat capacity model is one of the commonly-used mathematical models for predicting the phase change heat transfer (PCHT) process. In the model, the phase change process is converted into a nonlinear heat transfer problem with “single phase”. Therefore, PCM's phase state, sensible and latent heat will not be distinguished, and the PCM's effective heat capacity value changing with its temperature is used in calculations [1]. Some researchers analyzed the PCHT problems with the effective heat capacity based models. Borreguero et al. [2] obtained a reasonable agreement between computational and experimental results if PCM's latent heat (equivalent heat capacity) had been properly described. The thermal behavior of the PCM component only depended on the thermophysical properties of the used materials. Bowman and Brown [3] built a new type of effective heat capacity model which was verified to be more accurate than other models. In the model, the effective heat capacity was taken as the function of the temperature, percent of phase change material, and rate of energy addition. Heim and Clarke [4] embedded effective heat capacity method into ESP-r software and reached acceptable results. Stathopoulos et al. [5] experimentally
E-mail address: [email protected]. https://doi.org/10.1016/j.icheatmasstransfer.2019.104278
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verified the established new mathematical model by combining thermal balance model and sensible heat capacity model, and reached good agreements with experimental data. Furthermore, others explored new methods to precisely determine the thermal properties of PCM. Kravvaritis et al. [6] presented a thermal delay method by which the phase change temperature range, latent heat and effective heat capacity of PCM can be precisely determined for the engineering usage. LosadaPérez et al. [7] precisely obtained PCM's equilibrium enthalpy and effective heat capacity by Adiabatic Scanning Calorimetry. Also, Mandilaras et al. [8] proposed a new method for determining the effective heat capacity of shape stabilized PCM. Meanwhile, people found that deviation existed when using original effective heat capacity model, and the accurate results cannot be obtained in some cases. This question was paid more attention in the past years, and some impact factors had been analyzed. Gong and Mujumdar [9] analyzed the reasons of the non-convergence and calculation error, and presented a new optimized equation scheme and eliminated some disadvantages of the original effective heat capacity model such as without the limitation of the phase change temperature interval. Yang and He [10] composited effective heat capacity method with element-free Galerkin method, and compared the results of the new model with analytical solution. They found the numerical oscillation was avoided by the new model, which made the results more accurate. Yao and Chait [11,12] successfully built a new equivalent heat capacity model for improving the strict limitation of
International Communications in Heat and Mass Transfer 108 (2019) 104278
Y. Zhang
Nomenclature
Nsta,i T Tx,y,τ Tx,y,τ+1 x y
area represented by PCM's total latent heat, m2 scanned area from the beginning of a melting process to the end, m2 scanned area from the beginning of a solidification process to the end, m2 area represented by PCM's absorbed latent heat, m2 specific heat, kJ·kg−1·K−1 effective heat capacity of PCM, kJ·kg−1·K−1 enthalpy of PCM, kJ·kg−1 sequence number PCM's total latent heat, kJ·kg−1 PCM's absorbed latent heat, kJ·kg−1 mass of PCM, kg PCM's liquid mass, kg PCM's total mass, kg data amount ith datum, °C
A A' A" A1 c ceff H i L L' m ml M n Ni
Greek symbols
σsta σmax
50
o
Effective heat capacity (kJ/(kgK))
Heat flow (mW)
works can be summarized as two aspects: the measurement of PCM's parameters and the numerical simulation method. However, there still be other assumptions in PCM's thermo-physical parameters and heat transfer process in original model that result in deviations. For example, the different melting and solidification TRs, multiple temperature turning points when melting and solidification, and PCM's liquid fraction during phase change process. However, the study of these factors is insufficient. Author's previous publications [17,18] have briefly referred to part of related contents, such as the TRs and liquid fraction, but more detailed analysis is still needed and the original computational method is waiting for modifications for the more accurate results. In the first part of the paper, based on the original model, some modified computational methods when using two-dimensional effective heat capacity model were presented. In the new methods, different melting and solidification temperatures, multiple temperature turning points, and PCM's liquid fraction during phase change process, were properly considered which were neglected in the original models. Subsequently, a serious of validation experiments was conducted. Finally, the accuracy of the presented methods together with the original methods using enthalpy model and effective heat capacity model were verified and compared based on the experimental data.
Peak=33.0 C
Delta H=138.04J/g
10
density, kg·m−3 time, s thermal conductivity, W·m−1·K−1 liquid fraction, % liquid fraction at the beginning of a melting process, % liquid fraction at the beginning of a solidification process, % standard deviation, °C maximum deviation, °C
ρ τ λ η η' η″
time discretization in original model. Ye et al. [13] analyzed the melting process of a paraffin-based stabilized PCM using effective heat capacity model. They found accurate results can be obtained if PCM's effective heat capacity values were calculated from the data of differential scanning calorimetry (DSC). Zhou et al. [14] found that the calculation became simpler and the numerical singular problem when the phase change temperature was a single value or narrow temperature range (TR) can be effectively resolved by using unit equivalent heat capacity. Following with Zhou's work, He and Yang [15] evaluated the influence on the calculation results accuracy produced by the finite element grid density, numerical integration scheme of heat capacity matrix and relevant parameters of Sigmoid function. Zsembinszki et al. [16] found that PCM's small time step and large number of grid produced accurate calculation results while accompanied by significantly computing time. The computational efficiency was enhanced by using PCM's specific heat obtained by the last iterative computation and constant physical parameters of heat transfer fluid. However, it produced slight deviations. Besides, the inlet temperature of heat transfer fluid, PCM's melting temperature, density and specific heat influenced the calculation results, while PCM's thermal conductivity, flow rate, convective heat transfer coefficient of heat transfer fluid, wall's thermal conductivity had little impact. The factors producing the computation deviations in the above 20
ith standard datum, °C temperature, °C temperature of point (x,y) at time τ, °C temperature of point (x,y) at time τ + 1, °C direction of thickness, m direction of width, m
0
-10 Delta H=-150.97J/g -20
Melting curve Solidification curve
40 30 20 10 0 -10 -20 -30 -40 Melting curve Solidification curve
-50 -60
o
Peak=26.4 C
-70
-30 0
10
20
30
40
50
60
0
70
10
20
30
40
50
60
o
Temperature ( C)
o
Temperature ( C)
a. DSC curves of PCM
b. Effective heat capacity curves of PCM
Fig. 1. DSC and effective heat capacity curves of the experimental PCM. 2
70
International Communications in Heat and Mass Transfer 108 (2019) 104278
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2. Modified computational methods using effective heat capacity model
In the presented model, the thermo-physical parameters, such as PCM's density and thermal conductivity, can be different when the PCM is solid, solid and liquid coexisting state and liquid. Furthermore, the thermal properties of each spatial point and time were all calculated using PCM's liquid fraction.
2.1. Assumptions Some assumptions were made in the presented methods. (1) All the materials were considered to be thermally homogeneous and isotropic. (2) With the effective restraint of the convection effect of liquid PCM with gypsum, the thermal conductivity of PCM was considered to be constant. (3) PCM's volume change and sub-cooling effects during phase change process were ignored. (4) The heat transfer in the computational domain was simplified to be two-dimensional heat transfer process (the heat transfer in vertical direction was ignored). (5) The equivalent thermal conductivity was applied for the closed air layers [19].
2.3. Initial and boundary conditions The initial temperature inside all the materials was considered to be the same. For the exterior and interior surfaces, the heat convection thermal boundary condition was applied. Adiabatic condition was applied on the width boundaries. 2.4. Different melting and solidification temperatures Like the different phase change TRs of the PCM in Fig. 1 and Table 1, PCM's melting and solidification temperature ranges are often not the same. If they are considered as the same range, deviations will occur. The proposed methods can be used to calculate the heat transfer of the PCM with the same or different melting and solidification temperature ranges.
2.2. Governing equation The governing equation of the presented model is shown in Eq. (1). PCM's effective heat capacity values changed with temperatures were obtained from the DSC measurement. In the model, sensible heat and latent heat were not distinguished, and PCM's specific heat was represented by the effective heat capacities. Because the PCM capric acid was used in the experiments, the referred PCM should be the capric acid in the following text.
ceff
T
=
2T
x2
+
2.5. Temperature turning points during phase change 2.5.1. Introduction Different phase change processes are shown in Fig. 2 in which the PCM holds different melting and solidification TRs. Like the Process 1, if PCM always transforms from the complete solid to the complete liquid and returns to the complete solid, the solid, solid and liquid coexisting, and liquid PCM can be respectively calculated according to the phase change TRs and thermo-physical parameters. This situation is simple. However, most phase change processes are not always like this. For instance, PCM transforms from the complete liquid to solid and liquid coexisting state, and returns to the complete liquid (Process 2 in Fig. 2) or another solid and liquid coexisting state (Process 3 in Fig. 2). Hence, the actual case is more complex than the ideal one. Multiple temperature turning points usually exist during phase change process.
2T
y2
(1)
where ρ, λ are the density and thermal conductivity of PCM, respectively. ceff is the effective heat capacity of PCM which is calculated by Eq. (2). T and τ are temperature and time. The thickness and width directions are represented by x and y.
ceff =
dH dT / m d d
(2)
where H is the enthalpy of PCM. dH/dτ is heat flow rate. m is the mass of PCM. dT/dτ is the velocity of PCM's temperature rise or fall. To know the thermo-physical parameters of PCM, nitrogen protected DSC device was used. This test was conducted with the temperature change velocity of 5 °C·min−1. The DSC curves of the PCM sample (4.980 mg) are shown in Fig. 1a. The curve with the positive values is the melting curve, while the negative one is the solidification curve. It is observed that the peak temperatures were 33.0 °C (melting) and 26.4 °C (solidification), and the latent heat were 138.04 J·g−1 (melting) and 150.97 J·g−1 (solidification). As a result of the slight unavoidable deviation when determining the PCM's phase change temperature ranges, the two latent heats were slightly different. The thermal properties of PCM are shown in Table 1. The effective heat capacity of PCM was calculated using the specific heat of sapphire (taken as the standard data) and Eq. (2), shown in Fig. 1b. It can be seen that the trend of the DSC curves and effective heat capacity curves are almost the same. The PCM's peak specific heat were 46.9 J·g−1·K−1 (melting) and 69.0 J·g−1·K−1 (solidification), respectively. In calculations, the specific heat of each spatial point inside the PCM was obtained by the interpolation of the two closest effective heat capacity values based on the temperature of the point. The interpolation was conducted using the proportion of the distance between the temperature of this point and the two closest temperatures in the obtained temperature-effective heat capacity data pairs. Hence, the sensible, latent heat and phase state of the PCM were not needed to be distinguished. The actual phase change process can be precisely described if the amounts of PCM's temperature-effective heat capacity data pairs were abundant. Enough data pairs can be obtained with small temperature intervals (e.g. 0.01 or 0.02 °C) in DSC measurement.
2.5.2. Methodology in the presented methods In the presented methods, when the temperature inside the PCM was increasing and already in PCM's melting TR, it was considered that the solid part was melting. At this time, only the conversion of the sensible heat took place in the already existed liquid section without phase change. The PCM's thermal properties were obtained by weighting the proportion of (1) the thermal properties of the solid PCM in which the phase change was taking place, and (2) the parameters of the liquid part without phase change. And the proportion was the mass of the solid to the liquid at the start of this phase change process. There are two types of the special processes. The first one is: when PCM is melting, the two phase change processes before and after this melting process are both solidification processes. That is, the whole process is from the solidification to melting and to the solidification again (Process 4 in Fig. 2). The second is that there are two melting processes divided by several temperature swings without phase transition (Process 5 in Fig. 2). Namely, following the first melting process, PCM's temperature decreases but does not enter into its solidification Table 1 Thermo-physical properties of the PCM (capric acid).
3
Density (kg·m−3)
Thermal conductivity (W·m−1·K−1)
Melting TR (°C)
Solidification TR (°C)
886 (Liquid) 1004 (Solid)
0.152
31.4–34.5
24.6–27.1
International Communications in Heat and Mass Transfer 108 (2019) 104278
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Fig. 2. PCM's phase change processes.
TR, and then increases again. After several temperature swings like this, the temperature enters its melting TR and the PCM melts again. The process of the first case is simple; while the second is not. As was introduced above, the PCM's parameters of the second phase change process were determined by the state at the end of the first one. For example, PCM solidifies and its temperature decreases to a value that remains in its solidification TR; after that, the decline pauses for a period of time and then another temperature decrease begins (Process 6 in Fig. 2). In the new methods, the two temperature fall and solidification periods were sequential and taken as one phase change process. Nevertheless, there is another example which is different. PCM solidifies to a temperature that remains in its solidification TR; then its temperature rises to another one which is still in its solidification TR but not in the melting TR, and followed by another temperature fall (Process 7 in Fig. 2). In this case, the PCM's liquid fraction at the end of each phase change process would be recalculated, and the second temperature fall was considered to be another solidification process. The liquid fraction in the second process was the one calculated at the end of the first process, but PCM's temperature and effective heat capacities at the start of the second process did not follow the values at the end of the first one. When the calculated PCM has different melting and solidification TRs, the process like this case probably takes place.
effective heat capacity curve, the vertical line of PCM's temperature, and the horizontal axis, to the encircled area by PCM's effective heat capacity curve and the horizontal axis. Namely, it is the percentage of the shaded area to the encircled area by the curve and the horizontal axis. Therefore, PCM's liquid fraction can be equivalent to the percentage of the area of the absorbed latent heat to the one of the total latent heat (Eq. (5)). The areas in the figure were calculated by the integral method. Therefore, to collect enough data and make the calculation result more precise, the data pairs of the temperature-heat flow rate should be obtained as many as possible in the DSC measurement. If the PCM's effective heat capacity data are sufficient, the area can be calculated by the sum of the product of each effective heat capacity value and its small temperature interval. While if the data are not very abundant but adequate, the effective heat capacity curve can be fitted out. Followed by an integral computation, the corresponding areas can be obtained.
=
2.6.2. Special situations If a melting process began with the solid and liquid coexisting state, only the solid part melted. The liquid fraction was obtained by Eq. (6). Similarly, if a solidification process began with the solid and liquid coexisting state, only the liquid part solidified. The liquid fraction was obtained by Eq. (7).
2.6.1. Methodology PCM's liquid fraction is the ratio of the liquid mass to the total mass, with the value ranging from 0 to 100% (Eq. (3)).
ml 100% M
(3)
where η is PCM's liquid fraction, ml is PCM's liquid mass, and M is PCM's total mass. In fact, it is quite difficult to obtain PCM's liquid mass at a certain time. In the presented methods, PCM's liquid fraction was equivalent to the percentage of the absorbed latent heat to the total latent heat. That is, the increase or decrease of the liquid mass was considered to change following the absorbed or released latent heat with the same ratio and velocity. If PCM's temperature increased from the complete solid and then began to melt, the liquid fraction of each point within the PCM was calculated by Eq. (4).
=
L 100% L
(5)
where A1 is the area represented by PCM's absorbed latent heat, and A is the area represented by PCM's total latent heat. The PCM's initial liquid fraction was calculated using the PCM's initial temperature and melting TR. If the PCM began from the complete solid or liquid, its initial liquid fraction was considered as 0 or 100%.
2.6. Liquid fraction
=
A1 100% A
=
+ (1
)
A 100% A
(4)
'
where L is PCM's absorbed latent heat, and L is PCM's total latent heat. However, it is still difficult to acquire PCM's absorbed latent heat at a certain time. The absorbed or released heat does not change with PCM's temperature in a linear manner. In fact, the heat changes with PCM's effective heat capacity curves. As is shown in Fig. 3, PCM's liquid fraction should be the percentage of the encircled area by PCM's
Fig. 3. Schematic diagram of PCM's liquid fraction. 4
(6)
International Communications in Heat and Mass Transfer 108 (2019) 104278
Y. Zhang
where η' is the liquid fraction at the beginning of the melting process. A′ is the scanned area from the beginning of this melting process to the end.
=
A 100% A
in spatial grid while a fully implicit finite difference scheme and forward differences were applied in time grid. The step size of spatial grid was 0.005 m and time grid was 300 s. Gauss-Seidel scheme was employed in the point iterations. Eq. (10) is the convergence criterion. The numerical calculations were conducted using a MATLAB code [23].
(7)
max |Tx , y,
where η″ is the liquid fraction at the beginning of the solidification process. A″ is the scanned area from the beginning of this solidification process to the end. For instance, the amount of the liquid PCM at the end of a melting process was obtained by the sum of the two parts: (1) the original liquid PCM at the start of this process, (2) the liquid PCM converted from the original solid part at the end of this process. Therefore, the old liquid fraction was then replaced by the new calculated one. PCM's liquid fraction was calculated using the temperature-effective heat capacity data. Thus, PCM's DSC measurement must be accomplished.
+
c = ( 1 c1 +
2 c2 )/2
3 c3
+
4 c4 )/4
10
10
(10)
3. Experimental setup 3.1. Hollow block A concrete hollow building block (Fig. 5) wall equipped with PCM was used and various experiments were conducted under periodical temperature conditions. The hollow block had three cavity rows, in which the thickness of the middle one was two times as the side one. The thermo-physical parameters of the block materials are shown in Table 2 [19]. The PCM was incorporated into different rows of cavities (Block 1, 2, and 3) and forming different experimental walls (Fig. 5c–e). Even though under the same boundary condition, the PCMs locating in different rows of cavities meant the different temperature conditions to them which influenced the effect of the phase change. Therefore, the walls with the PCM incorporated in different rows of cavities were actually different walls. To maintain the comparability among different hollow block walls, the PCM with the same mass (0.314 kg) was incorporated in the walls. Because the thickness of the middle cavities was larger than the side cavities, it was pure PCM filled in the side cavities (Fig. 5c–e), while the PCM mixed with a certain amount of gypsum and water was filled into the middle cavities (Fig. 5d).
2.7.1. Grid node of the interface For the grids at the interface between different materials, its center node located exactly on the interface [20]. In Fig. 4, four equal parts with their own thermo-physical properties constituted one internal grid. Therefore, it can be one to four kinds of materials within one internal grid. The volume specific heat of each grid was calculated by Eq. (8). The size of the boundary grids was half of the internal grids with their center nodes locating on the boundary lines. Each boundary grid consisted of the two parts with their own thermo-physical parameters. Therefore, one or two kinds of materials were included in the boundary grids. The volume specific heat was obtained by Eq. (9). The corner grid size of the internal grid with only one material inside each. 2 c2
+ 1|
where Tx,y,τ and Tx,y,τ+1 are the temperatures of the point (x,y) at the iteration step of τ and τ + 1 for each time step.
2.7. Numerical methodology
c = ( 1 c1 +
Tx, y,
(8)
3.2. Experimental device and methodology
(9)
where ρc is node's volume specific heat. ρ1c1, ρ2c2, ρ3c3, ρ4c4 are the volume specific heat for different parts of node, respectively.
The method of experiments was temperature-change hot chamber method [24]. Fig. 6 is the used building envelope thermal performance testing device (Type JW-I). Thermal insulation material was filled into the frames of the chambers. The air temperature was controlled by the heat and cold sources in the left chamber, while the natural air condition was maintained in the right chamber. The walls for testing were
2.7.2. Numerical calculation settings The governing equation and boundary condition were discretized by the Finite Difference Method [21,22]. Central differences were applied
Fig. 4. Distributions of grid nodes. 5
International Communications in Heat and Mass Transfer 108 (2019) 104278
Y. Zhang
190 40
25 20 30
30 40
140
390
140
40
30 20 25
(a) Photo of hollow block
Outside
Inside
(b) Block size
Outside
Inside
(c) Block 1
Outside
(d) Block 2 Fig. 5. Hollow blocks.
Table 2 Thermo-physical parameters of the block materials. Thermo-physical parameters
Density −3
Block materials
(kg·m
Lightweight concrete Air layer (20 mm thick) Air layer (40 mm thick)
1600 1.2 1.2
Table 3 Boundary conditions of different walls.
Specific heat )
−1
(J·kg
600 1005 1005
·K
−1
)
Thermal conductivity −1
(W·m
·K
−1
Boundary TRs
)
0.62 0.143 (Equivalent) 0.267 (Equivalent)
Shared boundary TRs
Added boundary TRs for Block 1 wall
built between the two chambers. All of the experimental walls were measured under the boundary conditions shown in Table 3.
Added boundary TRs for Block 2 and 3 walls
3.3. Uncertainty of testing device The air temperature swing and heat loss of chambers were lower than 0.5 °C and 5%. The thermocouples were T type. The testing range was from −50 °C to 100 °C, and the accuracy was ± 0.5 °C or 0.4% (the
TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR5 TR6 TR7 TR8
TRs of the left side air
Spans of the left side air's TRs
(°C)
(°C)
19.9–43.2 1.2–67.2 3.9–72.6 1.0–83.4 27.0–34.5 23.5–36.9 20.9–40.4 17.5–42.7 17.6–42.4 11.6–48.0 10.7–54.1 11.9–60.3
23.3 66 68.7 82.4 7.5 13.4 19.5 25.2 24.8 36.4 43.4 48.4
Thermal Insulation Material Aluminium Plate
Left Chamber
Cold Source
Right Chamber
Fan Aluminium Plate
Heat Source
Fan
Experimental Wall
(a) Photo of experimental device
(b) Schematic diagram Fig. 6. Experimental devices. 6
Specimen Bracket
International Communications in Heat and Mass Transfer 108 (2019) 104278
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Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
70
o
Temperature ( C)
o
35 30 25 20
50 40 30 20 10
15
0 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h) Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
70
60
70
80
90
100 110 120
(b) TR2
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
80
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
70
Temperature ( C)
60 50
60
o
o
50
Time (h)
(a) TR1
Temperature ( C)
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
60
40
Temperature ( C)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
40 30 20
50 40 30 20 10
10
0 0 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
45
70
80
90
100 110 120
(d) TR4
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
40 o
o
Temperature ( C)
40
Temperature ( C)
60
Time (h)
(c) TR3 Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
50
35 30 25 20
35 30 25 20
15
15 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
50
60
70
80
90
100 110 120
Time (h)
(f) TR6 Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
40 o
o
Temperature ( C)
40
Temperature ( C)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
35 30 25 20
35 30 25 20
15
15 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
50
60
70
Time (h)
(g) TR7
(h) TR8
Fig. 7. Results and calculation deviations of Block 1 wall.
7
80
90
100 110 120
International Communications in Heat and Mass Transfer 108 (2019) 104278
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accuracy should be subject to the larger one).
It is observed from Figs. 7–9 that the deviations produced by the original enthalpy model and effective heat capacity model were usually greater than the presented methods. Based on these results, it can be known that the estimations without considering PCM's different melting and solidification temperature ranges, multiple temperature turning points and liquid fraction during phase change process, would produce large deviations away from the actual cases. For the presented methods, the standard and maximum deviations became relatively large with large boundary temperature swings (TR2 TR4). The standard deviations σsta were from 0.36 °C to 1.37 °C, and the maximum deviations σmax were 0.72 °C–2.75 °C. Nevertheless, the deviations reduced under the narrower boundary TRs (TR1, TR5 - TR8), with the σsta and σmax ranging from 0.09 °C to 0.80 °C and from 0.19 °C to 1.37 °C, respectively. From these cases, it can be found that the standard and maximum deviations increased with the span of boundary TRs. However, the trend for the relative errors was basically adverse. The relative errors reduced when the boundary TRs turned wider (TR2 TR4). The relative errors of σsta were from 0.5% to 2.0%, and that of σmax located in 1.1% - 3.7%. Instead, the relative errors were usually large under the narrow boundary TRs (TR1, TR5 - TR8). The relative errors of σsta lay between 0.4% and 2.7%, and the relative errors of σmax were from 0.8% to 5.2%. It is summarized that the relative errors usually decrease with the increasing span of boundary TRs. This can be explained by the thermal inertia of wall, which generates the surface temperature swing and makes it lower than the surrounded air's. In addition, it can be observed that the temperature decrement increased with the boundary temperature swings. Also, the temperature swings on the left surface were larger than the right surface, making the deviations on the left surface relatively larger. The σsta and σmax on the two sides of wall were lower than 1.37 °C and 2.75 °C. Also, in most cases, the two values were lower than 0.70 °C
4. Deviation evaluation indices The used indices standard deviation and maximum deviation were calculated using Eqs. (11) and (12). n sta
=
Nsta, i ) 2 / n
(Ni
(11)
i=1 max
= max(|N1
Nsta,1 |, |N2
Nsta,2| |Nn
(12)
Nsta, n|)
where σsta and σmax are the standard deviation and maximum deviation, n is data amount, i is sequence number, Ni and Nsta,i are the ith datum and ith standard datum. 5. Discussions The cases for validations are the temperatures of wall under the periodical thermal boundary conditions in experiments. Each case had been conducted for several periods in experiments and calculations to eliminate the impact of the initial condition. The experimental results together with the deviations of the computational results are shown in Figs. 7–9. Meanwhile, the corresponding calculation results of the original enthalpy model and effective heat capacity model are also added for the comparisons. In the figures, the temperatures and the deviations on the exterior and interior surfaces of wall are portrayed. The “Ent”, “Eff”, and “New” are the result curves of the original enthalpy model, original effective heat capacity model, and the presented modified computational methods using effective heat capacity model. The computational and experimental results are not listed in Figs. 8 and 9, only remaining the deviation curves. Left surface (Ent) Left surface (Eff) Left surface (New)
Left surface (Ent) Left surface (Eff) Left surface (New)
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Maximum deviation ( C)
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(k) Relative errors of standard deviation
(l) Relative errors of maximum deviation
Fig. 7. (continued) 8
International Communications in Heat and Mass Transfer 108 (2019) 104278
Y. Zhang
Left surface (Ent) Left surface (Eff) Left surface (New)
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(a) Standard deviations Relative errors of maximum deviations ( C)
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(c) Relative errors of standard deviation
(d) Relative errors of maximum deviation
Fig. 8. Calculation deviations of Block 2 wall.
and 1.30 °C. The relative errors of σsta and σmax were smaller than 2.7% and 5.2%. Furthermore, most of the corresponding relative errors were smaller than 1.5% for the σsta and 2.8% for the σmax, respectively. The reasons for the deviations can be summarized as follows. (1) It was not the absolutely adiabatic frames for the experimental device, while it was ideal thermal insulator in mathematical models. (2) Uncertainty unavoidably existed in the experimental results because of the devices and operations. (3) The mathematical models cannot perfectly and completely describe the actual heat transfer process. (4) The used thermo-physical parameters of the experimental materials in mathematical models cannot perfectly describe the performances of the actually used materials. In fact, the temperature swing on the exterior surface is usually lower than 20 °C throughout a day. Under this condition, the σsta and σmax should be lower than 0.25 °C and 0.55 °C, and the maximum relative errors of σsta and σmax should be under 2.5% and 4.0%. Therefore, excellent agreements were reached between the calculation results of the modified computational methods using effective heat capacity model and the experimental data.
comparisons, various experiments under periodical boundary temperatures were performed using a concrete hollow building block wall with the PCM incorporated into its different cavities. Furthermore, the results of the original enthalpy model and effective heat capacity model were also added for comparisons. It was found that the deviations produced by the original enthalpy model and effective heat capacity model were usually greater than the presented methods using effective heat capacity model. For the presented methods, the standard and maximum deviations increased with the span of boundary temperature ranges, while the relative errors usually decreased when the span of boundary temperature ranges rose. Also, the deviations on the exterior side of wall were often larger than the interior side. It indicated that the standard and maximum deviations of the presented methods were lower than 1.37 °C and 2.75 °C. Meanwhile, the maximum relative errors of σsta and σmax were smaller than 2.7% and 5.2%. In the most cases of the calculations when comparing with the experimental data, the σsta and σmax were lower than 0.70 °C and 1.30 °C. In addition, most of the corresponding relative errors were smaller than 1.5% for the σsta and 2.8% for the σmax. Under general condition, the σsta and σmax should be usually lower than 0.25 °C and 0.55 °C, and the maximum relative errors of σsta and σmax should be under 2.5% and 4.0%. Excellent agreements have been reached between the calculation results of the presented methods and the experimental data. It demonstrates that the proposed methods together with the effective heat capacity model satisfies the required accuracy and is qualified for the calculation of the heat transfer process of PCM outfitted building envelopes.
6. Conclusions Some modified computational methods when using two-dimensional effective heat capacity model for calculating the phase change heat transfer in building envelopes has been put forward in this paper. Based on the original effective heat capacity model, several important improvements, e.g. considering different melting, solidification temperature, and multiple temperature turning points, and improving the determination of PCM's liquid fraction during phase change process, were conducted in the new methods. For the validations and 9
International Communications in Heat and Mass Transfer 108 (2019) 104278
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Right surface (Ent) Right surface (Eff) Right surface (New)
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(a) Standard deviations Left surface (Ent) Left surface (Eff) Left surface (New)
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(c) Relative errors of standard deviation
(d) Relative errors of maximum deviation
Fig. 9. Calculation deviations of Block 3 wall.
Acknowledgements
(6) (1997) 565–579. [10] H. Yang, Y. He, Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods, Int. Commun. Heat Mass Transf. 37 (4) (2010) 385–392. [11] M. Yao, A. Chait, Alternative formulation of the apparent heat capacity method for phasechange problems, Numer. Heat Transf. Part B 24 (3) (1993) 279–300. [12] M. Yao, A. Chait, Conservative formulation of the apparent heat capacity method for heat transport problems with phase change, Proceedings of the 6th AIAA/ASME Thermophysics and Heat Transfer Conference, ASME, Colorado Springs, CO, 1994, pp. 35–42. [13] H. Ye, H. He, X. Ge, et al., Comparative numerical investigations on the melting process of form-stable phase change material using enthalpy formulation method and effective heat capacity formulation method, Acta Energ. Solar. Sin. 25 (4) (2004) 488–491. [14] Y. Zhou, Z. Guan, Y. Gu, Equivalent heat capacity method for solution of heat transfer with phase change, J. Chem. Ind. Eng. 55 (9) (2004) 1428–1433. [15] Y. He, H. Yang, Solving heat transfer problems with phase change via sigmoid function based smoothed effective heat capacity and finite element method, J. Basic Sci. Eng. 19 (5) (2011) 817–829. [16] G. Zsembinszki, P. Moreno, C. Solé, et al., Numerical model evaluation of a PCM cold storage tank and uncertainty analysis of the parameters, Appl. Therm. Eng. 67 (1–2) (2014) 16–23. [17] Y. Zhang, K. Du, Y. He, et al., Impact factors analysis of the enthalpy method and the effective heat capacity method on the transient nonlinear heat transfer in phase change materials (PCMs), Num. Heat Transf. Part A 65 (1) (2014) 66–83. [18] Y. Zhang, K. Du, M.A. Medina, et al., An experimental method for validating transient heat transfer mathematical models used for phase change materials (PCMs) calculations, Phase Transit. 87 (6) (2014) 541–558. [19] Ministry of Housing and Urban-Rural Development of the People’s Republic of China, Code for Thermal Design of Civil Building (GB50176–2016), China Architecture & Building Press, Beijing, 2016. [20] Y. Zhang, G. Wang, K. Du, et al., Comparative analysis of the mesh treating methods in the numerical heat transfer calculation of non-homogeneous building components, Heat Transf. Eng. 37 (16) (2016) 1357–1368. [21] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, New York, 1980. [22] W.Q. Tao, Numerical Heat Transfer, second ed., Xi’an Jiaotong University Press, Xi’an, 2001. [23] B.Z. Liu, Y.H. Su, H.L. Zhang, MATLAB 7.0 from Entry to Master, Posts & Telecom Press, Beijing, 2010. [24] Y. Zhang, K. Du, Y. He, et al., Impact factors analysis on the thermal performance of hollow block wall, Energy Build. 75 (2014) 330–341.
This work was financially supported by the National Natural Science Foundation of China (No.51508232), the Natural Science Foundation of Jiangsu Province (No.BK20150521), China Postdoctoral Science Foundation (No.2016M600371), Jiangsu Province Postdoctoral Science Foundation (No.1501106B), Jiangsu Government Scholarship for Overseas Studies and Scientific Research, 2016 Excellent Young Key Teachers Training Project of Jiangsu University, and the Scientific Research Starting Foundation for Advanced Talents of Jiangsu University (No.15JDG049). References [1] R.Y. Zhang, Phase Change Material and Phase Change Energy Storage Techniques, Science Press, Beijing, 2009. [2] A.M. Borreguero, M. Luz Sánchez, J.L. Valverde, et al., Thermal testing and numerical simulation of gypsum wallboards incorporated with different PCMs content, Appl. Energy 88 (3) (2011) 930–937. [3] W.J. Bowman, D.A. Brown, Heat capacity model for a mixture containing a phase change material, Proceedings of the 7th Joint Thermophysics and Heat Transfer Conference, AIAA, Albuquerque, NM, 1998, pp. 1–9. [4] D. Heim, J.A. Clarke, Numerical modelling and thermal simulation of PCM–gypsum composites with ESP-r, Energy Build. 36 (8) (2004) 795–805. [5] N. Stathopoulos, M.E. Mankibi, M. Santamouris, Numerical calibration and experimental validation of a PCM-Air heat exchanger model, Appl. Therm. Eng. 114 (2017) 1064–1072. [6] E.D. Kravvaritis, K.A. Antonopoulos, C. Tzivanidis, Experimental determination of the effective thermal capacity function and other thermal properties for various phase change materials using the thermal delay method, Appl. Energy 88 (12) (2004) 4459–4469. [7] P. Losada-Pérez, C.S.P. Tripathi, J. Leys, et al., Measurements of heat capacity and enthalpy of phase change materials by adiabatic scanning calorimetry, Int. J. Thermophys. 32 (5) (2011) 913–924. [8] I.D. Mandilaras, D.A. Kontogeorgos, M.A. Founti, A hybrid methodology for the determination of the effective heat capacity of PCM enhanced building components, Renew. Energy 76 (2015) 790–804. [9] Z.X. Gong, A.S. Mujumdar, Non-convergence versus non-conservation in effective heat capacity methods for phase change problems, Int. J. Numer. Methods Heat Fluid Flow 7
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Modified computational methods using effective heat capacity model for the thermal evaluation of PCM outfitted walls
T
Yuan Zhang School of Energy and Power Engineering, Jiangsu University, 301, Xuefu Road, Zhenjiang, Jiangsu 212013, China
ARTICLE INFO
ABSTRACT
Keywords: Phase change material (PCM) Effective heat capacity model Modified computational methods Building envelope
Some modified computational methods when using two-dimensional effective heat capacity model for predicting the transient heat transfer process of the building envelopes outfitted with phase change materials (PCMs) were presented. In the methods, some important improvements were performed, such as considering different melting and solidification temperature ranges, multiple temperature turning points and PCM's liquid fraction during phase change process. The experiments were conducted to verify the accuracy of the proposed methods. Based on the experimental data, the comparative analysis of the presented methods together with the original methods using enthalpy model and effective heat capacity model were conducted. The deviations produced by the original methods were found to be markedly larger than the presented ones. The standard and the maximum deviations of the modified methods were less than 0.25 °C and 0.55 °C, and the corresponding relative errors were under 2.5% and 4.0%.
1. Introduction As one of the important approaches to improve the energy conservation level and indoor thermal comfort of building envelope, more and more studies have been conducted on the thermal performance of the building envelope equipped with phase change materials (PCMs). Effective heat capacity model is one of the commonly-used mathematical models for predicting the phase change heat transfer (PCHT) process. In the model, the phase change process is converted into a nonlinear heat transfer problem with “single phase”. Therefore, PCM's phase state, sensible and latent heat will not be distinguished, and the PCM's effective heat capacity value changing with its temperature is used in calculations [1]. Some researchers analyzed the PCHT problems with the effective heat capacity based models. Borreguero et al. [2] obtained a reasonable agreement between computational and experimental results if PCM's latent heat (equivalent heat capacity) had been properly described. The thermal behavior of the PCM component only depended on the thermophysical properties of the used materials. Bowman and Brown [3] built a new type of effective heat capacity model which was verified to be more accurate than other models. In the model, the effective heat capacity was taken as the function of the temperature, percent of phase change material, and rate of energy addition. Heim and Clarke [4] embedded effective heat capacity method into ESP-r software and reached acceptable results. Stathopoulos et al. [5] experimentally
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verified the established new mathematical model by combining thermal balance model and sensible heat capacity model, and reached good agreements with experimental data. Furthermore, others explored new methods to precisely determine the thermal properties of PCM. Kravvaritis et al. [6] presented a thermal delay method by which the phase change temperature range, latent heat and effective heat capacity of PCM can be precisely determined for the engineering usage. LosadaPérez et al. [7] precisely obtained PCM's equilibrium enthalpy and effective heat capacity by Adiabatic Scanning Calorimetry. Also, Mandilaras et al. [8] proposed a new method for determining the effective heat capacity of shape stabilized PCM. Meanwhile, people found that deviation existed when using original effective heat capacity model, and the accurate results cannot be obtained in some cases. This question was paid more attention in the past years, and some impact factors had been analyzed. Gong and Mujumdar [9] analyzed the reasons of the non-convergence and calculation error, and presented a new optimized equation scheme and eliminated some disadvantages of the original effective heat capacity model such as without the limitation of the phase change temperature interval. Yang and He [10] composited effective heat capacity method with element-free Galerkin method, and compared the results of the new model with analytical solution. They found the numerical oscillation was avoided by the new model, which made the results more accurate. Yao and Chait [11,12] successfully built a new equivalent heat capacity model for improving the strict limitation of
International Communications in Heat and Mass Transfer 108 (2019) 104278
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Nomenclature
Nsta,i T Tx,y,τ Tx,y,τ+1 x y
area represented by PCM's total latent heat, m2 scanned area from the beginning of a melting process to the end, m2 scanned area from the beginning of a solidification process to the end, m2 area represented by PCM's absorbed latent heat, m2 specific heat, kJ·kg−1·K−1 effective heat capacity of PCM, kJ·kg−1·K−1 enthalpy of PCM, kJ·kg−1 sequence number PCM's total latent heat, kJ·kg−1 PCM's absorbed latent heat, kJ·kg−1 mass of PCM, kg PCM's liquid mass, kg PCM's total mass, kg data amount ith datum, °C
A A' A" A1 c ceff H i L L' m ml M n Ni
Greek symbols
σsta σmax
50
o
Effective heat capacity (kJ/(kgK))
Heat flow (mW)
works can be summarized as two aspects: the measurement of PCM's parameters and the numerical simulation method. However, there still be other assumptions in PCM's thermo-physical parameters and heat transfer process in original model that result in deviations. For example, the different melting and solidification TRs, multiple temperature turning points when melting and solidification, and PCM's liquid fraction during phase change process. However, the study of these factors is insufficient. Author's previous publications [17,18] have briefly referred to part of related contents, such as the TRs and liquid fraction, but more detailed analysis is still needed and the original computational method is waiting for modifications for the more accurate results. In the first part of the paper, based on the original model, some modified computational methods when using two-dimensional effective heat capacity model were presented. In the new methods, different melting and solidification temperatures, multiple temperature turning points, and PCM's liquid fraction during phase change process, were properly considered which were neglected in the original models. Subsequently, a serious of validation experiments was conducted. Finally, the accuracy of the presented methods together with the original methods using enthalpy model and effective heat capacity model were verified and compared based on the experimental data.
Peak=33.0 C
Delta H=138.04J/g
10
density, kg·m−3 time, s thermal conductivity, W·m−1·K−1 liquid fraction, % liquid fraction at the beginning of a melting process, % liquid fraction at the beginning of a solidification process, % standard deviation, °C maximum deviation, °C
ρ τ λ η η' η″
time discretization in original model. Ye et al. [13] analyzed the melting process of a paraffin-based stabilized PCM using effective heat capacity model. They found accurate results can be obtained if PCM's effective heat capacity values were calculated from the data of differential scanning calorimetry (DSC). Zhou et al. [14] found that the calculation became simpler and the numerical singular problem when the phase change temperature was a single value or narrow temperature range (TR) can be effectively resolved by using unit equivalent heat capacity. Following with Zhou's work, He and Yang [15] evaluated the influence on the calculation results accuracy produced by the finite element grid density, numerical integration scheme of heat capacity matrix and relevant parameters of Sigmoid function. Zsembinszki et al. [16] found that PCM's small time step and large number of grid produced accurate calculation results while accompanied by significantly computing time. The computational efficiency was enhanced by using PCM's specific heat obtained by the last iterative computation and constant physical parameters of heat transfer fluid. However, it produced slight deviations. Besides, the inlet temperature of heat transfer fluid, PCM's melting temperature, density and specific heat influenced the calculation results, while PCM's thermal conductivity, flow rate, convective heat transfer coefficient of heat transfer fluid, wall's thermal conductivity had little impact. The factors producing the computation deviations in the above 20
ith standard datum, °C temperature, °C temperature of point (x,y) at time τ, °C temperature of point (x,y) at time τ + 1, °C direction of thickness, m direction of width, m
0
-10 Delta H=-150.97J/g -20
Melting curve Solidification curve
40 30 20 10 0 -10 -20 -30 -40 Melting curve Solidification curve
-50 -60
o
Peak=26.4 C
-70
-30 0
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Temperature ( C)
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Temperature ( C)
a. DSC curves of PCM
b. Effective heat capacity curves of PCM
Fig. 1. DSC and effective heat capacity curves of the experimental PCM. 2
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2. Modified computational methods using effective heat capacity model
In the presented model, the thermo-physical parameters, such as PCM's density and thermal conductivity, can be different when the PCM is solid, solid and liquid coexisting state and liquid. Furthermore, the thermal properties of each spatial point and time were all calculated using PCM's liquid fraction.
2.1. Assumptions Some assumptions were made in the presented methods. (1) All the materials were considered to be thermally homogeneous and isotropic. (2) With the effective restraint of the convection effect of liquid PCM with gypsum, the thermal conductivity of PCM was considered to be constant. (3) PCM's volume change and sub-cooling effects during phase change process were ignored. (4) The heat transfer in the computational domain was simplified to be two-dimensional heat transfer process (the heat transfer in vertical direction was ignored). (5) The equivalent thermal conductivity was applied for the closed air layers [19].
2.3. Initial and boundary conditions The initial temperature inside all the materials was considered to be the same. For the exterior and interior surfaces, the heat convection thermal boundary condition was applied. Adiabatic condition was applied on the width boundaries. 2.4. Different melting and solidification temperatures Like the different phase change TRs of the PCM in Fig. 1 and Table 1, PCM's melting and solidification temperature ranges are often not the same. If they are considered as the same range, deviations will occur. The proposed methods can be used to calculate the heat transfer of the PCM with the same or different melting and solidification temperature ranges.
2.2. Governing equation The governing equation of the presented model is shown in Eq. (1). PCM's effective heat capacity values changed with temperatures were obtained from the DSC measurement. In the model, sensible heat and latent heat were not distinguished, and PCM's specific heat was represented by the effective heat capacities. Because the PCM capric acid was used in the experiments, the referred PCM should be the capric acid in the following text.
ceff
T
=
2T
x2
+
2.5. Temperature turning points during phase change 2.5.1. Introduction Different phase change processes are shown in Fig. 2 in which the PCM holds different melting and solidification TRs. Like the Process 1, if PCM always transforms from the complete solid to the complete liquid and returns to the complete solid, the solid, solid and liquid coexisting, and liquid PCM can be respectively calculated according to the phase change TRs and thermo-physical parameters. This situation is simple. However, most phase change processes are not always like this. For instance, PCM transforms from the complete liquid to solid and liquid coexisting state, and returns to the complete liquid (Process 2 in Fig. 2) or another solid and liquid coexisting state (Process 3 in Fig. 2). Hence, the actual case is more complex than the ideal one. Multiple temperature turning points usually exist during phase change process.
2T
y2
(1)
where ρ, λ are the density and thermal conductivity of PCM, respectively. ceff is the effective heat capacity of PCM which is calculated by Eq. (2). T and τ are temperature and time. The thickness and width directions are represented by x and y.
ceff =
dH dT / m d d
(2)
where H is the enthalpy of PCM. dH/dτ is heat flow rate. m is the mass of PCM. dT/dτ is the velocity of PCM's temperature rise or fall. To know the thermo-physical parameters of PCM, nitrogen protected DSC device was used. This test was conducted with the temperature change velocity of 5 °C·min−1. The DSC curves of the PCM sample (4.980 mg) are shown in Fig. 1a. The curve with the positive values is the melting curve, while the negative one is the solidification curve. It is observed that the peak temperatures were 33.0 °C (melting) and 26.4 °C (solidification), and the latent heat were 138.04 J·g−1 (melting) and 150.97 J·g−1 (solidification). As a result of the slight unavoidable deviation when determining the PCM's phase change temperature ranges, the two latent heats were slightly different. The thermal properties of PCM are shown in Table 1. The effective heat capacity of PCM was calculated using the specific heat of sapphire (taken as the standard data) and Eq. (2), shown in Fig. 1b. It can be seen that the trend of the DSC curves and effective heat capacity curves are almost the same. The PCM's peak specific heat were 46.9 J·g−1·K−1 (melting) and 69.0 J·g−1·K−1 (solidification), respectively. In calculations, the specific heat of each spatial point inside the PCM was obtained by the interpolation of the two closest effective heat capacity values based on the temperature of the point. The interpolation was conducted using the proportion of the distance between the temperature of this point and the two closest temperatures in the obtained temperature-effective heat capacity data pairs. Hence, the sensible, latent heat and phase state of the PCM were not needed to be distinguished. The actual phase change process can be precisely described if the amounts of PCM's temperature-effective heat capacity data pairs were abundant. Enough data pairs can be obtained with small temperature intervals (e.g. 0.01 or 0.02 °C) in DSC measurement.
2.5.2. Methodology in the presented methods In the presented methods, when the temperature inside the PCM was increasing and already in PCM's melting TR, it was considered that the solid part was melting. At this time, only the conversion of the sensible heat took place in the already existed liquid section without phase change. The PCM's thermal properties were obtained by weighting the proportion of (1) the thermal properties of the solid PCM in which the phase change was taking place, and (2) the parameters of the liquid part without phase change. And the proportion was the mass of the solid to the liquid at the start of this phase change process. There are two types of the special processes. The first one is: when PCM is melting, the two phase change processes before and after this melting process are both solidification processes. That is, the whole process is from the solidification to melting and to the solidification again (Process 4 in Fig. 2). The second is that there are two melting processes divided by several temperature swings without phase transition (Process 5 in Fig. 2). Namely, following the first melting process, PCM's temperature decreases but does not enter into its solidification Table 1 Thermo-physical properties of the PCM (capric acid).
3
Density (kg·m−3)
Thermal conductivity (W·m−1·K−1)
Melting TR (°C)
Solidification TR (°C)
886 (Liquid) 1004 (Solid)
0.152
31.4–34.5
24.6–27.1
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Fig. 2. PCM's phase change processes.
TR, and then increases again. After several temperature swings like this, the temperature enters its melting TR and the PCM melts again. The process of the first case is simple; while the second is not. As was introduced above, the PCM's parameters of the second phase change process were determined by the state at the end of the first one. For example, PCM solidifies and its temperature decreases to a value that remains in its solidification TR; after that, the decline pauses for a period of time and then another temperature decrease begins (Process 6 in Fig. 2). In the new methods, the two temperature fall and solidification periods were sequential and taken as one phase change process. Nevertheless, there is another example which is different. PCM solidifies to a temperature that remains in its solidification TR; then its temperature rises to another one which is still in its solidification TR but not in the melting TR, and followed by another temperature fall (Process 7 in Fig. 2). In this case, the PCM's liquid fraction at the end of each phase change process would be recalculated, and the second temperature fall was considered to be another solidification process. The liquid fraction in the second process was the one calculated at the end of the first process, but PCM's temperature and effective heat capacities at the start of the second process did not follow the values at the end of the first one. When the calculated PCM has different melting and solidification TRs, the process like this case probably takes place.
effective heat capacity curve, the vertical line of PCM's temperature, and the horizontal axis, to the encircled area by PCM's effective heat capacity curve and the horizontal axis. Namely, it is the percentage of the shaded area to the encircled area by the curve and the horizontal axis. Therefore, PCM's liquid fraction can be equivalent to the percentage of the area of the absorbed latent heat to the one of the total latent heat (Eq. (5)). The areas in the figure were calculated by the integral method. Therefore, to collect enough data and make the calculation result more precise, the data pairs of the temperature-heat flow rate should be obtained as many as possible in the DSC measurement. If the PCM's effective heat capacity data are sufficient, the area can be calculated by the sum of the product of each effective heat capacity value and its small temperature interval. While if the data are not very abundant but adequate, the effective heat capacity curve can be fitted out. Followed by an integral computation, the corresponding areas can be obtained.
=
2.6.2. Special situations If a melting process began with the solid and liquid coexisting state, only the solid part melted. The liquid fraction was obtained by Eq. (6). Similarly, if a solidification process began with the solid and liquid coexisting state, only the liquid part solidified. The liquid fraction was obtained by Eq. (7).
2.6.1. Methodology PCM's liquid fraction is the ratio of the liquid mass to the total mass, with the value ranging from 0 to 100% (Eq. (3)).
ml 100% M
(3)
where η is PCM's liquid fraction, ml is PCM's liquid mass, and M is PCM's total mass. In fact, it is quite difficult to obtain PCM's liquid mass at a certain time. In the presented methods, PCM's liquid fraction was equivalent to the percentage of the absorbed latent heat to the total latent heat. That is, the increase or decrease of the liquid mass was considered to change following the absorbed or released latent heat with the same ratio and velocity. If PCM's temperature increased from the complete solid and then began to melt, the liquid fraction of each point within the PCM was calculated by Eq. (4).
=
L 100% L
(5)
where A1 is the area represented by PCM's absorbed latent heat, and A is the area represented by PCM's total latent heat. The PCM's initial liquid fraction was calculated using the PCM's initial temperature and melting TR. If the PCM began from the complete solid or liquid, its initial liquid fraction was considered as 0 or 100%.
2.6. Liquid fraction
=
A1 100% A
=
+ (1
)
A 100% A
(4)
'
where L is PCM's absorbed latent heat, and L is PCM's total latent heat. However, it is still difficult to acquire PCM's absorbed latent heat at a certain time. The absorbed or released heat does not change with PCM's temperature in a linear manner. In fact, the heat changes with PCM's effective heat capacity curves. As is shown in Fig. 3, PCM's liquid fraction should be the percentage of the encircled area by PCM's
Fig. 3. Schematic diagram of PCM's liquid fraction. 4
(6)
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where η' is the liquid fraction at the beginning of the melting process. A′ is the scanned area from the beginning of this melting process to the end.
=
A 100% A
in spatial grid while a fully implicit finite difference scheme and forward differences were applied in time grid. The step size of spatial grid was 0.005 m and time grid was 300 s. Gauss-Seidel scheme was employed in the point iterations. Eq. (10) is the convergence criterion. The numerical calculations were conducted using a MATLAB code [23].
(7)
max |Tx , y,
where η″ is the liquid fraction at the beginning of the solidification process. A″ is the scanned area from the beginning of this solidification process to the end. For instance, the amount of the liquid PCM at the end of a melting process was obtained by the sum of the two parts: (1) the original liquid PCM at the start of this process, (2) the liquid PCM converted from the original solid part at the end of this process. Therefore, the old liquid fraction was then replaced by the new calculated one. PCM's liquid fraction was calculated using the temperature-effective heat capacity data. Thus, PCM's DSC measurement must be accomplished.
+
c = ( 1 c1 +
2 c2 )/2
3 c3
+
4 c4 )/4
10
10
(10)
3. Experimental setup 3.1. Hollow block A concrete hollow building block (Fig. 5) wall equipped with PCM was used and various experiments were conducted under periodical temperature conditions. The hollow block had three cavity rows, in which the thickness of the middle one was two times as the side one. The thermo-physical parameters of the block materials are shown in Table 2 [19]. The PCM was incorporated into different rows of cavities (Block 1, 2, and 3) and forming different experimental walls (Fig. 5c–e). Even though under the same boundary condition, the PCMs locating in different rows of cavities meant the different temperature conditions to them which influenced the effect of the phase change. Therefore, the walls with the PCM incorporated in different rows of cavities were actually different walls. To maintain the comparability among different hollow block walls, the PCM with the same mass (0.314 kg) was incorporated in the walls. Because the thickness of the middle cavities was larger than the side cavities, it was pure PCM filled in the side cavities (Fig. 5c–e), while the PCM mixed with a certain amount of gypsum and water was filled into the middle cavities (Fig. 5d).
2.7.1. Grid node of the interface For the grids at the interface between different materials, its center node located exactly on the interface [20]. In Fig. 4, four equal parts with their own thermo-physical properties constituted one internal grid. Therefore, it can be one to four kinds of materials within one internal grid. The volume specific heat of each grid was calculated by Eq. (8). The size of the boundary grids was half of the internal grids with their center nodes locating on the boundary lines. Each boundary grid consisted of the two parts with their own thermo-physical parameters. Therefore, one or two kinds of materials were included in the boundary grids. The volume specific heat was obtained by Eq. (9). The corner grid size of the internal grid with only one material inside each. 2 c2
+ 1|
where Tx,y,τ and Tx,y,τ+1 are the temperatures of the point (x,y) at the iteration step of τ and τ + 1 for each time step.
2.7. Numerical methodology
c = ( 1 c1 +
Tx, y,
(8)
3.2. Experimental device and methodology
(9)
where ρc is node's volume specific heat. ρ1c1, ρ2c2, ρ3c3, ρ4c4 are the volume specific heat for different parts of node, respectively.
The method of experiments was temperature-change hot chamber method [24]. Fig. 6 is the used building envelope thermal performance testing device (Type JW-I). Thermal insulation material was filled into the frames of the chambers. The air temperature was controlled by the heat and cold sources in the left chamber, while the natural air condition was maintained in the right chamber. The walls for testing were
2.7.2. Numerical calculation settings The governing equation and boundary condition were discretized by the Finite Difference Method [21,22]. Central differences were applied
Fig. 4. Distributions of grid nodes. 5
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190 40
25 20 30
30 40
140
390
140
40
30 20 25
(a) Photo of hollow block
Outside
Inside
(b) Block size
Outside
Inside
(c) Block 1
Outside
(d) Block 2 Fig. 5. Hollow blocks.
Table 2 Thermo-physical parameters of the block materials. Thermo-physical parameters
Density −3
Block materials
(kg·m
Lightweight concrete Air layer (20 mm thick) Air layer (40 mm thick)
1600 1.2 1.2
Table 3 Boundary conditions of different walls.
Specific heat )
−1
(J·kg
600 1005 1005
·K
−1
)
Thermal conductivity −1
(W·m
·K
−1
Boundary TRs
)
0.62 0.143 (Equivalent) 0.267 (Equivalent)
Shared boundary TRs
Added boundary TRs for Block 1 wall
built between the two chambers. All of the experimental walls were measured under the boundary conditions shown in Table 3.
Added boundary TRs for Block 2 and 3 walls
3.3. Uncertainty of testing device The air temperature swing and heat loss of chambers were lower than 0.5 °C and 5%. The thermocouples were T type. The testing range was from −50 °C to 100 °C, and the accuracy was ± 0.5 °C or 0.4% (the
TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR5 TR6 TR7 TR8
TRs of the left side air
Spans of the left side air's TRs
(°C)
(°C)
19.9–43.2 1.2–67.2 3.9–72.6 1.0–83.4 27.0–34.5 23.5–36.9 20.9–40.4 17.5–42.7 17.6–42.4 11.6–48.0 10.7–54.1 11.9–60.3
23.3 66 68.7 82.4 7.5 13.4 19.5 25.2 24.8 36.4 43.4 48.4
Thermal Insulation Material Aluminium Plate
Left Chamber
Cold Source
Right Chamber
Fan Aluminium Plate
Heat Source
Fan
Experimental Wall
(a) Photo of experimental device
(b) Schematic diagram Fig. 6. Experimental devices. 6
Specimen Bracket
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Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
70
o
Temperature ( C)
o
35 30 25 20
50 40 30 20 10
15
0 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h) Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
70
60
70
80
90
100 110 120
(b) TR2
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
80
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
70
Temperature ( C)
60 50
60
o
o
50
Time (h)
(a) TR1
Temperature ( C)
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
60
40
Temperature ( C)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
40 30 20
50 40 30 20 10
10
0 0 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
45
70
80
90
100 110 120
(d) TR4
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
40 o
o
Temperature ( C)
40
Temperature ( C)
60
Time (h)
(c) TR3 Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
50
35 30 25 20
35 30 25 20
15
15 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
50
60
70
80
90
100 110 120
Time (h)
(f) TR6 Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
45
Right surface (Exp) Right surface (Ent) Right surface (Eff) Right surface (New)
40 o
o
Temperature ( C)
40
Temperature ( C)
Left surface (Exp) Left surface (Ent) Left surface (Eff) Left surface (New)
35 30 25 20
35 30 25 20
15
15 0
10
20
30
40
50
60
70
80
90
100 110 120
0
10
20
30
40
Time (h)
50
60
70
Time (h)
(g) TR7
(h) TR8
Fig. 7. Results and calculation deviations of Block 1 wall.
7
80
90
100 110 120
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accuracy should be subject to the larger one).
It is observed from Figs. 7–9 that the deviations produced by the original enthalpy model and effective heat capacity model were usually greater than the presented methods. Based on these results, it can be known that the estimations without considering PCM's different melting and solidification temperature ranges, multiple temperature turning points and liquid fraction during phase change process, would produce large deviations away from the actual cases. For the presented methods, the standard and maximum deviations became relatively large with large boundary temperature swings (TR2 TR4). The standard deviations σsta were from 0.36 °C to 1.37 °C, and the maximum deviations σmax were 0.72 °C–2.75 °C. Nevertheless, the deviations reduced under the narrower boundary TRs (TR1, TR5 - TR8), with the σsta and σmax ranging from 0.09 °C to 0.80 °C and from 0.19 °C to 1.37 °C, respectively. From these cases, it can be found that the standard and maximum deviations increased with the span of boundary TRs. However, the trend for the relative errors was basically adverse. The relative errors reduced when the boundary TRs turned wider (TR2 TR4). The relative errors of σsta were from 0.5% to 2.0%, and that of σmax located in 1.1% - 3.7%. Instead, the relative errors were usually large under the narrow boundary TRs (TR1, TR5 - TR8). The relative errors of σsta lay between 0.4% and 2.7%, and the relative errors of σmax were from 0.8% to 5.2%. It is summarized that the relative errors usually decrease with the increasing span of boundary TRs. This can be explained by the thermal inertia of wall, which generates the surface temperature swing and makes it lower than the surrounded air's. In addition, it can be observed that the temperature decrement increased with the boundary temperature swings. Also, the temperature swings on the left surface were larger than the right surface, making the deviations on the left surface relatively larger. The σsta and σmax on the two sides of wall were lower than 1.37 °C and 2.75 °C. Also, in most cases, the two values were lower than 0.70 °C
4. Deviation evaluation indices The used indices standard deviation and maximum deviation were calculated using Eqs. (11) and (12). n sta
=
Nsta, i ) 2 / n
(Ni
(11)
i=1 max
= max(|N1
Nsta,1 |, |N2
Nsta,2| |Nn
(12)
Nsta, n|)
where σsta and σmax are the standard deviation and maximum deviation, n is data amount, i is sequence number, Ni and Nsta,i are the ith datum and ith standard datum. 5. Discussions The cases for validations are the temperatures of wall under the periodical thermal boundary conditions in experiments. Each case had been conducted for several periods in experiments and calculations to eliminate the impact of the initial condition. The experimental results together with the deviations of the computational results are shown in Figs. 7–9. Meanwhile, the corresponding calculation results of the original enthalpy model and effective heat capacity model are also added for the comparisons. In the figures, the temperatures and the deviations on the exterior and interior surfaces of wall are portrayed. The “Ent”, “Eff”, and “New” are the result curves of the original enthalpy model, original effective heat capacity model, and the presented modified computational methods using effective heat capacity model. The computational and experimental results are not listed in Figs. 8 and 9, only remaining the deviation curves. Left surface (Ent) Left surface (Eff) Left surface (New)
Left surface (Ent) Left surface (Eff) Left surface (New)
5
Maximum deviation ( C)
2.5
o
o
Standard deviation ( C)
3.0
Right surface (Ent) Right surface (Eff) Right surface (New)
2.0 1.5 1.0 0.5 0.0
4
3
2
1
0 1
2
3
4
5
6
7
8
1
2
3
Left TRs
5
3
2
1
0 2
3
4
5
6
Left surface (Ent) Left surface (Eff) Left surface (New)
o
4
1
5
6
7
8
7
8
(j) Maximum deviations
Right surface (Ent) Right surface (Eff) Right surface (New)
Relative errors of maximum deviations ( C)
o
Left surface (Ent) Left surface (Eff) Left surface (New)
4
Left TRs
(i) Standard deviations Relative errors of standard deviations ( C)
Right surface (Ent) Right surface (Eff) Right surface (New)
7
8
8
Right surface (Ent) Right surface (Eff) Right surface (New)
7 6 5 4 3 2 1 0
Left TRs
1
2
3
4
5
6
Left TRs
(k) Relative errors of standard deviation
(l) Relative errors of maximum deviation
Fig. 7. (continued) 8
International Communications in Heat and Mass Transfer 108 (2019) 104278
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Left surface (Ent) Left surface (Eff) Left surface (New)
Left surface (Ent) Left surface (Eff) Left surface (New)
5
Maximum deviation ( C)
2.5
o
o
Standard deviation ( C)
3.0
Right surface (Ent) Right surface (Eff) Right surface (New)
2.0 1.5 1.0 0.5 0.0
4
3
2
1
0 1
2
3
4
5
6
7
8
1
2
3
Left TRs
8
6 5 4 3 2 1 0 2
3
4
5
6
Left surface (Ent) Left surface (Eff) Left surface (New)
o
7
1
5
6
7
8
7
8
(b) Maximum deviations
Right surface (Ent) Right surface (Eff) Right surface (New)
Relative errors of standard deviations ( C)
o
Left surface (Ent) Left surface (Eff) Left surface (New)
4
Left TRs
(a) Standard deviations Relative errors of maximum deviations ( C)
Right surface (Ent) Right surface (Eff) Right surface (New)
7
8
5
Right surface (Ent) Right surface (Eff) Right surface (New)
4
3
2
1
0 1
2
Left TRs
3
4
5
6
Left TRs
(c) Relative errors of standard deviation
(d) Relative errors of maximum deviation
Fig. 8. Calculation deviations of Block 2 wall.
and 1.30 °C. The relative errors of σsta and σmax were smaller than 2.7% and 5.2%. Furthermore, most of the corresponding relative errors were smaller than 1.5% for the σsta and 2.8% for the σmax, respectively. The reasons for the deviations can be summarized as follows. (1) It was not the absolutely adiabatic frames for the experimental device, while it was ideal thermal insulator in mathematical models. (2) Uncertainty unavoidably existed in the experimental results because of the devices and operations. (3) The mathematical models cannot perfectly and completely describe the actual heat transfer process. (4) The used thermo-physical parameters of the experimental materials in mathematical models cannot perfectly describe the performances of the actually used materials. In fact, the temperature swing on the exterior surface is usually lower than 20 °C throughout a day. Under this condition, the σsta and σmax should be lower than 0.25 °C and 0.55 °C, and the maximum relative errors of σsta and σmax should be under 2.5% and 4.0%. Therefore, excellent agreements were reached between the calculation results of the modified computational methods using effective heat capacity model and the experimental data.
comparisons, various experiments under periodical boundary temperatures were performed using a concrete hollow building block wall with the PCM incorporated into its different cavities. Furthermore, the results of the original enthalpy model and effective heat capacity model were also added for comparisons. It was found that the deviations produced by the original enthalpy model and effective heat capacity model were usually greater than the presented methods using effective heat capacity model. For the presented methods, the standard and maximum deviations increased with the span of boundary temperature ranges, while the relative errors usually decreased when the span of boundary temperature ranges rose. Also, the deviations on the exterior side of wall were often larger than the interior side. It indicated that the standard and maximum deviations of the presented methods were lower than 1.37 °C and 2.75 °C. Meanwhile, the maximum relative errors of σsta and σmax were smaller than 2.7% and 5.2%. In the most cases of the calculations when comparing with the experimental data, the σsta and σmax were lower than 0.70 °C and 1.30 °C. In addition, most of the corresponding relative errors were smaller than 1.5% for the σsta and 2.8% for the σmax. Under general condition, the σsta and σmax should be usually lower than 0.25 °C and 0.55 °C, and the maximum relative errors of σsta and σmax should be under 2.5% and 4.0%. Excellent agreements have been reached between the calculation results of the presented methods and the experimental data. It demonstrates that the proposed methods together with the effective heat capacity model satisfies the required accuracy and is qualified for the calculation of the heat transfer process of PCM outfitted building envelopes.
6. Conclusions Some modified computational methods when using two-dimensional effective heat capacity model for calculating the phase change heat transfer in building envelopes has been put forward in this paper. Based on the original effective heat capacity model, several important improvements, e.g. considering different melting, solidification temperature, and multiple temperature turning points, and improving the determination of PCM's liquid fraction during phase change process, were conducted in the new methods. For the validations and 9
International Communications in Heat and Mass Transfer 108 (2019) 104278
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o
Standard deviation ( C)
3.0
Right surface (Ent) Right surface (Eff) Right surface (New)
2.5 2.0 1.5 1.0 0.5 0.0 1
2
3
4
5
6
Left surface (Ent) Left surface (Eff) Left surface (New)
o
Relative errors of maximum deviations ( C)
Left surface (Ent) Left surface (Eff) Left surface (New)
7
8
8 7 6 5 4 3 2 1 0 1
2
3
Left TRs
3
2
1
0 2
3
4
5
6
Left surface (Ent) Left surface (Eff) Left surface (New)
o
4
1
5
6
7
8
7
8
(b) Maximum deviations
Right surface (Ent) Right surface (Eff) Right surface (New)
Relative errors of standard deviations ( C)
o
Maximum deviation ( C)
5
4
Left TRs
(a) Standard deviations Left surface (Ent) Left surface (Eff) Left surface (New)
Right surface (Ent) Right surface (Eff) Right surface (New)
7
8
5
Right surface (Ent) Right surface (Eff) Right surface (New)
4
3
2
1
0 1
2
3
4
5
6
Left TRs
Left TRs
(c) Relative errors of standard deviation
(d) Relative errors of maximum deviation
Fig. 9. Calculation deviations of Block 3 wall.
Acknowledgements
(6) (1997) 565–579. [10] H. Yang, Y. He, Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods, Int. Commun. Heat Mass Transf. 37 (4) (2010) 385–392. [11] M. Yao, A. Chait, Alternative formulation of the apparent heat capacity method for phasechange problems, Numer. Heat Transf. Part B 24 (3) (1993) 279–300. [12] M. Yao, A. Chait, Conservative formulation of the apparent heat capacity method for heat transport problems with phase change, Proceedings of the 6th AIAA/ASME Thermophysics and Heat Transfer Conference, ASME, Colorado Springs, CO, 1994, pp. 35–42. [13] H. Ye, H. He, X. Ge, et al., Comparative numerical investigations on the melting process of form-stable phase change material using enthalpy formulation method and effective heat capacity formulation method, Acta Energ. Solar. Sin. 25 (4) (2004) 488–491. [14] Y. Zhou, Z. Guan, Y. Gu, Equivalent heat capacity method for solution of heat transfer with phase change, J. Chem. Ind. Eng. 55 (9) (2004) 1428–1433. [15] Y. He, H. Yang, Solving heat transfer problems with phase change via sigmoid function based smoothed effective heat capacity and finite element method, J. Basic Sci. Eng. 19 (5) (2011) 817–829. [16] G. Zsembinszki, P. Moreno, C. Solé, et al., Numerical model evaluation of a PCM cold storage tank and uncertainty analysis of the parameters, Appl. Therm. Eng. 67 (1–2) (2014) 16–23. [17] Y. Zhang, K. Du, Y. He, et al., Impact factors analysis of the enthalpy method and the effective heat capacity method on the transient nonlinear heat transfer in phase change materials (PCMs), Num. Heat Transf. Part A 65 (1) (2014) 66–83. [18] Y. Zhang, K. Du, M.A. Medina, et al., An experimental method for validating transient heat transfer mathematical models used for phase change materials (PCMs) calculations, Phase Transit. 87 (6) (2014) 541–558. [19] Ministry of Housing and Urban-Rural Development of the People’s Republic of China, Code for Thermal Design of Civil Building (GB50176–2016), China Architecture & Building Press, Beijing, 2016. [20] Y. Zhang, G. Wang, K. Du, et al., Comparative analysis of the mesh treating methods in the numerical heat transfer calculation of non-homogeneous building components, Heat Transf. Eng. 37 (16) (2016) 1357–1368. [21] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, New York, 1980. [22] W.Q. Tao, Numerical Heat Transfer, second ed., Xi’an Jiaotong University Press, Xi’an, 2001. [23] B.Z. Liu, Y.H. Su, H.L. Zhang, MATLAB 7.0 from Entry to Master, Posts & Telecom Press, Beijing, 2010. [24] Y. Zhang, K. Du, Y. He, et al., Impact factors analysis on the thermal performance of hollow block wall, Energy Build. 75 (2014) 330–341.
This work was financially supported by the National Natural Science Foundation of China (No.51508232), the Natural Science Foundation of Jiangsu Province (No.BK20150521), China Postdoctoral Science Foundation (No.2016M600371), Jiangsu Province Postdoctoral Science Foundation (No.1501106B), Jiangsu Government Scholarship for Overseas Studies and Scientific Research, 2016 Excellent Young Key Teachers Training Project of Jiangsu University, and the Scientific Research Starting Foundation for Advanced Talents of Jiangsu University (No.15JDG049). References [1] R.Y. Zhang, Phase Change Material and Phase Change Energy Storage Techniques, Science Press, Beijing, 2009. [2] A.M. Borreguero, M. Luz Sánchez, J.L. Valverde, et al., Thermal testing and numerical simulation of gypsum wallboards incorporated with different PCMs content, Appl. Energy 88 (3) (2011) 930–937. [3] W.J. Bowman, D.A. Brown, Heat capacity model for a mixture containing a phase change material, Proceedings of the 7th Joint Thermophysics and Heat Transfer Conference, AIAA, Albuquerque, NM, 1998, pp. 1–9. [4] D. Heim, J.A. Clarke, Numerical modelling and thermal simulation of PCM–gypsum composites with ESP-r, Energy Build. 36 (8) (2004) 795–805. [5] N. Stathopoulos, M.E. Mankibi, M. Santamouris, Numerical calibration and experimental validation of a PCM-Air heat exchanger model, Appl. Therm. Eng. 114 (2017) 1064–1072. [6] E.D. Kravvaritis, K.A. Antonopoulos, C. Tzivanidis, Experimental determination of the effective thermal capacity function and other thermal properties for various phase change materials using the thermal delay method, Appl. Energy 88 (12) (2004) 4459–4469. [7] P. Losada-Pérez, C.S.P. Tripathi, J. Leys, et al., Measurements of heat capacity and enthalpy of phase change materials by adiabatic scanning calorimetry, Int. J. Thermophys. 32 (5) (2011) 913–924. [8] I.D. Mandilaras, D.A. Kontogeorgos, M.A. Founti, A hybrid methodology for the determination of the effective heat capacity of PCM enhanced building components, Renew. Energy 76 (2015) 790–804. [9] Z.X. Gong, A.S. Mujumdar, Non-convergence versus non-conservation in effective heat capacity methods for phase change problems, Int. J. Numer. Methods Heat Fluid Flow 7
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