Expanding Heisler chart to characterize heat transfer phenomena in a building envelope integrated with phase change materials

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Expanding Heisler chart to characterize heat transfer phenomena in a building envelope integrated with phase change materials Arash Bastani, Fariborz Haghighat ∗ Department Building, Civil and Environmental Engineering, Concordia University, Montreal, Canada

a r t i c l e

i n f o

Article history: Received 9 February 2015 Received in revised form 19 April 2015 Accepted 22 May 2015 Available online xxx Keywords: Building envelope PCM Heisler chart Transient heat transfer Demand shift Net zero energy building

a b s t r a c t Building envelope integrated with phase change material (PCM) can provide thermal energy storage (TES) distributed in its entire surface area and inhibit the need for enhanced thermal mass in lightweight buildings. Selecting the most appropriate PCM wallboard based on its thickness and thermo-physical properties is the main challenge in the design of net-zero energy buildings and high performance buildings; yet, there is a lack of an appropriate design tool. To develop a design tool, characterizing transient heat transfer phenomena of wallboards impregnated with PCM during charging procedure is required. Accordingly, this study focuses on the characterizing heat transfer of PCM wallboards, and to identify the influential parameters on the charging procedure of a PCM wallboard. The non-dimensionalized analysis was conducted, and the dimensionless numbers influencing the thermal behavior of a PCM wallboard were identified. Moreover, the correlations between the dimensionless parameters and the performance of the PCM wallboard were determined through a comprehensive parametric study. Consequently, a procedure was developed to expand Heisler chart application to study thermal behavior of PCM wallboards. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Among different sectors of the primary energy users, building sector is one of the main users. According to the Natural Resources Canada [1] more than 30% of the total secondary energy was used by residential and commercial/institutional buildings. This fact implies that the building sector contributes largely in total energy consumption. Particularly, in extremely cold/hot climate areas, space heating/cooling results in high energy consumption. The data from the National Resources Canada [1] also shows that space heating accounts up to 63% of the total energy used in non-industrial buildings. Extracted from this data, electricity is an exclusive source of energy for space conditioning in buildings. The electrical energy demand has a daily variation due to the combination of activities in industrial, commercial, and residential sectors. This results in peak (mostly in early morning) and off-peak periods. In Quebec, Canada, 70% of residential buildings and 60% of the commercial and institutional sector utilize electricity for space conditioning [2]. 80% of the total load of all-electric households during peak hours results from space conditioning. Also, the combined space

Abbreviations: TES, thermal energy storage; PCM, phase change material. ∗ Corresponding author. Tel.: +1 514 848 2424x3192; fax: +1 514 848 7965. E-mail address: [email protected] (F. Haghighat).

heating load of the non-industrial buildings accounts for almost 40% of the total electric utility peak during winter peak hours [3]. According to Hydro Quebec [4], during the peak period in winter the electricity cost for the supply side is 10 $/kW and it is increased to 100 $/kW in 2015. Taking this information into account, shifting a significant portion or the entire space conditioning energy consumption to off-peak periods would have significant economic impact on both the supply and demand sides. The shifting of the demand from peak periods to off peak periods can result in a significant reduction of a building’s operational costs of the demand side [5]. Meanwhile, the capital investment in the equipment that generates power in peak periods may reduce on the supply side. The shifting can be accomplished by storing energy during offpeak periods to be utilized during peak ones and other time of a day. Building envelope and building central heating system have been used as thermal energy storage (TES) [6,7]. Moreover, the application of phase change materials (PCMs) as latent heat thermal storage draws interests due to its high energy density. The application of PCM as a TES in buildings was reviewed in details by previous researchers [8–13]. Building envelope impregnated with PCM can provide latent heat TES distributed in the whole building envelope surface area and evade the need for enhanced thermal mass in lightweight buildings. Nevertheless, there is a lack of a general framework to select and size a PCM wallboard, which can be

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Nomenclature T temperature (◦ C) T = (Tm − Ts ) melting range (◦ C)  density (kg m−3 ) specific heat capacity (J kg−1 K−1 ) C t time (s) k conductivity (W m−1 K−1 ) space coordinate (m) x L latent heat (J kg−1 ) ˛ thermal diffusivity (m2 s) h heat transfer film coefficient (W m−2 K−1 ) l wallboard thickness (m) f liquid fraction slope SL A1 constant 1 constant Dimensionless parameters X dimensionless space coordinate  dimensionless temperature dimensionless temperature constant  Fo Fourier number Stefan number Ste Bi Biot number Subscript fusion f s solid/solidification li liquid m melting fully liquefied fl op operating temperature indicator pc PCM Low Op.Temp. the lowest operating temperature High Op.Temp. the highest operating temperature reference ref i initial ambient ∞ in indoor

employed along with an appropriate control strategy to shift and shave the peak load. Although there are a number of simulations tools to simulate the thermal performance of a building with PCM wallboard (e.g. TRNSYS, Energy Plus, and ESPR), a simple design tool can help to size the optimum thickness of a required PCM wallboard quickly. Bastani et al. [14] highlighted the requirement of a design tool based on dimensionless parameters and identified the effective dimensionless numbers that influence the performance of a PCM wallboard. To effectively employ the latent heat storage of a PCM wallboard, they defined the design objective as to have a fully charged PCM within given charging time. Thus, Fofl as the Fourier number calculated for a fully charged PCM wallboard and the effect of Biot number (Bi) and Stefan number (Ste) on the performance of PCM wallboard was characterized. The outcome was a chart, which can be used to calculate the time that a PCM wallboard has a temperature over than a design value. Therefore, the concept of the chart is similar to Heisler chart [15] for transient heat transfer of conventional building material. In addition to Bi and Ste, other dimensionless parameters were introduced as the effective parameters in sizing a PCM wallboard. These parameters are resulted from the phase transition nature of a PCM in room air temperature. Characterizing the impact of those parameters on the charging performance of a

PCM wallboard results in a framework to select and size a PCM wallboard to be applied in a building envelope. Moreover, illustrating the correlation between those dimensionless parameters creates a number of charts, which can be considered as the expansion of Heisler chart from conventional materials to PCMs. This study reports the development of a procedure to characterize the transient thermal behavior of PCM wallboards. A non-dimensional approach is used to characterize the impact of effective parameters on the charging performance of a PCM wallboard. 2. Expanding Heisler chart One concern in building operation is the peak-demand shifting which is beneficial for both supply and demand sides. Installing PCM wallboard in building envelope provides the required medium to store relatively large amount of energy in the form of latent heat during off-peak hours. Consequently, longer shift of energy is achieved and less frequent air conditioner is needed during day. The stored energy is in its highest possible value when PCM is fully charged. A PCM wallboard is called fully charged when its phase state is completely changed from one phase to another (i.e., solid to liquid for heating application, liquid to solid for cooling application). To evaluate the phase status of a wallboard, its temperature profile across its thickness is required. Considering the transient nature of the heat transfer to/from the wallboard, the temperature profile is changing as a function of time and location. Transient heat conduction analysis in a non-PCM1 plane wall has been discussed in details in heat transfer textbooks and its non-dimensionalized one-dimension analysis defines temperature within inside a wallboard as a function of three independent variables [15]:  = f (X, Bi, Fo)

(1)

Here,  is the dimensionless temperature inside a non-PCM wallboard as a function of time and location. (X, Fo) =

T (x, t) − T∞ Ti − T∞

(2)

The other variables are Fo and dimensionless spatial coordinate as follow: x Dimensionless spatial coordinate : X = (3) l Fourier number : Fo =

˛t l2

(4)

where  is the transient dimensionless temperature profile in a non-PCM wall exposed to convective heat transfer on both surfaces. Therefore, it has a non-zero heat transfer boundary condition (Fourier boundary condition) on its surfaces and a zero heat transfer boundary condition (Neumann boundary condition) in the center due to the symmetrical boundary condition.The approximate analytical solution for  is as follows: =

T (x, t) − T∞ = A1 exp(−21 Fo) cos Ti − T∞

 x 1

L

(5)

and for the center of the wallboard:  (0, Fo) = A1 exp (−21 Fo)

(6)

where the constant A1 and 1 are functions of the Bi only and their values are presented in Table 4-2 in Ref. [15] against the Bi. Also, the graphical solution is presented in a number of charts which are called Heisler charts. The equations and the relative charts can be

1

In this manuscript, the conventional building materials are called non-PCM.

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used to calculate the temperature distribution within the wallboard as function of time. Eventually, it makes it possible to calculate the appropriate thickness of the wallboard in a way to have it at a specific temperature after a predetermined period of time. Bastani et al. [14] analyzed heat transfer through PCM using non-dimensional analysis considering that PCM wall is exposed to a convective heat transfer on inner surface and insulation on the outer surface. The heat transfer in a PCM wallboard is characterized as follows: C

∂T ∂f ∂T ∂2 T = ks/li 2 + L ∂t ∂T ∂t ∂x

(7)

Here,  is the density (kg m−3 ), C is the heat capacity (J kg−1 K−1 ), T is the temperature (K), t is the time (s), k is the conductivity in solid/liquid (W m−1 K−1 ), L is latent heat (J kg−1 ), and f is the liquid fraction which express as follows:

f (T ) =

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨

[0, 1]

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1

T < Ts

Solid

∂f =0 ∂T

Ts < T < Tm

Mushy

∂f = / 0 ∂T

T > Tm

Liquid

∂f =0 ∂T

 ∂f 1− Ste ∂pc



(8)

(9)

∂pc (0, Fo) = Bi{[in (Fo)op + pc ] − pc (0, Fo)} ∂X

T (x, t) − Ts Tm − Ts

op =

Ts − TLow Op.Temp. Tm − Ts THigh Op.Temp. − TLow Op.Temp. Tm − Ts

rin =

kin−1 t 2 x (Cp + Lf  )

(18)

fin−1 − fin−2 Tin−1 − Tin−2

(19)

3.1. The effect of Ste

(12)

In this study, the dimensionless temperature is defined based on the solidification temperature (Ts ) and melting temperature (Tm ). The additional other effective parameters are:

pc =

(17)

(14)

(11)

where  pc is the dimensionless PCM wallboard temperature.

1◦ C (Tm − Ts )

n n + 2(1 − rin )Tin + rin Ti+1 = (rin )Ti−1

(13)

(10)

pc = f (Bi, Fo, X, Ste, , op , pc )

 =

n+1 n+1 −(rin )Ti−1 + 2(1 + rin )Tin+1 − (rin )Ti+1

First and second order partial derivatives are selected in terms of central differences for internal nodes, while at boundaries the forward and backward differences were applied with second order approximation. MATLAB [16] was used to solve the system of equation. Further information about the model development and validation can be found in Ref. [14]. The goal is to have a PCM wallboard fully charged in predetermined time duration (charging period). As the heat transfer phenomena occurs between the room air and the wallboard, the charging is completed when the outer surface of the wallboard has a temperature over Tm (melted for heating purpose) or under Ts (solidified for cooling purpose). Therefore, Fo is calculated at the time when the PCM wallboard is fully charged. Here, to limit the number of simulations and, also be consistent with the Heisler chart, only the heating process is considered and Fo was calculated when the PCM is fully melted. The adopted control strategy is to increase room set-point temperature from 20 ◦ C to 25 ◦ C during charging period/night time and then set it back to 20 ◦ C at the start of peak hours and for the rest of the day. As the operational temperature of the building was assumed to be consistent for all the cases (20–25 ◦ C), the effect of  op has not been investigated in this study.

Eqs. (9) and (10) indicate that a PCM wallboard dimensionless temperature is a function of seven independent variables, See Eq. (11). This equation shows that four more additional parameters are involved in characterizing PCM temperature. They are Ste,  ,  pc , and  op .

CT L

To investigate the effect of each parameter, parametric study was conducted by carrying out a number of simulation utilizing Eq. (7). It was solved numerically using finite difference technique. The Crank–Nicolson implicit numerical scheme is adopted to solve the parabolic partial differential heat equation. For the node i in the time step n, the equation becomes:

f =

The boundary condition of the wallboard in non-dimensional format is expressed as follows:

Stefan number = Ste =

3. Results and discussion

and,

∂pc ∂2 pc = ∂Fo ∂X 2

Dimensionless PCM temperature : pc (X, Fo) =

(non-PCM wallboards). It has Robin boundary condition on one side and Neumann boundary condition on the other side. Therefore, the Heisler chart can be considered as the base chart and then be expanded to cover PCM wallboards as well.

where:

Finally, they non-dimensionalized the equation as follows:



3

(15) (16)

Phase transition occurs in PCM wallboards in the range of room operational temperature, and the value of Ste affects the heat transfer phenomena within PCM wallboards.  ,  pc , and  op present the effect of melting range, wallboard phase status at the start of t charging process and the relationship between the melting range of the PCM wallboard and the room operational temperature, respectively. Bastani et al. [14] assumed a PCM wallboard installed on inner surface of a building envelope which has similar boundary conditions as the cases considered in the development of Heisler chart

The effect of Ste was investigated by carrying out a number of simulations with different Bi and Ste [14]. In the case of nonPCM wallboard, Ste−1 = 0 while for PCM wallboard it has a non-zero value. The parametric study on the effect of Ste provided a chart for a specific fusion temperature and melting range as presented in Fig. 1. Fofl is the Fourier number calculated at the time when the PCM wallboard is fully melted. This chart is called the reference chart as it was developed for materials with reference fusion temperature and melting range ( ref. and  pc,ref. ). Here, the values are for PCM wallboards which have Ts = 17 ◦ C and Tm = 23 ◦ C. Ten design cases were considered with various Bi and Ste to evaluate the accuracy of the chart in calculating Fofl . Thus, Fofl for those cases were calculated numerically using Eq. (9) and then the calculations were compared with the reference chart

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400

Bi=0.01

Bi=0.02

Bi=0.04

Bi=0.03

Table 2 Fofl calculated in testing scenarios either with simulation or the chart.

Bi=0.05

350 300

Bi=0.1

Fofl

250 200

Bi=0.15

150

Bi=0.2

100 Bi=0.5

50

Bi=1 Bi=1.5 Bi=2

0 0

50

100

150

200

1/(Ste) Fig. 1. Fofl of PCM wallboard as a function of Bi and (Ste)−1 : reference chart from Ref. [14].

(Fig. 1). The thickness and the thermo-physical properties of the PCM wallboards in those design cases were selected randomly for the following ranges; thickness (0.005–0.05 m), heat capacity (500–3000 J kg−1 K−1 ), conductivity (0.1–1.2 W m−1 K−1 ), and latent heat (50,000–400,000 J kg−1 ). The aforementioned ranges were selected based on the available materials in literatures. Table 1 gives the specifications of the PCM wallboards used for those cases. Table 2 gives calculated Fofl by simulation, from the reference chart, and the relative error: the mean average error for the testing scenarios was less than 3%. Moreover, each trend-line presented in the reference chart was validated employing the approximate analytical solution and the Heisler chart. The y-intercepts of each trend-lines in the reference chart (Fig. 1) presents Fo calculated at the time the temperature of the outer surface reaches to a specific value for a non-PCM (Ste−1 = 0). For instance according to simulations, the temperature at the outer surface of the wallboard was equal to 23 ◦ C which is equivalent to  = 0.4 in Heisler chart considering Ti and T∞ were at 20 ◦ C and 25 ◦ C, respectively [14]. Table 3 compares the calculated Fo, and it shows there is no significant difference between the Fo predicted from both methods. 3.2. Effect of  In the former section, the effect of Ste on the charging performance of a PCM wallboard was characterized and the Heisler chart was expanded to include the effect of Ste. As a result, the reference chart (Fig. 1) was built for a reference fusion temperature and melting range ( ref. and  pc,ref. ) to calculate Fofl . To expand the results for different melting ranges, a number of simulations were conducted to study the impact of  on Fofl .  represents the range and the magnitude of the temperature in which PCM

Testing Case

Ste−1

Bi

a

T-Case 1 T-Case 2 T-Case 3 T-Case 4 T-Case 5 T-Case 6 T-Case 7 T-Case 8 T-Case 9 T-Case 10

94 57 116 62 45 191 136 22 189 184

0.86 0.88 0.53 0.11 0.09 0.26 0.22 0.33 0.49 0.44

23.14 14.59 41.52 94.70 84.00 123.21 105.30 14.81 69.33 75.86

a b

b

Fofl

Fofl

Error (%)

22.81 14.11 41.82 88.75 80.20 121.58 99.75 15.16 71.14 75.45

1.42 3.26 0.73 6.28 4.52 1.32 5.27 2.34 2.62 0.54

Fofl calculated by simulation. Fofl calculated by the reference chart (Fig. 1).

Table 3 Comparison between the reference chart @ Ste−1 = 0 and Heisler chart. Bi

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2.00

a

9.63 10.02

5.06 5.51

3.53 3.32

2.76 2.83

2.30 2.79

2.00 2.12

1.78 1.74

1.62 1.68

1.49 1.49

1.39 1.34

0.93 0.83

Fo b Fofl a b

Fo @ x = 0 and  = 0.4 calculated by Eq. (5) (Heisler chart). Fofl calculated by reference chart (Fig. 1).

undergoes phase transition. To have a constant  pc , the simulation cases were designed in a way to yield a similar mean temperature in their melting range (similar fusion temperature). To study the effect of  , nine design cases were considered with different Bi and Ste. Each of them was modeled with three different melting ranges, while the mean temperature of their melting range (fusion temperature) was kept constant at 20 ◦ C. Therefore, 27 further simulations were carried out to investigate the impact of  on the time required to completely liquefy a PCM wallboard for a given Bi, Ste and  pc . The specifications of those 27 simulated cases are presented in Table 4. Like the former section, all the simulations were conducted for a room operating within 20–25 ◦ C temperature. Table 5 gives the calculated Fofl of PCM wallboards for all the considered design cases. Table 5 shows the outcome simulation for three different values Bi, Ste and  (three different  express three different melting ranges). To confirm the outcome of the former section on the effect of Bi and Ste, the change of Fofl as a function of Bi and Ste for different melting ranges was investigated and illustrated in Figs. 2 and 3. The figures show the changes of Fofl as a function of Ste−1 for  = 0.25 and  = 0.5, respectively, and they show Fo is changing linearly as a function of Ste for different values of  . Similar to the cases with  = 0.17 (see Ref. [14]), the y-intercepts of the trend-lines in Figs. 2 and 3 were compared with the analytically calculated Fo for the cases with (Ste)−1 = 0 utilizing Eq. (5) (the Heisler chart). The mean average error was less than 8%.

Table 1 The parameters of PCM in the testing cases. Testing case

Thickness (l) (m)

Film coef. (h) (W m−2 K−1 )

Conductivity (k) (W m−1 K−1 )

Density () (kg m−3 )

Heat capacity (C) (J kg−1 K−1 )

Latent (L) (J kg−1 )

Ste−1

Bi

T-Case 1 T-Case 2 T-Case 3 T-Case 4 T-Case 5 T-Case 6 T-Case 7 T-Case 8 T-Case 9 T-Case 10

0.017 0.042 0.031 0.007 0.008 0.018 0.016 0.011 0.018 0.035

18 9 16 11 13 12 8 15 14 10

0.35 0.43 0.93 0.71 1.10 0.82 0.58 0.50 0.52 0.80

1320 700 1220 990 1100 990 840 1300 1150 980

2070 1640 1240 1360 2790 1940 2610 2940 620 860

195000 93000 144000 84000 126000 370000 355000 66000 117000 158000

94 57 116 62 45 191 136 22 189 184

0.86 0.88 0.53 0.11 0.09 0.26 0.22 0.33 0.49 0.44

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Table 4 The parameters of the simulations cases for various  . Case

l (m)

h (W m−2 K−1 )

k (W m−1 K−1 )

 (kg m−3 )

C (J kg−1 K−1 )

L (J kg−1 )

Ts (◦ C)

Tm (◦ C)

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 Case 24 Case 25 Case 26 Case 27

0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175 0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175 0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175

10 8 10 5 10 12 15 10 10 10 8 10 5 10 12 15 10 10 10 8 10 5 10 12 15 10 10

0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25 0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25 0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25

1100 1400 1100 1400 700 1100 700 1100 1100 1100 1400 1100 1400 700 1100 700 1100 1100 1100 1400 1100 1400 700 1100 700 1100 1100

2500 1000 1500 2000 2000 1000 2700 900 1700 2500 1000 1500 2000 2000 1000 2700 900 1700 2500 1000 1500 2000 2000 1000 2700 900 1700

75000 50000 105000 60000 100000 70000 81000 45000 119000 75000 50000 105000 60000 100000 70000 81000 45000 119000 75000 50000 105000 60000 100000 70000 81000 45000 119000

17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 19

23 23 23 23 23 23 23 23 23 22 22 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21

l: thickness, h: film coefficient, k: thermal conductivity, : density, C: specific heat capacity, L: latent heat, Ts : solidification temperature, Tm : melting temperature. Table 5 Measured Fofl for simulation cases with different Bi, Ste, and  .

30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70

Bi

 pc

0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7 0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7 0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7

−0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

Fofl 29.45 45.55 60.10 15.92 24.29 32.73 9.95 15.43 20.91 23.09 36.57 47.47 12.65 19.28 26.18 7.83 11.94 17.02 18.54 29.88 35.86 10.49 15.36 20.73 6.28 10.31 14.36

Fofl = 0.64*(Ste)-1 + 3.42 R² = 0.98

50

Fofl =

+ 1.80 R² = 0.98 Fofl = 0.22*(Ste)-1 + 1.00 R² = 0.97

40 Fofl

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15 Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23 Case 24 Case 25 Case 26 Case 27

Ste−1

30

0.35*(Ste)-1

bi=0.2 bi=0.4 bi=0.7

20 10 0 0

20

40 Ste-1

60

80

Fig. 2. Fofl as a function of (Ste)−1 for different Bi,  = 0.25 and  pc = −0.50.

Completely Liquefied PCM, Ψ=0.50, Ψpc=-0.50 40

Fofl = 0.50*(Ste)-1 + 2.49 R² = 0.93

35

Fofl =

+ 1.22 R² = 0.98 Fofl = 0.19*(Ste)-1 + 0.48 R² = 0.98

30 25 Fofl

Case

Completely Liquefied PCM, Ψ=0.25, Ψpc=-0.50 60

20

0.28*(Ste)-1

bi=0.2 bi=0.4 bi=0.7

15 10 5 0

The effect of  on Fofl is illustrated for each set of (Bi, Ste) in Fig. 4. The figure shows the change of log (Fofl ) as a function of log ( ) for constant  pc . Regarding the curve fitting of the data in Fig. 4, log (Fofl ) is a linear function of log ( ), and the slope remains constant (−0.40) for the entire range of Bi and Ste. the change of log (Fofl ) is linear as a result of a change of log ( ) with a similar slope of almost (−0.40) for various sets of Bi and Ste. Employing this relationship between Fofl and  , any reference chart such as the one presented in Fig. 4, can be modified to calculate Fofl for a material with different melting range (different  ).

0

20

40 (Ste)-1

60

80

Fig. 3. Fofl as a function of (Ste)−1 for different Bi,  = 0.50 and  pc = −0.50.

3.3. Effect of  pc  pc relates the fusion temperature, the melting range of a PCM wallboard and the operational room air temperature (see Eq. (13)). This equation has the melting range of the PCM wallboard

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Effect of Ψ on Fofl 2

log(Fofl) = -0.41*log(Ψ)+1.06 R² = 0.98

Bi=0.4 1/Ste=50

1.6

log(Fofl) = -0.39*log(Ψ)+1.36 R² = 0.97

Bi=0.2 1/Ste=50

1.4

log(Fofl) = -0.38*log(Ψ) + 0.90 R² = 0.96

Bi=0.7 1/Ste=50

log(Fofl) = -0.37*log(Ψ) + 0.90 R² = 0.93 log(Fofl) = -0.41*log(Ψ)+ 1.14 R² = 0.97

Bi=0.4 1/Ste=30

log(Fofl) = -0.41*log(Ψ) + 0.67 R² = 0.97

Bi=0.7 1/Ste=30

log(Fofl) = -0.41*log(Ψ)+ 1.19 R² = 0.9813 log(Fofl) = -0.44*log(Ψ)+1.41 R² = 0.97 log(Fofl) = -0.36*log(Ψ) + 1.05 R² = 0.96

Bi=0.4 1/Ste=70

1.8

Log(Fofl)

1.2 1 0.8 0.6 0.4 0.2 0

-0.8

-0.6

-0.4 Log(Ψ)

-0.2

Bi=0.2 1/Ste=30

Bi=0.2 1/Ste=70 Bi=0.7 1/Ste=70

0

Fig. 4. Effect of  (melting range) on the Fofl for different Bi and Ste having similar  pc .

Table 6 The parameters of the simulations cases for various  pc . Case

l (m)

h (Wm2 K−1 )

k (W m−1 K−1 )

 (Kg m−3 )

C (J Kg−1 K−1 )

L (J Kg−1 )

Ts (◦ C)

Tm (◦ C)

Case 28 Case 29 Case 30 Case 31 Case 32 Case 33 Case 34 Case 35 Case 36 Case 37 Case 38 Case 39 Case 40 Case 41 Case 42 Case 43 Case 44 Case 45 Case 46 Case 47 Case 48 Case 49 Case 50 Case 51 Case 52 Case 53 Case 54 Case 55 Case 56 Case 57 Case 58 Case 59 Case 60 Case 61 Case 62 Case 63

0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175 0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175 0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175 0.0100 0.0175 0.0060 0.0320 0.0100 0.0150 0.0117 0.0245 0.0175

10 8 10 5 10 12 15 10 10 10 8 10 5 10 12 15 10 10 10 8 10 5 10 12 15 10 10 10 8 10 5 10 12 15 10 10

0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25 0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25 0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25 0.50 0.70 0.30 0.40 0.25 0.45 0.25 0.35 0.25

1100 1400 1100 1400 700 1100 700 1100 1100 1100 1400 1100 1400 700 1100 700 1100 1100 1100 1400 1100 1400 700 1100 700 1100 1100 1100 1400 1100 1400 700 1100 700 1100 1100

2500 1000 1500 2000 2000 1000 2700 900 1700 2500 1000 1500 2000 2000 1000 2700 900 1700 2500 1000 1500 2000 2000 1000 2700 900 1700 2500 1000 1500 2000 2000 1000 2700 900 1700

75000 50000 105000 60000 100000 70000 81000 45000 119000 75000 50000 105000 60000 100000 70000 81000 45000 119000 75000 50000 105000 60000 100000 70000 81000 45000 119000 75000 50000 105000 60000 100000 70000 81000 45000 119000

20 20 20 20 20 20 20 20 20 21 21 21 21 21 21 21 21 21 19 19 19 19 19 19 19 19 19 18 18 18 18 18 18 18 18 18

23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 22 22 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21

l: thickness, h: film coefficient, k: thermal conductivity, : density, C: specific heat capacity, L: latent heat, Ts : solidification temperature, Tm : melting temperature.

in its denominator. Moreover, it has the difference between the solidification temperature and the lower bond of thermal comfort temperature as the lowest limit of the operational room air temperature in its numerator. Thus, the numerator of this ratio presents the

phase status of the PCM wallboard when the charging procedure is initiated. In addition, different PCM wallboards with similar melting range may have different  pc value when they have different solidification temperature (Ts ).

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A. Bastani, F. Haghighat / Energy and Buildings xxx (2015) xxx–xxx Table 7 Measured Fofl for simulation cases with different Bi, Ste, and  pc . Case

Ste−1

Bi



 pc

Fofl

Case 28 Case 29 Case 30 Case 31 Case 32 Case 33 Case 34 Case 35 Case 36 Case 37 Case 38 Case 39 Case 40 Case 41 Case 42 Case 43 Case 44 Case 45 Case 46 Case 47 Case 48 Case 49 Case 50 Case 51 Case 52 Case 53 Case 54 Case 55 Case 56 Case 57 Case 58 Case 59 Case 60 Case 61 Case 62 Case 63

30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70 30 50 70

0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7 0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7 0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7 0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 -0.33 -0.33 -0.33 -0.33 -0.33 -0.33 -0.33 -0.33 -0.33 -0.67 -0.67 -0.67 -0.67 -0.67 -0.67 -0.67 -0.67 -0.67

53.64 85.88 114.65 28.14 45.36 62.18 17.97 28.92 39.81 82.73 131.92 177.27 43.20 70.36 96.54 27.73 44.29 60.85 29.82 47.67 62.18 16.31 25.00 34.18 10.05 16.20 22.31 12.91 20.41 24.75 7.30 10.54 14.36 4.44 7.31 9.82

The magnitude of  pc may have three states depending on the solidification temperature of a PCM wallboard: either positive, negative or zero. The sign of  pc value implies the state of PCM at the time when the charging process is initiated: positive magnitude indicates that PCM is at the solid phase, while the negative magnitude indicates it is at the mushy/liquid phase. Consecutively, zero  pc points out that the PCM wallboard phase change is happening. Like the procedure applied for the melting range, the effect of  pc (fusion temperature) on the Fofl (performance of the PCM wallboard) is investigated. For this purpose 36 simulation cases were designed with nine different sets of (Bi, Ste), similar  , and four

different  pc . The specifications of those 36 simulated cases are presented in Table 6. Table 7 gives the calculated Fofl from those considered design cases. As presented in Table 7, these cases consist of three different Bi, and Ste and four different  pc. To confirm the outcome of the former section on the effect of Bi and Ste, the change of Fofl as a function of Bi and Ste for other melting ranges was investigated as illustrated in Figs. 5–8. The fitted curves presented in Figs. 5–8 confirm that Fo changes linearly as a function of Ste, and it is independent of the melting ranges and fusion temperatures. Similar to the cases with  pc = −0.5, the y-intercepts of the trend-lines in Figs. 5–8 were compared with the analytically calculated Fo of the cases for (Ste)−1 = 0 utilizing the equation (5) (Heisler chart). Based on their Tm , the y-intercepts of the trend-lines in Fig. 5 need to be similar to the ones in Table 3. Although those cases have different fusion temperatures and melting ranges, they become completely melted when the temperature across the wallboard exceeds 23 ◦ C. The materials with (Ste)−1 = 0 have the same temperature profile after the same time period; it is independent of their melting range and fusion temperature. Considering this argument, the y-intercepts of the trend-lines in Figs. 7 and 8 need to be close to the ones in Figs. 2 and 3, respectively. Moreover, for the cases in Fig. 6 with Tm = 24 ◦ C, the y-intercepts are compared with Fo calculated by Eq. (5) at x = 0 and  = 0.2. The trivial difference between the predicted Fo by both methods validated the outcome of curve fitting with analytical solution for the cases with (Ste)−1 = 0 (mean average error less than 5.5%). The effect of  pc on Fo is illustrated for each set of (Bi, Ste) in Fig. 9. This figure shows the change of Fo when PCM is completely melted as a function of  pc for constant  . The figure shows that Fo varies linearly for each set of (Bi, Ste). To map the result from one available  pc to another, the correlation between the slope (SL ) of the fitted curves, Bi and Ste needs to be studied. For the simulated sets of Ste and Bi, the magnitudes of the SL of the fitted curves are tabulated in Table 8. Figs. 10 and 11 illustrate the correlations between SL , Ste, and Bi, respectively. Like Fo, SL increases linearly as a function of (Ste)−1 and log (SL ) decreases linearly by increasing log (Bi). Moreover, like the graph developed for Fo for different Ste and Bi (Fig. 1), the correlations obtained from Figs. 10 and 11 are utilized to build up a graph to calculate SL for any set of (Bi, Ste). Considering the reference chart (Fig. 1), the slopes calculated from the impact of  and  pc on Fofl are applied to map the reference chart from one fusion temperature and melting range to other fusion temperature and melting range.

Completely Liquefied PCM, Ψ=0.33, Ψpc=0.00

120

Fo = 1.57*(Ste)-1 + 5.81 R² = 0.98 Fo = 0.85*(Ste)-1 + 2.74 R² = 0.99

100 80 Fofl

7

60

Fo = 0.54*(Ste)-1 + 1.74 R² = 0.99

40

Bi=0.2 Bi=0.4 Bi=0.7

20 0 0

20

40 (Ste)-1

60

80

Fig. 5. PCM completely liquefied Fofl as a function of (Ste)−1 in different Bi,  = 0.33 and  pc = 0.

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Fofl

Completely Liquefied PCM, Ψ=0.33, Ψpc=0.33 200 180 160 140 120 100 80 60 40 20 0 0

20

40 (Ste)-1

60

Fo = 2.42*(Ste)-1 + 9.59 R² = 0.98

Bi=0.2

Fo = 1.31*(Ste)-1 + 4.42 R² = 0.98

Bi=0.4

Fo = 0.83*(Ste)-1 + 2.98 R² = 0.98

Bi=0.7

80

Fig. 6. PCM completely liquefied Fofl as a function of (Ste)−1 in different Bi,  = 0.33 and  pc = 0.33.

Completely Liquefied PCM, Ψ=0.33, Ψpc=-0.33 70 60

Fofl

50 40 30

Fo = 0.85*(Ste)-1 + 3.61 R² = 0.97

Bi=0.2

Fo = 0.46*(Ste)-1 + 1.88 R² = 0.98

Bi=0.4

Fo = 0.30*(Ste)-1 + 1.01 R² = 0.99

Bi=0.7

20 10 0 0

20

40 (Ste)-1

60

80

Fig. 7. PCM completely liquefied Fofl as a function of (Ste)−1 in different Bi,  = 0.33 and  pc = −0.33.

Completely Liquefied PCM, Ψ=0.33, Ψpc=-0.67 30

Fofl

20

10

Fo = 0.34*(Ste)-1 + 2.07 R² = 0.95

Bi=0.2

Fo = 0.19*(Ste)-1 + 1.04 R² = 0.98

Bi=0.4

Fo = 0.13*(Ste)-1 + 0.52 R² = 0.99

Bi=0.7

0 0

20

40 (Ste)-1

60

80

Fig. 8. PCM completely liquefied Fofl as a function of (Ste)−1 in different Bi,  = 0.33 and  pc = −0.67.

Finalizing the effect of  pc , the Heisler chart is updated to cover wallboards impregnated with PCM. In fact, the effect of Ste,  , and  pc on the charging procedure of a PCM wallboard was investigated and a number of charts were developed to characterize the transient heat transfer of a PCM wallboard. Here, the Heisler chart is presented as the base chart to determine the time required to heat up (or cool down) a material to a specific temperature. Bi is

the only effective variable in the Heisler chart for a given material and given time. Then, the effect of Ste,  , and  pc on the charging procedure of a PCM wallboard was added to expand the application of Heisler chart to PCM wallboard. Implementing the effect of Ste for PCM wallboard, the reference chart (Fig. 1) was developed for a reference fusion temperature and melting range ( ref. and  pc,ref. ). To cover other values of  and  pc , the reference chart needs to

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9

Effect of Ψpc on Fofl 200

Fofl = 70.00*Ψpc + 56.50 R² = 0.97

180

Fofl = 111.84*Ψpc + 90.20 R² = 0.97 Fofl = 153.04*Ψpc + 120.35 R² = 0.95

160 140

Fofl

120 100 80 60

20 0 -1

-0.5

0

0.5

Bi=0.2 1/Ste=50 Bi=0.2 1/Ste=70

Fofl = 59.95*Ψpc + 47.86 R² = 0.97

Bi=0.4 1/Ste=50

Fofl = 82.37*Ψpc + 65.61 R² = 0.97

Bi=0.4 1/Ste=70

Fofl = 23.34*Ψpc + 18.96 R² = 0.97

Bi=0.7 1/Ste=30

Fofl = 37.27*Ψpc + 30.38 R² = 0.97 Fofl = 51.19*Ψpc + 41.77 R² = 0.97

40

Bi=0.2 1/Ste=30

Fofl = 35.87*Ψpc + 29.74 R² = 0.97

Bi=0.7 1/Ste=50 Bi=0.7 1/Ste=70 Bi=0.4 1/Ste=30

Ψpc Fig. 9. Effect of  pc on Fo in different Bi and Ste with similar  .

SL VS Bi

Table 8 SL of linear fitted curve in the simulated set of (Bi, Ste).

2.4

Ste−1

Bi

Slope (SL )

Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 Set 9

30 50 70 30 50 70 30 50 70

0.2 0.2 0.2 0.4 0.4 0.4 0.7 0.7 0.7

70.00 111.84 153.04 35.87 59.95 82.37 23.34 37.27 51.19

180 160 140

SL

120

Bi=0.2

SL = 1.16*(Ste)-1 + 1.27 R² = 0.97

Bi=0.4

SL =

100 80

R² = 0.99

+ 2.45

1.8 1.6

1/Ste=30 1/Ste=50 1/Ste=70

1.4 1.2 1 -0.5 log(Bi)

0

Fig. 11. SL as a function of Bi for different (Ste)−1 .

SL = 2.08*(Ste)-1 + 7.83 R² = 0.98

0.70*(Ste)-1

log(SL) = -0.88*log(Bi) + 1.43 R² = 0.98 log(SL) = -0.88*log(Bi) + 1.22 R² = 0.97

2

-1

SL VS (Ste)

log(SL) = -0.88*log(Bi) + 1.57 R² = 0.98

2.2

log(SL)

Set

Bi=0.7

60 40 20 0 0

20

40 (Ste)-1

60

80

Fig. 10. SL as a function of (Ste)−1 for different Bi.

be modified using the correlations obtained from Fig. 4 for  , and from Figs. 9 and 12 for  pc . Fig. 13 presents a three-step flowchart developed in this study to expand the Heisler chart to cover PCM. To check the accuracy of the expansion of the Heisler chart for PCM wallboards, the testing cases designed and utilized in Table 1 are employed for different Ts and Tm in order to compare the Fofl of the PCM wallboard acquired by simulation with the one calculated with the modified Heisler chart as presented in Fig. 13. Here, the testing cases presented in Table 1 have  and  pc

Fig. 12. SL calculator for different Bi and Ste.

equivalent to the reference magnitude in the reference chart (Fig. 1). To consider the effect of  and  pc , the melting and solidification temperatures (Ts and Tm ) of those testing cases were modified to have different  and  pc than the reference

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Fig. 13. Flowchart to expand Heisler chart for PCM.

magnitudes as presented in Table 9. To calculate Fofl , the modified Heisler chart was employed as presented in three steps in Fig. 13. Therefore, for each testing case, at first, the base chart (Fig. 1) was modified through step-2 and step-3 as presented in Fig. 13 regarding the effect of  and  pc . Finally, based on their Ste and Bi, Fofl of the testing PCM wallboard was calculated. Table 9 gives the calculated Fofl for these cases either by simulation or from the charts and the relative error: the mean average error for the testing scenarios was less than 5%. According to Fig. 13, for the conventional building materials with Ste−1 = 0, Heisler chart can be utilized to analyze its transient heat transfer. For a PCM wallboard with Ste−1 = / 0, the reference chart can be used to analyze transient heat transfer if its  =  ref. and  pc =  pc,ref. Otherwise, step two or step three or both steps needs to be followed to update the reference chart for an appropriate magnitude of  and  pc .

Table 9 Fofl calculated in testing scenarios with both simulation and the charts (effect of  and  pc ). Testing case

Ts (◦ C)

Tm (◦ C)



 pc

a

T-Case 1 T-Case 2 T-Case 3 T-Case 4 T-Case 5 T-Case 6 T-Case 7 T-Case 8 T-Case 9 T-Case 10

18 19 20 19 18 20 18 19 20 20

23 23 23 22 21 22 23 23 23 22

0.20 0.25 0.33 0.33 0.33 0.50 0.20 0.25 0.33 0.50

−0.40 −0.25 0.00 −0.33 −0.66 0.00 −0.40 −0.25 0.00 0.00

22.81 14.11 41.82 88.75 80.20 121.58 99.75 15.16 71.14 75.45

a b c

Fofl

b

Fofl

26.85 20.57 84.50 106.29 229.29 238.00 119.25 21.20 144.97 146.02

c

Fofl

27.44 21.08 80.47 99.01 217.29 227.43 125.35 20.54 135.73 136.52

Error (%) 2.14 2.42 5.01 7.35 5.52 4.65 4.87 3.23 6.81 6.96

The base Fofl (Fig. 1). Fofl after the effect of  and  pc (Fig. 13). Fofl calculated by simulation.

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4. Conclusions The need for an efficient tool to size a PCM wallboard guided this research work to characterize heat transfer phenomena inside a PCM wallboard. Due to the fact that the focus is to charge and store thermal energy in a PCM wallboard during off-peak periods, the design tool consists of the relationship between the design parameters and the time required to fully melt a PCM. Time required to charge a PCM wallboard is the key for an effective design of a building envelope: the wallboard must be fully charged during off-peak hours. Eventually, heat transfer inside a PCM wallboard was characterized graphically to correlate the design parameters of a PCM wallboard (its thickness and thermos-physical properties) to the charging duration. Consequently, Heisler chart was expanded to have a similar graph to calculate the charging period and the thickness of a PCM wallboard. Those graphs and their sequences are summarized in Fig. 13. By analyzing the expansion of Heisler chart, the following outcomes need to be highlighted: • Fo increases linearly by decreasing Ste when the remaining dimensionless parameters are kept constant. Lower magnitude of Ste means lower ratio of sensible over latent heat storage which infers domination of the latent over the sensible heat. Thus, a higher Fo of a fully melted PCM is expected for the lower magnitude of Ste. • Utilizing the correlations between Fofl , Ste and Bi for  ref. and  pc,ref. the reference chart was developed. • The developed chart is the extension of Heisler chart for PCMs. As the y-intercepts of the correlations are the calculated Fo for materials with Ste−1 = 0, they were compared with the analytical calculation of Fo and were in good agreement. • Similar to Heisler Chart, Fo calculated by the expansion of Heisler chart in Ste−1 = 0 is only a function of the temperature and Bi of the wallboard. • The precision of the reference chart to calculate Fofl for a PCM wallboard with  ref. and  pc,ref. was evaluated for 10 design cases; the mean average error was less than 3%. • The effect of the melting range and the fusion temperature were studied by characterizing the change of Fofl as a function of  and  pc . As the essential elements, the correlations regarding these two dimensionless parameters are required to map the reference chart from  ref. and  pc,ref. to another  and  pc . This transformation helps to take into account all the physical properties and characteristics of PCM wallboards and develop a universal chart. • Based on the parametric study of the impact of  , it was found that the log (Fofl ) is a linear function of log ( ) with the slope of (−0.40), and it is independent of Bi and Ste. Therefore, Fofl calculated by the reference chart for all PCM wallboards can be extrapolated to another  than  ref. . • The effect of fusion temperature and its relationship to the room operational temperature were characterized by conducting parametric study on the effect of  pc on Fofl . The conducted

11

parametric study provided an auxiliary chart to calculate the required slope to map the reference chart to another  pc than  pc,ref. . The slope depends on the Bi and Ste of the PCM wallboard. As the reference chart needs to be mapped to another  and  pc than  ref. and  pc,ref. , the precision of the mapped chart to calculate Fofl in PCM wallboard was evaluated for 10 testing cases, and the mean average error was less than 5%. The recommended modified Heisler chart has the same limitations as the original Heisler chart. Acknowledgements The authors would like to express their gratitude to the Public Works and Government Services Canada and Concordia University for their support. References [1] Natural Resources Canada, Energy Efficiency Trends in Canada 1990 to 2009, 2011, Cat. No. M141-1/2009E-PDF (Online), Available at: HTTP://OEE.NRCAN. GC.CA/CORPORATE/STATISTICS/PUBLICATIONS. ISSN 1926-8254. [2] HydroQuebec, Demande R-3648-2007. Hydro Quebec Distribution-1, 2007. [3] M.-A. Leduc, A. Daoud, L.B. Celyn, Developing winter residential demand response strategies for electric space heating, in: Proceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association, Sydney, 14–16 November, 2011. [4] HydroQuebec, Demande R-3776-2011. Hydro Quebec Distribution-2, 2011. [5] A.M. Khudhair, M.M. Farid, A review on energy conservation in building applications with thermal storage by latent heat using phase change materials, Energy Convers. Manag. 45 (2004) 263–275. [6] A. El-Sawi, F. Haghighat, H. Akbari, Assessing long-term performance of centralized thermal energy storage system, Appl. Therm. Eng. 62 (2014) 313–321, http://dx.doi.org/10.1016/j.applthermaleng.2013.09.047 [7] D.N. Nkwetta, P.-E. Vouillamoz, F. Haghighat, M. El-Mankibi, A. Moreau, A. Daoud, Impact of phase change materials types and positioning on hot water tank thermal performance: using measured water demand profile, Appl. Therm. Eng. 67 (2014) 460–468, http://dx.doi.org/10.1016/j.applthermaleng. 2014.03.051 [8] Y. Dutil, D.R. Rousse, N.B. Salah, S. Lassue, L. Zalewski, A review on phase-change materials: mathematical modeling and simulations, Renew. Sustain. Energy Rev. 15 (2011) 112–130. [9] M.M. Farid, A.M. Khudhair, S.A.K. Razack, S. Al-Hallaj, A review on phase change energy storage: materials and applications, Energy Convers. Manag. 45 (2004) 1597–1615. [10] F. Kuznik, D. David, K. Johannes, J.-J. Roux, A review on phase change materials integrated in building walls, Renew. Sustain. Energy Rev. 15 (2011) 379–391. [11] P.A. Mirzaei, F. Haghighat, Modeling of phase change materials for applications in whole building simulation, Renew. Sustain. Energy Rev. 16 (2012) 5355–5362, http://dx.doi.org/10.1016/j.rser.2012.04.053 [12] P. Verma, Varun, S.K. Singal, Review of mathematical modeling on latent heat thermal energy storage systems using phase-change material, Renew. Sustain. Energy Rev. 12 (2008) 999–1031. [13] Y. Zhang, G. Zhou, K. Lin, Q. Zhang, H. Di, Application of latent heat thermal energy storage in buildings: state-of-the-art and outlook, Build. Environ. 42 (2007) 2197–2209. [14] A. Bastani, F. Haghighat, J. Kozinski, Designing building envelope with PCM wallboards: design tool development, Renew. Sustain. Energy Rev. 31 (2014) 554–562, http://dx.doi.org/10.1016/j.rser.2013.12.031 [15] Y.A. Cengel, A.J. Ghajar, Heat and Mass Transfer; Fundamentals and Applications, 4th ed., The McGraw-Hill Companies, Inc., New York, NY, 2011. [16] Matlab, R2010b Math Work Documentation, Math Works Inc., 2010.

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