Thermal performance analysis of a roof with a PCM-layer under Mexican weather conditions

Journal Pre-proof Thermal performance analysis of a roof with a PCM-layer under Mexican weather conditions J. Xamán, A. Rodriguez-Ake, I. Zavala-Guillén, I. Hernández-Pérez, J. Arce, D. Sauceda PII:

S0960-1481(19)31956-1

DOI:

https://doi.org/10.1016/j.renene.2019.12.084

Reference:

RENE 12793

To appear in:

Renewable Energy

Received Date: 16 August 2019 Revised Date:

15 December 2019

Accepted Date: 19 December 2019

Please cite this article as: Xamán J, Rodriguez-Ake A, Zavala-Guillén I, Hernández-Pérez I, Arce J, Sauceda D, Thermal performance analysis of a roof with a PCM-layer under Mexican weather conditions, Renewable Energy (2020), doi: https://doi.org/10.1016/j.renene.2019.12.084. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Centro Nacional de Investigación y Desarrollo Tecnológico “2019, Año del Caudillo del Sur, Emiliano Zapata”

09/November/2019

Dr. Soteris Kalogirou Editor in Chief Renewable Energy Cyprus University of Technology, Limassol, Cyprus

Dear Prof. Kalogirou:

Author contributions are the following: J. Xamán: Supervision, Methodology, Conceptualization, Writing - Review & Editing A. Rodriguez-Ake: Software, Validation, Writing - Original Draft I. Zavala-Guillén: Writing - Original Draft, Writing - Review & Editing, Formal analysis I. Hernández-Pérez: Writing - Original Draft, Writing - Review & Editing, Formal analysis J. Arce: Visualization, Investigation, Conceptualization D. Sauceda: Visualization, Investigation, Conceptualization

Sincerely Prof. J. Xamán CENIDET - TecNM - SEP. Mech. Engin. Department Email: [email protected]

Interior Internado Palmira S/N, Col. Palmira, C. P. 62490, Cuernavaca, Morelos. Tel. (01) 777 3 62 77 70, ext. 4201, e-mail: [email protected] www.tecnm.mx | www.cenidet.edu.mx

Thermal performance analysis of a roof with a PCM-layer under Mexican weather conditions J. Xam´ana,∗, A. Rodriguez-Akea,b , I. Zavala-Guill´enb , I. Hern´andez-P´erezc , J. Arcea , D. Saucedab a

Centro Nacional de Investigaci´ on y Desarrollo Tecnol´ ogico CENIDET-TecNM-SEP, Prol. Av. Palmira S/N. Col. Palmira. Cuernavaca, Morelos, CP. 62490, M´exico b Centro de Educaci´ on Cient´ıfica y de Educaci´ on Superior de Ensenada (CICESE), Carretera Ensenada-Tijuana No.3918, Zona Playitas, Ensenada 22860, Baja California, M´exico c Universidad Ju´ arez Aut´ onoma de Tabasco, Divisi´ on Acad´emica de Ingenier´ıa y Arquitectura (DAIA-UJAT), Carretera Cunduac´ an-Jalpa de M´endez km. 1, Cunduac´ an, Tabasco, CP 86690, M´exico

Abstract The thermal performance of a concrete roof with a phase change material (PCM) layer on its interior surface under a Mexican warm weather (Merida) is presented. We analyzed a roof with three types of PCM: Paraffin wax - MG29 (R-PCM1), N-Eicosane (R-PCM2), and Salt Hydrates (R-PCM3). We also considered different thickness of the PCM layer. A conventional concrete roof (R-C) was considered as a reference to compare the results. The numerical simulations were conducted during the warmest and the coldest days of the year. A numerical in-house code was developed, and it was verified by solving reference solutions, obtaining good agreement. The results indicate that the case R-PCM1 with 2 cm of PCM layer had the lowest values of thermal load during the coldest (204.5 W h m−2 ) and the warmest day (610.7 W h m−2 ); such values are up to 57% lower than the thermal load corresponding to the R-C. The use of R-PCM1 with 2 cm of PCM-layer in Merida city will have a payback period of 12.18 years, Taking into account that buildings in Mexico have a 30-year ordinary service life, the use of these materials is cost-effective. Therefore, it is recommended the R-PCM1 to improve the thermal behavior of buildings located in Merida.

∗

Corresponding author Email addresses: [email protected] (J. Xam´an), [email protected] (A. Rodriguez-Ake), [email protected] (I. Zavala-Guill´en), [email protected] (I. Hern´ andez-P´erez), [email protected] (J. Arce), [email protected] (D. Sauceda)

Preprint submitted to Renewable Energy

December 23, 2019

Keywords: Phase change material, Roof, Mexican weather Nomenclature Cp

specific heat, J/kg·K

fl

liquid fraction

G

solar radiation, W/m2

h

heat transfer coefficient, W/m2 K

Hc

thickness of roof, m

hls

Specific enthalpy of fusion, J/kg

HP CM thickness of PCM layer, m L

length, m

q

heat flux, W/m2

Sg

extinction coefficient, 1/m

T

temperature, ◦ C

t

time, s

vwind wind speed, m/s x, y

coordinates, m

Greek λ

thermal conductivity, W/m·K

ρ

density, kg/m3

σ

Stefan–Boltzmann constant (5.67×10−8 W/m2 · K4 )

2

Subscripts ave

average

c

concrete

conv convective ef f

effective

in

inside

m

melting

out

outside

rad

radiative

s, in

inner surface

s, out outer surface

1. Introduction In recent years, the growing energy demand for buildings in Mexico has increased because of the widespread use of mechanical systems for space conditioning, which is attributed to the fact that a significant fraction of the population lives in urban areas and 5

spends a lot of time inside buildings where they develop different activities. Particularly, in the territory of Mexico prevails a warm climate because of its geographical position and of course, the centers for economic activity are located in this climate. In such a way that the concentration of population in urban zones increases the energy consumption for the conditioning of spaces, both in the residential and commercial and services sectors. In

10

this sense, the National Commission for the Efficient Use of Energy (CONUEE-Mexico) points out that from 2012 to 2016 the number of users of electric energy in those sectors 3

increased by 12%, while the total average consumption per user in the sector did not grow; i.e., the energy consumption increased as the number of users raised. However, it shows that the average consumption per user for thermal comfort increased about 22% 15

from 2012 to 2016, the use for thermal comfort include both cooling and heating energy demand. The increase in the average consumption for thermal comfort is closely related to the fact that 63% from the users located in areas with warm weather had an increase of over 50% [1]. In Mexico, a user in a warm climate region consumes an average of twice as much electricity as one in a temperate climate [1]. The regions with a warm weather have

20

a high incidence of solar irradiation, a high temperature, and some cases high relative humidity content in the environment.

However, weather is not the only factor that influences the thermal comfort of a building. There are other factors such as the components of the building envelope and their 25

respective construction materials that also influence the thermal behavior of a building. In particular, the roof is one component that significantly influences the thermal behavior of a building because when it is located in a region with warm weather, the roof can contribute with up to 50% of the building thermal load [2]. For this reason, different alternatives have been studied to improve the thermal performance of the roofs of buildings, such as

30

reflective coatings (cool roofs) [3, 4], the use of vegetation (green roofs)[5, 6], integrating heat exchangers (heat exchanger roof), ventilated roofs [7–10], and recently the use of phase change materials (PCM) in different roof configurations.

Among the studies with macro-encapsulated PCM introduced in the construction 35

material, there is a research reported by Alawadhi and Alqallaf [11, 12]. Who carried out the numerical and experimental study of a building roof with macro-encapsulated PCM, considering conical and cylindrical geometries for the PCM. They found that the heat flow to the indoor space was reduced up to 39% with the N-Eicosane and a conical geometry. The results of his research showed that the A32, and a configuration with

40

0.1979 m in diameter and 0.0383 m in height, are the best in terms of the heat gain. The authors concluded that this roof configuration reduced the heat flux to the interior

4

between 9 and 17.26%.Another study available in the literature is the research presented by Toku¸c et al. [13]. They developed numerical simulations with FLUENT software to evaluate the thermal performance of a roof with PCM under the weather conditions of 45

four regions of Turkey. The results showed that in May, the PCM with a thickness of 5 cm reduced the cooling loads by 48.2, 56.6, 78.7, and 99.1% for Izmir, Istanbul, Ankara, and Erzurum, respectively.

Later, Panayiotou et al. [14] evaluated theoretically the thermal performance of a 50

macro encapsulated PCM, which was applied in a typical Cyprus apartment studied under Mediterranean weather conditions. The authors considered four different configurations of roof and walls, which were analyzed using model Type1270 of the TRNSYS. The results showed that the combined configuration (PCM and insulation) achieves energy savings of up to 67.6% compared to the base configuration. Hamza et al. [15] investigated the

55

thermal performance of adding a PCM in a building roof. The authors evaluated three different locations and six PCMs: S19, Clim Sel C21, S23, Clim Sel C24, RT25-RT30, and Polyethylene glycol 900.

The most frequent practice of adding a PCM to roofs is in panels to form multilayer 60

configurations as shown by Pasupathy and Verlaj [16]. Who carried out the numerical and experimental study of a multilayer roof with PCM under climatic conditions of the city of Chennai, India. The authors concluded that the configuration of a slab with a thickness of 12 cm, a PCM1 (Salt Hydrate) panel 2.5 cm thick, a PCM2 panel (Climsel) 4 cm thick, and another slab with a thickness of 10 cm had the best thermal performance.

65

Furthermore, Mushtaq et al.[17] experimentally studied the effect of incorporating a PCM into a roof with a cooling system, which was subjected to summer weather conditions in Baghdad, Iraq. They used a PCM with a melting temperature of 37 ◦ C and found that the roof with PCM reduced heat transfer up to 46.71% compared to the reference roof. Yu et al. [18] numerically investigated the thermal performance of a roof with a PCM

70

under climatic conditions in five regions in China. The PCM was a mixture of paraffin and high-density polyethylene. They observed that the maximum indoor temperatures were

5

reduced to 4 ◦ C compared with a roof without PCM. They also found that the decrement factor was reduced to 88.79% due to the PCM.

In 2013, Chou et al. [19] conducted a numerical-experimental study to test the thermal

75

behavior of a roof consisting of two corrugated metal layers, a layer of phase change material (PCM) and a polyurethane layer. They studied the case of a roof of corrugated metal sheets. The results demonstrated that the configuration with PCM reduced the heat flow towards the indoor, and they showed that this configuration reduced 52.7% the 80

electricity consumption compared to the roof of corrugated metal sheets. Later, Li et al. [20] carried out a numerical study to evaluate the thermal behavior of a roof formed by four layers: 2 mm of aluminum alloy plate, 20 mm of cement, 100 mm of fine stone concrete and PCM, and 100 mm of reinforced concrete. The authors used FLUENT software to carry out the study. They found that the PCM retards the peak temperatures

85

of the base layer on the roofs by more than 3 h compared to a conventional roof. Guichard et al. [21] conducted a numerical and experimental study to evaluate a roof formed by a corrugated galvanized steel layer 1 mm thick, an air cavity of 280 mm, a layer of PCM Energain 5.26 mm, and a layer of 12.55 mm of plasterboard. They used the software ISOLAB to simulate the thermal behavior using climatic conditions of Reunion Island,

90

France. Their results showed that the PCM was able to reduce the temperature by 2.4 ◦

C compared to a roof without PCM.

Other studies reported the addition of the PCM to the internal surface of the roof, as Kong et al. [22]. Who conducted an experimental study to evaluate the thermal 95

performance of the walls and roof of a building, in which they adhered PCM panels on their inside surface (PCMIW) and on their outside surface (PCMOW). Their results showed that the panels of PCM used in the rooms reduced between 1 and 2 ◦ C the indoor air temperature compared with a room without panels of PCM. The PCM delayed the maximum temperature of around 2 to 3 hours. Kong et al. [22] conducted a numerical

100

study of the same cases using the FLUENT 6.3 software. With this study they found that the PCMIW configuration (1-Dodecanol) reduced 2.4 ◦ C in the indoor temperature,

6

while PCMOW (capric acid) reduced it only 0.6 ◦ C.

The literature review shows that the addition of phase change materials in a roof 105

considerably improves its thermal behavior and, therefore, it also reduces the cooling energy load of buildings.

To obtain favorable results, it is important to select the

PCM according to the climatic conditions of the location where the building is situated. Moreover, it is clear that the energy-saving potential of PCM integrated into a roof has been reported under different climates of the world [16, 17, 21–25]. However, in countries 110

such as Mexico, the deployment of PCM is emerging and one cannot find evidence of its effectiveness to improve the thermal behavior of building roofs. Further, because the thermal behavior of the PCMs is strongly related to its thermophysical properties. Some studies showed that the thermal conductivity, specific heat capacity and the latent heat of PCM could increase the temperature time-lag and, hence, improve the thermal

115

performance of systems with PCM [9, 26]. On the other hand, it can be appreciated that detailed numerical models for system-scale thermal simulations of PCM-enhanced envelope, such as building roofs, can give more detail about the physics of heat transfer process [11, 15]; and usually, this numerical modeling is the first step in the thermal design of the PCM-enhanced building system. This process frequently includes decisions about

120

the following: (i) sequence of materials, (ii) location of PCM, and (iii) amount of PCM [27]. Therefore, in this research, we address the numerical modeling of a conventional concrete roof system with a PCM layer adhered to its interior surface to its thermal evaluation under warm weather conditions of Merida , Mexico; the location of PCM was

125

selected taking into account that it could be easily installed in operating buildings. To select the most suitable material for this weather, three PCMs are evaluated (Paraffin wax - MG29, N-Eicosane, and Salt Hydrates); all of them with different thermophysical properties in order to observe its effect on the heat transfer process. Furthermore, the effect of thickness in the PCMs on the energy-saving potential in terms of heat flow

130

reduction is presented. The case of the conventional concrete roof (without PCM) is also presented as a reference case to compare the results.

7

2. Physical and mathematical model 2.1. Physical model 135

Fig 1 shows the physical model that represents the roof with a PCM layer adhered to the inner surface of the roof (R-PCM). It is a roof with 0.12 m in thickness (Hc ) and an additional PCM layer of thickness HP CM , which ranges from 0.5 to 2 cm. Both the R-C and the R-PCM have a length L of 1 m, and both are exposed to the outdoor environment in the outer surface and interacting with the indoor conditions in the inner surface.

140

The roof is considered as a conductive opaque wall, that receives solar radiation and is exposed to an ambient temperature (Tout ), therefore the convective loss was considered. Moreover, it is considered a radiative exchange between the roof and the sky-dome at sky temperature (Tsky ). The roof reflects and absorbs solar radiation. Therefore, due to the 145

increase/decrease energy in the roof, this increases/decreases its temperature causing a heat flux transferred toward/from the inner surface to the indoor (qin ). In the case of the R-PCM, the heat flux at the inner surface of the Roof is transferred toward/from the PCM layer; and due to high latent heat of the PCM materials, it acts as an element of heat storage that delays or avoids the heat flux to the indoor environment. The total heat

150

flux at the inner surface of the R-C and R-PCM layer is formed by the convective and radiative heat fluxes, while the temperature at the indoor condition (Tin ) was considered constant at 24◦ C.

The thermophysical properties of the materials were considered constant. Table 1 155

shows the thermophysical properties of the concrete roof and of the PCM materials considered in this study: Paraffin wax - MG29, N-Eicosane , and Salt Hydrates (NA2 HPO4 , 12H2 O). PCM was considered homogeneous and isotropic, and the convection within the PCM layer in the liquid state was neglected. The thermal properties of the paraffin wax MG 29 were assumed identical for melting and freezing processes because of its nucleate

160

without noticeable subcooling [27]. Most of the PCM layers are encapsulated in different

8

𝑇𝑜𝑢𝑡 𝑞𝑟𝑎𝑑,𝑜𝑢𝑡

G

Roof 𝑞𝑐𝑜𝑛𝑣,𝑜𝑢𝑡

α𝐺

Hc HPCM

𝑞𝑐𝑜𝑛𝑣,𝑖𝑛

𝑞𝑟𝑎𝑑, 𝑖𝑛

𝑇𝑖𝑛

PCM

L Figure 1. Physical model.

plaster forms, and the contact thermal resistance due to the shell material is neglected.

2.2. Mathematical model 2.2.1. Roof The equation that governs the conductive heat flux and temperature distribution in transient state of the roof of concrete is given by: ∂ ∂(ρc Cpc Tc ) = ∂t ∂x 165

    ∂Tc ∂ ∂Tc λc + λc ∂x ∂y ∂y

(1)

where the subscript c indicates that each variable corresponds to the portion of concrete in the R-C and the R-PCM. 2.2.2. Phase change material layer The governing differential equation of the thermal diffusion in a phase change material is given by: ∂ ∂(ρP CM Cpef f T ) = ∂t ∂x

    ∂T ∂ ∂T λP CM + λP CM ∂x ∂y ∂y

(2)

The mathematical model of the PCM was solved by the effective heat capacity method [32–34], which corresponds to the group of fixed domain methods. 9

The effective capacity method takes into account the phase change phenomenon into

170

the heat capacity term, through by the addition of the enthalpy change (hls ) to the specific heat during the solid-liquid phase change. Therefore, the effective heat capacity includes the storage energy and the latent heat of the phase change. Then, the effective heat capacity for each phase change state is:

Cpef f

175

  Cps    Cps + Cpl hls = +  2 2∆T    Cp l

T < Tm − ∆T

f or

solid phase

f or (Tm − ∆T ) ≤ T ≤ (Tm + ∆T ) mushy phase f or

T > Tm + ∆T

(3)

liquid phase

where Cpl and Cps are the specific heats of the liquid and solid phases, Tm is the melting point temperature. Tl and Ts are defined as the lower and higher limit of the melting temperature range, respectively; and 2∆T = Tl − Ts is the freezing interval. For an isothermal phase change (∆T = 0), a small finite freezing interval is usually assumed [35]. The moisture transfer is neglected in the modeling because the encapsulating material

180

of the PCM has high moisture resistance such as silicates and aluminum (Pendyala S., 2012). 2.3. Boundary conditions and initial condition In both cases (R-C and R-PCM), the outer surface (y = Hc ) is exposed to outdoor climatic conditions, therefore the outer boundary condition consider the amount of solar

185

radiation absorbed, as well as the convective and radiative losses. This boundary condition is mathematically written as:

−λ

∂Tc 4 4 = αG + hout (Ts,out − Tout ) + εσ(Ts,out − Tout ) ∂x

(4)

where hout is the convective heat transfer, considered here as a function of the wind speed (vwind ) according to the relation hout = 2.8 + 3.0vwind [36]. Tc,out is the temperature of the outer surface of the roof; G is the solar irradiance received by the roof and α is the 190

absorptivity of the roof, in such way that the term αG is the energy absorbed by outer 10

surface of the roof. ε is the emissivity of the outer surface, σ is the Stefan–Boltzmann constant and Tsky is the sky temperature given by [37] as Tsky = 0.0552(Tout )1.5 . It was considered that the inner surface of the roof (y = 0) exchanges heat by convection and radiation with the indoor air which remains with a constant temperature Tin , and also it exchanges heat by radiation with the other indoor surface. Thus, the boundary condition of the inner surfaces is expressed as: −λ

∂T 4 4 = hin (Ts,in − Tin ) + εσ(Ts,in − Tin ) ∂x

(5)

where the subscript s, in indicates that the variable corresponds to the inner surface, which depending on the case, it can be the inner surface of the concrete for the R-C or the inner 195

surface of the PCM layer for the R-PCM. The indoor convective heat transfer coefficient hin is 9.24 W/m2 K when the heat flux is from the indoor toward the outdoor (Ts < Tin ), and 6.13 W/m2 K when the heat flux is from the outdoor toward the indoor (Ts > Tin ) [28].

The two vertical boundary conditions, at x = 0 and x = L, were assumed adiabatic 200

for the two cases of study. In addition, the initial temperature of the domain was the ambient temperature at 00:00 h for both the warmest day and coldest day.

3. Weather data The study was carried out with weather conditions corresponding to a typical warm-subhumid 205

weather in Mexico, which is represented by the city of Merida, Yucatan. Merida is located at the latitude 20◦ 58’ 04” N and in the longitude 89◦ 37’ 18” O. For the modeling were selected the climatic conditions of the warmest and the coldest day of 2018 based on its ambient temperature. These days correspond to April 07th and January 23th, respectively. The climatologic variables considered were solar irradiance, ambient

210

temperature, and wind speed, they were obtained from the measurements performed every 10 minutes by a meteorological station. These data were provided by the National Meteorological Service-National Commission of Water (SMN-CONAGUA-M´exico) for each 10 minutes. Figure 2 shows the ambient temperature, the solar irradiance received 11

4 5

C o ld e s t d a y W a rm e st d a y

T (°C )

3 6 2 7 1 8

C o ld e s t d a y W a rm e st d a y

7 5 0

G (W /m

2

)

1 0 0 0 5 0 0 2 5 0 0

C o ld e s t d a y W a rm e st d a y

v

w in d

(m /s )

6 4 2 0 0 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

2 4 :0 0

t(h ) Figure 2. Behavior of the ambient temperature, the solar irradiance, and the wind velocity.

by a horizontal surface, and the wind velocity for the coldest day; the variables show 215

maximum values of 821 W/m2 , 29 ◦ C, and 4.7 m/s, respectively.

Figure 2 also shows the ambient temperature, the solar irradiance received by a horizontal surface, and the wind speed for the warmest day; the variables show maximum values of 1001 W/m2 , 41.7 ◦ C, y 5.0 m/s, respectively. The climate of Merida is a warm 220

climate during the whole year and its ambient temperature remain above to the comfort temperature. Therefore, is important to avoid any heat gains in buildings located in Merida. It is important to mention that, based on the weather data available, correlations for the climatic variables was made to use time-step intervals shorter than 10 minutes in the modeling.

225

12

4. Numerical procedure and validation A numerical code based on the finite volume method [38] was developed using the Fortran computational language to solve the mathematical models. The temporal term of the equations was discretized applying the totally implicit scheme, while the diffusive term 230

was approximated with the central difference scheme. The system of algebraic equations obtained from the discretization was solved by using the line by line method (LBL) with the alternating direction implicit scheme (ADI). The effective specific heat methods solve the phase change problems basically as single-domain problems with temperature-dependent specific heat. In addition, for the coupling of the roof and the PCM layer, an energy

235

balance at the interface of these domains was made, using the corresponding and updated conductive heat fluxes in each main iteration. Global convergence was achieved when the residual for every variable reached a value of 10−10 in each time-step.

A temporal and spatial grid independence analysis was performed for the R-C and the 240

R-PCM cases in the warmest day for the period with the higher solar radiation, which was from 11:00 to 13:00 h. A time-step analysis was carried out considering intervals of 1, 3, 5, 10, 15, 20, 30 and 40 s to observe the accuracy of temporal numerical resolution. It was performed considering a spatial mesh of 101× 41 nodes for the R-C. It was found that a time-step of 30 s was enough to move along time because with this time-step the heat

245

flux toward indoor and temperature at the inner surface showed a maximum percentage deviation of ≈1%. To ensure that the accuracy of the results was independent of the mesh, numerous tests based on the effect of the mesh size were carried out. From this study, it can be concluded that starting from 61×31 nodes gives a mesh independent solution for the R-C, which showed a non-significant difference (≈1%). In the same way,

250

for the R-PCM, it was found that a time-step of 10 s was enough to move along time, and a mesh of 61×41 nodes for the PCM layer, resulting in a total mesh of 61×71 nodes; with maximum differences of ≈1% in both studies. Maximum deviation of grid independence and temporal independence tests are shown in Figure 3.

255

The validation of the transient conduction in a roof with convective and radiative losses

13

0 .0 6

M a x im u m

d e v ia tio n (% )

0 .0 5

0 .0 4

0 .0 3

0 .0 2

0 .0 1

0 .0 0 6 1 x 3 1

6 1 x 4 1

6 1 x 5 1

6 1 x 6 1

6 1 x 7 1

N o d e s

(a)

(b)

Figure 3. Maximum deviation of a) grid independence and b) temporal independence tests.

was presented by the authors in a previous work [39]. In addition, the numerical code was also verified comparing our results against the results of two-phase Stefan problem in a slab reported by Solomon [40] and Arici et al. [41]. In which the melting process of N-Eicosane Paraffin wax in a slab with constant temperature in one side was performed, 260

while the others sides of the slab were considered isolated and an initial condition of 21 ◦ C was used. Table 2 shows the comparison of the temperature obtained in the slab against the analytical and numerical solution reported in [40] and [41], respectively; that results were obtained after 3600 s of the melting process applying a time step of 1 s. The table indicates that the results obtained with our numerical model show a good agreement with

265

the results reported in the literature, with a maximum deviation ≤1.3% (0.52◦ C) with respect to Solomon [40] and 10.5% (3.84◦ C) with respect to [41].

14

5. Results To analyze the transient thermal performance of different PCMs, this section presents effect of the melting process, as well as the effect of the PCM-layer thickness on the 270

thermal behavior of the R-PCM. The average heat fluxes from the roof to the indoor environment for each hour of the day, the average thermal load and its cost analysis are also presented.

With the aim to facilitate the analysis and comparison of results, the following cases 275

of R-PCM are introduced: • R-PCM1: R-PCM with a PCM layer of Paraffin wax - MG29. • R-PCM2: R-PCM with a PCM layer of N-Eicosane. • R-PCM3: R-PCM with a PCM layer of Salt Hydrates (Na2 HPO4 , 12H2 O). 5.1. Effect of the melting process on the thermal behavior

280

Because of the melting process, the time within the temperature of PCM remains almost constant is proportional to the time within the PCM is in the mushy phase. To understand the effect of the melting process on the thermal behavior, we show the case of R-PCM1 with 2.0 cm and 0.5 in thickness. First, we show the liquid fraction and its relation with the thermal behavior during the melting process. Additionally, we discuss

285

the importance of the optimal thickness in the system. Figure 4 (a) shows the Ts,in and Ts,out of the R-PCM1 with HP CM =2 cm different HP CM and the R-C configurations for the coldest day. Figure 4 (b) shows the liquid fraction (fl ) field in the PCM-layer for HP CM =2 cm from 09:00 to 22:00 h. First, the R-C reach Ts,in up to 33.4 ◦ C; therefore, it is clear that the PCM-layer was can be able to decrease

290

the temperature of R-PCM1 significantly. It can be observed that from approximately 9:00 h until the end of the day, the R-PCM1 with HP CM of 2.0 cm had values of Ts,in close to the melting temperature of the PCM (Tm = 27-29 ◦ C); which indicates that in this time interval the PCM was storing latent heat and at the same time it was changing phase. The aforementioned fact is shown in Figure 4(b) for the HP CM =2.0 cm, where is 15

295

observed that the melting process begins in the upper region of the PCM-layer at 09:00 h because of on the upper boundary of the PCM-layer is transferring the energy absorbed by the roof and it is stored in the upper zone of the PCM-layer; therefore the PCM-layer presents values of fl = 0.1, which indicates that the PCM is in the mushy zone. After that, the phase change is going on and the PCM in the mushy region increases, and the

300

PCM-layer presents values of fl between 0 and 0.8 at 10:00 h. Later, at 13:00 h the PCM-layer presents values of fl up to 1 in the upper region and it shows values of fl = 0 below half of the HP CM , in such way that at this time the PCM-layer presents the liquid and solid phase, as well as the mushy region. Subsequently, it can be observed that the thickness of the liquid phase (fl =1) in the PCM-layer is increased, as the Ts,in increased,

305

and the thickness of solid phase decreased (fl =0). However, even at the 16:00 h when Ts,in reaches its maximum value (Ts,in ' Tm ), the liquid phase does not present in the entire PCM-layer. At 19:00 h, without solar radiation when the ambient temperature (Tout ) decreases and is lower than the Ts,out , part of the stored heat by the PCM-layer is transferring to the ambient, therefore solidification process is carried out from the

310

horizontal boundaries to the inside of the PCM-layer; in consequence, the value of fl near to the horizontal boundaries is up to 0.2 and in the middle of the layer is 0.8. Finally, Figure 4(b) shows that at the 22:00 h the solidification process occurring with values of fl between 0.4 and 0; therefore the PCM is still in the mushy region even without solar radiation. On his hand, the R-C reach Ts,in up to 33.4

315

circ

C; therefore, it is clear that the

PCM-layer was can be able to decrease the temperature of R-PCM1 significantly. Additionally, Figure 5 (a) shows the Ts,in and Ts,out of the R-PCM1 with HP CM =2.0 cm for the warmest day, and the results are compared to the obtained for the R-C ; further, to observe the distribution of the liquid fraction in the PCM-layer, Figure 5(b) shows the fl for HP CM = 2 cm from 08:00 to 22:00 h. The relationship between the

320

constant behavior of the Ts,in and the fl of the PCM-layer of HP CM =2.0 cm is observed in Figure 5; for instance at 08:00 h when the Ts,in began to remain constant, the fl in the upper zone of the PCM-layer is up to 0.2, which indicates that the PCM is in the mushy region. As the PCM-layer is storing latent heat, the liquid fraction increases, in such way that the upper zone of the PCM-layer reaches the liquid phase (fl =1) at 10:00 h; it also

16

Frame 001  18 Jun 2019  Thermal

Frame 001  20 Oct 2019  Thermal

Frame 001  10 Jun 2019  Thermal

FLIQ

Frame 001  10 Jun 2019  Thermal

solid

Phase: mushy liquid

FLIQ: 0.0 0.1 0.2 0.4 0.6 0.8 1.0 09:00 h 0.1

0.1

0.0

Frame 001  10 Jun 2019  Thermal

0.6

0.10.0

0.2

0.2

1.0 0.8 0.6 0.4 0.2 0.0

0.4 0.0

Frame 001  10 Jun 2019  Thermal

0.0

H

P C M

:

0.0

2 .0 c m C -R o o f

3 6

T

FLIQ

13:00 h

o u t

1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.6 Frame 001  10 Jun 2019  Thermal

0.0

0.8

0.4

0.2

0.0

3 2

T s ,in t ( ° C )

T

FLIQ

16:00 h m

2 8

1.0 0.8 0.6 0.4 0.2 0.0

1.0 1.0

0.8

0.6 0.2

0.4 0.0

2 4

FLIQ

19:00 h 0.6

0.4

1.0 0.8 0.6 0.4 0.3 0.2 0.1 0.0

0.6

0.8

2 0

0.6 0.4 0.2

22:00 h

1 6 0 0 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

2 4 :0 0

0.1

0.0 0.3

0.4 0.3

t (h )

0.1

(a) Thermal behavior

0.2 0.4 0.2

0.0

(b) fl in the PCM-layer for HP CM =2cm

Figure 4. (a)Thermal behavior (Ts,in ) in the R-C and the R-PCM1, and the (b) Liquid fraction (fl ) field in the PCM-layer for HP CM = 2 cm along the coldest day.

325

shows values of fl = 0 near to the middle of the layer, in such way that in this hourly the PCM-layer presents the liquid and solid phase, as well as the mushy region. Later, at 13:00 h the liquid phase increases considerably in the upper zone and it reaches the entire thickness of the PCM-layer until ≈ 16:00 h that correspond to a value of Ts,in higher than Tm . Approximately at 22:00 h, the PCM-layer is in the mushy region again with a fl

330

between 0.4 and 0.9; on his hand the Ts,in is very close to the Tm (27◦ C) and it remains almost constant until the end of the day, this behavior indicates that the PCM-layer is in the mushy zone until the end of the day. On the other hand, the time interval in which the PCM-layer is completely in the liquid phase is inversely proportional to the value of HP CM . For instance, the aforementioned

335

fact occurred from 08:00 to ≈22:00 h (14 h) when HP CM = 0.5 cm, while for HP CM = 2.0 cm it occurred during ≈11 h, from ≈13:00 to 24:00 h. These results occurred because the increase in HP CM resulted in an increase in the volume of the PCM, increasing its 17

1.0 0.8 0.6 0.4 0.2 0.0

FLIQ

10:00 h 0.2

1.0 0.8 0.6 0.4 0.2 0.1 0.1 0.0 FLIQ

Frame 001  18 Jun 2019  Thermal

Frame 001  20 Oct 2019  Thermal

Frame 001  09 Jun 2019  Thermal

FLIQ

Frame 001  09 Jun 2019  Thermal

Phase: mushy liquid

solid

FLIQ:h 0.0 0.1 0.2 0.4 0.6 0.8 1.0 08:00 0.2

0.2

0.1

0.0

0.0

Frame 001  09 Jun 2019  Thermal

0.1

0.6

0.2

0.20.8

H

P C M

:

Frame 001  09 Jun 2019  Thermal

2 .0 c m

0.0

C -R o o f

T

0.0

FLIQ

13:00 h

o u t

1.0 0.9 0.8 0.6 0.4 0.2 0.0

4 0 Frame 001  09 Jun 2019  Thermal

T s , in ( ° C )

1.0

0.8

0.6

0.4

0.2

3 6

FLIQ

16:00 h 1.0

3 2

T

1.0 1.0

m

2 8

1.0

0.9

0.9

0.8

0.8

22:00 h

2 0

0.6

0.6 0.8

0 0 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

0.8

2 4 :0 0

0.9 0.8

t(h ) (a) Thermal behavior

0.4

0.8 0.6

0.4

0.6

(b) fl in the PCM-layer for HP CM =2cm

Figure 5. (a)Thermal behavior (Ts,in ) in the R-C and the R-PCM1, and the (b) Liquid fraction (fl ) field in the PCM-layer for HP CM = 2 cm along the warmest day.

heat storage capacity and, therefore, it was necessary a longer time of solar irradiance to store the energy necessary to change phase. Moreover, compared to R-PCM1 to the 340

R-C, the PCM-layer delays and decreases the peak of the Ts,in ; thus, the adding of PCM increases the thermal inertia/mass of the R-PCM1, therefore, it increases its capacity to store energy. On the other hand, it is important to take into account that the thermal resistance of the R-PCM1 depends on the thermophysical properties and thickness of the PCM-layer,

345

the low thermal conductivity of the PCM is not able to improve the thermal resistance of the system. To explain this effect, we show the case of R-PCM1 with 0.5 cm in thickness compared to the R-C case during the warmest day; this HP CM present Ts,in even higher than the R-C. First, Figure 6 shows the solar radiation (G) along the day, the behavior of the average heat flux at the outer surface of the roof, qout , and the average heat flux at

350

the inner surface of the roof, qin , for both cases. In such a way, the negative values of qin 18

1.0 0.9 0.8 0.6 0.4 0.2 0.0 FLIQ

19:00 h

2 4

1.0 0.8 0.6 0.4 0.2 0.0

1.0 0.9 0.8 0.6 0.4 0.2 0.0

1.0

0.0

4 4

FLIQ

FLIQ

10:00 h 0.2

1.0 0.8 0.6 0.4 0.2 0.1 0.0

1.0 0.9 0.8 0.6 0.4 0.2 0.0

indicate that the heat flux is going from the indoor ambient to the roof. Therefore, Figure 7 a) indicates that during the first hours of the day, from 00:00 to 07:15 h, the R-PCM1 gained energy by the inner and outer surface. It is worth to mention that during most time in this period, the solar radiation is absent; hence, the roof only exchanges energy 355

with the indoor and outdoor ambient. According to the aforementioned fact, in Figure 7 we observed that the value of the average temperature at the inner surface (Ts,in ) of the R-PCM1 is lower than the indoor temperature (Tin =24◦ C), and in the same way, the average temperature at the outer surface of the R-PCM1 (Ts,out ) is lower than the outdoor ambient temperature (Tout ). In consequence, the roof gains and stores a certain amount

360

of energy in the first hours of the day; in such a way that the temperature increases up to a similar value to the Tm (Ts,in ≈ 27 ◦ C) at 07:15 h when the roof received the first amount of solar radiation. Therefore, at 07:15 h, the PCM layer started the melting process, and, hence, it was in the mushy phase until approximately 09:45 h. Comparing the Figure 6 a) and b), and taking into account that the energy stored can be appreciated

365

like the area between the curves of qin and qout , it can be noted that the R-PCM1 stored energy in a higher quantity than R-C. However, during the melting process, Ts,in and qin remain almost constant (see Figure 6 a) because the PCM layer is storing energy as latent heat; subsequently, when the PCM layer is in its liquid phase, the value of Ts,in increases considerably because at this stage the PCM stores energy as sensible heat. On his hand,

370

the Ts,in of the R-C increases proportionally to its stored energy, which is smaller than that of R-PCM1; such that the maximum value of Ts,in of R-PCM1 is presented first than that of R-C (see Figure 7). 5.2. Effect of the PCM-layer thickness The thickness of the PCM-layer into the R-PCM was modeled with thicknesses of 0.5,

375

1.0, 1.5, and 2.0 cm to determine its influence on the thermal performance of the system. It is worthy to mention that the inner surface of the PCM layer is exchanging heat with the indoor air, therefore, influence directly to the thermal comfort of the indoor space. Therefore, Ts,in and Ts,out for each time step modelled of the days are present.

380

Figure 8 shows the Ts,in and Ts,out of the a) R-PCM1, b) R-PCM2 and c) R-PCM3 19

0 9 :4 5 h

7 5 0

q

o u t

q

in t

7 5 0

G

(W /m 2)

0 7 :1 5 h

o u t

q

in t

G 0 7 :1 5 h

5 0 0

a v e

5 0 0

q

q

q a v e (W /m 2 )

0 9 :4 5 h

1 0 0 0

1 0 0 0

2 5 0

2 5 0

0

0 0 0 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

0 0 :0 0

2 4 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

2 4 :0 0

t (h )

t(h )

(a) R-PCM1

(b) R-C

Figure 6. Solar radiation (G), average heat flux at the outer surface of the roof (qout ) and the average heat flux at the inner surface of the roof (qin ) for the a)R-PCM1 and b)R-C. 7 0

R - P C M 1 ( 0 .5 c m ) R -C T

T s ,o u t ( ° C )

6 0

o u t

5 0 4 0 3 0 2 0 R - P C M 1 ( 0 .5 c m ) R -C

4 4 T

T s , in (° C )

4 0

o u t

m u sh y p h a se

3 6

T

3 2

m

2 8 2 4 2 0 0 0 :0 0

0 4 :0 0

0 8 :0 0

1 2 :0 0

1 6 :0 0

2 0 :0 0

2 4 :0 0

t(h )

Figure 7. Outdoor ambient temperature (Tout ), average temperature at the inner surface (Ts,in ) and average temperature at the outer surface of the R-PCM1 (Ts,out ) for the R-PCM1 and R-C case.

with different HP CM , as well as the R-C for the coldest day. This Figure indicates that regardless of the value HP CM , the behavior of Ts,out was qualitatively similar to the solar irradiance because this temperature presented its maximum values around 13:00 h and remained almost constant during the night. Besides, the time interval in which the 385

PCM-layer is in the liquid phase completely was inversely proportional to the value of HP CM . For R-PCM1 the maximum value of Ts,out was ≈47 ◦ C, while the maximum Ts,out for the R-C was ≈ 43 ◦ C. On the other hand, the maximum Ts,in was 34.9, 32.6, 27.9 and 20

27 ◦ C for HP CM of 0.5, 1.0, 1.5 and 2.0 cm, respectively. While the maximum Ts,in of the R-C was 33.4 ◦ C. 390

Figure 8 (b) shows that the behavior of the Ts,out of the R-PCM2 was similar for all HP CM considered, and it had a maximum value of ≈47 ◦ C at approximately 13:10 h. Thus, it is clear that in the case of R-PCM2, the HP CM had no effect on the Ts,out . Furthermore, the peak of the curve corresponding to the Ts,out of the R-C occurred later, 395

at 1:45 pm, with a value up to 10% lower than the R-PCM2. On the other hand, the Ts,in of the R-PCM2 had values of up to 35.3, 34, 33 and 32 ◦ C for the HP CM of 0.5, 1.0, 1.5 and 2.0 cm, respectively. The Ts,in for an HP CM = 2.0 cm was up to 4.4% lower than corresponding to the R-C (33.5 ◦ C); while the Ts,in for the 0.5 cm thickness was up to 5.4% greater than the R-C case. Therefore, the Ts,in was smaller than the melting

400

temperature of the N-Eicosane (Tm = 37 ◦ C) in all cases, which means that the PCM layer did not completely melt in the coldest day. However, the figure shows a characteristic behavior of a PCM during the phase change because the Ts,in of the R-PCM2 remained almost constant when it reached maximum values in the interval of 12:00 h at 15:00 h; this behavior occurs because the upper surface of the PCM is kept at constant Tm when

405

it stored latent heat for the phase change.

In Figure 8 c), the Ts,in and the Ts,out of the R-PCM3 had a similar behavior to the R-PCM2, with Ts,out of up to 47 ◦ C at 13:30 h, and it remained constant during the night. As the value of HP CM increased, the maximum Ts,in diminished. In the coldest day, Ts,in 410

had maximum values between 33.8 and 35.5 ◦ C, such values were slightly greater that the corresponding to the reference case R-C. Thus, the maximum Ts,in of the R-PCM3 with HP CM =2.0 cm was just ≈ 2% greater than the maximum Ts,in of the R-C. Another important factor in the applications with PCM is the time lag. For the R-PCM1 case with a thickness of 2 cm the time lag was approximately 4 h 44 min, while the time lag

415

of the R-C was approximately 1 h 37 min; on his hand,the R-PCM2 and R-PCM3 with HP CM =2.0 cm had a time lag of 1 h 13 min and 13 min, respectively. Therefore, the R-PCM1 considerably improved the time lag compared to the R-C.

21

T S , o u t ( ° C )

T S , in ( ° C )

2 0

2 4

2 8

3 2

3 6

1 0

2 0

3 0

4 0

5 0

0 0 :0 0

H

H

P C M

R -C

T

m

T

out

0 4 :0 0

0 .5 c m

0 .5 c m

1 .0 c m

1 .0 c m

t(h )

1 2 :0 0

2 .0 c m

2 .0 c m

a) R-PCM1

0 8 :0 0

1 .5 c m

1 .5 c m

1 6 :0 0

2 0 :0 0

2 4 :0 0

16

20

24

28

32

36

40

44

10

20

30

40

50

0 0 :0 0

H P C M

H

:

:

R -C

P C M

R -C

0 4 :0 0

0 .5 c m

0 .5 c m T ou t

1 .0 c m

1 .5 c m

1 .5 c m

t (h )

1 2 :0 0

2 .0 c m

2 .0 c m

b) R-PCM2

0 8 :0 0

1 .0 c m

1 6 :0 0

2 0 :0 0

T m

2 4 :0 0

20

25

30

35

40

10

20

30

40

50

H

0 0 :0 0

H

P C M

P C M

:

: R -C

R -C

T ou t

0 4 :0 0

0 .5 c m

0 .5 c m

T m

0 8 :0 0

1 .5 c m

1 .5 c m

t (h )

1 2 :0 0

2 .0 c m

2 .0 c m

c) R-PCM3

1 .0 c m

1 .0 c m

Figure 8. Behavior of the Ts,out and the Ts,in along the coldest day of the R-C and the a) R-PCM1, b) R-PCM2 and c) R-PCM3.

:

R -C

P C M

:

T s , o u t ( ° C ) ( ° C ) T s , in

T s , o u t ( ° C ) ( ° C ) T s ,in

22

1 6 :0 0

2 0 :0 0

2 4 :0 0

Figure 9 shows the Ts,in and Ts,out of the a) R-PCM1, b) R-PCM2 and c)R-PCM3 and the R-C along the warmest day; the HP CM of the R-PCM ranges between 0.5 and 2.0 cm. 420

The behavior of the Ts,out during the warmest day was qualitatively similar to that of the coldest day, but in Figure 9 a) reached maximum values of up to 65 ◦ C. The Ts,out of the R-C had a maximum value of ≈56 ◦ C. For the R-PCM1, the maximum Ts,in were 42.6, 40.6, 38.7 and 35.7 ◦ C for HP CM of 0.5, 1.0, 1.5 and 2.0 cm, respectively. Therefore, for HP CM > 1.5 cm, the maximum Ts,in reached values up to 4.7% above the R-C (40.7 ◦ C).

425

Figure 9 (a) also shows that in the warmest day, all HP CM had Ts,in values above the Tm (27-29 ◦ C) at a certain period of the day, which means that the PCM reached the liquid phase in all cases. Furthermore, due to the process of phase change occurs at a constant temperature of Tm , it can be observed that this process started approximately at 08:00 h in all cases of R-PCM1, since the behavior of the Ts,in remained almost constant because,

430

as was observed in the last section, a fraction of the PCM was storing energy as latent heat.

In Figure 9 (b) it can be observed that the behavior of the Ts,out of the R-PCM2 with all the different HP CM was qualitatively similar; however, the value of Ts,out was directly proportional to the value of the HP CM , as the HP CM became thicker the maximum Ts,out 435

increased in a range between 65 and 68 ◦ C. On his hand, the R-C had a maximum Ts,out of 56 ◦ C at ≈ 13:30 h; therefore, the maximum Ts,out reached by R-C was lower than that obtained by the R-PCM2, and occurred ≈ 20 min later than the maximum Ts,out corresponding to the R-PCM2. On the other hand, the Ts,in of the R-PCM2 system reached values of up to 36.5 ◦ C for an HP CM = 2.0 cm for approximately 30 min from

440

13:30 to 14:00 h, which is ≈ 10% lower than the maximum Ts,in of R-C (≈41 ◦ C), and just 0.5 ◦ C lower than the melting temperature. As was shown in the case of R-PCM1, the nearly constant behavior of Ts,in for a time interval indicates that part of the PCM layer was in the melting process, however, the value of Ts,in below the Tm reveals that the PCM layer did not completely change of phase.

445

Below, in Figure 9 (c) the behavior of the Ts,out and the Ts,in of the R-C and the R-PCM3 with different HP CM during for the warmest day is shown.

23

There are no

significant differences in the behavior of Ts,out for the different HP CM of the R-PCM3, with a maximum value of ≈64 ◦ C for HP CM = 2.0 cm at 13:06 h. Figure 9 (c) also shows 450

that the maximum Ts,in of R-PCM3 with thicknesses HP CM ≤ 1.5cm was up to 6% higher than the maximum obtained with the reference case R-C (≈41 ◦ C). While the maximum Ts,in for an HP CM = 2.0 cm was just 0.5 ◦ C lower than that reached by the R-C. The Ts,in of the R-PCM3 quickly reached the melting temperature (Tm =35 ◦ C) independently of the HP CM , and did not remain constant for prolonged time intervals. It is worth mention

455

that the Tm of Hydrated Salt is similar to that of N-eicosane (Tm = 37 ◦ C), however, as observed in both cases, R-PCM2 and R-PCM3, they exhibit behavior totally different from Ts,in under the conditions of the warmest day. These behaviors occurred because of the thermal conductivity of the Hydrated Salt is greater than that of the N-Eicosane and, therefore, it rapidly transfers the heat through the PCM layer, causing the phase change

460

process to be present in an almost imperceptible time interval compared to the R-PCM2 case under the same conditions. The aforementioned behavior is shown for instance by the R-PCM3 with HP CM =1.5 cm; in which the PCM layer initiated the phase change process at 10:13 h and it stores latent heat for 20 min until it completely melts; later, it increased its temperature due to the sensible heat that it absorbed as it receives energy.

465

Later, from 16:13 h, solidification of the PCM begins and, once solidified, it remains so until the end of the day. The aforementioned results indicate that during the warmest day, the R-PCM1 with 2.0 cm HP CM reduces the Ts,in considerably improved the thermal behavior compared to the rest of the cases; which is confirmed by the time lag, because the R-PCM1 has values of 1h 46min, contrary to the R-PCM2 and R-PCM3 which have

470

only 25 min and 40min.

In this section it was shown that the cases R-PCM1 and R-PCM2 reduced the Ts,in with respect to the R-C, and the R-PCM1 shows a higher time lag during both days; however, to quantify this decrease more accurately, the following section shows the thermal load of 475

the the R-PCM1, R-PCM2 and R-C throughout each day.

24

T s , o u t ( ° C )

( ° C )

T s , in

0 0 :0 0

m

0 4 :0 0

T

out

1 .0 c m

1 .0 c m

0 8 :0 0

1 .5 c m

1 .5 c m

1 2 :0 0

2 .0 c m

2 .0 c m

1 6 :0 0

2 0 :0 0

2 4 :0 0

32

36

40

44

20

30

40

50

60

70

0 0 :0 0

H P C M

H

:

T

P C M

m

R -C

: R -C

T

0 4 :0 0

0 .5 c m

0 .5 c m out

1 .0 c m

0 8 :0 0

1 .5 c m

1 .5 c m

t(h )

1 2 :0 0

2 .0 c m

2 .0 c m

b) R-PCM2

1 .0 c m

1 6 :0 0

2 0 :0 0

2 4 :0 0

2 0

2 4

2 8

3 2

3 6

4 0

4 4

4 8

2 0

3 0

4 0

5 0

6 0

7 0

0 0 :0 0

H

P C M

H P C M

:

R -C

R -C

: T

T

out

0 4 :0 0

0 .5 c m

0 .5 c m

m

0 8 :0 0

1 .5 c m

1 .5 c m

t (h )

1 2 :0 0

2 .0 c m

2 .0 c m

c) R-PCM3

1 .0 c m

1 .0 c m

Figure 9. Behavior of the Ts,out and the Ts,in along the warmest day of the R-C and the a) R-PCM1, b) R-PCM2 and c) R-PCM3.

a) R-PCM1

t (h )

20

T

0 .5 c m

0 .5 c m

2 0

R -C

R -C

24

:

:

2 4

P C M

P C M

28

H

H

2 8

3 2

3 6

4 0

4 4

2 0

3 0

4 0

5 0

6 0

7 0

T s , o u t ( ° C ) ( ° C ) T s ,in

T s , o u t ( ° C ) ( ° C ) T s , in

25

1 6 :0 0

2 0 :0 0

2 4 :0 0

5.3. Thermal load To determine the configuration of R-PCM that is able to diminish significantly the energy gained respect to it get by R-C, we calculated the average thermal load during the hours with solar irradiance by numerical integration. The average heat fluxes from the 480

R-PCM to the indoor environment for each hour of the day are also presented.

Table 3 and Figure 10 shows the average heat flux (qave,in ) for the R-PCM1 and the R-PCM2 with HP CM of 1.5 and 2.0 cm, and they are compared to the qave,in obtained for R-C during the coldest and the warmest day. The negative values in the table indicate 485

that qave,in goes from the indoor to the outdoor environment, and the positive values indicate the energy gained by the indoor environment. It is important to mention that R-PCM3 was not included in the Table 3 nor in Figure 10, because in the previous section it was observed that it does not improve the thermal performance of the roof.

490

In general, during the first hours of the day (07:00-10:00 h) negative values of qave,in are shown, which means that the flux of heat goes to the outdoor environment; however, from ≈10:00 h the indoor environment started to have heat gains. It can be observed in Figure 10 that the R-PCM1 shows the lowest values in both days; but it is clear that in the coldest day the qave,in had a behavior almost constant during most hours of the

495

day. The R-PCM1 shows the values between a range of -45.94 ≤ qave,in ≤ 44.81 W/m2 . Therefore, the lowest thermal loads correspond to the R-PCM1 with values of 276.7 and 204.5 W·h·m−2 for HP CM of 1.5 and 2.0 cm, respectively; such thermal load values are up to 57% lower than the thermal load corresponding to the R-C (485.4 W·h·m−2 ). In contrast, the R-PCM2 with HP CM = 2.0 cm shows a thermal load of 589.1 W·h·m−2 ,

500

which is 21% higher than the R-C case.

Figure 10 also shows that during the warmest day, the R-PCM1 had heat gains in most hours of the day with average values between -11.52 and 172.86 W/m2 ; the above results in thermal loads of 861.1 and 610.7 W·h·m−2 for HP CM of 1.5 and 2.0 cm, respectively. 505

In such a way that, the R-PCM1 with HP CM = 2.0 cm reduced the thermal load of the

26

day up to 53% with respect to that obtained by the R-C case (1,316.07 W·h·m−2 ). On the other hand, the R-PCM2 with HP CM of 1.5 and 2.0 cm shows thermal loads 12 and 5% lower than the R-C case, respectively. Therefore, based on the previous discussion, we conclude that the R-PCM1 with HP CM = 2.0 cm considerably reduces the thermal 510

load with respect to the reference case (R-C) in both of the considered days. 1 2 0

2 0 0 1 0 0 8 0

1 5 0

q a v e ,in ( W /m 2 )

q a v e ,in ( W /m 2 )

6 0 4 0

1 0 0

2 0 0

5 0

-2 0 -4 0

0 R -C R - P C M 1 ( 1 .5 c m ) R - P C M 1 ( 2 .0 c m )

-6 0

R -C R - P C M 1 ( 1 .5 c m ) R - P C M 1 ( 2 .0 c m )

R - P C M 2 ( 1 .5 c m ) R - P C M 2 ( 2 .0 c m )

-8 0

R - P C M 2 ( 1 .5 c m ) R - P C M 2 ( 2 .0 c m )

-5 0 0 7 :0 0

0 9 :0 0

1 1 :0 0

1 3 :0 0

1 5 :0 0

1 7 :0 0

1 9 :0 0

0 6 :0 0

t (h )

0 8 :0 0

1 0 :0 0

1 2 :0 0

1 4 :0 0

1 6 :0 0

1 8 :0 0

t (h )

(a) Coldest day

(b) Warmest day

Figure 10. Average heat flux (qave,in ) for the R-PCM1, the R-PCM2, and the R-C for the a) coldest and b) warmest day.

To conduct the economic evaluation, this research utilizes the Static Payback Period (SP P ) method. Then, an economic analysis was performed to calculate the simple payback period for the R-PCM1 for HR−P CM 1 of 1.5 and 2.0 cm. The static payback period (SP P ) estimates the number of years required for a project’s savings to equal the 515

investment. This would allow the investors to determine total gains at the end of each time period. This method is simple and therefore widely used in many technical economic analyses [42–45]. For PCM application, the SP P is given by the following equation:

SP P =

CP CM S

(6)

where, SP P is a Static Payback Period in years, CP CM is the extra initial investment for PCM application (USD) and S is the annual energy cost savings (USD/year). 520

For computing the SP P , all of the data used has been obtained from the US Department 27

for Energy’s review into ‘Cost Analysis of Simple Phase Change Material-Enhanced Building Envelopes’ [46] and and considering that each square meter of PCM by increasing its thickness from 1.5 to 2 cm its price increases ≈ 33% and the PCM cost per kg is ≈ 2.0 USD/kg (1700 USD/m3 ). The installation cost is the same per square meters when 525

installing panels of 1.5 mm and panels of 20 mm. Moreover, the simple installation of the PCM panels inside the constructive system makes the maintenance cost negligible during the lifetime period [43].

Energy savings in USD can be calculated by using the current price authorized by 530

the Mexican Commission of Electricity (CFE) for Merida city. In Mexico, residential electricity costs are divided into seven different rates (1, 1A, 1B, 1C, 1E, 1D and 1F) [47], each one of them with different prices according to the quantity of energy consumption (basic, intermediate and surplus). The rate that the CFE assigns to every city is based on the daily minimum average temperature during summer. For this analysis, the rate

535

used in Merida city according to CFE was 1D. The annual electricity saving obtained by the use of R-PCM1 was computed under the following consideration: (1) warm (summer) and cold (winter) weather conditions, (2) for summer conditions May to October, (3) for winter conditions November to April were used for the calculus and (4) a 3m×3m roof (A = 9 m2 ) were considered for the analysis.

540

The electric power consumption for summer and winter conditions was determined by multiplying the energy fare by the heat load, taking into account the following consideration: in summer a heat pump is used and in winter an electrical resistive heater instead, which is considered 100% energy efficient, consequently all the incoming electric energy is converted into heat. To determine the electrical power required by air conditioning, the heat load

545

was divided by 3, which is the coefficient of performance, a value commonly found in air conditioning systems, this factor might change depending on the characteristics of the equipment to be used [48]. For the hypothetical case in the present article, the way to calculate the SP P does not consider the inflation rate and the respective interest rate, the constant cost was considered during the period. Based on Eq. 6 and the energy cost

550

savings listed in Table 4, the use of PCMs in Merida city for HR-PCM1 of 1.5 and 2.0

28

cm will have a payback period of 13.54 and 12.18 years, respectively. As a consequence, in the Mexican construction industry with a 30 year common service life of buildings, the use of these materials is cost-effective. 6. Conclusions 555

We carried out the numerical modeling of a conventional concrete roof system with a PCM layer adhered to its interior surface (R-PCM). The thermal evaluation of the R-PCM was performed under the warm weather conditions of Merida , Mexico; the use of PCM is emerging and one cannot find evidence of its effectiveness to improve the thermal behavior of building roofs located in Mexico. To select the most suitable material for this

560

weather, three PCMs were analized, using a conventional concrete roof (without PCM) as a reference case to compare the results. The cases were: case R-C (concrete roof), R-PCM1 (R-PCM with Paraffin wax - MG29), R-PCM2 (R-PCM with N-Eicosane) and R-PCM3 (R-PCM with Salt Hydrates). Based on the numerical results for the transient thermal performance of a roof with a

565

PCM layer under warm weather conditions of Mexico, the following is concluded:

The R-PCM1 with HP CM = 2.0 cm showed the highest time lag, with values of 4 h 44 min and 1 h 50 min for the coldest and warmest day, respectively. On the contrary, the time lag of the R-C was 1 h 37 min and 1 h 46 min for coldest and warmest day, 570

respectively. On his hand, the R-PCM2 and R-PCM3 with HP CM =2.0 cm had a time lag of 1 h 13 min and 13 min, respectively. Therefore, it is concluded that the R-PCM1 significantly improved the time lag compared with the R-C.

Regarding the thermal load per day, the case R-PCM1 showed the lowest values during 575

the coldest and the warmest day. For the coldest day, the R-PCM1 with HP CM of 1.5 and 2.0 showed values of total thermal load of 276.7 and 204.5 W·h·m−2 for cm, respectively; and the R-PCM2 with HP CM of 1.5 and 2.0 cm had a total thermal load of 641.8 and 589.1 W·h·m−2 . For the warmest day, the R-PCM1 with HP CM of 1.5 and 2.0 cm reached up to 61.1 and 610.7 W·h·m−2 , respectively; on his hand, the R-PCM2 with HP CM of

580

1.5 and 2.0 cm had a total thermal load of 1239.8 and 1156.6 W·h·m−2 , respectively. In 29

such way that the total thermal loads corresponding to the R-PCM1 with HP CM = 2.0 are up to 57% lower than the thermal load corresponding to the R-C, which were 485.4 and 1,316.0 W·h·m−2 for the coldest and warmest day, respectively. But, the increasing of R-PCM1 thickness from 1.5 to 2 cm raises by 33% its price. However, the use of 585

R-PCM1 in Merida city for HP CM of 1.5 and 2.0 cm will have a payback period of 13.54 and 12.18 years, respectively. Taking into account that buildings in Mexico have a 30-year common service life, the use of these materials is cost-effective. Therefore, it is concluded that R-PCM1 is suitable to improve the thermal behavior of the roof in buildings under weather conditions of Merida, Yucatan.

590

It was observed that the thermal resistance of the R-PCM depends on the thermophysical properties and thickness of the PCM-layer; besides, the behavior of the R-PCM also depends on the weather conditions. Therefore is important to conduct parametric studies of PCMs with different properties as different layer thickness for the different climates on 595

the country. Future analysis will focus on the annual thermal load of a whole-building using the case R-PCM1 with and without reflective coating. Furthermore, the extend of the numerical system scale modeling will permit extending the current model to multidimensional and other passive alternatives, such as a cover to form a ventilated channel in conjunction with the ceiling, and it will also allow to explore the advantage of

600

nano particles-PCM in the system. Acknowledgements A. Rodriguez-Ake thank Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT) for the financial support given through its masters’ scholarship program. Besides, the authors are grateful to Servicio Meteorol´ogico Nacional-Comisi´on Nacional del Agua

605

(SMN-CONAGUA) for providing the weather data used in this research.

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35

36 37 35

Salt Hydrates[31]

27-29

–

Tm (◦ C)

Eicosane [30]

Paraffin wax - MG29 [29]

Concrete [28]

Material

0.514

0.39

0.21

1.7

Solid

0.476

0.157

0.21

–

Liquid

λ (W/m·K)

1,700

1,900

850

880

Solid

1,950

2,200

850

–

Liquid

ρ (kg/m3 )

1,520

810

2,230

2,240

Solid

1,442

770

2,230

–

Liquid

Cp (J/kg·K)

Table 1. Thermophysical properties of the concrete and of the PCMs.

281,000

241,000

205,000

–

hls (J/kg)

Table 2. Comparison results between the present study and the reference solution of Solomon [40] and Arici et al. [41].

x(m)

T(◦ C) Arici et al. [41]

Solomon [40]

Present study

0.000

95.00

95.00

95.00 ∗ (0.00)

0.002

86.60

86.78

86.77 ∗ (0.01)

0.004

78.21

78.60

78.60 ∗ (0.00)

0.006

69.84

70.52

70.57 ∗ (0.06)

0.008

61.51

62.58

62.72 ∗ (0.23)

0.010

53.22

54.83

55.11 ∗ (0.51)

0.012

44.95

47.29

47.73 ∗ (0.94)

0.014

36.70

40.02

40.54 ∗ (1.30)

0.016

35.32

35.72

36.04 ∗ (0.88)

0.018

33.96

34.18

34.50 ∗ (0.94)

0.020

32.65

32.73

33.04 ∗ (0.95)

0.022

31.39

31.38

31.68 ∗ (0.95)

0.024

30.20

30.13

30.42 ∗ (0.97)

0.026

29.09

28.99

29.27 ∗ (0.98)

0.028

28.07

27.95

28.23 ∗ (0.99)

0.030

27.13

27.01

27.28 ∗ (0.99)

0.032

26.29

26.17

26.42 ∗ (0.97)

0.034

25.54

25.42

25.66 ∗ (0.94)

0.036

24.87

24.75

24.98 ∗ (0.92)

0.038

24.28

24.17

24.38 ∗ (0.86)

0.040

23.77

23.66

23.85 ∗ (0.80)

0.042

23.32

23.21

23.39 ∗ (0.78)

0.044

22.93

22.83

22.99 ∗ (0.72)

0.046

22.60

22.51

22.65 ∗ (0.64)

0.048

22.32

22.23

22.37 ∗ (0.61)

0.050

22.08

22.00

22.12 ∗ (0.55)

0.052

21.88

21.81

21.92 ∗ (0.50)

0.054

21.71

21.65

21.75 ∗ (0.46)

0.056

21.57

21.52

21.61 ∗ (0.44)

0.058

21.45

21.41

21.50 ∗ (0.44)

0.060

21.36

21.32

21.42 ∗ (0.47)

0.062

21.28

21.25

21.36 ∗ (0.50)

0.064

21.22

21.20

21.31 ∗ (0.53)

0.066

21.17

21.15

21.29 ∗ (0.64)

0.068

21.13

21.12

21.29 ∗ (0.80)

∗

(%) Absolute differences regards analytical solution [40]

37

38

276.7

109.55

106.58

89.13

60.08

485.4

15:00

16:00

17:00

18:00 R 18:00 q(t)dt 6:00

(W·h·m−2 )

42.88

101.22

14:00

44.54

44.81

40.62

34.80

31.07

27.71

83.82

25.01

22.70

13:00

-2.48

10:00

19.58

59.52

-36.08

09:00

-13.89

12:00

-56.83

08:00

-45.94

29.49

-60.73

07:00

–

204.5

34.51

34.00

32.77

30.63

27.97

25.31

22.90

20.92

19.00

14.23

-19.32

-42.56

–

641.8

-16.31

11.54

58.04

95.43

104.96

104.94

104.93

96.90

69.85

36.51

-9.38

-51.88

–

589.1

-12.45

17.40

60.46

91.42

94.92

94.89

94.89

88.59

61.40

28.44

-14.92

-49.59

–

2.0

1.5

1.5

2.0

HR−P CM 2 (cm)

Coldest day HR−P CM 1 (cm)

11:00

–

R-C

06:00

t(h)

1,316.0

150.04

180.85

197.28

198.18

185.63

161.57

131.31

97.63

62.38

30.43

5.14

-8.67

-7.92

R-C

qave,in (W·m−2 )

861.1

53.81

127.38

127.38

158.48

172.86

149.72

39.97

31.14

26.72

23.90

21.26

5.38

-11.52

1.5

610.7

54.84

85.61

122.33

138.04

103.12

35.37

29.49

25.26

22.17

20.07

17.67

1.20

-10.35

2.0

HR−P CM 1 (cm)

Warmest day

Table 3. Average heat flux (qave,in ) for the R-PCM1, the R-PCM2, and the R-C.

1,239.8

41.71

86.36

128.55

148.59

153.28

153.28

146.91

132.05

104.93

92.21

54.53

10.25

-13.37

1.5

1,156.6

45.90

89.33

124.72

141.36

147.66

142.87

133.40

120.44

99.65

82.56

45.47

5.92

-12.50

2.0

HR−P CM 2 (cm)

39

0.052

Cold

0.0922

0.2870 0.0682

0.2036

2.0 cm

45.69

0.0754

0.1741

R-C

16.94 13.54

S

SPP

28.74

0.0430

0.1139

1.5 cm

229.5

Static Payback Period- SPP (Years)

0.1618

0.4387

1.5 cm

R-PCM1

12.18

25.07

305.24

20.62

0.0318

0.0808

2.0 cm

R-PCM1

(US$-kWh / day)

Cost Energy = FE x AEMC x A

CP CM

Annual

0.044

R-C

comfort (AEMC )

(FE)(US$-kWh) (kWh/m2 )

Added Energy to maintain

Fare Energy

Warm

Condition

Table 4. Economic analysis for PCM application with A = 9m2

Highlights 1. A paraffin wax MG29-layer in a concrete roof improves its thermal behavior in Mérida. 2. Melting cycles of the PCM-layer improves the thermal performances of the roof. 3. A Concrete roof with a paraffin wax MG29-layer shows a time lag up to 4h 44min 4. A paraffin wax MG29-layer in a concrete roof diminish the thermal load up to 57%

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: