*Article* **Phase Change Material Integration in Building Envelopes in Different Building Types and Climates: Modeling the Benefits of Active and Passive Strategies**

**Francesco Carlucci 1,\*, Alessandro Cannavale 2,3, Angela Alessia Triggiano 1, Amalia Squicciarini <sup>1</sup> and Francesco Fiorito <sup>1</sup>**


**Featured Application: Reduction of energy consumption in residential and office buildings through the improvement of latent heat storage in active and passive strategies.**

**Citation:** Carlucci, F.; Cannavale, A.; Triggiano, A.A.; Squicciarini, A.; Fiorito, F. Phase Change Material Integration in Building Envelopes in Different Building Types and Climates: Modeling the Benefits of Active and Passive Strategies. *Appl. Sci.* **2021**, *11*, 4680. https://doi.org/ 10.3390/app11104680

Academic Editors: Tiziana Poli, Andrea Giovanni Mainini, Mitja Košir, Juan Diego Blanco Cadena and Gabriele Lobaccaro

Received: 8 April 2021 Accepted: 17 May 2021 Published: 20 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Abstract:** Among the adaptive solutions, phase change material (PCM) technology is one of the most developed, thanks to its capability to mitigate the effects of air temperature fluctuations using thermal energy storage (TES). PCMs belong to the category of passive systems that operate on heat modulation, thanks to latent heat storage (LHS) that can lead to a reduction of heating ventilation air conditioning (HVAC) consumption in traditional buildings and to an improvement of indoor thermal comfort in buildings devoid of HVAC systems. The aim of this work is to numerically analyze and compare the benefits of the implementation of PCMs on the building envelope in both active and passive strategies. To generalize the results, two different EnergyPlus calibrated reference models the small office and the midrise apartment—were considered, and 25 different European cities in different climatic zones were selected. For these analyses, a PCM plasterboard with a 23 ◦C melting point was considered in four different thicknesses—12.5, 25, 37.5, and 50 mm. The results obtained highlighted a strong logarithmic correlation between PCM thickness and energy reduction in all the climatic zones, with higher benefits in office buildings and in warmer climates for both strategies.

**Keywords:** phase change material; thermal energy storage; energy efficiency; passive strategies; active strategies; adaptive envelopes

#### **1. Introduction**

Thermal comfort and reduction of energy consumption are consolidated topics in scientific literature. Currently, an increasing interest in these fields is related to the benefits of adaptive technologies. These technologies, when applied to building envelopes, allow buildings to adjust their characteristics, in a reversible way, in response to external stimuli. As a result, an adaptation of their behavior to climate fluctuations is achieved, and, consequently, users' comfort requirements can be more efficiently met [1,2]. Currently, the most promising results of adaptive envelopes [3] are related to wall-integrated PCMs [4,5], switchable glazing [6–9], adaptive solar shadings [10–12], dynamic insulation [13,14], and multifunctional facades [15,16].

Among this wide range of technologies, PCMs have constantly grown their importance in recent years, thanks to broad experimentation and diffusion in different scientific fields [17,18]. These include mainly the aerospace industry [19], the design of low-energy buildings [20], the preservation of products [21], the electronic industry [22], and waste heat

recovery systems [5]. The spread of PCMs in all these fields is related to the advantages of latent heat storage (LHS), which allows the storage and release of naturally available heat in low-volume elements, increasing, therefore, the energy storage density of the material [23].

In the building design domain, the importance of PCMs is related mainly to two different topics usually investigated in scientific literature: the reduction of HVAC energy consumption (i.e., active strategies) and the reduction of local and global thermal discomfort (i.e., passive strategies). Many studies have highlighted that PCMs can reduce the energy demand of HVAC systems by up to 30% if applied as a retrofit solution in residential buildings located in cold climates [24]. Moreover, thanks to the stabilization of the indoor radiant temperature, a significant reduction of thermal discomfort can be obtained [25]. Starting from these two important results, the aim of this paper is to merge and generalize these studies by considering: (i) a broader range of climates, (ii) both passive and active strategies, (iii) different building types, and (iv) different thicknesses of PCMs. Therefore, the benefits of PCM implementation are assessed through numerical analyses.

To design and model a PCM-integrated building element correctly and maximize its benefits, it is fundamental to understand the functioning of this technology and the different products available. The functioning is strictly related to the different ways in which materials can store or release heat: sensible heat storage (SHS), latent heat storage (LHS), and thermochemical heat storage. While SHS and LHS are applicable to buildings, thermochemical heat storage technologies are currently not applied in the civil field [23]. Regarding SHS, the heat absorbed/released is related to the increase/decrease of temperature in relation to the mass of the body (*m*), the specific heat (*c*), and the variation of temperature (*dT*), as described in Equation (1).

$$Q\_S = \int\_{T1}^{T2} m \text{ c } dT \tag{1}$$

During LHS, the heat absorbed or released leads to a change of phase—for example, a melting from solid to liquid—without changing the temperature of the body. In this case, the heat stored depends on the mass, the fraction melted (*fm*), and the variation of enthalpy of fusion per unit mass (Δ*hm*) (Equation (2)).

$$Q\_L = mf\_m \Delta h\_m \tag{2}$$

Therefore, generalizing a phenomenon with SHS and LHS for a gypsum-PCM board, the heat exchanged corresponds to the total enthalpy variation Δ*Htot* that depends on the enthalpy variation of each material contained in the board, according to Equations (3)–(5) [26]:

$$
\Delta H\_{tot} = m \cdot \left[ (1 - f) \cdot \Delta h\_{\text{\textdegree} \text{\textdegree Sum}} + f \cdot \Delta h\_{\text{PCM}} \right] \tag{3}
$$

where, considering the complete melting of the PCM (*fm* = 1), the partial enthalpies are:

$$
\Delta h\_{\text{gypsum}} = (1 - f) \int\_{T1}^{T2} mc\_{\text{S}} \, dT \tag{4}
$$

$$
\Delta h\_{PCM} = f \left( \int\_{T1}^{Tm} c\_S \, dT + \Delta h\_{\text{ff}} + \int\_{Tm}^{T2} c\_L \, dT \right) \tag{5}
$$

with *f* being the mass fraction of the PCMs in gypsum, *cS* and *cL* representing, respectively, the specific heat of the solid state and the liquid state, and *Tm* the melting temperature of the PCMs.

For this reason, in building applications, a PCM with a melting range within the thermal comfort range (20 ◦C–30 ◦C) can take advantage of LHS [27]. This is due to the storage of a good amount of heat in low-volume elements without increasing the surface temperature and therefore, without affecting thermal comfort. While, theoretically, PCMs work on four different possible changes of phase for building applications, namely, solid–solid, solid–liquid, gas–liquid, gas–solid [28], solid–liquid PCMs are usually considered.

PCMs can be classified into three main categories: organic, inorganic, and eutectic. Organic PCMs are composed of paraffins, fatty acids, fatty-acid esters, and sugar alcohols and, in general, can be classified as paraffin or non-paraffin. One of the main advantages of organic PCMs is that repeated melting–freezing does not lead to phase segregation; moreover, they are slightly affected by supercooling. Nevertheless, considering that paraffinic PCMs are derived from oil refining, organic PCMs have low ignition resistance, and, for this reason, envelope applications can be problematic. Organic PCMs can have a broad range of possible melting temperatures (−57 ◦C; +187 ◦C) with melting latent heat ranging from 85 to 300 J/g [29–31]; therefore, they should be accurately chosen to optimize their functioning. Inorganic PCMs are classified as salt-hydrates or metallic, and, despite having an enthalpy per mass similar to organic PCMs, they can reach higher melting latent heat per unit volume thanks to their higher density. Moreover, they have higher conductivity, can reach higher melting points, and are less expensive and less flammable than organic PCMs. However, inorganic PCMs present some limitations, such as supercooling, phase segregation, and corrosion. Lastly, eutectic PCMs are composed of at least two PCMs with the same melting point and can be classified as organic–organic, inorganic–inorganic, and organic–inorganic, depending on the types of PCMs used.

Often PCMs are encapsulated, primarily to hold both solid and liquid phases and to protect the PCMs from harmful interactions with the environment and other building materials. Encapsulation can also reduce phase segregation and corrosion, provide easier handling, and increase the heat transfer area [32,33]. Encapsulations can be classified, depending on capsule size, into macroencapsulation (d > 1 mm), microencapsulation (1 μm<d<1 mm), or nanoencapsulation (d < 1 μm). Moreover, they can be made of different materials, e.g., aluminum, plastic, polyolefin, rubber, polymers, in different containers, such as balls, tubes, plates, and boxes [33].

Many studies have been performed to deepen the possible applications of PCMs to buildings, mainly classified into active storage systems and passive storage systems.

Active systems are characterized by heat exchangers and forced convection and can be, in turn, classified into direct and indirect systems. In direct systems, the heat transfer fluid is also the storage element of the system, while in indirect systems, the fluid serves as the transfer medium and another material is used as the storage element [34]. Active systems have been studied [35] and applied to suspended ceilings [36,37], ventilation systems [38,39], external double-skin façades [40], solar collectors [41,42], heat storage water tanks [43,44], integrated photovoltaics [45,46], and building cores enhanced with PCMs activated through the use of ducts or pipes [47].

When considering passive systems, there is no forced convection, and, in this case, these systems can be classified according to the way the PCMs are embedded in the building element: inside the material, as a new layer, and in windows or as sun protection [48]. Considering the PCMs embedded inside the material, encapsulated PCMs can be easily added to other construction materials such as concrete [49], plaster [50], cellulose, or glass fiber [51]. Another diffuse solution to applying PCMs in buildings is to add a new layer to increase the thermal inertia of lightweight constructions. The most common application of PCMs as a new layer is the PCM-enhanced gypsum plasterboard; many products, such as the Alba Balance (Rigips-Saint Gobain), are already commercially available. Other applications of PCMs as new layers are PCM sandwich panels [52] and macroencapsulated PCMs in plates or bags, such as the Delta-Cool24 (Dorken) or the Energain (Dupont). Lastly, PCMs can be used for both sun protection (for example, in internal blinds [53]) and inside windows (for example, with an extra air gap, behind the inner glass [54]).

Considering all these available PCMs, particular attention should be paid to their melting point because, due to the broad variability of this parameter, each application has its own more suitable range. Different studies concerning building energy performance [28,55–57] have identified that the most suitable melting points for cooling are up to 21 ◦C, while for heating, they are 22 ◦C or, in general, 2 ◦C higher than the heating setpoint temperature. Moreover, suitable melting point ranges for thermal comfort are between 22 and 28 ◦C; for hot water

applications, they are between 29 and 60 ◦C. Finally, higher melting points—between 61 and 120 ◦C—are suitable for waste heat recovery applications. For the current study, a PCM layer was considered the inner layer of external walls, roofs, and floors, implementing a paraffinic microencapsulated organic PCM (Micronal) embedded in gypsum plasterboard with a melting point of 23 ◦C, the characteristics of which are described in the following sections. Considering that the models were run for active and passive strategies and for heating and cooling systems, the choice of this transition temperature allows us, as described before, to take advantage of heating, cooling, and thermal comfort applications.

#### **2. Materials and Methods**

Energy and comfort analyses were performed using EnergyPlus v.9.3. The main advantage of EnergyPlus [58] is that it can guarantee multidomain integration and physical interaction accounting for thermal, visual, mass-flow, and building service interactions [3,59]. A good starting point for energy-efficiency-oriented research in EnergyPlus [60] is offered by the U.S. Department of Energy (DOE), which developed 16 reference building models in different climatic zones. These models have been calibrated using several references such as the Commercial Buildings Energy Consumption Survey (CBECS), ASHRAE Standards 90.1, and many other academic sources [61].

Offices and apartments represent two of the most diffused building types in Europe, and they respectively account for 6% and 75% of total European building stock [62]. Therefore, two representative reference models, the small office and the midrise apartment, were chosen for this study. The geometrical data and main characteristics of the two reference buildings are summarized in the table and pictures below (Table 1, Figure 1).


**Table 1.** Main characteristics of reference buildings from the DOE.

**Figure 1.** Reference buildings chosen from the DOE: small office (**a**) and mid-rise apartment (**b**).

The HVAC equipment of the reference buildings is modeled in accordance with ASHRAE Standard 90.1 and consider, for both the small office and the midrise apartment, a boiler for heating generation, a packaged air conditioning unit for cooling generation, and a single-zone constant-volume system for air distribution.

Particular attention was paid to the reliability of consumption and, considering that the analyses were run for different building types and different locations, to the independence of the results from the HVAC system sizing process. Therefore, the original HVAC systems were substituted with properly calibrated ideal-load air systems. These systems are similar to a traditional HVAC unit in EnergyPlus; the main difference is that it is not connected to a central system, and each ideal-load air system provides heating or cooling to satisfy the zone setpoint [63]. Starting from the ideal heating and cooling loads, the energy consumption is calculated by dividing the loads by an ideal energy efficiency ratio (EER) equal to 3 in cooling mode and by a coefficient of performance (CoP) equal to 3 in heating mode, which can be considered average values for packaged direct expansion air conditioning systems.

Existing HVAC systems are characterized by a series of controls, such as the management of outdoor air and a night-cycle availability manager. Therefore, to guarantee the proper behavior of the ideal-load system, an energy management system (EMS) program was written in order to account for these controllers and obtain results close to the ones of the original DOE reference building.

To confirm the reliability of the calibrated ideal-load model, a statistical analysis was performed and values of mean bias error (MBE) and cumulative variation of root mean squared error (CVRMSE) were obtained from hourly values of heating and cooling energy consumption of the reference and modified models. The analysis was performed for each thermal zone, starting from three different reference models, one for each of the chosen locations (Miami—ASHRAE Zone 1A, Chicago—ASHRAE Zone 5A, Fairbanks—ASHRAE Zone 8), for both small office and midrise apartment.

Table 2 summarizes the results of these analyses, considering the average of whole thermal zones and cities and excluding only the zones where the HVAC system is off for at least 95% of total hours since the thermal behavior of these zones is not considered representative of the behavior of the entire building.


**Table 2.** Average values of MBE and CVRMSE for the considered models, excluding the results of zones with a system functioning period of less than 5% of total hours.

<sup>1</sup> Without considering Fairbanks' cooling consumption, the cooling average CVRMSE is 19.6%.

Values of MBE and CVRMSE were then compared with reference values from ASHRAE guideline 14 [64,65], which gives the hourly calibration criteria for real building modeling. Indeed, most of the thermal zones in all the configurations meet the ASHRAE hourly criteria, with |MBE| ≤ 10% and CVRMSE ≤ 30% [64,65]. All the results meet the criteria described before, except for the cooling CVRMSE of the small office, which was slightly over the threshold; in this case, the results were negatively influenced by the Fairbanks result, which can be considered nonrepresentative as it has a reduced functioning period (19% of total hours for small office, 17% for midrise apartment) and very low cooling energy consumption. Indeed, considering the Fairbanks yearly global HVAC consumption, the differences obtained by comparing reference and ideal-load models are very low: 1.1% and 1.5% for the office and apartment models, respectively. For these reasons, the model can be considered reliable.

The performance assessment of the integration of PCMs in buildings was carried out considering a whole year. In order to properly account for the transient behavior of PCMs, each simulation hour was divided into 20 analysis timesteps. Moreover, 25 different European cities were considered in order to assess the benefits of the integration of PCMs into buildings. To define a broad and nonredundant city sample, different European cities most of them characterized by common EU directives—were studied, selecting cities with very different climates, as described in Figure 2 and Table 3.

**Figure 2.** Individuation of the sample of 25 cities used for the analyses.


**Table 3.** Characteristics of the 25 European cities considered (data from epw weather file).

In order to adapt the models to the European climates, all thermal properties were adjusted to meet the different energy requirements [66–70]. The selected cities were then grouped in six different climatic zones—B, C, D, E, F, and G, from warmer to colder—in accordance with their values of heating degree days (HDDs). Hence, the thermal properties of roofs, walls, slabs, and windows were changed according to the climatic zones, as described in Table 4.


**Table 4.** Characteristics of the envelope properties for each climatic zone.

To understand energy demand variations, an ideal-load HVAC system was considered for active strategies, while, for passive strategies, no HVAC system was implemented and natural ventilation strategies were adopted. HVAC operational schedules were kept as standard, and setpoints of 21 ◦C in heating operational mode and 24 ◦C in cooling mode were considered.

With regard to passive strategies, a supplementary EMS program was developed in order to calculate the adaptive thermal comfort optimal range, in accordance with current European recommendations (EN 16798-1:2019 [71]). This EMS program calculates, starting from the temperature of the previous 7 days (*θed–i*), the running mean outdoor temperature (*θrm*, Equation (6)) and the optimal operative temperature (*θc*, Equation (7)). Then, considering the first comfort class, the program calculates the operative temperature (*TOP*) comfort range (Equation (8)), as described in the following equations:

$$\theta\_{rm} = \frac{\left(\theta\_{\rm cl-1} + 0.8\,\theta\_{\rm cl-2} + 0.6\,\theta\_{\rm cl-3} + 0.5\,\theta\_{\rm cl-4} + 0.4\,\theta\_{\rm cl-5} + 0.3\,\theta\_{\rm cl-6} + 0.2\,\theta\_{\rm cl-7}\right)}{3.8} \,\mathrm{[^{\circ}\mathrm{C}]},\tag{6}$$

$$
\theta\_c = 0.33\,\theta\_{rm} + 18.8\,\mathrm{[^{\circ}C]},\tag{7}
$$

$$
\theta\_{\mathfrak{c}} - 3 \le |T\_{\text{OP}}| \le |\theta\_{\mathfrak{c}} + 2| \, [^{\circ}\text{C}].\tag{8}
$$

Hence, the results of these calculations were used for both the definition of the number of discomfort hours—comparing the operative temperature with the optimal range in Equation (6)—and the definition of a natural ventilation schedule. Indeed, in the passive strategy case, natural ventilation is activated only when the operative temperature of the thermal zone oversteps the higher boundary of the optimal range and the outdoor dry-bulb air temperature is lower than the indoor dry-bulb air temperature. This choice is due to the behavior of the building users, who generally open windows only for warm thermal discomfort.

For both strategies, a linearly dimmable artificial lighting system was implemented, controlled by an illuminance sensor placed in the middle of each thermal zone, with a target illuminance of 300 lux in the apartments, 100 lux in the corridors, and 500 lux in the offices. Internal gains, occupancy, and indoor appliances were left in accordance with the original reference models. According to the ASHRAE Standards [72], indoor appliances consume a maximum of 10.76 W/m<sup>2</sup> in small offices and 5.38 W/m2 in midrise apartments, with a radiant fraction equal to 0.5. An occupancy density of 0.054 persons/m2 was considered for the offices, while a maximum number of 2.5 people was assumed for the apartments. In both cases, the metabolic emission rate was considered equal to 120 W/person.

Another fundamental step in energy analyses is to properly model the PCM in EnergyPlus. To that end, a commercial PCM-enhanced plaster panel—the Alba Balance by Saint-Gobain Rigips—was considered as a reference. Two phase-transition temperatures, 23 and 26 ◦C, are commercially available for these boards. In this study, a 23 ◦C phase transition with a latent heat of 300 kJ/m<sup>2</sup> is considered, as shown in Table 5, which reports the producer-declared technical data.

**Table 5.** Characteristics of PCM-enhanced plaster panel.


In order to suitably model this material in EnergyPlus, an enthalpy table, ranging from −30 to +100 ◦C, was calculated, starting from the areal density, the specific heat, the latent heat, and the following enthalpy equation.

$$H = \mathfrak{mc} \,\,\Delta T \,\,\tag{9}$$

Once the material reaches the melting point (23 ◦C), it stores latent heat (300 kJ/m<sup>2</sup> = 12 kJ/kg). This heat storage is equally distributed as an increase of enthalpy at three temperature steps within the melting range (21, 22, and 23 ◦C). The plot of the enthalpy curve is included in Figure 3.

Finally, these calculated values were inputted in EnergyPlus, and, to model the behavior of the PCM properly, the heat balance algorithm was changed from the Conduction Transfer Function (CTF) to the Conduction Finite Difference (ConFD) algorithm, increasing the timesteps from 6 to 20 [73]. These settings significantly increase the running time but discretize the surfaces depending on the thermal diffusivity of the material and the time step, allowing us to model particular materials, such as PCMs or variable thermal conductivity materials [63]. This timestep setting has been largely validated and verified [63,73] and guarantees a good adherence of the numerical model with measured data. Further detailed hysteresis analysis would have led to errors of about 1% instead of a just-few-times-higher error [74], which can be considered acceptable for annual analyses.

**Figure 3.** Enthalpy curve for the PCM-enhanced plasterboard.

The chosen plasterboards can be used in both walls and ceilings, and, for these analyses, they were considered the inner layer of all external walls and roofs. Despite Zwanzig et al. stating that in a multilayer wall, centrally located PCM better reduces heat fluxes [75], other studies have confirmed that PCMs located as an inner layer improve thermal comfort, thanks to the stabilization of radiant temperature [25]. Moreover, to understand how the thicknesses of the boards influence the results, four different boards, with thicknesses of 12.5, 25, 37.5, and 50 mm, were considered and modeled.

#### **3. Results**

The benefits of implementing PCMs were assessed by considering the reduction of energy demand in the active strategy and the reduction of total discomfort hours in the passive strategy. The following subparagraphs show the different results obtained for each strategy considered.

#### *3.1. Active Strategies: Energy Demand Reduction*

The reduction of energy demand is shown in the following figures (Figures 4 and 5), where the results are grouped with reference to the climatic zones, showing the changes of the control parameter as a change of PCM thickness. The contribution of PCMs in the reduction of heating and cooling energy consumption was split so as to differentiate the strategies. The percentage of energy reduction (*ER%*) refers to the difference between the total energy demand in the baseline case (*ET baseline*)—without PCM—and the total energy demand in each PCM-implemented case (*ET PCM*). This difference was then divided by the total energy demand of the baseline case, as described in the following equation (Equation (10)):

$$ER\_{\%} = 100 \ast \frac{E\_{T \text{ baseline}} - E\_{T \text{ PCM}}}{E\_{T \text{ baseline}}} \tag{10}$$

**Figure 4.** Small office, total energy demand reduction for the analyzed cities: (**a**) Climatic Zone B, (**b**) Climatic Zone C, (**c**) Climatic Zone D, (**d**) Climatic Zone E, (**e**) Climatic Zone F, and (**f**) Climatic Zone G.

Overall, it can be easily argued that the type and magnitude of energy reduction are strictly dependant on building typology, and outcomes show opposite trends when comparing the two models. On the one hand, in the small office model, the benefits of PCMs are clearer and increase in colder climates, with an average that rises from 2.57% for Zone B to 5.13% for Zone G. On the other hand, benefits for the midrise apartment model are lower and countertrend, considering that the main contribution is related to heating demand reduction and that the zone average decreases (with a less evident trend) in colder zones. However, both models confirm that an increase in PCM thickness corresponds to an increase in benefits, with a strong logarithmic correlation between thickness and energy reduction (Figure 6).

**Figure 5.** Midrise apartment, total energy demand reduction for the analyzed cities: (**a**) Climatic Zone B, (**b**) Climatic Zone C, (**c**) Climatic Zone D, (**d**) Climatic Zone E, (**e**) Climatic Zone F, and (**f**) Climatic Zone G.

**Figure 6.** Logarithmic correlation between zone average and PCM thickness for (**a**) small office and (**b**) midrise apartment in the active strategies case.

#### *3.2. Passive Strategies: Thermal Discomfort Reduction*

In this second set of analyses, the index of the advantages related to the implementation of PCMs is the reduction of total discomfort hours (*DHT*) in baseline (*DHT Baseline*) and PCM (*DHT PCM*) models, calculated by comparing the operative temperature of the thermal zones with the acceptance limits expressed in Equation (8). Analogously to the ER% described in the previous section, the percentage of discomfort hours reduction (*DHR%*) was calculated as described in Equation (11), and the results are reported in Figures 7 and 8.

$$DHR\_{\%} = 100 \ast \frac{DH\_{T\\_Baseline} - DH\_{T\\_PCM}}{DH\_{T\\_Baseline}} \tag{11}$$

**Figure 7.** Small office, total discomfort hours reduction for the analyzed cities: (**a**) Climatic Zone B, (**b**) Climatic Zone C, (**c**) Climatic Zone D, (**d**) Climatic Zone E, (**e**) Climatic Zone F, and (**f**) Climatic Zone G.

Analogies and differences with results obtained for the active strategies can be easily noted from the graphs. Firstly, these simulations confirm that the implementation of PCMs is more effective in small offices than in midrise apartments.

With regards to the small office model, a clear different behavior can be identified when comparing the climatic zones: warmer climates (Zones B, C, and D) show higher benefits, mainly related to the reduction of cold thermal discomfort, while colder zones (E, F, and G) show lower benefits related to the reduction of warm thermal discomfort. This different behavior is also confirmed by the zone averages that fluctuate between 4.5% and 9% for the first three zones while oscillating around 1.2% for the other zones.

On the contrary, in midrise apartments, there is hardly ever a reduction of cold thermal discomfort; the results are mainly pushed by the reduction of warm thermal discomfort. In this case, the major advantages are registered in the intermediate climates—Zones C

and D—with average values of 3.45% and 2.15%, respectively, while for other zones, the average benefits are lower and range from 0.73% to 1.57%.

**Figure 8.** Midrise apartment, total discomfort hours reduction for the analyzed cities: (**a**) Climatic Zone B, (**b**) Climatic Zone C, (**c**) Climatic Zone D, (**d**) Climatic Zone E, (**e**) Climatic Zone F, and (**f**) Climatic Zone G.

Finally, comparing the effect of different PCM thicknesses, the comfort analysis confirms that thicker panels show higher advantages (Figure 9).

**Figure 9.** Logarithmic correlation between zone average and PCM thickness for (**a**) small office and (**b**) midrise apartment in the passive strategies case.

#### **4. Discussion**

The analyses performed point out a complex frame of results, where a few trends can be deduced, in general, for PCMs while others should be referred to each specifically analyzed case.

Firstly, one of the clearest conclusions that can be drawn is that the benefits of PCMs, integrated into the envelope's inner face, depend logarithmically on the thickness of the boards. The presented outcomes confirm that this trend is common to all analyses and does not depend on the climatic zones, building typology, or the comfort strategy adopted. Therefore, it can be stated that increasing board thickness is particularly advantageous, considering layers up to 50 mm; further increases lead to gradually lower improvements that may not be affordable. This behavior can be explained by considering the relatively low thermal conductivity of the boards (0.27 W/mK). In this case, increasing board thickness adds a further shift in the load peaks related to the thermal conduction inside the board that gradually reduces the effectiveness of the PCM.

Referring to different building typologies, the small office model shows higher benefits than the midrise apartment model, regardless of the climatic zone, PCM thickness, and comfort strategy adopted. The main reason for this trend can be identified in the different occupancy rates (35.3 m2/person for the midrise apartment and 18.6 m2/person for the small office) and the different internal loads of equipment and lights, which are sensibly higher in the office model. Considering the case of PCM boards located on the envelope's inner face, increasing the internal loads can increase the number of melting cycles, improving energy and comfort, especially in cooling-dominated climates.

Lastly, considering the climatic zones analyzed, it can be stated in general that the advantages of PCMs are higher in warm climates—Zones B, C, and D—with the only countertrend exception represented by the energy demand of the small office model in active strategies, where colder zones correspond to higher benefits, thanks to a constant increase in heating energy reduction.

Beyond these general considerations on the results obtained, referring to each analyzed case, on the one hand, the small office model has shown:


Comparing these results with those reported in similar studies, the main differences obtained can be explained by the positioning of the boards and the use of different building types, occupancy, and system availability schedules that can increase or decrease PCM effectiveness. Moreover, in this case, a PCM-embedded gypsum board was considered for adaptive implementation while, in other studies, pure PCM materials were applied to the envelope. Finally, as described by this study, the different locations can significantly change the effectiveness of the implementation of this technology.

#### **5. Conclusions**

The analyses presented in this paper are focused primarily on the effect of the PCMenhanced boards studied, considering different boundary conditions, in order to define

general trends and the specific behavior of this technology. To this end, the simulations considered different building typologies, climates, thicknesses, and active/passive strategies. Considering all the differences that can arise with this broad number of variables, the methodology adopted to create the energy building model tries to reduce all the fluctuations related to different national laws or the specific HVAC system in order to have comparable results.

Overall, this study highlights both general and specific trends for this technology, with interesting results. Undoubtedly, further studies could expand this research to other calibrated reference models or different climates in order to further generalize or specify the trends described in this paper. In this first work, our attention was focused on the European geographic area in order to analyze cities with similar energy containment directives. Future developments could expand this area—changing the envelope characteristics and also considering tropical and dry climates.

The economic aspects of PCM implementation are out of the boundaries of the current study as there are very different compositions of energy and energy prices in the considered countries. Future works could focus on smaller groups of locations and estimate a payback period on selected groups of cities with similar economic and energetic backgrounds.

Another interesting development of this first study could consider a simulation in future climatic scenarios. As shown in our research, office models are characterized by a strong cooling reduction in the active strategy and mainly by cold thermal discomfort in passive strategies. Considering future scenario simulations—characterized by warmer boundary conditions—the energy benefits should be higher than those obtained in our active strategy simulations while, on the contrary, in the passive strategy, the benefits could be lower. Finally, with regard to the midrise apartment model, the PCM benefits are related mainly to heating consumption reduction (active strategy) and the reduction of warm discomfort hours (passive strategy). Therefore, the benefits of PCM implementation could probably be reduced in future scenarios.

**Author Contributions:** Conceptualization, F.F. and A.C.; methodology, F.C., F.F. and A.C.; software, F.C.; validation, F.C. and F.F.; formal analysis, F.C.; investigation, A.S. and A.A.T.; resources, A.C.; data curation, F.C.; writing—original draft preparation, F.C.; writing—review and editing, F.F., A.C., A.A.T. and A.S.; visualization, F.C. and F.F.; supervision, F.F. and A.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


### *Article* **Influence of Traditional and Solar Reflective Coatings on the Heat Transfer of Building Roofs in Mexico**

**Iván Hernández-Pérez**

División Académica de Ingeniería y Arquitectura (DAIA-UJAT), Universidad Juárez Autónoma de Tabasco, Carretera Cunduacán-Jalpa de Méndez km. 1, Cunduacán, Tabasco CP 86690, Mexico; ivan.hernandezp@ujat.mx; Tel.: +52-777-227-4111

**Abstract:** Building roofs are sources of unwanted heat for buildings situated in zones with a warm climate. Thus, reflective coatings have emerged as an alternative to reject a fraction of the solar energy received by roofs. In this research, the thermal behavior of concrete slab roofs with traditional and solar reflective coatings was simulated using a computational tool. The studied slab configurations belong to two groups, non-insulated and insulated roofs. In the second group, the thermal insulation thickness complies with the value recommended by a national building energy standard. Weather data from four cities in Mexico with a warm climate were used as boundary conditions for the exterior surface of the roofs. The computational tool consisted of a numerical model based on the finite volume method, which was validated with experimental data. A series of comparative simulations was developed, taking a gray roof as the control case. The results demonstrated that white roofs without insulation had an exterior surface temperature between 11 and 16 ◦C lower than the gray roof without insulation. Thus, the daily heat gain of these white roofs was reduced by a factor ranging between 41 and 54%. On the other hand, white roofs with insulation reduced the exterior surface temperature between 17 and 21 ◦C compared to the gray roof with insulation. This temperature reduction caused insulated white roofs to have a daily heat gain between 37 and 56% smaller than the control case. Another contribution of this research is the assessment of two retrofitting techniques when they are applied at once. In other words, a comparison between a non-insulated gray roof and an insulated white roof revealed that the latter roof had a daily heat gain up to 6.4-times smaller than the first.

**Keywords:** solar reflective coatings; heat transfer; daily heat gains; cool roofs

#### **1. Introduction**

The buildings sector used 36% of the total final energy around the world and had 39% of the energy-related CO2 emissions in 2018 [1]. Because the population is expected to increase by 2.5 billion people by 2050, the energy use in the building sector is set to rise sharply. In warm locations, the energy consumption from air conditioners is high due to the heat flow received by buildings situated in these zones. The building envelope plays a vital role in the thermal interaction between the outdoor and indoor environments. Thus, it is important to minimize the energy gain from the building envelope to avoid the excessive use of electricity for comfort purposes.

Today, several technologies are available for building energy retrofitting. There are advanced facades [2], highly insulated windows [3], high insulation levels for roofs and walls [4], reflective coatings [5], phase change materials [6], and well-sealed structures [7], to mention a few. In particular, reflective surfaces are becoming popular for two main reasons. First, the most direct way to reduce the incident solar energy is to reflect it. When new, solar reflective coatings can reflect to the sky up to 90% of the solar energy received by a surface. The second reason is that applying reflective coatings to opaque building components is probably the most simple passive measure because most of these coatings can be installed in the same way as ordinary paint [8]. These coatings are usually used

**Citation:** Hernández-Pérez, I. Influence of Traditional and Solar Reflective Coatings on the Heat Transfer of Building Roofs in Mexico. *Appl. Sci.* **2021**, *11*, 3263. https:// doi.org/10.3390/app11073263

Academic Editor: Tiziana Poli

Received: 28 February 2021 Accepted: 2 April 2021 Published: 6 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

on building roofs because these components are subject to solar radiation for more hours than other building envelope components [9]. Therefore, when a roof is retrofitted with a reflective coating, it is known as a cool roof.

Several researchers around the world have studied the benefits that cool roofs can bring to buildings in different types of climates using a number of approaches. Several experimental studies are available in the literature. For instance, Pisello and Cotana [10] performed a two-year monitoring campaign to test cool roofs in a residential building located in Italy. The results showed that the cool roof solution reduced the peak temperature of the air in the attic by around 5 ◦C. Further, the cool roof provided an overall year-round energy savings that corresponded to 14 kWh/m<sup>2</sup> per year. In China, Quin et al. [11] used a building cell to test different samples of concrete tiles with different solar reflectance. The authors performed a series of experiments and found that the tiles with a reflective coating reached lower interior surface temperatures than the control tile. Moreover, an additional 2–6 ◦C cooler temperature around the noon time was achieved by installing at the bottom of the tile a low-emissivity sheathing. Hernández-Pérez et al. [12] evaluated several building roofs with different coatings using two outdoor test cells located in Cuernavaca, Mexico. They showed that in summer, the white roof was 29 ◦C cooler than the black roof and just 1.5 ◦C warmer than the ambient air. Further, the average daily energy gain of the white roof was 73% less than the black roof.

Other studies aiming to investigate the year-round energy savings from cool roofs by using building energy simulation tools are available. Algarni [13] studied the influence of solar reflective roofs on the energy consumption of residential buildings of Saudi Arabia. The author used eQuest building simulation software to perform simulations of a building prototype using weather data of 13 major cities of this country. The researcher found that a reflective roof reduced the annual energy consumption required for building cooling by between 110.3 and 181.9 kWh/m2. The maximum increase in annual energy consumption due to winter heating was only 4.4 kWh/m2. Piselli et al. [14] assessed the effectiveness of cool roofs with optimal insulation levels in different weather conditions worldwide. The authors coupled a dynamic energy simulation tool with an optimization technique to find the best-combined building roof thermal insulation and solar reflectance and minimize the annual energy consumption. The results showed that a high value of solar reflectance reduced the annual energy consumption for most of the analyzed climate zones. Thus, they concluded that the optimal roof configuration must have high solar reflectance and no/a low insulation level. Dominguez-Delgado et al. [15] developed an energy and economic life cycle assessment of cool roofs applied to residential buildings in Southern Spain. The simulations demonstrated that the cool roof decreased the annual energy consumption. The maximum decrease found by the authors was 32%, and it was obtained when a roof with a solar reflectance of 0.1 was retrofitted with a reflective coating with a solar reflectance of 0.9. The LCA analysis reported that savings were 18.33 e/m2, and the payback period was around thee years.

Some researchers have used validated models to predict the benefits of cool roofs. For instance, Tong et al. [16] studied the thermal behavior of ventilated and non-ventilated roofs during a typical day in Singapore. The authors conducted experiments to validate the concrete roof models. Compared with the roofs with a solar reflectance of 0.1, each 0.1 increase in reflectance reduced the daytime heat gain by 11% on the ventilated roofs and the non-ventilated roofs. The reflective coatings application reduced the daily heat gain by 234 and 135 Wh/m2 in the ventilated and non-ventilated roofs, respectively. Further, they indicated that compared to non-ventilated roofs (both reflective and non-reflective), the individual use of ventilation and 2.5 cm of expanded polystyrene (EPS) reduced the heat gain by 42 and 68%, respectively. Zingre et al. [17] developed a model to study heat transfer in roofs. They used a spectral approximation method to solve the unsteady one-dimensional heat conduction equation. Furthermore, the model was validated using experimental data obtained from measurements made in two identical apartments with concrete roofs located in Singapore. Model predictions showed that on a sunny day, the

reflective coating reduced the maximum roof temperature, indoor air temperature, and daytime heat gain by 14.1 ◦C, 2.4 ◦C, and 0.66 kWh/m2 (54%), respectively.

As shown in the literature review, reflective or cool roofs have brought a series of benefits to the buildings in which they are installed. Thus, it is essential to understand how a cool roof behaves in a particular area. For this purpose, it is necessary to have tools capable of predicting the thermal behavior of a cool roof before its installation to determine if it is feasible from an energy point of view. This work presents the development of a computational tool for modeling the heat transfer of concrete slab-type roofs with traditional and solar reflective coatings. This tool is then used to estimate the temperature reductions and, consequently, the ability of reflective roofs to modulate the heat gains by implementing this technology in buildings situated in four different warm climates in Mexico.

#### **2. Physical Model**

Figure 1 shows the physical model of the roof, which is a concrete slab with traditional or solar reflective coating. Two configurations were considered: (a) a single roof with a thickness of 10 cm and (b) an insulated roof with a thickness of 13 cm. The insulated roof was made of concrete and polystyrene, and a thin plaster layer of concrete protected the polystyrene because it should not be exposed to solar radiation. The polystyrene had a thickness equal to 2.5 cm, and the plaster layer had a thickness of 0.5 cm. The width of the roof (*W*) was considered 1 m. Both configurations were studied with traditional and solar reflective coatings. The thickness of the coatings was considered negligible. The materials of both roof configurations were considered homogeneous. The solar absorptance and thermal emissivity of the roofs were also regarded as constant. Further, it was considered that the side surfaces of the two roof configurations were adiabatic. Finally, it was supposed that solar radiation had a normal incidence on the roof and that this was a gray body that only absorbed the incident solar radiation.

**Figure 1.** Physical model. (**a**) Single roof; (**b**) insulated roof.

#### **3. Mathematical Model**

The partial differential equation for transient heat conduction of a two-dimensional solid is [18]:

$$\frac{\partial(\rho c\_p T)}{\partial t} = \frac{\partial}{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) \tag{1}$$

where *λ* is the thermal conductivity, *ρ* is the density, and *cp* is the specific heat. The boundary conditions used to solve in the physical model were as follows: The exterior surface (*y* = *y*1) is in contact with the outdoor environment or environmental conditions of different cities of Mexico. Therefore, the roof has convective and radiative exchange with the surroundings. The lateral surfaces are thermally insulated (*x* = 0 and *x* = *W*), and the interior surface (*y* = 0) also exchanges heat by convection and radiation with the indoor environment. Mathematically, the boundary conditions can be expressed as:

$$-\lambda \frac{\partial T}{\partial y} = -aG + h\_{\rm out}(T - T\_{\rm out}) + \sigma \varepsilon (T^4 - T\_{\rm sky}^4) \quad \text{for} \quad y = y\_1, \quad 0 \le x \le \mathcal{W} \tag{2}$$

$$-\lambda \frac{\partial T}{\partial y} \quad = \quad h\_{in}(T - T\_{in}) \quad \text{for} \quad y = 0, \quad 0 \le x \le W \tag{3}$$

$$\frac{\partial T}{\partial \mathbf{x}} \quad = \quad 0 \quad \text{for} \quad \mathbf{x} = \mathbf{0}, \quad 0 < y < y\_1 \tag{4}$$

$$\frac{\partial T}{\partial \mathbf{x}} \quad = \quad 0 \quad \text{for} \quad \mathbf{x} = \mathcal{W}\_{\prime} \quad 0 < \mathbf{y} < \mathbf{y}\_{1} \tag{5}$$

In Equation (2), *G* is the solar radiation received by the roof, *α* is the solar absorptance of the coating, *ε* is the thermal emittance, and *σ* is the Stefan–Boltzmann constant. The beam-solar radiation received by roofs has different angles during the day. The solar radiation data used for the simulations were obtained from measurements performed using weather stations, and these stations used pyranometers for measuring total (beam plus diffuse) radiation over a horizontal surface. The detectors of the pyranometers had a response that did not depend on radiation wavelength over the solar spectrum. Further, these devices had a response independent of the angle of incidence of solar radiation. Thus, it can be supposed that the roofs received total radiation in a perpendicular direction. To calculate the outdoor convective heat transfer coefficient (*hout*) in Equation (2), the following empirical correlation was used [19]:

$$h\_{\rm out} = 2.8 + 3.0\upsilon\tag{6}$$

where *v* is the wind speed in m/s and *hout* is the outdoor convective heat transfer coefficient in W/m2K. The value of *Tsky* is calculated with the following expression [19]:

$$T\_{sky} = 0.0552 T\_{out}^{1.5} \tag{7}$$

In Equation (7), *Tout* is the outdoor air temperature, and both temperatures *Tsky* and *Tout* are expressed in K. The heat transfer coefficient in the interior environment *hin* in Equation (3), which considers both convection and thermal radiation, is equal to 6.13 W/m<sup>2</sup> when the heat flux heat goes to the indoor air and 9.26 W/m2K when the heat flux goes from the indoor air to the interior surface [20]. For the roof configuration (b) of Figure 1, due to the different materials involved and the thermophysical properties' changes across the interface between concrete and thermal insulation, the harmonic mean was used to calculate these properties in the interface. Furthermore, perfect contact between both materials was considered, such that heat flux through the interface was the same for the materials involved. The properties of the concrete used for the simulations were *ρ* = 2400 kg/m3, *cp* = 1080 J/(kg·K), and *λ* = 1.8 W/(m·K). On the other hand, the properties of the insulation were *<sup>ρ</sup>* = 28 kg/m3, *cp* = 1800 J/(kg·K), and *<sup>λ</sup>* = 0.033 W/(m·K).

#### **4. Solution Methodology for the Roof Model**

This section present the methodology followed to numerically solve the heat conduction Equation (1). The following steps were developed:


The discretization technique used in this work was the finite volume method (FVM), and the following subsections describe each of these steps.

#### *4.1. Generation of the Computational Mesh*

This step consisted of dividing the domain into small control volumes where the nodes were situated, and the value of the temperature *T*(*x*, *y*) was determined. In this analysis, the main node *P* was located in the center of the control volume (Figure 2), so that the its interface (point *w*, *e*, *n*, and *s*) was at the middle from one node to another.

**Figure 2.** A control volume in the computational mesh.

Equation (8) describes the distribution of the nodes in the mesh in the direction of *x*:

$$\mathbf{x}(i) = \frac{\mathcal{W}}{N\_x - 1}(i - 1) \quad \text{ for} \quad i = 1, 2, 3, \dots, N\_{\mathbf{x}} \tag{8}$$

where *Nx* is the number of nodes in the *x* direction and *W* is the width of the building roof (Figure 1). This equation provides the direction coordinate of the nodes *x*. For the nodes in direction *y*, the coordinate is given by:

$$y(j) = \frac{y\_1}{N\_y - 1}(j - 1) \quad \text{ for} \quad j = 1, 2, 3, \dots, N\_y \tag{9}$$

where *Ny* is the number of nodes in the *y* direction and *y*<sup>1</sup> is the height of the building roof (Figure 1). It is also necessary to know the thickness of the control volume in both directions, and Equation (10) is used to calculate such thicknesses:

$$
\Delta \mathbf{x} = \frac{\mathcal{W}}{N\_{\mathbf{x}} - 1} \qquad \qquad \Delta y = \frac{y\_1}{N\_{\mathbf{y}} - 1} \tag{10}
$$

#### *4.2. Discretization of the Mathematical Model*

Discretization consists of applying a technique to transform the partial differential equation of the mathematical model into a set of algebraic expressions to facilitate its solution. For the internal nodes, the Equation (1) governs their behavior, and this equation is known as the general differential equation for diffusion processes and can be represented by:

$$\frac{\partial \rho \phi}{\partial t} = \frac{\partial}{\partial x} \left( \Gamma \frac{\partial \phi}{\partial x} \right) + \frac{\partial}{\partial y} \left( \Gamma \frac{\partial \phi}{\partial y} \right) \quad \text{for} \quad 0 < x < W; \quad 0 < y < y\_1$$

where: *φ* = *T*, Γ = *λ*/*cp*. Taking the domain of interest, for each term of the previous equation, between the points *w* and *e* in the *x* direction of the control volume in Figure 2 and between the points *n* and *s* in the *y* direction and, furthermore, considering the time step, a domain of interest is taken from an earlier time (*t*0) to a later time (*t* = *t*<sup>0</sup> + Δ*t*):

$$\begin{aligned} \int\_{t\_0}^{t} \int\_{s}^{n} \int\_{w}^{\epsilon} \frac{\partial \rho \Phi}{\partial t} \, dx \, dy \, dt &= \int\_{s}^{n} \int\_{w}^{\epsilon} \frac{\partial}{\partial x} \left( \Gamma \frac{\partial \phi}{\partial x} \right) dx \, dy + \int\_{s}^{n} \int\_{w}^{\epsilon} \frac{\partial}{\partial y} \left( \Gamma \frac{\partial \phi}{\partial y} \right) dx \, dy \\ &+ \quad \int\_{s}^{n} \int\_{w}^{\epsilon} S \, dx \end{aligned}$$

Integrating in the space the previous equation over the control volume, the following is obtained:

$$
\left[\frac{\partial\overline{\rho\Phi}}{\partial t}\right] \Delta x \Delta y = \left[\left(\Gamma \frac{\partial\phi}{\partial x}\right)\_e - \left(\Gamma \frac{\partial\phi}{\partial x}\right)\_w\right] \Delta y + \left[\left(\Gamma \frac{\partial\phi}{\partial y}\right)\_n - \left(\Gamma \frac{\partial\phi}{\partial y}\right)\_s\right] \Delta x + \overline{S} \Delta x \Delta y
$$

Now, using an implicit interpolation scheme for time:

 (*ρφ*)*<sup>t</sup> <sup>P</sup>* − (*ρφ*) *t*0 *P* Δ*t* Δ*x*Δ*y* = <sup>Γ</sup> *∂φ ∂x t e* − <sup>Γ</sup> *∂φ ∂x t w* Δ*y* + <sup>Γ</sup> *∂φ ∂y t n* − <sup>Γ</sup> *∂φ ∂y t s* Δ*x* + *S*Δ*x*Δ*y*

Because the conditions at the interfaces were unknown and the centered scheme interpolation was used, using the known values of the nodes adjacent to that control volume interface (nodes E, W, N, and S), the following was obtained:

$$
\left[\frac{(\rho\Phi)\_P^t - (\rho\phi)\_P^{t\_0}}{\Delta t}\right] \Delta x \Delta y = -\Gamma\_t \left(\frac{\phi\_E - \phi\_P}{\delta x\_{PE}}\right) \Delta y - \left(\frac{\phi\_P - \phi\_W}{\delta x\_{PW}}\right) \Delta y + \Gamma\_n \left(\frac{\phi\_N - \phi\_P}{\delta y\_{PN}}\right) \Delta x \Delta y$$

$$

It is convenient to group the terms of the equation into coefficients as follows:

$$\begin{aligned} \label{2} \phi\_{P} \underbrace{\left(\left(\frac{\rho \Delta x \Delta y}{\Delta t}\right) + \left(\frac{\Gamma\_{\varepsilon}}{\delta \mathbf{x}\_{PE}} + \frac{\Gamma\_{w}}{\delta \mathbf{x}\_{PW}}\right) \Delta y + \left(\frac{\Gamma\_{n}}{\delta \mathbf{y}\_{PN}} + \frac{\Gamma\_{s}}{\delta \mathbf{y}\_{PS}}\right) \Delta x\right)}\_{a\_{P}} = \phi\_{E} \underbrace{\left(\frac{\Gamma\_{\varepsilon} \Delta y}{\delta \mathbf{x}\_{PE}}\right)}\_{a\_{E}} \\\\ \cline{2} + \phi\_{W} \underbrace{\left(\frac{\Gamma\_{w} \Delta y}{\delta \mathbf{x}\_{PW}}\right)}\_{a\_{W}} + \phi\_{N} \underbrace{\left(\frac{\Gamma\_{n} \Delta x}{\delta \mathbf{y}\_{PN}}\right)}\_{a\_{N}} + \phi\_{S} \underbrace{\left(\frac{\Gamma\_{s} \Delta x}{\delta \mathbf{x}\_{PS}}\right)}\_{a\_{S}} + \phi\_{P}^{b\_{0}} \underbrace{\left(\frac{\rho \Delta x \Delta y}{\Delta t}\right)^{t\_{0}}}\_{b\_{P}^{0}} + \overline{\mathsf{S}} \Delta x \Delta y \end{aligned}$$

Therefore:

$$a\_E = \frac{\Gamma\_e \Delta y}{\delta \mathbf{x}\_{PE}} \qquad a\_W = \frac{\Gamma\_w \Delta y}{\delta \mathbf{x}\_{PW}} \qquad a\_N = \frac{\Gamma\_n \Delta x}{\delta y\_{PN}} \qquad a\_S = \frac{\Gamma\_s \Delta x}{\delta \mathbf{x}\_{PS}}$$

$$a\_P^0 = \frac{\rho \Delta x \Delta y}{\Delta t}$$

$$b = S \Delta x \Delta y + a\_P^0 \phi\_P^0$$

*aP* = *aE* + *aW* + *aN* + *aS* + *a*<sup>0</sup> *<sup>P</sup>* − *S*Δ*x*Δ*y*

We obtain here Equation (11), which is the generative equation of the system of algebraic equations in the notation of grouped coefficients:

$$a\_P \phi\_P = a\_E \phi\_E + a\_W \phi\_W + a\_N \phi\_N + a\_S \phi\_S + b \tag{11}$$

It is also necessary to discretize the boundary nodes. According to the physical model, there is a third class condition (interior surface of the roof), two second class conditions at the left and right ends, and another third class with radiative loss or gain in the exterior surface.

#### *4.3. Solution of the System of Algebraic Equations*

If the system of algebraic equations resulting from the discretization of a two-dimensional model is adjusted in a matrix way, a matrix of pentagonal and diagonally dominant coefficients is obtained. For the one-dimensional case, the Thomas algorithm or tridiagonal matrix algorithm (TDMA) is applied as a direct method by the dominant tridiagonal matrix of coefficients; however, Thomas's algorithm can be used for the two-dimensional case by combining it with iterative equation solving methods. In this work, the system of algebraic equations was solved using the line-by-line Gauss–Seidel method with alternating directions (LGS-ADI).

#### **5. Verification and Validation of the Mathematical Model**

The mathematical model as mentioned above was solved using the finite volume method; thus, it was necessary to develop a computer simulation tool that could follow the solution methodology described in Section 4. The numerical code was developed in the Fortran programming language. First, the mathematical model was used to solve a reference exercise available in the literature to verify that it was able to provide reliable results. It was verified by comparing its results against the analytical solution for a composite solid reported by Chen and Paine [21]. Then, the mathematical model was subjected to another test known as validation, where the results of the model were compared with data from temperature measurements from a roof under outdoor ambient conditions. To validate the numerical model that solved the heat conduction equation, experimental data from a previous publication of the author were used [22]. The model was validated for two cases: a conventional gray roof and a white reflective roof. In the experiment performed by Hernández-Pérez et al. [22] with two test cells, the exterior surface temperature of the roofs was measured every 10 min for five days. Figure 3 shows the temperatures obtained by solving the model and the experimental temperatures. This figure shows that the model satisfactorily reproduced the behavior of the gray roof and the white reflective roof. The maximum deviations of the temperatures obtained for the gray and white roofs were 5.5% and 4.6%, respectively. Therefore, this model can be used to study the thermal performance of concrete roof slabs in different weather conditions. One advantage that the current model brought is that most of the validated tools available were validated for a single day [17,23]. In contrast, the model presented by the author was validated by using data

from an experimental test performed for five consecutive days. In other words, this tool was validated by comparing its results with 720 temperature values for each type of roof.

**Figure 3.** Comparison of the experimental data from Hernández-Pérez et al. [22] and the data obtained with the numerical model.

#### **6. Weather Data**

Once the roof model was verified and validated, its thermal behavior was studied under the outdoor environment of different cities. Data from four representative towns in Mexico with warm weather were used. Table 1 shows the four major cities with their corresponding type of weather. The weather data used for the simulations were filtered from files provided by the National Meteorological Service-National Commission of Water (Servicio Meteorológico Nacional-Comisión Nacional del Agua (SMN-CONAGUA)), the public body responsible for providing information on the state of the weather that prevails or affects the territory of this country. The SMN-CONAGUA obtains the weather data files from weather stations situated in each city. The data files contained recorded several variables recorded every ten minutes for a whole year. Still, only solar radiation, wind speed, and air temperature for the simulations were used, as shown in the Results Section. The thermal behavior of the roofs was analyzed for the week with the highest temperatures of the year 2018.

**Table 1.** Representative cities of the zones with warm weather in Mexico.


#### **7. Properties of Traditional and Reflective Coatings**

Table 2 shows the values of solar absorptance and thermal emittance of the four coatings analyzed in this research. There were two traditional coatings, gray and terracotta, and two solar reflective coatings, White # 1 and White # 2. These optical properties of the coatings were measured in two previous works of the author [12,22]. The solar reflectance was measured using a spectrophotometer with an integrating sphere following the procedure indicated in the standard ASTM E 903-12. After obtaining the solar reflectance, the solar absorptance was calculated because this property was required in the boundary conditions of the model as presented above. A portable ambient temperature emissometer was used, according to the standard ASTM C1371 (2015): laboratory or field measurement of hemispherical thermal emittance with a portable emissometer (indirect technique using calibrated references) the thermal emittance obtained for all coating samples obtained at ambient temperature. The emittance of the coatings can be considered a constant value independent of temperature because the coatings were opaque materials that could be

considered as paint, and according to Duffie and Beckman [19], the emittance of these materials only changed slightly at very high temperatures.


**Table 2.** Optical properties of the coatings.

#### **8. Results**

This section presents the concrete roof's behavior with four coatings in terms of the temperature of the exterior surface, the temperature of the interior surface, and the heat flux traveling through the roofs. Further, the cumulative heat gain was obtained by determining the area under the heat flux curve for each day. This section shows first the results for the single roof configuration and then the corresponding results for the insulated roof configuration.

#### *8.1. Single Roof*

As mentioned above, the roofs' thermal behavior was analyzed using the weather data of the week with the warmest outdoor air temperatures of 2018 using weather data from four representative towns in Mexico with a warm climate. The detailed results of Hermosillo are shown here because it was the town with the highest outdoor temperature. At the end of this subsection, a summary table presents the results of the thermal evaluation of the single roof in all cities.

For Hermosillo Sonora, the days selected were from 30 May to 5 June 2018 because this was the week with the highest temperatures of the year. Figure 4a shows the solar irradiation and the wind speed during the seven days analyzed. The maximum solar radiation was around 1000 W/m2. According to the behavior of solar irradiance, all days selected were clear days without clouds. The maximum wind speed reached around 5 m/s. Figure 4b shows the air temperature; due to the type of weather of this city, the ambient air reached very high temperatures, with an average maximum temperature of 45 ◦C.

Figure 4b presents the temperature of the exterior surface of the roofs and the ambient air temperature during the selected week. Conventional roofs had a similar behavior, and on the other hand, reflective roofs maintained a similar behavior concerning the temperature of the exterior surface. This effect occurred because the solar reflectance of conventional coatings was very similar. The exterior surface of the roofs reached its maximum temperature between 14:30 and 15:00 h. The exterior surface of the single terracotta roof (STR), the single gray roof (SGR), Single White Roof #1 (SWR1), and Single White Roof #2 (SWR2) reached on average 61, 59, 45, and 43 ◦C, respectively. These values indicated that applying the terracotta coating, on average, increased the temperature of the exterior surface by 2 ◦C compared to the SGR. In contrast, SWR1 and SWR2 reduced the temperature of the exterior surface by on average 14 and 16 ◦C, respectively. Furthermore, if the average maximum temperatures of reflective roofs were compared with the average maximum temperature of ambient air (45 ◦C), SWR1 had the same maximum temperature as the ambient air. In contrast, SWR2 reached a temperature of 2 ◦C lower than the maximum air temperature. On the other hand, the SGR and STR reached a temperature of 14 and 16 ◦C higher than the ambient air temperature.

**Figure 4.** Thermal behavior of the single roof with traditional and solar reflective coatings in Hermosillo: (**a**) solar irradiance and wind speed; (**b**) temperature of exterior surface and outdoor air; (**c**) temperature of interior surface; (**d**) heat flux of the roofs.

Figure 4c presents the temperature of the single roofs' interior surface. The interior surface of the roofs reached their maximum temperature between 16:00 and 16:40 h. The temperature of the interior surface of the STR, SGR, SWR1, and SWR2 reached a maximum temperature of 51, 49, 38, and 37 ◦C, respectively. Therefore, the influence of a reflective coating on the interior surface temperature was obtained by comparing the previous temperature values. SWR1 decreased the temperature of the interior surface by around 10 ◦C compared to the SGR, while SWR2 reduced the temperature of this surface by around 11 ◦C.

Figure 4d shows the behavior of the heat flux of the roofs in Hermosillo during the seven days analyzed. The maximum heat flux traveling through the roofs occurred between 16:00 and 16:50 h. The STR, SGR, SWR1, and SWR2 had an average peak heat flux of 157, 150, 87, and 82 W/m2, respectively. These values indicated that the peak heat flux crossing SWR1 and SWR2 was 41% and 45% smaller than that corresponding to the SGR, while the heat flux of the STR was 5% greater than that of the SGR. The cumulative heat gain for one day or daily heat gain was obtained by calculating the area under the heat flux curve for each day. During the seven days analyzed, the STR had an average heat gain of 1793 W·h/(m2-day), the SGR a gain of 1675 W·h/(m2-day), the SWR1 of 1045 W·h/(m2-day), and the SWR2 roof of 993 W·h/(m2-day). Thus, by calculating the percentage difference between the average daily heat gain, it was found that the TSR located in Hermosillo had a 7% greater heat gain than the SGR, while the SWR1 and SWR2 roofs had 38 and 41% less heat gain than the gray roof.

A similar procedure for the other three cities was developed to perform the simulations; the week with the warmest outdoor air temperatures of 2018 was selected. The weather data from this week were introduced to the simulation tool. Table 3 presents a summary of the results obtained from the evaluation of the single roof with traditional and solar reflective coatings in the four cities of Mexico. The table shows the average peak surface temperatures (*Tes* and *Tis*), the average peak heat flux (*QR*), and the average daily heat gain of the roofs (*HG*). Taking the SGR as a reference, the percentage differences between the peak heat flux of this roof and the other cases are also given within parenthesis (%) in the table, and the same is done for the heat gain. Table 3 demonstrates that SWR1 and SWR2 were able to reduce *Tes* between 11 and 16 ◦C compared to the gray roof. Thus, the peak heat flux crossing the SGR can be shaved between 42 and 57% due to a white reflective coating application. Further, SWR1 and SWR2 reduced *HG* between 41 and 85%. On the other hand, the STR reached a maximum temperature 2 ◦C above the SGR temperature in all cities. These higher temperatures caused an increase in the daily heat gain between 7 and 11%.

**Table 3.** Summary of the thermal evaluation of a single roof in four cities of Mexico (Average peak values for the different variables).


#### *8.2. Insulated Roof*

The insulated roof was studied also in Hermosillo as it was the warmest city among the selected locations. As in the previous section, first, the detailed analysis of this city is presented, and then, the results for other cities are summarized at the end of this subsection.

Figure 5b shows the behavior of the temperature of the external surface of the insulated roofs and the temperature of the ambient air during the seven days considered. Insulated roofs with a conventional color had a similar behavior, and on the other hand, the insulated roofs with reflective coating maintained a similar behavior regarding the temperature of the exterior surface. The exterior surface of the roofs reached its maximum temperature between 13:30 and 14:00 h. The exterior surface of the insulated terracotta roof (ITR) reached on average 72 ◦C, the insulated gray roof (IGR) 69◦, Insulated White Roof #1 (IWR1) 50 ◦C, and Insulated White Roof #2 (IWR2) 49 ◦C. These temperatures indicated that ITR had on average a temperature of the exterior surface about 3 ◦C higher than the IGR, while IWR1 and IWR2 reduced the temperature of the exterior surface by 19 and 20 ◦C on average, respectively. If the average maximum temperatures of white reflective roofs were compared with the average maximum temperature of ambient air (45 ◦C), IWR1

had a maximum temperature of 5 ◦C above the maximum air temperature, while the IWR2 roof reached a temperature 4 ◦C higher than the maximum air temperature. On the other hand, the IGR and ITR reached a temperature of 23 and 26 ◦C higher than the maximum ambient air temperature, respectively.

**Figure 5.** Thermal behavior of the insulated roof with traditional and reflective coatings in Hermosillo: (**a**) solar irradiance and wind speed; (**b**) temperature of exterior surface and outdoor air; (**c**) temperature of interior surface; (**d**) heat flux of the roofs.

Figure 5c shows the temperature of the interior surface of the four roofs. Due to thermal insulation, the temperatures of the roofs' interior surface had a small oscillation between day and night compared to the cases without insulation. These surfaces reached their maximum temperature between 17:30 and 18:10 h. The temperature of the interior surface of the ITR and that of the IGR reached a maximum temperature of 29.3 ◦C and 29 ◦C, while the surface temperature of IWR1 and IWR2 reached 27.4 ◦C and 27.2 ◦C. This figure demonstrates that the insulation caused the interior temperature of the roofs to remain relatively constant.

Figure 5d shows the behavior of the heat flux of the insulated roofs in Hermosillo. The maximum heat flux traveling through the roofs occurred between 17:30 and 18:10 h. Insulated traditional roofs (ITR and IGR) had an average peak heat flux of 27 and 25.3 W/m2, while insulated white reflective roofs, IWR1 and IWR2, had a peak heat flux of 15.4 and 14.6 W/m2. As mentioned above, the total heat gain of the roof over a day was determined by calculating the area under the heat flux curve of each day. The ITR had an average heat gain of 304 W·h/(m2-day), the IGR of 273 W·h/(m2-day), IWR2 of 140 W·h/(m2-day), and IWR2 of 128 W·h/(m2-day). The ITR located in Hermosillo had a 6% higher heat gain than the IGR, while IWR1 and IWR2 had a 37 and 40% lower heat gain than the IGR.

Finally, the insulated roofs' thermal behavior was simulated for the remaining cities following the same procedure used for Hermosillo. Table 4 presents a summary of the results obtained from the evaluation of the insulated roofs in the four cities of Mexico. The table shows the values for the average peak temperature of the exterior (*Tes*) and the interior surface (*Tis*), the average peak heat flux of the roofs (*QR*), and the average daily heat gain (*HG*). Taking the IGR as the control case, the differences between the peak temperatures (*Tes* and *Tis*) were calculated, along with the peak heat flux (*QR*) and the heat gain (*HG*). The white reflective roofs were able to reduce *Tes* between 17 and 21◦C compared to the IGR. Therefore, they could reduce the *QR* that crossed the roofs by a factor ranging between 39 and 54%. Further, these roofs had an *HG* between 37 and 56% smaller than the ISG. On the other hand, the ITR reached a maximum temperature of 3 ◦C above the IGR. This temperature increment caused an increase of the *QR* between 7% and 15%. Moreover, the ITR increased the *HG* by about 11 and 33%.


**Table 4.** Summary of the thermal evaluation of insulated roof in four cities of Mexico (Average peak values for the different variables).

#### **9. Discussion**

This section discusses the comparison of some of the results obtained in this work with other research available in the literature. Alqalaf and Alawadhi [23] evaluated the thermal effectiveness of a white reflective coating on the exterior surface of a concrete roof in Kuwait. The authors of the previous research built a test cell to perform a series of experiments and then developed a numerical model validated with experimental data. They simulated white and gray roof thermal performance in the season with the highest solar radiation and outdoor air temperature. The results showed that the temperature of the interior surface of the white roof was 6 ◦C lower than the temperature of the gray roof. Further, the white roof caused a reduction of the heat flux of 50%. Because Kuwait's climate is warm and dry, such as the climate of Hermosillo, it is worth comparing the results of the current research with those obtained in [23]. Besides the type of climate, the indoor air temperature and the roofs' thermophysical properties studied in the previous research were very similar to the values used in this work. Table 3 indicates that the temperature of the interior surface of SWR1 was 10 ◦C lower than the temperature of the SGR. At the same time, the heat flux of SWR1 was 42% smaller than the corresponding to the SGR. Here, we mention only SWR1 because this roof had the same absorptance (*α* = 0.2) as the white roof studied in [23]. However, the solar absorptance of the gray roof analyzed by Alqalaf and Alawadhi (*α* = 0.8) was greater than the absorptance of the SGR considered here (*α* = 0.67). Therefore, this main factor for the reduction of the heat flux presented in [23] was more significant than the reduction of the heat flux presented in the current research. On the other hand, as can be noticed above, the temperature reduction presented

by [23] was smaller, but this difference occurred because the thickness of the roof analyzed in Alqalaf and Alawadhi was 0.15 m, which was greater than the thickness of the single roof; therefore, the thermal inertia of the roofs was the other factor that caused the slightly different results. Thus, the information presented above demonstrated that the findings of this research work were consistent with what other researchers have reported.

Another contribution of this research that is important to discuss is the influence of thermal insulation on the roofs' thermal performance. This effect can be obtained by comparing the results presented in Tables 3 and 4. Because the thermal insulation caused the roofs to have an indoor surface temperature with small oscillations, the heat flux crossing the insulated roof was very small compared to the flux of single roofs. Thus, the *HG* of the insulated roofs was around four-times smaller than that corresponding to single roofs regardless of the coating and the city. For instance, using the results for Hermosillo, by comparing the *HG* of the SGR with the *HG* of the IGR, it can be noticed that the first value was 3.8-times greater than the second value. Therefore, thermal insulation could have an essential contribution in reducing heat gains. On the other hand, another action that the results of this research can evaluate is the comparison between the SGR and IWR2; this is the comparison of the traditional roof configuration (SGR) with the roof configuration with two retrofitting techniques (thermal insulation and reflective coating). Using again Hermosillo as an example, the *HG* of SWR was equal to 1675 W·h/(m2-day) (Table 3), and the *HG* of IWR2 was equal to 262 W·h/(m2-day) (Table 4). Comparing the two previous values indicated that a roof with thermal insulation and a solar reflective coating could have a daily heat gain up to 6.4-times smaller than a gray roof without insulation. This result is important; however, thermal insulation installation could be more complex and more expensive than applying a reflective coating. Thus, a life cycle cost analysis is needed to find the more cost-effective configuration of roofs.

#### **10. Conclusions**

A computational tool was used to simulate the thermal behavior of insulated and non-insulated concrete slab roofs with traditional and solar reflective coatings in four cities with warm climates in Mexico. This simulation tool is a computer model based on the finite volume method that numerically solves the heat conduction equation in an unsteady state. The simulations were done using the weather data for the week with the highest outdoor air temperature. Two traditional and two solar reflective coatings installed on the exterior surface of the roofs were considered, and the following was concluded:

Regarding the simulation of the single roofs, SWR2 was the best configuration to minimize the heat transfer. Due to the small solar absorptance of the coating, SWR2 presented a peak exterior surface temperature up to 16 ◦C lower than the temperature of the SGR. Further, the peak interior surface temperature of SWR2 was up to 11 ◦C lower than the SGR. Thus, SWR2 diminished the heat flux and the daily heat gains up to 57% and 54%, respectively.

The insulated roofs simulations indicated that the surface temperature reduction of the exterior surface due to the reflective coatings was more significant than the single roofs. IWR2 was the configuration with the best thermal performance. The maximum temperature reduction provided by IWR2 was 19 ◦C lower than the temperature of the IGR. IWR2 provided a maximum interior surface temperature reduction 1.6 ◦C, which was very small. This effect occurred because the thermal insulation maintained the interior surface with small oscillations. IWR2 reduced the peak heat flux and the daily heat gain up to 54% and 15%, respectively.

This research highlights the importance of selecting the type of coating to be used in building roofs well. In the terracotta coating, this color had a solar absorptance equal to 0.7, which was just a little higher than the absorptance of the gray color of bare concrete (*α* = 0.67). The difference in the solar absorptance for these two roofs may seem insignificant, but as shown in the Results Section, the STR had a daily heat gain between 9 and 11% higher than the SGR. Similarly, the ITR had a daily heat gain between 5 and 12% higher than the IGR. These results demonstrated that even a small increment in the solar absorptance could cause a significant increment in the daily heat gain of the roofs.

Because in buildings situated in warm climates, the roof is a source of unwanted heat, applying a coating with a lower absorptance causes a lower amount of energy to be absorbed by the roof's exterior surface. Therefore, the heat traveling through the roof structure is reduced, and then, the roof exhibits better thermal performance. Since Solar Reflective Coating # 2 was the material with the smallest solar absorptance, this coating improved both the single and insulated roof thermal behavior. Finally, because most buildings in Mexico have bare gray or terracotta roofs, there is a great potential for using reflective coatings as a retrofitting technique in this country. This research demonstrated that white reflective coatings are an excellent alternative to improve the thermal performance of roofs, which could lead to energy savings and mitigating greenhouse gas emissions from buildings.

**Funding:** This research received no external funding.

**Acknowledgments:** The author acknowledges CONACYT-Mexico for the support given through the System of National Researchers program (Sistema Nacional de Investigadores, SNI). The author also acknowledges the Servicio Meteorológico Nacional-Comisión Nacional del Agua (SMN-CONAGUA) for providing the weather data used for the simulations. Finally, the author is grateful to Jesús Xamán from CENIDET for the Finite Volume Method course.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Abbreviations**


#### **Nomenclature**



#### **References**


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