Latent heat storage technologies that use phase change materials (PCM) embedded in lightweight building elements are considered an interesting alternative to sensible storage by heavyweight constructions [(Zalba et al., 2003); (Khudhair & Farid, 2004); (Mavrigiannaki & Ampatzi, 2016)]. These materials can undergo a phase change, typically melting or solidification, and therefore exchange more heat with the environment in terms of latent heat rather than sensible heat storage capacity. In recent times the attention has also been drawn by the coupling of PCMs with cool roofs, studied by either numerical simulation or experiments. Experimental results, by (Karlessi et al., 2011), demonstrated that PCM incorporated in building coatings yields lower surface temperatures than either common coatings or cool infrared-reflective coatings. The numerical investigation by (Aguilar et al., XXXX) extended previous work by considering the impact of PCM embedded in roofing module on cooling energy and showed that solar reflectance is the parameter with the biggest impact. Nonetheless, PCMs may be worthwhile in locations where the reflectance undergo a sharp decrease due to soiling. Moreover, they verified that simulation is a powerful tool for the involved multi-parameter analyses. (Roman et al., 2016) showed through simulation that a PCM allows a sharp decrease of through-roof heat gain at a wide range of albedo. In (Chou et al., 2013) the coupling of a metal sheet cool roofing structure with a PCM was studied in order to absorb the downward heat flow induced by incident solar radiation and then release it back to the environment by convection during the nocturnal cycle. Experimental and numerical analyses showed that the downward thermal flow through the roof into the house can be significantly reduced. Another study was aimed at the development and prototyping of a cool polyurethane-based membrane with PCM inclusion for roofing applications (Pisello et al., 2016). In another, PCM cool roof system was created using PCM doped tiles (Chung & Park, 2016). Experimental results showed that such tiles, during the summer, allow a decrease of surface temperature while maintaining low room temperatures.

Generally speaking, most of the studies have shown that PCM can even both positive and negative peaks of surface temperature and thus improve thermal comfort and reduce cooling energy demand. None, however, seem to have yet paid attention to the specific issue of using PCM to limit condensation and biological growth on cool roof surfaces. This is therefore the topic of investigation in this work.

In order to parametrically investigate the thermal behavior of a cool roof coating coupled with a PCM, a one-dimensional mathematical model of the roof system was implemented in the Matlab programming environment. The model, based on a finite-volume approach with implicit discretization of the time derivatives, takes into account the cyclical variability of the boundary conditions (temperature, solar irradiance, heat transfer coefficients) and the temperature dependence on the thermophysical properties of materials. It has been used to identify the time periods during which the external surface temperature falls below the dew-point and the risk of humidity condensation occurs. In this way, a comparison of the situation with and without a PCM layer below the cool roof surface has been carried out considering surfaces with relatively low thermal emittance.

A flat roof was considered in this study. Hourly weather data on air temperature, sky temperature, wind velocity, solar irradiance and dew-point temperature were obtained over the whole year from the TRNSYS programming environment (TMY data). Representative time evolution patterns are shown in Figure 2.

The heat transfer process is depicted in Figure 3. More specifically, the boundary condition at the external surface of the solid matter (x=0), delimiting the external environment from the waterproofing layer, is expressed as

(1)

where the so-called sol-air temperature Tsol/air (^◦C) is calculated as follows: from the effective external temperature Te (^◦C) and the absorbed fraction of the solar irradiance Isol (W/m2) that results from the surface solar reflectanceρ_sol (0 < ρ_sol < 1):

(2)

The effective external temperature T_e (^◦C) is in turn the average of air and sky temperatures, T_air and T_sky (^◦C), weighted by the external convective and radiative heat transfer coefficients, h_ce and h_re (W/(m^2◦C)):

(3)

Air temperature and sky temperature, as well as the solar irradiance and, consequently, T_e and T_sol/air, are function of the time t (s). The convective heat transfer coefficient h_ce is evaluated from the wind velocity v_wind (m/s) according to ISO 6946 (ISO, 2007a):

(4)

The radiative heat transfer coefficient h_re is again evaluated according to ISO 6946 at the mean absolute external temperature T_me (K), also considering the thermal emittance e_ter (0<e_ter<1), as

(5)

where 0 is the Stefan-Boltzmann constant ( 0=5.67×10⁻⁸ W/(m²K⁴)), and

(6)

An explicit approach is generally followed to evaluate the radiative heat transfer coefficient from the calculated surface temperature.

Concerning thermal interaction between the roof structure and the inhabited space below, the indoor temperature T_i (^◦C) was assumed to be controlled by an appropriate air conditioning system and kept constant at a level adequate to thermal comfort, e.g. T_i=27^◦C for the summer period in the analyses presented here. The boundary condition at the internal surface (x=L), delimiting the roof structure from the indoor space, is expressed as

(7)

Constant values obtained from ISO 6946 were used for both the internal convective coefficient h_ci, taken equal to 0.7 W/(m^2◦C), and the internal radiative coefficient h_ri, evaluated equal to 5.5 W/(m^2◦C) by a formula analogous to eq. (5) for the typical value of the thermal emittance of inner building surfaces, equal to 0.9, and an internal absolute mean temperature of 27^◦C = 300 K.

The same model used in a previous work was exploited for the heat transfer in the solid matter comprised between the external surface (x=0) and the internal one (x=L), including the PCM layer (Barozzi et al., 2009). In fact, the thermal behaviour of a PCM can be mathematically modeled through different approaches. The one adopted here is based on the definition of a fictitious equivalent material whose specific heat c (J/(kg×◦C)) is a function of temperature. Heat absorbed or released by such a fictitious material must be equal to that absorbed or released by the actual material for the same increase or decrease of temperature. This seems consistent with the literature which shows that currently available PCM do not change phase at a precise temperature level, but rather over a more or less narrow temperature interval [(Carbonari et al., 2006); (Farid et al., 2004); (Schossig et al., 2005); (Tyagi & Buddhi, 2007); (Zalba et al., 2003)]. Moreover, the slope of heat absorption or release in the phase change interval is often similar to a gaussian distribution about the central temperature. In the industrial practice, either the central value or the amplitude of the phase change interval are modulated through an appropriate formulation of the PCM (ex: the phase change temperature of paraffin depends on the length of the molecular chains, which can be modulated in terms of statistical distribution about an assigned average value in order to obtain the desired properties).

Going into detail, the equivalent specific heat c of a PCM with latent heat q_sl (J/kg), specific heat c_s (J/(kg×◦C)) of the solid phase (or the low temperature phase in solid-solid transitions) and specific heat c_l (J/(kg×◦C)) of the liquid phase (or the high temperature phase) is represented in this study by a gaussian distribution in a range with amplitude 2×DT_sl about the nominal phase change temperature T_sl (^◦C) (see Figure 3). This is described by the following relationship:

(8)

The thermal conductivity k (W/(m^◦C)) and other thermophysical properties which may vary significantly during the phase transition can also be modelled by assuming, over the phase change temperature interval, a linear variation between the values for the solid and the liquid phases, e,g. k_s and k_l (W/(m·◦C)):

(9)

Nonetheless, for the time being, certainly reliable data on the PCM are missing, so the properties of the solid and liquid phases are assumed to be equal for sake of simplicity, in view of their relatively low differences.

The properties of common building materials as reported in ISO 13786 (ISO, 2007b) for a thick concrete structure with thermal insulation were used for the different layers (see Table 1), whereas a commercial PCM board, DuPont Energain (DuPont, n.d.), was considered for the PCM to be introduced between the waterproof coating and the thermal insulation layer below with a thickness up to 10 mm. More specifically, a product developed by DuPont and called Energain^® was considered, that is a composite PCM wallboard constituted of 60% of microencapsulated paraffin included in a polymeric structure; such a mixture is laminated by aluminium. Since the literature raises doubts on the material specifications (DuPont, n.d.), these were integrated with experimental data from a third party (Le Du et al., 2012) (see Table 2).

The model has been used to identify the time periods during which the external surface temperature T_se (^◦C) falls below the dewpoint T_dp (^◦C), thus explaining the risk of humidity condensation. The process is depicted in Figure 5, where the surface temperature is plotted for a short mid-summer period in case of absence of PCM and presence of a PCM layer with 10 mm thickness.

The frequency of the risk of condensation is represented in Figure 6-a) for a cool roof with thermal insulation but without PCM below the surface. It is then shown in Figure 6-b) and Figure 6-c), respectively, that the risk becomes lower with a 5 mm PCM layer, and much lower with a 10 mm PCM layer. In Figure 6-d) the risk of condensation is represented for a cool roof with thermal insulation and without PCM, but with thermal emittance 0.6 instead of 0.9; in this case one can observe that the reduced thermal emittance limits heat loss toward the sky during the night-time and, consequently, yields a risk of condensation similar to that provided by a 10 mm thick layer of PCM. Nevertheless, it was also found that an always positive change Dq_i (W/m²) of the heat flow entering the inhabited space is obtained with respect to a cool roof surface with typical emittance of 0.9, that is an increased entering heat flow (see Figure 6), whereas a cool roof surface with emittance 0.9 coupled with PCM causes an oscillating change Dq_i of the entering heat flow but with null average.

The frequency of the risk of condensation is represented in Figure 6-a) for a cool roof with thermal insulation but without PCM below the surface. It is then shown in Figure 6-b) and Figure 6-c), respectively, that the risk becomes lower with a 5 mm PCM layer, and much lower with a 10 mm PCM layer. In Figure 6-d) the risk of condensation is represented for a cool roof with thermal insulation and without PCM, but with thermal emittance 0.6 instead of 0.9; in this case one can observe that the reduced thermal emittance limits heat loss toward the sky during the night-time and, consequently, yields a risk of condensation similar to that provided by a 10 mm thick layer of PCM. Nevertheless, it was also found that an always positive change Dq_i (W/m²) of the heat flow entering the inhabited space is obtained with respect to a cool roof surface with typical emittance of 0.9, that is an increased entering heat flow (see Figure 7), whereas a cool roof surface with emittance 0.9 coupled with PCM causes an oscillating change Dq_i of the entering heat flow but with null average.

In this work, it was shown that a PCM can actually increase the nighttime surface temperature of a thermally-insulated cool roof without affecting the overall performance of the cool roofing product. Heat is accumulated in the daytime by the PCM and then released in the night-time, increasing the minumum temperature and thus reducing the risk of falling below the dewpoint temperature, with the start of humidity condensation. Limiting humidity condensation my help preserves the cool roof surface from biological growth, especially for insulated roofs. A similar result can also be provided by a decrease of the thermal emittance at the external surface, but with a penalization of the cool roof performance in terms of heat transmitted to the inhabited space below and also maximum surface temperatures.

A relatively high mass fraction of PCM must be installed in the outer layer, either integrated in a waterproofing membrane or, as a board, placed just below the membrane, or below an outer metal sheet in insulated sandwich components. Effective results were in fact found only for a PCM board with thickness 10 mm and percent content of PCM around 60%. A lower amount of dispersed PCM may lead to an ineffective contribution.

In upcoming work, ambient data for different climatic conditions will be taken into account, considering either arid or very humid climates. Moreover, the analysis will focus on the choice of the phase change temperature, which must be slightly higher than the expected dew point temperature and should probably be optimized depending on the location. A more detailed model of the PCM is also under development, to be supported by experiments. Eventually, integration of the proposed solution in comprehensive dynamic models is a long-term objective.

The author wishes to acknowledge Giulia Santunione, Chiara Ferrari, Susanna Magli, Antonio Libbra and all the other researchers of the Dept. of Engineering “Enzo Ferrari” who provided support and data for this work.