Keywords

1 Introduction

Buildings consume about one-third of the total energy usage and emit around 24% in terms of CO2. In Indian building sector, residential and commercial buildings share 23 and 12% of the total electricity demand, respectively [1]. The electrical energy demand in the buildings will continue to rise, as it is estimated that the construction industry in the country is growing at a rapid growth rate of over 9% per year [2]. Therefore, energy conservation in buildings is a major challenge nowadays. To reduce heat ingress inside the building through building envelop, one of the important methods is incorporating thermal insulation [3, 4]. In similar manner, another method is utilization of latent heat thermal energy storage (LHTES) materials. LHTES using phase change materials (PCMs) can improve the energy performance and indoor thermal comfort of building.

PCMs provide a large heat capacity over a limited temperature range, and they can act like an almost isothermal reservoir of heat. Due to its ability to exhibit high enthalpy of fusion in a relatively small volume, PCMs store or release large amounts of energy as latent heat during melting and solidification. The ability of a material to store a large quantity of heat as latent heat without undergoing change in temperature promises wide use of PCM in construction industries. PCM included inside concrete walls and roofs of buildings can reduce fluctuation of room air temperature which brings thermal comfort inside the building.

Soares et al. [5] explored the evaluation of how and where PCMs are used in passive latent heat thermal energy storage (LHTES) systems and how these construction solutions contribute to building’s energy efficiency. It was concluded that improvement can be made in indoor thermal comfort using PCM in LHTES systems. Pomianowski et al. [6] studied the various PCM technologies integrated in building applications. The methods to determine the correct thermal properties of PCM materials were identified and discussed. Also, the procedures to determine the energy storage and saving potential were explored. Many authors have studied the different methods of incorporating PCM in building walls [7–12], and a detailed review of the applications of PCM in buildings can be found in the literature [13–16].

Kylili and Fokaides [17] investigated the effect of natural convection in the liquid phase of PCM as it melts under incident solar radiation. They also highlighted the importance of natural convection within the liquid PCM using numerical simulation. They identified several factors which can directly improve the thermal performance of the PCM. Waqas and Kumar [18] studied the use of PCM in hot and dry climate of South Asia region. It was reported that PCM can be used as better heat source for ambient air during hot daytime. The stored heat can be easily discharged due to cooler night temperatures.

Researchers have also conducted studies incorporating PCMs in the building roofs, which is major source of heat ingress inside the building. Alqallaf and Alawadhi [19] carried out study on the thermal performance of building roofs with cylindrical holes containing PCM. The objective of incorporating PCM in roof was to reduce heat gain inside a room during peak hours of electricity demand by utilizing its latent heat of fusion. A similar study was conducted by both of them [20] where building roof was constructed with vertical conical holes containing PCM. The PCM n-eicosane filled inside conical frustum showed improved performance with a heat flux inside the room reduced up to 39%.

Vakilaltojjar and Saman [21] developed a latent heat storage system encompassing two different PCMs which can be utilized for producing thermal comfort during both summer and winter months. Ravikumar and Sirinivasan [22] reported that heat ingress inside the room can be attenuated by about 50% when the PCM is integrated in the roof of a building. Hamza et al. [23] studied the consequences of integrating two different PCM layers in the roof of a building. They also evaluated the optimum thickness and location of the PCMs to minimize energy usage while maintaining thermal comfort inside the building.

Recently, melting and solidification phenomenon of the PCMs has been investigated experimentally [24–26] by some authors. Pasupathy et al. [27] conducted experimental investigation to analyze thermal performance of the roof utilizing PCM for Chennai City, India. They also carried detailed study on the effects of variation in the ambient condition through the year, variation in heat transfer coefficient on the outer surface of the roof, and the PCM panel thickness.

Several practical experiments have been carried out to determine thermal performances of different PCMs under different climatic conditions. But for estimating the performances of a large number of PCMs in different climatic conditions, a mathematical model will be very useful for quick assessment and comparison purposes. This study involves the development of such a mathematical model along with its application for computing and comparing the thermal performances of three different PCMs in three different climatic zones. Also the effects of different climatic parameters are combined into a single parameter which can determine the relative thermal performance and suitability of PCM in different climates.

2 Problem Statement and Modeling Method

The mathematical model and the numerical solution methodologies for a PCM-embedded concrete roof system are presented in this section.

2.1 Problem Statement

When solar radiation is incident on the PCM-embedded roof, it absorbs heat. At the same time, the PCM roof exchanges heat with the ambient and room interior by convection. If heat input to the PCM is more than heat lost by the PCM, then there is accumulation of heat in the PCM. This accumulated heat serves to change the phase of the PCM from solid to liquid while at constant temperature. If heat accumulation proceeds further, the whole PCM melts and after that sensible heating (superheating) of the liquid occurs.

If the heat input to the PCM is less than the heat loss by the PCM, then there will be reduction in the net heat stored in the PCM. This will facilitate phase change from liquid to solid. If the whole PCM solidifies and still the input radiation is less than the heat lost by PCM, there will be sensible cooling (subcooling) of the solid PCM.

It is desirable to minimize sensible heating and cooling of the PCM as the aim of incorporating PCM in concrete roofs is to keep inside temperature constant. Once sensible heating or cooling of PCM begins, its thermal behavior becomes similar to that of concrete material and it ceases to serve its purpose of latent heat storage.

A change in the geographical region results in variation of climatic parameters which alters the performance of PCM. The variation of performance of PCM with respect to variation of climate is discussed in the present study.

2.2 Model Geometry

The schematic of the geometry is presented in Fig. 1. The concrete properties are typically those of a reinforced concrete [28].

Fig. 1
figure 1

Dimensions of the PCM-embedded roof

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Following are dimensions of the roof containing PCM:

  • Thickness of outer concrete = 12 cm

  • Thickness of PCM = 2 cm

  • Thickness of inner concrete = 12 cm

  • Area = 2 m × 2 m

2.3 Mathematical Model

For the mathematical model of the above-stated problem, following assumptions are made:

  1. i.

    One-dimensional heat transfer (x-direction only).

  2. ii.

    Temperature is uniform over the PCM surface.

  3. iii.

    15% void is maintained initially in the space inside the concrete roof where the PCM is incorporated (to accommodate the volumetric changes in phase transition and possible superheating/subcooling of PCM).

  4. iv.

    Temperature inside the room is kept constant at 300 K by some air-conditioning unit. The purpose of incorporating PCM in the concrete roof is to minimize temperature fluctuation on the inner surface of the concrete roof so that variation in convective heat transfer between the inner surface of the concrete roof and room air is reduced.

The governing equation and the boundary condition are developed as below. Governing equation for heat transfer by conduction with no internal heat generation is given by

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\partial^{2} T}}{{\partial z^{2} }} = \rho C{\text{d}}T/k{\text{d}}t $$
(1)

where x, y, and z are three mutually perpendicular directions. ‘T’ is the temperature of the heat-conducting material having density ‘ρ,’ specific heat capacity ‘C,’ and thermal conductivity ‘k.’ ‘dT’ is the change in temperature in time ‘dt.’

Governing equations for latent and sensible heating of PCM can be expressed as

$$ \left( {Q_{\text{in}} - Q_{\text{out}} } \right){\text{d}}t = mL_{\text{f}} \quad \left( {{\text{Phase}}\,{\text{transformation}}} \right) $$
(2)
$$ \left( {Q_{\text{in}} - Q_{\text{out}} } \right){\text{d}}t = mC_{\text{l}} {\text{d}}T\quad \left( {{\text{Superheating}}\,{\text{of}}\,{\text{liquid}}} \right) $$
(3)
$$ \left( {Q_{\text{out}} - Q_{\text{in}} } \right){\text{d}}t = mC_{\text{s}} {\text{d}}T\quad \left( {{\text{Sub}}\,{\text{cooling}}\,{\text{of}}\,{\text{solid}}} \right) $$
(4)

where Q in is the heat input to the PCM and Q out is the heat going out of the PCM. The inner and outer concrete layers are subdivided into smaller nodes of size equal to that of the PCM. Basic energy conservation equation is applied to all the nodes considering all the thermal properties of the concrete and PCM.

The current heat transfer problem includes two independent variables x (along the direction of roof thickness) and t (time) and a dependent variable T (temperature).

$$ T = f\left( {x,t} \right) $$
(5)

For transient one-directional heat transfer, the time derivative of temperature is written in the following explicit form [29] as shown in Fig. 2.

Fig. 2
figure 2

Transient heat transfer along the thickness of the PCM-embedded roof

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$$ \frac{{{\text{d}}T}}{{{\text{d}}t}} = \frac{{T\left( {x,t +\Delta t} \right) - T\left( {x,t} \right)}}{{\Delta t}} $$
(6)

The nodal form of Eq. (6) is given as

$$ \frac{{{\text{d}}T}}{{{\text{d}}t}} = \frac{{T_{i,j + 1} - T_{i,j} }}{{\Delta t}} $$
(7)

where the nodal variable ‘j’ is defined by t = (j − 1) × Δt and j = 1 corresponds to t = 0.

The x (thickness) derivative of temperature is expressed by the following explicit equation.

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} = \frac{{T\left( {x -\Delta x,t} \right) - 2T\left( {x,t} \right) + T\left( {x +\Delta x,t} \right)}}{{\Delta x^{2} }} $$
(8)

The nodal form of Eq. (8) is as follows:

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} = \frac{{T_{i - 1,j} - 2T_{i,j} + T_{i + 1,j} }}{{\Delta x^{2} }} $$
(9)

Combining Eqs. (6) and (8), we get the explicit Eq. (10) which facilitates temperature calculation at any node at any time step.

$$ T\left( {x,t +\Delta t} \right) = \frac{{\alpha\Delta t}}{{\Delta x^{2} }}\left[ {T\left( {x -\Delta x,t} \right) + T\left( {x +\Delta x,t} \right)} \right] + \left[ {1 - 2\frac{{\alpha\Delta t}}{{\Delta x^{2} }}} \right]T\left( {x,t} \right) $$
(10)

The nodal form of Eq. (10) can be expressed as follows:

$$ T_{i,j + 1} = \frac{{\alpha\Delta t}}{{\Delta x^{2} }}\left[ {T_{i - 1,j} + T_{i + 1,j} } \right] + \left[ {1 - 2\frac{{\alpha\Delta t}}{{\Delta x^{2} }}} \right]T_{i,j} $$
(11)

For the first element (i = 1) which is exposed to ambient, following equation is used to compute the temperature at any time step t.

$$ T_{1,j + 1} = T_{1,j} \left[ {1 - 2F_{\text{o}} \left( {1 + B_{i} } \right)} \right] + 2F_{\text{o}} B_{\text{i}} T_{\text{o}} + 2F_{\text{o}} T_{2,j} $$
(12)

where F o is the Fourier number defined by \( F_{\text{o}} = \frac{{\alpha\Delta t}}{{\Delta x^{2} }} \) and B i is Biot number defined by \( B_{{i}} = \frac{{h\Delta x}}{k} \).

The case discussed in the study involves heat input to outer concrete by radiation, conduction of heat through PCM and concrete. Then, there is convective heat transfer between ambient and outer concrete surface and again between inner surface of concrete and room. Such boundary condition is known as Robin-type boundary condition. Since rate of heat transfer varies with time, the analysis is transient. For the inner concrete surface, the heat transfer coefficient for horizontal surface is taken as h i  = 7.7 W/m2-K from [28]. Considering the effect of wind velocity, average outer surface heat transfer coefficient is taken as h o = 22.7 W/m2 K. Also from [30], monthly average solar insolation (global radiation, I g and water precipitable, w) and ambient temperatures, Ta, of different cities of Bikaner (hot–dry), Chennai (warm-humid), and Hyderabad (composite) in India have been taken. Summer and monsoon months of May, June, and July are chosen for the assessment of thermal performance of the PCM.

Sol-air temperature for roof has been calculated from the collected solar insolation data for the different cities from the below expression [31]:

$$ T_{\text{sol}} = T_{\alpha } + \frac{{{\text{abs}}\,I_{g} }}{{h_{\text{o}} }} - \frac{{{ \in \Delta }R}}{{h_{\text{o}} }} $$
(13)

where T sol is sol-air temperature, abs is absorptivity of the surface (taken as 0.8 for the concrete roof), ϵ is emissivity of the surface (taken as 0.8 for concrete roof), and ΔR is long-wavelength radiation exchange (taken as 63 W/m2 for the horizontal surface) [28].

3 Methodology

The general procedure for the numerical simulation can be summarized in the following steps:

  1. 1.

    Data collection and material selection.

  2. 2.

    Validation of the numerical model.

  3. 3.

    Initial simulation and check for the cycle independence.

  4. 4.

    Final simulation run, save, and stop.

3.1 Data Collection and Material Selection

The numerical simulation requires solar irradiance on horizontal surface, ambient temperature, and precipitable water which has been presented in the Tables 1, 2, and 3. The data has been collected from the weather files available on the EnergyPlus Web site [30]. The weather data contains database of average of 25 years measured values for different cities.

Table 1 Monthly mean hourly global solar irradiance, ambient temperature, and precipitable water for Bikaner location
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Table 2 Monthly mean hourly global solar irradiance, ambient temperature, and precipitable water for Chennai location
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Table 3 Monthly mean hourly global solar irradiance, ambient temperature, and precipitable water for Hyderabad location
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Sol-air temperature for roof has been computed using the Eq. 13. Hourly variations of sol-air temperatures for roof along with ambient temperature for three consecutive days are depicted in Fig. 3.

Fig. 3
figure 3

Variation of ambient temperature and sol-air temperature for the selected locations (Bikaner, Chennai, and Hyderabad) w.r.t. time

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PCMs considered in the study have melting temperatures in the range of 302–309 K. Reason for selecting the PCMs in this temperature range is that their melting temperatures are closer to the ambient temperature in the daytime. Therefore, the PCMs can be easily get melted by absorbing heat during daytime and getting have enough time to release the heat during nighttime. If we choose PCMs having lower range of melting temperature, most of the PCMs will easily get melted to liquid state in short time and no more heat will be added. Also, because of high ambient temperature during nighttime, they will not be able release heat and will remain in liquid state. Other important properties which influenced the results are latent heat of fusion (L f), thermal conductivity (k), and specific heat (C). The properties of PCM considered in the study are listed in Table 4 [26, 16].

Table 4 Properties of PCMs
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3.2 Numerical Model Validation

The results of the numerical model are validated by comparing with the results of the experimental work carried out by Pasupathy et al. With similar input data such as PCM properties, incident solar radiation, ambient temperature variation, the numerical model can successfully replicate the temperature variation profile of PCM for a diurnal cycle as obtained by Pasupathy et al. (Fig. 4). Slight disconformities occur because the incident solar radiation values obtained from EnergyPlus may deviate to some extent from the actual values of incident solar radiation during the experiment of Pasupathy et al.

Fig. 4
figure 4

Validation of the numerical model

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4 Results and Discussion

The numerical simulation yields the values of net heat gain into the room through the roof for the months of May, June, and July in the cities Bikaner, Chennai, and Hyderabad. A few sample plots of net heat gain versus time have been depicted in Figs. 5, 6, and 7. From net heat gain data, average daily heat gain through the roof is calculated for the three cities and for the three months (Tables 5, 6, 7, 8, 9, and 10). Two different cases are studied for the three cities. First is without considering the effect of wind (still air), and the second includes the effect of wind on heat gain and PCM performance.

Fig. 5
figure 5

Net heat gain through the roof in the presence of wind (Bikaner, June)

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Fig. 6
figure 6

Net heat gain through the roof in the absence of wind (Hyderabad, July)

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Fig. 7
figure 7

Net heat gain through the roof in the presence of wind (Chennai, May)

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Table 5 Average daily heat gain in the absence of wind (Bikaner)
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Table 6 Average daily heat gain in the absence of wind (Chennai)
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Table 7 Average daily heat gain in the absence of wind (Hyderabad)
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Table 8 Average daily heat gain in the presence of wind (Bikaner)
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Table 9 Average daily heat gain in the presence of wind (Chennai)
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Table 10 Average daily heat gain in the presence of wind (Hyderabad)
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4.1 Thermal Performance of PCM Roof in the Absence of Wind

The heat transfer coefficient for a horizontal surface in contact with still air can be conveniently taken as 7.7 W/m2K. Under such condition, the performances of the three PCMs (BioPCM, OM32, and OM35) are evaluated in three different places which belong to three different climatic zones of India. All the three PCMs give maximum average daily heat gain for the month of May (Tables 5, 6, 7, 8, 9, and 10). It is observed that the amount of precipitable water quantity is minimum in May (Tables 1, 2, and 3) which is an indicator of clear sky with more incident solar radiation on the roof. Ambient temperature is also high in May. All these factors lead to increased heat gain into the room. In May, BioPCM gives minimum average daily heat gain into the room and OM32 gives maximum heat gain into the room out of the three PCMs studied. In June and July, the precipitable water quantity is greater than in the month of May. This indicates a relatively overcast weather in June and July resulting in less amount of solar radiation incident on the roof. As a result, heat gain into the room reduces. A change in the trend of relative performances of PCM is observed in June and July. Now, BioPCM gives minimum heat gain into the room and OM35 gives maximum heat gain. Heat gain for OM32 lies in between that for BioPCM and OM35. The heat gain values through all PCMs are less than the corresponding values for a bare concrete roof.

4.2 Thermal Performance of PCM Roof in the Presence of Wind

The effect of wind is primarily to increase heat transfer rate from the roof to the air resulting in a decrease in the average daily heat gain to the room. It is observed that the relative performances of all the three PCMs follow the same trend in the presence of wind. Here, BioPCM gives the minimum average daily heat gain and OM35 gives the maximum average daily heat gain (Tables 8, 9, and 10). The performance of OM32 falls between BioPCM and OM35. The heat gain values through all PCMs are less than the corresponding values for a bare concrete roof.

Referring to Tables 5, 6, and 7, it is observed that the relative performances of the three PCMs change above and below a critical value of average daily heat gain through the bare concrete roof. If we look specifically in Table 5, we can observe the shift in the trend of relative performances of PCM. In third column (June month), the average daily heat gain through bare concrete is 16,748 kJ/day and OM35 gives maximum heat gain through the roof out of the three PCMs. But if we look at the 3 months’ average column where the average daily heat gain through bare concrete is 17,154 kJ/day, then OM32 gives the maximum heat gain. Therefore, the critical value lies between 16,748 and 17,154 kJ/day. And similar trend can be observed in all other tables. Whenever the value of average daily heat gain through bare concrete is above 17,000 kJ/day, the trend of relative performances of PCMs is different than the cases with average daily heat gain through bare concrete being lower than 17,000 kJ/day. It is difficult to pinpoint the exact value in the interval between 16,748 and 17,154 kJ/day in Table 5 where the shift in the trend occurs primarily because average daily heat gain cannot be varied as an independent variable in the mathematical model as it is the combined effect of all the climatic and roof configuration parameters. Therefore, a ballpark number of 17,000 kJ/day can be considered for the shift in relative performances of the PCMs.

Thus, the combined effect of variation in incident solar radiation, ambient temperature, precipitable water quantity, and wind speed for different climatic zones on the relative performances of PCMs can be summarized into a single factor which takes into account all these climatic factors. This single factor is average daily heat gain into the room. Firstly, average daily heat gain into the room through bare concrete roof is computed. If average heat gain through the concrete roof is below 17,000 kJ/day, then BioPCM is the most suitable PCM out of the three for all the climatic zones studied, while OM35 is the least suitable one. If the value exceeds 17,000 kJ/day, then BioPCM is the most suitable PCM and OM32 is the least suitable PCM. This conclusion is supported by the fact that the three PCMs’ relative performance depended on this threshold value of average daily heat gain in all the three different climatic zones. In order to account for roofs of different dimensions, heat gain per unit area of the roof can be considered. For a horizontal plane roof of any surface area, if the average daily heat gain through unit surface area of the bare concrete roof is more than 4250 kJ, then OM32 is the least suitable PCM and if the average daily heat gain per unit area of the bare concrete roof is less than 4250 kJ, OM35 is the least suitable PCM (BioPCM being the best PCM in both the cases).

Another round of computation is performed by taking PCM thickness of 4 cm (in place of 2 cm PCM thickness in the previous cases thereby increasing the total thickness of the PCM roof by 2 cm) and in the absence of wind. The results obtained from the numerical model are presented in Tables 11, 12, and 13. As can be seen from the new results, the critical value of average daily heat gain through bare concrete still lies within the interval of 16,748–17,154 kJ/day as determined from Table 5. Table 11 further shows that the trend shift occurs somewhere near 16,835 kJ/day which is close to 17,000 kJ/day as determined before. Changing PCM and roof thickness may affect the value of average daily heat gain for a particular place in a particular month, but the critical value where the shift in the trend of relative performances of the PCMs occurs remains unaltered.

Table 11 Average daily heat gain in the absence of wind (Bikaner) with 4 cm PCM thickness
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Table 12 Average daily heat gain in the absence of wind (Chennai) with 4 cm PCM thickness
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Table 13 Average daily heat gain in the absence of wind (Hyderabad) with 4 cm PCM thickness
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5 Conclusion

The comparative study demonstrates that heat gain through the roof can be considered as a deciding factor which takes into account several climatic factors affecting the performance of PCM in building roofs. Suitability of PCM can be determined by computing the heat gain through the bare concrete roof. The temperature of the inner surface of the roof can be measured by simple temperature measuring instruments, and applying convective heat transfer equation between inner surface of roof and room interior (kept at constant temperature by air conditioner), one can compute the heat gain. After this, a suitable PCM can be selected based on the computed value. This method is an easy alternative to determine relative performances of PCMs in different geographical regions rather than considering the effect of each climatic factor (such as ambient temperature, wind, precipitable water quantity) individually on the performance of PCM and then drawing an overall conclusion.