Keywords

1 Introduction

Within Morocco, the construction industry comprises 33% of the overall final energy consumption and faces a considerable yearly increase in energy needs [1]. This upswing is predominantly ascribed to the considerable external thermal fluctuations experienced by building exteriors in warm climates, resulting in substantial energy consumption to maintain a satisfactory internal thermal comfort level. Despite the construction sector occupies a crucial role in the broader energy consumption scenario, it is also offers substantial opportunities to enhance thermal efficiency. This improvement can be realized by elevating the energy performance of construction materials within building envelopes. Furthermore, fired clay bricks continue to be the predominant construction elements in contemporary buildings [2]. This material offers numerous advantages, such as an improved thermal insulation properties attributed to the good insulation capacity of the air confined in the hole [3]. Nevertheless, its effectiveness is constrained in areas characterized by extreme climatic conditions [4]. To address this limitation, the incorporation of PCMs into bricks is suggested. To achieve this, the principal goal of the study is to evaluate the advantages of utilizing PCMs in brick walls. Exploring this integration is being studied as a strategy to decrease the energy needed for cooling buildings in the specific climate of Fez.

Several published papers [5,6,7,8,9,10,11] have highlighted the positive effects of incorporating PCMs into bricks, showcasing their ability to mitigate internal temperature fluctuations. Jia et al. [5] investigated the simultaneous incorporation of Insulation Material (IM) and PCM in bricks. Their results highlighted the distinct thermal adjustment mechanisms of IM and PCM, showcasing improvements in both steady and transient thermal performance, respectively. Furthermore, the outcomes outlined in [6] indicated that the optimal improvements in thermal inertia characteristics and thermal resistance of hollow clay walls were achieved by filling the central holes of hollow bricks with PCM32, coupled with the application of low emissivity paint or the introduction of IM into both internal and external holes. Rehman et al. [7] presented a dual-layer PCMs arrangement designed for walls in diverse climatic conditions, encompassing both hot and cold environments. Their study revealed that using double PCMs effectively upheld a stable inner temperature swings throughout both summer and winter, resulting in decreased energy needs in Islamabad.

In this paper, the main objective is to identify the appropriate type of PCM for integrating into bricks, with the aim of reducing overall cooling building loads in the climate of Fez.

2 Numerical Analysis of the Thermal Behavior of the PCM-Brick

2.1 Physical Model

The numerical study concentrated on evaluating the thermal characteristics of hollow bricks with 12 perforations incorporated with PCMs. The clay brick under examination is equipped with twelve air holes, each surrounded by cylindrical PCM capsules with a diameter of four millimeters and a mass fraction of 20% [8]. Figure 1 depicts the 2D configuration of the modeled brick, encompassing both PCM capsules and air holes. To fulfill the goal of this study, specific simplifications were incorporated.

  • The examination of heat transfer is confined to a two-dimensional analysis.

  • Convection is excluded from consideration within the PCM capsule.

  • The Boussinesq formulation [12] is utilized to approximate the air density.

  • The energy equation neglects the contribution of viscous heat dissipation.

  • In this study, The thermal characteristics of the materials remain unchanged, except for air density (refer to Table 1).

Fig. 1.
figure 1

Schematic illustration of the brick embedding PCMs

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Table 1. The thermal characteristics of the utilized materials
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The equations below delineate the mathematical model grounded on the previously mentioned assumptions:

Coupled heat transmission, involving conduction and convection, is described as follow:

$$ \frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} + \frac{{\partial {\text{v}}}}{{\partial {\text{y}}}} = 0 $$
(1)
$$ \frac{{\partial {\text{u}}}}{{\partial {\text{t}}}} + {\text{u}} \cdot \frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} + {\text{v}}\frac{{\partial {\text{u}}}}{{\partial {\text{y}}}} = - \frac{{\partial {\text{P}}}}{{\partial {\text{x}}}} +\upmu _{{\text{f}}} \left( {\frac{{\partial^{2} {\text{u}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{y}}^{2} }}} \right) $$
(2)
$$ \frac{{\partial {\text{v}}}}{{\partial {\text{t}}}} + {\text{u}} \cdot \frac{{\partial {\text{v}}}}{{\partial {\text{x}}}} + {\text{v}}\frac{{\partial {\text{v}}}}{{\partial {\text{y}}}} = - \frac{{\partial {\text{P}}}}{{\partial {\text{y}}}} +\upmu _{{\text{f}}} \left( {\frac{{\partial^{2} {\text{v}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{v}}}}{{\partial {\text{y}}^{2} }}} \right) -\uprho _{{\text{f}}} {\text{G}}\upbeta _{{\text{T}}} \left( {{\text{T}} - {\text{T}}_{0} } \right) $$
(3)
$$ \frac{{\partial {\text{T}}}}{{\partial {\text{t}}}} + {\text{u}} \cdot \frac{{\partial {\text{T}}}}{{\partial {\text{x}}}} + {\text{v}}\frac{{\partial {\text{T}}}}{{\partial {\text{y}}}} = \frac{{\uplambda _{{\text{f}}} }}{{\uprho _{{\text{f}}} {\text{c}}_{{\text{f}}} }}\left( {\frac{{\partial^{2} {\text{T}}}}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{T}}}}{{\partial {\text{y}}^{2} }}} \right) $$
(4)

The heat transmission through the PCM can be simulated using the following equations:

$$\uprho _{{{\text{pcm}}}} {\text{C}}_{{{\text{p}},{\text{pcm}}}} \frac{{\partial {\text{T}}_{{{\text{pcm}}}} }}{{\partial {\text{t}}}} =\uplambda _{{{\text{pcm}}}} \left( {\frac{{\partial^{2} {\text{T}}_{{{\text{pcm}}}} }}{{\partial {\text{x}}^{2} }} + \frac{{\partial^{2} {\text{T}}_{{{\text{pcm}}}} }}{{\partial {\text{y}}^{2} }}} \right) $$
(5)

where:

$$ \left\{ \begin{gathered}\uprho _{{{\text{pcm}}}} = {\uptheta \rho }_{{\text{s}}} + \left( {1 -\uptheta } \right)\uprho _{l} \hfill \\ {\text{C}}_{{{\text{p}},{\text{pcm}}}} = \frac{1}{\uprho }\left( {{\uptheta \rho }_{{\text{s}}} {\text{C}}_{{{\text{p}},{\text{s}}}} + \left( {1 -\uptheta } \right)\uprho _{{\text{l}}} {\text{C}}_{{{\text{p}},{\text{l}}}} } \right) + {\text{L}}_{{\text{f}}} \frac{{\partial \upalpha _{{\text{m}}} }}{{\partial {\text{T}}}} \hfill \\\uplambda _{{{\text{p}},{\text{pcm}}}} = \uptheta \uplambda _{{\text{s}}} + \left( {1 -\uptheta } \right)\uplambda _{{\text{l}}} \hfill \\ \end{gathered} \right. $$
(6)
$$ \left\{ {\begin{array}{*{20}l} {\uptheta = 1;} \hfill & {{\text{if}}\;{\text{T}}_{{{\text{pcm}}}} \le {\text{T}}_{{\text{m}}} - \frac{{\Delta {\text{T}}}}{2}} \hfill \\ {\uptheta = \frac{1}{{\Delta {\text{T}}}}\left( {{\text{T}}_{{\text{m}}} + \frac{{\Delta {\text{T}}}}{2} - {\text{T}}_{{{\text{pcm}}}} } \right);} \hfill & {{\text{if}}\;{\text{T}}_{{\text{m}}} - \frac{{\Delta {\text{T}}}}{2} \le {\text{T}}_{{{\text{pcm}}}} \le {\text{T}}_{{\text{m}}} + \frac{{\Delta {\text{T}}}}{2}} \hfill \\ {\uptheta = 0;} \hfill & {{\text{if}}\;{\text{T}}_{{\text{m}}} + \frac{{\Delta {\text{T}}}}{2} \ge {\text{T}}_{{{\text{pcm}}}} } \hfill \\ \end{array} } \right. $$
(7)
$$ \upalpha _{{\text{m}}} = \frac{1}{2}\frac{{\left( {1 -\uptheta } \right)\uprho _{{\text{l}}} - {\uptheta \rho }_{{\text{s}}} }}{{{\uptheta \rho }_{{\text{s}}} + \left( {1 -\uptheta } \right)\uprho _{{\text{l}}} }} $$
(8)

Now, let's establish the initial and boundary conditions to solve the mathematical model:

$$\uplambda _{{\text{M}}} \frac{{\partial {\text{T}}\left( {{\text{x}},{\text{y}},{\text{t}}} \right) \, }}{{\partial {\text{x}}}}\left| {_{{{\text{x}} = 0}} } \right. = {\text{h}}_{{1}} \cdot \left[ { - {\text{T}}\left( {{\text{x}} = 0,{\text{y}},{\text{t}}} \right) + {\text{T}}_{{{\text{sa}}}} \left( {\text{t}} \right)} \right]\quad \forall {\text{y}} \in \left[ {0,{\text{H}}} \right] $$
(9)
$$\uplambda _{{\text{M}}} \frac{{\partial {\text{T}}\left( {{\text{x}},{\text{y}},{\text{t}}} \right) \, }}{{\partial {\text{x}}}}\left| {_{{{\text{x}} = {\text{L}}}} } \right. = {\text{h}}_{2} \left[ {{\text{T}}\left( {{\text{x}} = {\text{L}},{\text{y}},{\text{t}}} \right) - {\text{T}}_{{{\text{in}}}} } \right]\quad \forall {\text{y}} \in \left[ {0,{\text{H}}} \right] $$
(10)

With

$$ T_{in} = 26\,^\circ {\text{C}} $$
(11)

and

$$ {\text{T}}_{{{\text{sa}}}} \left( {\text{t}} \right) = {\text{T}}_{{{\text{amb}}}} \left( {\text{t}} \right) + \frac{{\upalpha _{{\text{s}}} }}{{{\text{h}}_{1} }}{\text{I}}\left( {\text{t}} \right) - \frac{{\upxi \Delta {\text{R}}}}{{{\text{h}}_{1} }} $$
(12)

For vertical elements, \(\frac{{\upxi \Delta {\text{R}}}}{{{\text{h}}_{1} }} = 0\) [16].

In warmer day in the Fez region, the ambient temperature \({\text{T}}_{{{\text{amb}}}} \left( {\text{t}} \right)\) is written as follows:

$$ T_{amb} \left( t \right) = 9.8 \times Sin\left( {\frac{2\pi t}{{86400}} - \frac{\pi }{2}} \right) + 35.5 $$
(13)

The solar irradiation \({\text{I}}\left( {\text{t}} \right)\) is formulated as follows:

$$ I\left( t \right) = 695.e^{{ - 0.06\left( {\frac{t}{86400} - 12} \right)^{2} }} $$
(14)

Ultimately, \({\text{T}}_{{{\text{sa}}}} \left( {\text{t}} \right)\) is given by the following equation:

$$ T_{sa} \left( t \right) = 35.5 + 9.8 \times Sin\left( {\frac{2\pi t}{{86400}} - \frac{\pi }{2}} \right) + 28 \times \alpha_{s} \times e^{{ - 0.06\left( {\frac{t}{86400} - 12} \right)^{2} }} $$
(15)

The numerical solution of the mathematical model was conducted using the Galerkin finite element approach with the assistance of COMSOL software.

2.2 Numerical Validation

A comparative analysis is conducted, employing results extracted from the work [17], to assess the precision of the adopted solution procedure. Lachheb et al. [17] computed the temperature swings of the plaster-PCM composite using the finite volume method to solve the energy equation.

Table 2 offers a quantitative comparison of the peaks in the temperature swings of the plaster-PCM composite. The difference between the numerical data extracted from the literature [17] and the findings of this study is observed to be less than 1%.

Table 2. Transient temperature for different wallboard-PCM thickness
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Furthermore, to affirm the precision of the numerical approach employed in this study for investigating conjugate heat transmission, an assessment was conducted a physical model encompassing natural convection flows in a square cavity [18].

Figure 2 illustrates the qualitative comparison of the dimensionless temperature distribution within the cavity. Notably, the data from the literature [18] and the results obtained through our simulations exhibit a remarkable similarity, as depicted in Fig. 2. Thus, the numerical technique utilized in this study demonstrates sufficient precision.

Fig. 2.
figure 2

The dimensionless temperature for Ra = 105 and Pr = 0.2: (a) current simulation. (b) the outcomes documented in [18].

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3 Results and Discussion

To evaluate the impact of PCM type, numerical simulations were conducted on PCMs with three distinct melting temperatures (32 ℃, 37 ℃, and 42 ℃). Figure 3 illustrates the transient internal temperature of the brick for various PCM type.

Fig. 3.
figure 3

Temperature swings for different PCM types

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Fig. 4.
figure 4

The time lag for different PCM types

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The findings reveal that the energy efficiency of the three PCM-based bricks surpasses that of the standard hollow brick without PCM. In comparison to the brick without PCM, bricks featuring a PCM mass fraction of 20% (PCM37 and PCM42) exhibit a minor decrease in temperature peak and a low phase shift. The PCM32-based brick shows a significant decrease in peak temperature and a substantial time lag (see Fig. 4) when compared to bricks without PCM. Considering indoor boundary conditions, the optimal Tm should closely align with the average outdoor thermal temperature. This alignment boost the benefit of a PCM's latent heat capacity, enhancing the potential to minimize total cooling loads by augmenting the wall's thermal inertia.

In conclusion, under the studied climatic conditions, PCM32 emerges as the most suitable choice when compared to the other two PCMs (PCM37 and PCM42). It effectively reduces the oscillations of the indoor thermal wave while maintaining a satisfactory level of comfort.

Figure 5 depicts the heat flux swings for the three types of PCM integrated into the bricks. It is evident that incorporating any PCM into brick permits attenuate the inner heat flux swings. Nevertheless, this decrease remains below 12% for the brick with PCM42 when compared to the brick without PCM. This outcome can be attributed to the low molten fraction of PCM42, as the external temperature consistently stays below its melting point for most of the time, rendering its latent heat capacity practically inactive.

On the other hand, it is evident that utilizing PCM32 permits attenuate the inner heat flux swings, amounting to approximately 34% compared to the brick without PCM.

Fig. 5.
figure 5

Heat flux swings for different PCM types

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Analyzing the variations in total liquid fraction enhances the comprehension of the results obtained in this section. Figure 6 illustrates that PCM42 has a low molten fraction (total liquid fraction less than 0.15), rendering its latent heat storage practically inactive. Consequently, incorporating PCM42 into bricks yields the lowest improvement compared to the other two PCMs. In contrast, PCM32 and PCM37 remain partially solid (partially liquid), activating their capacity for latent heat storage. Moreover, the molten percentage of PCM32 surpasses that of PCM37, indicating that the stored latent heat in the PCM32-based brick is higher.

Fig. 6.
figure 6

Liquid fraction for different PCM types

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4 Conclusion

This paper underscored the benefits of incorporating a PCM into bricks, with a specific PCM recommendation tailored to the Fez climate. The findings demonstrated that integrating a 20% mass fraction of PCM with a Tm of 32 ℃ resulted in a substantial 34% reduction in inner heat flux swings and a 2.5-h delay in the infiltration time of the external temperature swings into the indoor space compared to the brick without PCM. Furthermore, the outcomes suggested that the utilization of PCM significantly mitigates fluctuations in interior temperature. Consequently, the incorporation of PCMs in hollow bricks proves effective in maintaining satisfactory internal thermal comfort while concurrently reducing the energy required for cooling buildings.