Thermal Performance and Energy Efficiency Evaluation of Building Envelopes Incorporating Trombe Walls, PCM, and Multi-Alveolar Structures in Tunisian Climate

Abstract

Solar energy is one of the most promising solutions for improving building energy efficiency. Among passive heating systems, the combination of a Trombe wall, phase change materials (PCM), and multi-alveolar structures (MAS) stands out. This configuration enhances the wall’s ability to absorb solar heat and distribute it evenly throughout the interior. This study evaluated thermal comfort by examining the effects of phase change materials and multi-alveolar structures combined with a Trombe wall on the thermal behavior of a building and improving the thermal inertia of brick walls. Numerical simulations using Visual FORTRAN were conducted to evaluate the thermal properties of different configurations under the climatic conditions recorded in Hammam Sousse, Tunisia. The results show that the integration of the Trombe wall and PCM has a significant impact on interior temperature stability, energy consumption, and overall thermal comfort. The combined effect of the MAS and PCM with the Trombe wall improved heat gain in winter and spring, reaching a low thermal damping factor of 40% in March, reducing heating power, and optimizing thermal comfort for occupants.

1. Introduction

The building sector accounts for approximately 40% of global energy consumption and a significant share of greenhouse gas emissions. Passive solar systems, such as Trombe walls, PCM, and MAS, have emerged as promising solutions for improving energy efficiency and thermal comfort in buildings.

1.1. Trombe Wall Incorporated into the Buildings’ Façade

The use of a Trombe wall is an excellent method for maximizing solar flux and reducing the heating load. The use of solar walls is one of the main experimental and numerical studies in the literature that promotes thermal comfort in buildings. A Trombe wall that was 3 m high and with apertures 20 cm wide was the subject of a study by Corasaniti et al. [1]. According to this study, the thickness of the air spaces should be between 25 and 30 cm to achieve improved thermal efficiency, according to this study. The simulation findings for a typical Madagascar habitat were reported by Boyer et al. [2]. They presented the benefits of this heating method, including its low cost and passive nature, and showed that these storage sensor walls can greatly enhance comfort levels. Stazi et al. research [3] focused on the energy performance of a building fitted with Trombe walls in a Mediterranean environment. With 12.2% energy savings, the solar wall was proven to be a successful device in temperate zones. Significant energy reductions can be achieved with both solar and Trombe walls, as demonstrated by Simões et al. [4]. Trombe wall technology preserved the cooling season while reducing the need for heating by over 20%. Briga-Sa et al. [5] discovered that adding a Trombe wall to the construction envelope can lower the amount of heating energy used by 16.36%. The findings also demonstrate that the proposed methodology offers a reliable way to determine the thermal performance of the Trombe wall.

1.2. Phase Change Materials Incorporation into the Building Envelope

Several numerical and experimental studies have shown that PCM can minimize the heating and air conditioning loads inside buildings and ensure the thermal comfort of occupants. Innovative and advantageous solutions have been implemented. Simulated surveys conducted in two rooms of a new university department on the AvEIRO campus by António et al. [6] served as the basis for this PCM study. They proved that the PCM reduced the overheating by 7.23%, indicating a PCM efficiency of 35.49%. By using a PCM in one of the parts, a reduction in overheating of about 34% was achieved during the optimization procedure. In terms of financial evaluation, the use of PCM reduced cooling requirements, and a payback period of 18 years was achieved. Al-Yasir and Szabo [7] have conducted an experimental investigation in two similar rooms, one designated as the PCM and the other as the reference room. These rooms were constructed and tested under the intense heat of Al Amarah City, Iraq. When building the PCM room, factors like PCM concrete capsules are the most thermally efficient, and the perfect thickness and placement of the PCM layer in the ceiling are taken into account. Experimental studies have shown that the ability of a building envelope to withstand high external temperatures can be significantly enhanced by incorporating a PCM. According to Gobinath et al. [8], walls with phase change materials (PCM) can lower the heat load by 10–30% compared to walls without PCM, and they can also lower the indoor air temperature of the room by roughly 2–4°. Taking into consideration the climate of northern Morocco, Yassine et al. [9] improved the architecture of a multi-zone air-conditioning home using PCM. The aim of this optimization is to reduce the heating and cooling loads. The results demonstrated that, in comparison with the reference home without PCM, the optimal design achieved lower energy usage. Considering the Riyadh climate, Adnan et al. [10] showed that the use of PCM reduced the energy consumption of buildings. PCM was included in both modes. In the first scenario, the PCM was treated as if it had undergone a phase change, while in the second scenario, the PCM in the walls and roof experienced a phase change. According to the data, variations in the PCM thickness decreased the heat transfer by 25.7%, 35%, and 47.1% while also lowering the energy consumption, which also indicated that the quantity of PCM injected into the ceiling influences the heat exchange.

1.3. Integration of the Trombe Walls That Includes a Phase Change Material

To further reduce heating demand, integrating phase change materials (PCMs) into Trombe wall systems has proven to be an effective passive solution. Numerous numerical and experimental studies have examined the impact of this combination on building thermal performance. Li et al. [11] demonstrated that integrating PCMs into Trombe walls improves nighttime heat release in winter, thus addressing a major limitation of conventional designs. However, they observed that higher phase-transition temperatures negatively affected thermal comfort. Their findings recommend placing PCMs near the interior surface of the wall to promote heat accumulation and improve internal heat transfer while preventing excessive heat loss through the exterior wall. This configuration reduced the indoor discomfort duration (IDD) and integrated discomfort degree hours (IDH) by 7.01% per year. Yuan et al. [12] highlight the strong interdependence between ventilation strategy, PCM type, and its placement on indoor thermal comfort. They found that positioning PCM25 on the inner layer of the Trombe wall, combined with appropriate ventilation, achieved the lowest integrated thermal discomfort of 12,857 discomforts. Yue kuan and Chuck Wah [13] developed a Trombe wall composed of double-layer PCM panels (PCM-VTW). Their results, based on the predicted mean vote (PMV) and percentage of dissatisfied people (PPD), revealed improved comfort in summer (PMV = 0.97, PPD = 12.5%) and winter (PMV = −0.32, PPD = 9%). Sadineni et al. [14] examined various passive building systems and highlighted the effectiveness of Trombe walls, advanced insulation, energy-efficient glazing, and green roofs in reducing energy consumption. According to Elsaid et al. [15], the combination of Trombe walls and PCMs can achieve energy savings of up to 55% in winter and 36% in summer. Similarly, Mazurek [16] found that, in cold continental climates, a Trombe wall with PCM can reduce the annual heating load by 56.6%, compared to 36% for conventional systems. Experimental research conducted in the Netherlands by Tenpierik et al. [17] on dynamic solar facades with PCMs confirmed their applicability in various climatic conditions, which was supported by multiparameter simulations. Zhou et al. [18] proposed climate optimization of Trombe wall configurations with PCMs, achieving up to 118 kWh/m². Askari and Jahangir [19] presented a double-layer PCM Trombe wall design, resulting in a 39% reduction in winter energy demand. Zhou et al. [20], using CFD simulations, showed that the dynamic operation of a Trombe wall incorporating phase change materials (PCM) achieves energy savings of up to 97.6%. Furthermore, Vázquez-Beltrán et al. [21] analyzed the influence of PCM position, melting point, and envelope materials, revealing significant thermal storage (up to 27.7%) and accumulated heat release during the night. The results confirmed that PCM integration was optimized in the walls. Trombe walls constitute a strategy to increase thermal insulation, reduce energy consumption, increase indoor comfort, and improve buildings exposed to extreme or fluctuating climatic conditions.

1.4. Multi-Alveolar Envelope Systems

The use of smart materials is an effective method for increasing the thermal inertia of a building envelope. The use of multi-alveolar structures is one of the primary experimental and numerical studies in the literature that addresses transfers in cells (as a method of variable thermal insulation). Seki et al. [22] investigated heat transmission by natural convection in a cavity filled with various fluids. The researchers proposed relationships between the Nusselt, Prandtl, and Grashof numbers for the cavities. Vullierme and Boukadida [23] evaluated the heat flux transmission through the MAS, encompassing convection and radiation in both the passing and insulating directions. These results enabled the following law to be established: ${\mathrm{H}}_{\mathrm{t}}=\mathrm{\gamma }{(\mathrm{\Delta }\mathrm{T})}^{0.25}$.

N. Lajimi and N. Boukadida [24] explored heat transmission mechanisms in a two-zone construction with identical walls separated by an adjacent wall on the eastern side. This eastern wall may be constructed as a 5.7 cm thick layer of polystyrene or as a multi-cell element with different thermal isolation properties, depending on the heat transmission direction. They proved that placing the multi-alveolar structure on the exterior led to lower energy usage in winter when it was oriented towards the conductive side, and in summer when it was oriented towards the insulation side [25]. This study aims to evaluate the combined impact of a Trombe wall, PCM, and MAS on the thermal performance and energy efficiency of buildings in Tunisia’s Mediterranean climate, a region with high solar potential but significant daily temperature variations. While previous research has examined these technologies individually, their synergistic integration remains underexplored, particularly in climates where passive heating and thermal regulation are critical.

This study advances building energy efficiency research by comprehensively evaluating the synergistic integration of the Trombe wall, PCM, and MAS, a combined approach rarely examined in previous work. Unlike conventional studies that assess these technologies in isolation, our research demonstrates how their unified application in Tunisia’s Mediterranean climate optimizes heat storage, distribution and insulation within a single building envelope. This work provides crucial climate-specific insights for a region with high solar potential but significant diurnal temperature variations, addressing a gap in the literature dominated by temperate or extreme climate studies.

2. Materials and Methods

The increase in the demand for energy and thermal comfort in buildings requires innovative approaches to passive heating and insulation. In traditional buildings, the walls often fail to fit the required thermal regulations, leading to excessive energy use for heating and cooling. This research focuses on the thermal behavior of a building (surface area of 30 m² and volume of 300 m³) based on the analysis of four different wall models, with the incorporation of advanced materials and technologies to optimize the thermal comfort and energy efficiency.

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Model 1: Corresponds to the reference model: traditional wall with a layer of bricks and plaster (Figure 1a).

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Model 2: Consists of a wall integrating a Trombe wall on the south face to enhance passive solar heating (Figure 1b).

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Model 3: Consists of a wall featuring a MAS designed to improve thermal insulation and inertia (Figure 2a).

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Model 4: Consists of a wall incorporating PCM to store and release heat to stabilize indoor temperatures (Figure 2b).

Figure 1. Vertical wall constitutions in Model-1 and Model-2.

Figure 1. Vertical wall constitutions in Model-1 and Model-2.

Figure 2. Vertical wall constitutions in Model-3 and Model-4.

Figure 2. Vertical wall constitutions in Model-3 and Model-4.

Table 1. Thermophysical characteristics of the building walls.

Walls	Components (from Inside to Outside)	Thickness (m)	Density (kg/m3)	Specific Heat (kJ/kg·K)	Thermal Conductivity (kJ/h·m·K)
Components of vertical walls of Mode 1	Plaster coating [26]	0.005	1500	1	1.264
	Brick [27]	0.126	1800	1	3.2
Components of the vertical walls of Model 2	Plaster coating	0.005	1500	1	1.264
	Brick	0.126	1800	1	3.2
	Air gap	0.013	1	1.227	0.289
	Glass	0.013	2490	0.83	4.188
Components of the vertical walls of Model 3	Plaster coating	0.005	1500	1	1.264
	PCM [28]	0.02	814	2.15	0.35
	Brick	0.126	1800	1	3.2
	Air gap	0.013	1	1.227	0.289
	Glass	0.013	2490	0.83	4.188
Components of the vertical walls of Model 4	Plaster coating	0.005	1500	1	1.264
	PCM	0.02	814	2.145	0.04
	Brick	0.126	1800	1	3.2
	MAS [24]	0.057	2490	0.83	4.188
	Glass	0.013	2490	0.83	4.188
Roof	Concrete	0.240	2400	0.8	7.56
	Insulation	0.16	40	0.8	5
Floor	Floor	0.005	800	1	0.252
	Stone	0.06	2000	1	5
	Concrete	0.240	2400	0.8	7.56
	Insulation	0.080	40	0.8	5

presents a summary of the thermophysical properties of the four wall models and the roof and floor components.

2.1. Characteristics of PCM

The RT25 PCM was selected for its unique balance between thermodynamic properties and climate adaptability. Its phase transition at 20 °C coincides with winter comfort temperatures in Tunisia, and its chemical stability and high latent heat (223 kJ/kg) make it a reliable solution for everyday energy storage. These characteristics are listed in

Table 2. Thermo-physical properties of the studied PCM.

State	$\mathit{\lambda }\left(\mathbf{kJ}/\mathbf{hm}\text{·}\mathbf{K}\right)$	$\mathit{C}\mathit{p}(\mathbf{J}/\mathbf{k}\mathbf{g}.\mathbf{K})$	$\mathit{\rho }(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$	$\mathrm{\Delta }\mathit{H}(\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g})$
Solid PCM	1.26	2150	814	-
Liquid PCM	0.54	2150	750	223

and validated by similar applications in comparable climates [11,28].

The apparent method of thermal capacity using a regular and fixed mesh is the most commonly utilized among PCM. Numerical simulations were performed using Visual FORTRAN with a time step of Δt = 1 h. Based on the above presumptions, the equation for enthalpy is as follows:

$${\displaystyle \frac{\partial \mathrm{h}\left(\mathrm{x},\mathrm{t}\right)}{\partial \mathrm{t}}}={\mathrm{\lambda }}_{\mathrm{MCP}}\left(\mathrm{T}\right){\displaystyle \frac{{\partial }^2\mathrm{T}\left(\mathrm{x},\mathrm{t}\right)}{\partial {\mathrm{x}}^2}}$$

(1)

λ_MCP is the thermal conductivity of PCMs

$$\mathrm{h}\left(\mathrm{x},\mathrm{t}\right)=\left\{\begin{array}{l}{\mathrm{\rho }}_{\mathrm{s}}{\mathrm{c}}_{\mathrm{s}}\left(\mathrm{T}\left(\mathrm{x},\mathrm{t}\right)-{\mathrm{T}}_{\mathrm{F}}\right)\mathrm{si}\mathrm{T}\le {\mathrm{T}}_{\mathrm{F}}\\ {\mathrm{\rho }}_{\mathrm{l}}{\mathrm{c}}_{\mathrm{l}}\left(\mathrm{T}\left(\mathrm{x},\mathrm{t}\right)-{\mathrm{T}}_{\mathrm{F}}\right)+{\mathrm{\rho }}_{\mathrm{l}}{\mathrm{L}}_{\mathrm{F}}\mathrm{si}\mathrm{T}\ge {\mathrm{T}}_{\mathrm{F}}\end{array}\right.$$

(2)

The apparent heat capacity is expressed as

$${\mathrm{K}}_{\mathrm{app}}\left(\mathrm{T}\right)={\displaystyle \frac{\mathrm{dh}}{\mathrm{dT}}}={\mathrm{\rho }}_{\mathrm{s}}{\mathrm{c}}_{\mathrm{s}}+\mathrm{\Delta }\left(\mathrm{\rho }\mathrm{c}\right){\mathrm{f}}_{\mathrm{1}}+{\displaystyle \frac{\mathrm{d}{\mathrm{f}}_1}{\mathrm{dT}}}\left({\mathrm{\rho }}_{\mathrm{l}}{\mathrm{L}}_{\mathrm{F}}+\mathrm{\Delta }\left(\mathrm{\rho }\mathrm{c}\right)\left(\mathrm{T}\left(\mathrm{x},\mathrm{t}\right)-{\mathrm{T}}_{\mathrm{F}}\right)\right)$$

(3)

K_app (T): Heat capacity apparent of the PCM (J/m³K)

ρ_s: Solid density

ρ_l: Liquid density

f_l: Liquid fraction

L_f: Phase change enthalpy [kJ·kg⁻¹]

2.2. Characteristics of Multi-Alveolar Envelopes

Figure 3 shows the geometry of the MAS. The figure shows the air movement generated by the temperature gradient between the two walls (T1 and T2). Air circulation increases heat transfer, particularly when the MAS is positioned to pass heat, making it advantageous during the cold winter season. The structure is identified by the angle of inclination (α) and the form ratio (Rp = D/H), where D is the gap between two vertical walls and H is the height of the alveolar structure.

2.3. Heat Transfer Coefficient in Inclined Cavities of MAS

This coefficient was experimentally determined [23], which incorporates both convection and radiation. This study was analytically validated in previous works [24,25].

$${\mathrm{H}}_{\mathrm{t}}=\mathrm{\gamma }{\left(\mathrm{\Delta }\mathrm{T}\right)}^{0.25}$$

(4)

• γ: coefficient influenced by several factors, including the direction of heat transfer, the angle at which the slats are inclined relative to the horizontal, and the emissivity characteristics of the slat surfaces, whether they are low- or high-emissive.
• $\mathsf{\Delta }\mathrm{T}$: Temperature gradients (°C)

2.4. Mathematical Formulation

The mathematical formulation used to analyze the thermal behavior of the building envelope is presented in this section. It is based on some hypotheses that allow the simplification of the numerical model and the accuracy of the results.

2.4.1. The Hypotheses

The working hypotheses are as follows:

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Heat transmission occurs in only one direction.

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The thermophysical characteristics of the materials are the same in both zones and remain stable.

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Each layer is homogeneous and isotropic.

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The layers that make up the envelope walls are perfectly in contact with each other.

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There are no thermal bridges.

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Within these two zones, the air temperature remains constant.

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The incident solar flux is constant over the entire face.

2.4.2. Formulation and Discretization of the Problem

To estimate the heat transfer of various envelope types in multilayer wall buildings as a function of temperature. The mathematical model is based on the following balance sheet equation:

$$\left[{\left(\mathrm{mc}\right)}_{\mathrm{i}}+\left({\mathrm{K}}_{\mathrm{app}}\left(\mathrm{T}\right)\times {\mathrm{V}}_{\mathrm{i}}\right)\right]{\displaystyle \frac{\partial {\mathrm{T}}_{\mathrm{i}}}{\partial \mathrm{t}}}={\displaystyle \sum _{\mathrm{j}=1,\mathrm{n}}\left({\mathrm{C}}_{\mathrm{i},\mathrm{j}}+\left(1-{\mathrm{f}}_{\mathrm{l}}\right)\mathrm{C}{\mathrm{s}}_{\mathrm{i},\mathrm{j}}+{\mathrm{f}}_{\mathrm{l}}\mathrm{C}{\mathrm{l}}_{\mathrm{i},\mathrm{j}}\right)}\left({\mathrm{T}}_{\mathrm{j}}-{\mathrm{T}}_{\mathrm{i}}\right)+{\displaystyle \sum _{\mathrm{j}=1,\mathrm{n}}{\mathrm{K}}_{\mathrm{i},\mathrm{j}}\left({\mathrm{T}}_{\mathrm{j}}^4-{\mathrm{T}}_{\mathrm{i}}^4\right)}+{\mathrm{P}}_{\mathrm{i}}$$

(5)

With:

(mc)_i: Element heat capacity (J/K)

V_i: Volume of the PCM layer that element I occupies (m³)

T_i: Temperature at instant t (K)

C_i,j: Coupling coefficient for heat transfer by convection and conduction between nodes i and j (J/h·K)

Cs_i,j: Thermal conduction conductance between elements i and j within the solid phase of the PCM (J/h·K)

Cl_i,j: Thermal conduction conductance between elements i and j within the liquid phase phase of the PCM (J/h·K)

K_i,j: Coefficient of radiative interaction between nodes i and j (J/h·K⁴)

P_i: At time t, node i, absorbs all external requests (J/h)

Using the implicit discretization in Equation (5) performed for every “i” element results in a linear equation system expressed in the matrix [25].

2.5. Conditions of Thermal Boundaries

2.5.1. External Boundary Conditions

Weather and Temperature Conditions

This study focuses on Hammam Sousse, a coastal city in eastern Tunisia, characterized by a Mediterranean climate. This region experiences hot, dry summers and wet winters. Weather data for this study were sourced from the weather station installed at the Higher School of Science and Technology of Hammam Sousse, Tunisia (latitude 35.86° N, longitude 10.59° E). The dataset includes hourly temperature, solar flux, and measurements for representative days in January (winter) and March (spring), as illustrated in Figure 4, Figure 5 and Figure 6.

Figure 4 illustrates the hourly variation in outdoor temperatures in Hammam Sousse, Tunisia, during January and March, showcasing distinct seasonal differences. In January, the winter period is marked by pronounced daily fluctuations, with temperatures ranging from approximately 7 °C at night to 18 °C during the day, highlighting the need for effective thermal insulation and heat retention solutions. Conversely, in March, early spring conditions bring higher baseline temperatures and less variation, with nighttime lows of around 10 °C and daytime peaks closer to 20 °C. These seasonal differences significantly impact the thermal performance of buildings, necessitating the use of adaptive systems like Trombe walls, PCM, and MAS to stabilize indoor temperatures, reduce energy consumption, and enhance thermal comfort in response to external climate variability.

Solar radiation is obtained as a total flux that includes the direct flux Φ_Dir and diffuse flux Φ_Diff. It is computed hourly for the 15th day of each month, which is a normal day.

$$\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{Diff}}+{\mathrm{\Phi }}_{\mathrm{Dir}}$$

(6)

Figure 5 shows the variations in the total solar flux density across different orientations throughout the year. The South orientation consistently captures the highest solar flux, particularly during the colder months, with peaks reaching approximately 18 MJ/m². This makes it an optimal choice for passive heating systems in winter, enhancing thermal comfort and reducing heating demand. Conversely, the North orientation receives the least solar radiation, with flux levels remaining below 5 MJ/m² year-round, making it less effective for solar energy applications. The east- and west-facing orientations showed higher solar flux during the summer months, around 15–17 MJ/m², which can increase cooling loads due to their exposure to morning and afternoon sunlight, respectively. These variations underline the importance of orientation in designing energy-efficient building envelopes, ensuring the optimal use of passive heating, and minimizing cooling requirements.

The Modeling of the Global Solar Radiation Density Was Validated

Figure 6 presents a monthly comparison of solar flux density across the North, South, and East-West orientations, comparing the simulation results with the ASHRAE clear-sky model adapted for Riyadh. The results show a strong agreement between the two models, validating the accuracy and reliability of the simulation for building energy-performance analysis. For the North orientation, the solar flux density remains low year-round, peaking at approximately 10 MJ/m² during summer, reflecting limited solar exposure. In contrast, the south orientation demonstrates significantly higher flux during winter and early spring, reaching around 15–18 MJ/m², highlighting its suitability for passive solar heating. The East-West orientation exhibits a symmetric pattern with peak flux levels of about 15 MJ/m² during summer, making it less effective for winter heating but more relevant for managing summer cooling loads. The close alignment of the simulation data with the ASHRAE predictions confirms the utility of this model for designing energy-efficient building systems and optimizing solar energy utilization based on orientation and seasonal conditions.

Performance indicators of the validation model

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum _{\mathrm{i}=1,\mathrm{N}}\frac{{\left(\mathrm{y}\mathrm{e}-\mathrm{y}\mathrm{s}\right)}^2}{\mathrm{N}}}}(\mathrm{Root\; Mean\; Square\; Error})$$

(7)

ye: Observed value (ASHRAE data)

ys: Value simulated by our model

N: Number of data points

$$\mathrm{Relative}\mathrm{error}(\%)={\displaystyle \frac{\mathrm{RMSE}}{\mathrm{y}"}}\times 100$$

(8)

External Thermal Convective Coefficient

The external convective exchange coefficients were determined for different wall orientations at a constant wind speed of 2 m/s. For the horizontal walls, the heat transfer coefficient was calculated as 50.4 kJ/(h·m²·K), while vertical walls showed a slightly lower coefficient of 39.6 kJ/(h·m²·K). These values reflect the orientation-dependent nature of convective heat transfer in the building envelope.

2.5.2. Internal Boundary Conditions

The coefficient of convective heat transfer within the space is determined using the classical standard correlations that provide the Nusselt number (Nu) in relation to the Prandtl (Pr) and Grashof (Gr) numbers:

$$\mathrm{Nu}=\mathrm{a}{\left(\mathrm{Gr}\cdot \mathrm{Pr}\right)}^{\mathrm{b}}$$

(9)

The coefficients (a, b) for the Nusselt number correlation Equation (9) vary according to the wall type. For vertical walls, the coefficients were a = 0.59 and b = 0.25, while for horizontal walls (facing upward), the coefficients were a = 0.54 and b = 0.25. These coefficients are essential for accurately modeling convective heat transfer in different building envelope configurations.

2.6. Comfort and Building Response

2.6.1. Building Thermal Comfort

One way to do this is to calculate the proportion of people who are not happy with the comfort level. This percentage is directly correlated with the average vote of the population.

Therefore, thermal comfort can be measured using the following two parameters:

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PMV: Average Predicted Vote, which uses a standardized scale in accordance with EN ISO 7730 [30] to measure the comfort level.

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PPD: estimates the proportion of dissatisfied individuals in a thermal environment [30].

2.6.2. Thermal Response of the Building

Building thermal sensitivity can be described by two factors [31]. The first phase shift φ represents the amount of time needed for the internal temperature $t_{Tint,max}$ to reach its maximum as soon as the external temperature reaches its peak $t_{Text,max}$. building inertia causes a change in the phase shift. The following relationship serves as an expression:

$$\mathsf{\Phi }=t_{Tint,max}+t_{Text,max}$$

(10)

The thermal damping factor f is the second element that determines the response of the building. This concerns the relationship between the amplitudes of the internal temperature ΔT_int and external temperature ΔT_ext. As the damping factor decreases, the internal temperatures decrease. The expression for this factor is as follows:

$$\mathrm{f}={\displaystyle \frac{\mathrm{\Delta }{\mathrm{T}}_{\mathrm{int}}}{\mathrm{\Delta }{\mathrm{T}}_{\mathrm{ext}}}}$$

(11)

3. Results and Discussion

3.1. Case of Free Indoor Air Temperature

Figure 7a,b depict the hourly temperature progression of the building’s external and interior surfaces (TSe, TSi) for the various models examined in March and January. Compared with models 1 and 2, models 3 and 4 had higher interior and exterior surface temperatures (TSe and TSi). Because of the thermal inertia in the walls in January (a) and March (b), the thermal amplitude is less noticeable during the day than at night. These figures show that the temperature of the interior surface (TSi) of model 1 decreases rapidly at 24 h, and compared to the other models, this temperature remains stable. At 2 p.m., the Trombe Wall outside surface temperature (TSe) peaks at about 30 °C for model 2. Due to air movement in the channel and natural convection, this increase causes heat to be released inside the building, increasing heat transfer from the outside to the inside and raising the interior surface temperature (TSi). In Model 3, the Trombe Wall releases a specific amount of heat to the MAS. The design of this structure, along with the channel containing heated air, leads to an increase in the temperature amplitude of the inner surface of the MAS. This increased amplitude is important in Model 4, where the PCM within the building helps stabilize thermal fluctuations. Research has shown that PCM enable greater heat storage during the day, which is then released at night. The heat is retained from the energy absorbed through the Trombe wall and the MAS. Furthermore, the temperatures of the external and interior surfaces (TSi and TSe) in March (Figure 7b) are higher than those in January, which is attributed to the solar flux data shown in Figure 5. Figure 7a,b show that in the Tunisian climate, there is a clear advantage in integrating advanced passive systems (Trombe walls, MAS, and PCM) into the building envelope to increase heat comfort and energy efficiency.

Figure 8 presents the hourly fluctuation of the interior air temperature (Tint) for models 2, 3, and 4, and the external air temperature (Tout). In January (Figure 8a), there is a significant fluctuation in the outdoor temperature, with a low temperature of 5 °C at night and a peak of 16 °C during the day. The indoor temperatures for all models are higher and more stable than the outdoor temperature, indicating the ability of the proposed models to ensure the insulation and regulation of the internal thermal conditions. Model-2, which incorporates the Trombe wall, has the lowest indoor temperatures (18 °C to 20 °C), which indicates an improvement in the thermal inertia but is still sensitive to external temperature changes. Model-3, corresponding to the use of the MAS, has better performance; in fact, the indoor temperature is kept between 20 °C and 22 °C, which is due to the enhancement of the thermal buffering provided by the MAS design. Model-4, which uses PCM, has the best performance by maintaining the most stable and highest indoor temperatures (22 °C to 25 °C). In fact, PCM can absorb heat during peak external temperatures and release it during cooler periods, thus minimizing indoor temperature fluctuations and leading to enhanced thermal comfort. Similar trends are also observed in March (Figure 8b), but the outdoor temperature has a lower amplitude of fluctuation (10 °C to 22 °C). Model-2 continues to be sensitive to external conditions, with temperatures fluctuating from 19 °C to 22 °C. Model-3 is more stable, with temperatures rising from 20 °C to 23 °C. Model-4 is the most effective, with indoor temperatures between 23 °C and 25 °C. Overall, it should be noted that PCM has a considerable influence on the optimization of indoor thermal comfort and minimization of temperature fluctuations. The MAS also enhances the thermal inertia, but it is not as effective as the PCM. This enhancement will contribute to reducing energy consumption and enhancing thermal stability in regions with substantial daily temperature variations, such as the Tunisian climate.

Based on Figure 8a,b,

Table 3. Comparative summary.

Criteria	Model-2 (Trombe Wall)	Model-3 (MAS)	Model-4 (PCM + MAS + Trombe)	Comparative Analysis
Average Temperature (°C)
January	18.2	19.8	21.4	Model-4 maintains a higher temperature (21.4 °C) compared to Model 2 (18.2 °C).
March	22.2	23.8	25.4	Model-4 reaches the highest temperature (25.4 °C) compared to Model2 (20.5 °C).

presents a comparative analysis of the thermal performance of the building envelope configurations

Table 4. Phase-shifting and thermal damping factors in January and March.

Months	January		March
	φ (h)	F (%)	φ (h)	F (%)
Model-1	3	75	4	80
Model-2	4.25	57	5	71
Model-3	5.12	50	5.5	66
Model-4	6.02	40	6.12	50

presents the thermal performances of the four wall models in terms of the phase shift (Φ) and damping factor (f) for January and March, calculated from the temperature profiles shown in Figure 7a,b. Model-4, which incorporates the Trombe Wall, MAS, and PCM, shows the highest phase shift (6 h in January and March) and the lowest damping factor (40% in January and 50% in March). This indicates an enhanced thermal inertia, where indoor temperature changes are delayed and fluctuations are minimized, providing superior thermal stability. In contrast, Model-1 (a traditional wall) has the shortest phase shift (3–4 h) and highest thermal damping factors (75–80%), signifying rapid and larger indoor temperature variations. The results affirm that integrating advanced materials like PCM and MAS significantly improves the thermal performance of building envelopes, reducing energy consumption and enhancing comfort by maintaining stable indoor conditions. These results demonstrate that the combined PCM-MAS-Trombe wall system (Model 4) outperforms the conventional design in three key aspects. First, it improves thermal stability by reducing daily temperature fluctuations by 40%, compared to 50–60% for standard walls [32,33,34]. Second, it delays heat transfer by 6 h, compared to 3–4 h for traditional walls. Third, it adapts perfectly to the Tunisian climate by effectively managing daily thermal amplitudes of 7–25 °C. These validated performances are explained by synergistic mechanisms: the PCM (RT25) stores the solar energy captured by the Trombe wall, while the MAS structure minimizes nighttime losses.

3.2. Situation Where the Temperature Inside Is Set

Figure 9 shows the required daily heat power for the four models throughout January and March to compare the energy efficiency of these configurations. In January, the traditional wall (Model-1) had the highest heat power demand (about 800 kJ/h), indicating its poor thermal performance and lack of insulation. In fact, due to the high heat loss, external heating is required. The use of the Trombe wall (Model-2) and MAS (Model-3) provide good improvements, where the required heat power is around 500–600 kJ/h. This enhancement is due to passive solar heating and improved thermal inertia, which leads to better heat retention and lower reliance on external heating sources. The most efficient model is Model-4; in fact, only around 400 kJ/h is required in January. The daytime heat stored by the PCM is released at night, leading to a constant indoor temperature and reducing the energy required to maintain thermal comfort. This same trend occurs in March, where Model-4 stands out, with only 300 kJ/h as heating requirements, indicating that the application of PCM has a beneficial effect on the improvement of energy efficiency by reducing heating requirements. The use of walls incorporating PCM can be considered an ideal solution that allows a decrease in energy consumption in buildings while enhancing thermal comfort in varying climates.

3.3. Impact of Occupant Presence

The PMV and PPD of the occupants must be consulted to evaluate the indoor thermal environment and quantify the sensation of comfort on a scale standardized according to the EN ISO 7730 standard [30]. Figure 10 and

Table 5. Comfort index for various configurations under investigation.

Configuration	Model-1	Model-2	Model-3	Model-4
PMV	−1.5	−1.2	−1	−0.5
PPD	51%	25%	20%	9.8%

demonstrate the relationship between the (PMV) Predicted Mean Vote and the (PPD)Predicted Percentage of Dissatisfied for different wall configurations based on EN ISO 7730 standards [30]. The results highlight that Model-4, which incorporates the Trombe wall, MAS, and PCM, achieves the best thermal comfort with a PMV of approximately −0.5 and a low PPD of 9.8%, reflecting near-optimal conditions for occupant satisfaction. Model-3, which integrates the MAS, has a PMV of −1 and PPD of 21%, indicating a sensation of coolness with slightly reduced comfort. Model-2, which feature the Trombe wall, has a PMV of −1.2 and PPD of 25%, indicating further discomfort. Model-1, the traditional wall, performs the poorest, with a PMV of −1.5 and a PPD of 51%, signifying a significant percentage of dissatisfied occupants due to thermal discomfort. These findings underscore the importance of advanced materials and systems like PCM and MAS, in achieving superior indoor thermal comfort, minimizing dissatisfaction, and optimizing energy efficiency. This study evaluated thermal comfort through PMV (predicted mean vote (PMV) and percentage of dissatisfied people (PPD) indices, considering key factors such as radiant temperature, ventilation, and humidity. This study evaluated thermal comfort through PMV (predicted mean vote (PMV) and percentage of dissatisfied people (PPD) indices, considering key factors such as radiant temperature, ventilation, and humidity. The results show that the integrated system (Trombe-PCM-MAS wall) significantly improves comfort, with a PMV close to zero (−0.5) and a reduced PPD (9.8%), in accordance with ISO 7730 standards. These observations align with the work of Al-Homoud et al. [35], who highlighted the importance of phase change materials and thermal inertia in stabilizing indoor temperatures in Mediterranean climates. However, the impact of humidity and summer periods requires further investigation for a comprehensive analysis.

Table 6. Comparison works.

Configuration	Reference Room Zhou and Yu [13]	PCM-VTW Zhou and Yu [13]	Model-1	Model-4
PMV	−1.71	−0.32	−1.5	−0.5
PPD	29.8%	9.6%	51%	9.8%

compares the thermal comfort indices (PMV and PPD) of the configurations studied with the findings of Zhou and Yu [13]. The reference room in Zhou and Yu’s study exhibited a PMV of −1.71 and PPD of 29.8%, indicating significant discomfort during winter. The test room, equipped with a PCM-VTW (Phase Change Material-Ventilated Trombe Wall), significantly improved these metrics, achieving a PMV of −0.32 and a PPD of 9.6%, reflecting substantial thermal comfort enhancements. Similarly, in the current study, Model-4, which combines a Trombe wall, MAS, and PCM, shows a PMV of −0.5 and a PPD of 9.8%, comparable to the PCM-VTW configuration in Zhou and Yu’s work. In contrast, Model-1, representing a traditional wall, has a PMV of −1.5 and PPD of 51%, highlighting its poor performance in maintaining thermal comfort. The differences in PMV and PPD across studies and models are influenced by the configurations and climate conditions, underscoring the importance of advanced materials like PCM and Trombe walls in achieving superior indoor thermal environments. The effective implementation of passive strategies, such as the Trombe-PCM-MAS wall, requires close collaboration between universities, industries, and governments, following the Triple Helix model. As Sánchez et al. [36] demonstrate, this partnership accelerates technological innovation and facilitates transfer to concrete applications. Such synergy would make it possible to adapt these solutions to local contexts, establish construction standards, and financially support their large-scale deployment while improving the thermal comfort and energy efficiency of buildings. The optimal implementation of the Trombe wall-PCM-MAS integrated system is based on a model [37], where academic results (51% reduction in thermal fluctuations, 6 h phase shift) guide the industry towards cost-effective local solutions (e.g., prefabricated PCM-MAS panels) and inform public policies (PPD standards <10%, heating demand <400 kJ/h). This framework proposes targeted actions, including the certification of installers, subsidies for high-sunlight regions, and living laboratories to adapt solutions to Tunisian microclimates.

This research presents several major strengths: a novel evaluation of the Trombe wall-PCM-MAS synergy in a Mediterranean climate, quantified results (51% reduction in thermal fluctuations and 50% reduction in heating demand), and practical recommendations for energy policies (optimized materials and architectural orientations). However, some limitations should be noted: summer performance and humidity control were not evaluated, and an economic analysis is necessary. These results pave the way for effective passive solutions for sunny regions while clearly identifying areas for improvement in future research.

4. Conclusions

This study evaluated the thermal performance and energy efficiency of building envelopes incorporating Trombe walls, PCM, and MAS under Tunisian climatic conditions. The integrated system demonstrated significant improvements in thermal stability, energy efficiency, and occupant comfort compared with traditional wall configurations.

The key quantitative results for the compared models are as follows:

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Model 4 (Trombe wall + PCM + MAS) achieved optimal thermal stability, showing a 6-h phase shift and damping factors of 40% (January) and 50% (March). This configuration reduced indoor temperature fluctuations by 40% compared to Model 1 (conventional wall).

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Model 3 (MAS) demonstrated intermediate performance, with phase shifts of 5.1–5.5 h and damping factors of 50–66%, underscoring the contribution of alveolar structures to thermal inertia.

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The heating power demand of Model 4 was 400 kJ/h in January and 300 kJ/h in March, representing a 50% reduction compared to Model 1 (800 kJ/h in January).

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Model 2 (Trombe wall) reduced heating demand by 37.5% (500 kJ/h), highlighting the additional benefits of PCM and MAS integration.

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Thermal comfort in Model 4 (PMV: −0.5; PPD: 9.8%) complied with the ISO 7730 standards, whereas Model 1 exhibited inferior performance (PMV: −1.5; PPD: 51%).

The synergistic combination of these passive technologies effectively mitigated indoor temperature fluctuations, enhanced thermal inertia, and reduced the heating requirements. The results highlight the potential of these systems to optimize passive solar heating and thermal regulation in Mediterranean climates, where diurnal temperature variations are significant. By leveraging the complementary strengths of Trombe walls, PCM, and MAS, this research offers a sustainable solution for improving building performance without resorting to mechanical interventions. Future work should explore the system’s performance during summer and incorporate economic analyses to further validate its practical application.