The fight against water scarcity is an urgent cause, as currently, more than two billion individuals worldwide do not have access to drinking water [1] . To address this issue, the exploitation of ocean reservoirs, which are sources of almost unlimited water [4] and groundwater, presents an interesting solution. However, the use of saline and brackish water requires prior desalination treatments, such as reverse osmosis [5] , filtration [6] , electrodialysis [7] , distillation [2] , and more.

Generally, most arid regions, which face water scarcity, have significant solar radiation, a renewable energy source that can be used to carry out desalination processes [1] [8] . Thus, solar desalination, which involves transforming brackish or saline water into drinking water using solar energy, is an appropriate solution to meet the growing demand for water in these areas.

In isolated regions where the population is poor, the solar desalination process must be easy to design and use, economical, and require no energy sources other than the sun. Therefore, solar stills, which possess these advantages, are suitable for drinking water in these regions.

Solar distillation uses solar energy to evaporate water and condense the vapor, thereby producing fresh water. This process mimics the natural water cycle [4] , in which evaporation caused by the sun leads to precipitation.

A solar still is a sealed structure, typically in the shape of a trapezoid, that contains brackish water in a shallow basin. The device was designed to allow simultaneous evaporation of water and condensation of vapor. It is made from materials such as galvanized iron, wood, or concrete, with a transparent glass or plastic cover on top to allow sunlight to enter [2] . The interior and bottom were painted black to maximize heat absorption, and the sides were insulated to minimize thermal loss. Materials such as fiberglass, polyurethane foam, and sawdust are among the most commonly used insulating materials for solar stills [9] .

Solar radiation heats the water, causing it to evaporate, and then the vapor condenses on the cooler inner surface of the glass cover [3] where it is collected.

There are two major categories of solar stills.

Each category of solar stills (passive and active) can be classified into single- and multi-effect stills based on the number of glazing layers over the water surface [10] .

The design parameters of solar stills significantly influence their performance [2] . The essential parameters include the evaporation surface area, orientation of the glass cover for condensation, materials used for thermal storage, incorporated additives, and insulation level [11] . The productivity and intrinsic performance of solar stills are closely related to specific design parameters, such as the water depth in the basin, the tilt angle of the glass cover, and its thickness [12] .

According to existing literature, the main design parameters that have a substantial influence on solar distillation systems are as follows.

1) Thickness of the Glass Cover: Solar radiation plays a crucial role in distillation within solar stills [13] . Many studies have shown that the productivity of these devices increases with the intensity of solar radiation, particularly in summer and early afternoon [2] . The solar radiation incident on the still cover is a key factor that influences its efficiency. Research comparing four solar stills, three with glass covers of different thicknesses and one with plastic, revealed that the still with the thinnest glass cover produced the highest output, with a 15.5% increase [14] .

2) Choice of Materials: Evaluating the thermal properties of materials (thermal conductivity, absorptivity, and transmittance) is crucial for designing effective and durable solar stills [15] . The choice of materials for various components directly influences the efficiency [16] and longevity of the still [15] . Studies have indicated that aluminum offers better performance than stainless steel because of its good thermal conductivity [17] . Similarly, a copper basin allows for superior efficiency compared to galvanized sheet metal, with an increase of 80% [18] . In addition to efficiency, the long-term durability of stills depends on the use of corrosion-resistant materials [19] , which is essential for ensuring reliability, especially in rural and isolated areas [20] .

3) Water Depth in the box: A study analyzed the performance of stepped solar stills with different water depths (5, 7.5, and 10 mm), revealing that the efficiency decreases as the water depth increases [21] . Several researchers have confirmed the influence of water depth on the productivity of solar stills [11] [13] [22] [23] .

4) Tilt Angle of the Glass Cover: The tilt angle of the glass cover influences the condensation and movement of water on the inner surface [24] . Goshayeshi and Safaei (2019) [25] studied this effect on the performance of two types of stepped solar stills (flat and convex). Their study revealed that a still with a convex absorbing plate produced a higher average daily distilled water output than that with a flat absorbing plate. This is one of the most influential parameters, as it affects the dynamics of condensation and movement of water along the inner surface of the cover [25] .

5) Condensation Surface Area: Increasing the condensation surface area is an effective method for improving purified water output in solar stills [26] .

The average daily productivity of solar stills was low. This productivity is insufficient to meet the daily drinking water needs of the populations in arid regions. Therefore, these devices are the subject of several studies aimed at improving and optimizing their productivity. Table 1 lists the daily productivities of solar stills obtained from various studies.

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Modeling is essential in scientific and technical studies to analyze, understand, and predict the behavior of complex systems. This often relies on the equations that represent these systems. Currently, many engineering problems have been solved using computer-assisted numerical modeling [2] .

The efficiency of current solar stills is low, necessitating design improvements. Various studies have explored these enhancements by using numerical models and experimental analyses. For example, Khare et al. (2017) [23] developed a multiphase CFD model to simulate a simple solar still using ANSYS FLUENT, whereas Gnanavel et al. (2020) [31] enhanced the productivity of the still by integrating a phase-change material. Studies have compared simulations with experimental results to optimize performance using tools such as ANSYS CFD [32] , FLUENT, TRNSYS, MATLAB, and FORTRAN [33] .

Mathematical models are essential for evaluating the productivity of solar stills, as shown in Table 2 , which summarizes the various models used to estimate the distilled water production.

COMSOL® Multiphysics is a powerful simulation software that is widely used for modeling and solving scientific and engineering problems. This allows for the easy transformation of single-physics models into multiphysics models capable of simultaneously handling coupled phenomena [36] . The software automatically records each step of the modeling process, thereby facilitating the management and modification of simulations [2] .

This software also offers various specialized modules for different application areas, simplifying the creation and analysis of models. These modules include the application libraries and practical examples. One study utilized it for the parametric analysis of a solar desalination system, testing the effect of different materials and shapes of absorbers on the performance of passive and active solar stills [35] . The simulations enabled the modeling and optimization of the design parameters for solar stills [37] .

Although solar stills are the subject of several research studies [2] , further in-depth studies are still needed to optimize their productivity. This includes the following studies.

The objective of this study was to analyze the reliability and predictive accuracy of a numerical model of a solar still designed using COMSOL® Multiphysics. To achieve this, the results obtained from the numerical model are compared with those from two physical prototypes of cascade solar stills called “Pano Rano.” The difference between these two prototypes lies in the material used to construct their basin: one is made with a normal concrete basin ( Figure 1 ), and the other with a fused plastic basin ( Figure 2 ).

The components and their respective materials of these two physical prototypes of solar stills are presented in Table 3 .

The choice of these materials, normal concrete and fused plastic, is based on their favorable thermal properties. With relatively low thermal conductivities, ranging from 0.17 to 1.8 W/m∙K, these materials are capable of limiting heat losses, thereby promoting thermal retention within the still and improving the efficiency of the solar distillation process. Additionally, these materials are easy to work with and widely available locally, simplifying their implementation and reducing procurement costs.

Data collection is essential for evaluating the performance of the “Pano Rano” solar stills. During the experiments, various instruments were used to measure the necessary data for comparison with data from the numerical model. The measurement instruments are listed in Table 4 . Then, the following Figure 3 illustrates the studied “Pano Rano” cascade solar still model. Figures 4-6 show the measuring instruments used.

The configuration and resolution of a simulation in COMSOL follow these steps.

Step 1: Creation of a New Model

This step involved initiating a project by establishing a new model file. Once the software is launched, the model creation can begin. To achieve this, the following actions were performed.

Step 2: Creation of the Model Geometry

This step involves defining and drawing the structure of the model using geometry tools in COMSOL. The geometry of the Pano Rano consists of a combination of four blocks: a rectangular basin (box) and covering material (glass, water, and humid air). Table 6 lists the dimensions and coordinates of the solar still geometry.

Step 3: Definition of Material Properties

This step involved specifying the physical characteristics of the materials used in the model, namely, normal concrete, acrylic plastic, and glass. The materials in the geometry were defined as domains, as shown in Figure 7 . A domain represents a region of space in which the physical equations are solved, and in COMSOL®, the domains are defined by geometric shapes. Then, the characteristics of these materials are presented in Table 7 .

Step 4: Definition of physics and boundary conditions

This involves assigning the relevant physical laws to the model as well as defining the boundary conditions that govern the interactions and constraints applied at the boundaries of the model.

The “Surface-to-Surface Radiation” interface provides functionalities to account for thermal radiation as an energy transfer between boundaries and external heat sources [39] .

The total radiative flux (thermal) leaving the surface can be evaluated using Equation (1) [35] :

$Q_{ij}=A_i{F}_{ij}\left(J_i-J_j\right)$ (1)

where Q_ij, A_i, F_ij, J_i and J_j represent, respectively, the thermal power transmitted from body “i” to body “j,” the surface area of body “i,” the view factor from body “i” to body “j,” the total radiative flux leaving surface “i,” and the total radiative flux leaving surface “j.” All the surfaces and objects were considered to possess isothermal properties. The thermal radiation of a black body emitted by a surface is obtained using Equation (2) [35] :

$Q_i=\left(A_i{\epsilon }_i\left(\sigma T_i^4-J_i\right)\right)/\left(1-{\epsilon }_i\right)$ (2)

where Q_i, ε_i, σ and T_i represent the thermal energy leaving surface “i,” the thermal (infrared) emissivity of surface “i,” the Stefan-Boltzmann constant, and the temperature of surface “i,” the Stefan–Boltzmann constant, and the temperature of surface i, respectively. The radiative flux in equilibrium between two surfaces at different temperatures is given by Equation (3) [35] :

$Q_{ij}=\sigma A_i{\epsilon }_i\left(T_i^4-T_j^4\right)$ (3)

Furthermore, the view factor can be defined as the radiation from surface “i” intercepted by surface “j”:

$F_{ij}=\left(\text{radiation}\text{emitted}\text{by}{A}_i\text{and}\text{incident}\text{on}{A}_j\right)/\left(\text{total}\text{radiation}\text{emitted}\text{by}{A}_i\right)$(4)

The heat transfer interface in fluids solves Equation (5) [39] :

$\rho C_p\left(\partial T/\partial t+u\cdot \nabla T\right)+\nabla \cdot \left(q+q_r\right)={\alpha }_{p}T\left(\partial p/\partial t+u\cdot \nabla p\right)+\tau :\nabla u+Q$ (5)

With:

$\rho$: density (kg/m³)

$C_p$: specific heat capacity at constant pressure (J/(kg·K))

T: absolute temperature (K)

u: velocity vector (m/s)

q: heat flux by conduction (W/m²)

q_r: heat flux by radiation (W/m^²)

p: the pressure (Pa)

$\tau$: the viscous stress tensor (Pa)

Q: content of heat sources other than viscous dissipation (W/m³)

${\alpha }_p$: thermal expansion coefficient (L/K)

${\alpha }_p=-\partial \rho /\rho \partial T$ (6)

For ideal gases, the thermal expansion coefficient takes the simplified form ${\alpha }_p=1/T$:

Heat transfer through humid air is explained using Equation (7) [35] :

$\rho C_p\left(\partial T/\partial t+u\cdot \nabla T\right)+\nabla \cdot \left(q+q_r\right)={\alpha }_{p}T\left(\partial p/\partial t+u\cdot \nabla p\right)+\tau :\nabla u+Q_H+Q$ (7)

where $Q_H$ is the diffusive heat enthalpy flux due to the rate of change of air and vapor in humid air and is calculated using Equation (8) [35] :

$Q_H=-\left(C_{p,v}-C_{p,a}\right)g_w\nabla T$ (8)

where $C_{p,v}$, $C_{p,a}$ and $g_w$ represent the specific heat capacity at a constant vapor pressure, the specific heat capacity at constant air pressure, and the vapor flux by diffusion, respectively.

The “Transport of Moisture in Air” interface provides functionalities to model the transfer of moisture through liquid transport (capillary flow) and vapor diffusion [39] . The Moisture Transport in Air interface solves Equation (9), where the variation in moisture content is expressed through the transport of vapor concentration when the vapor concentration is low. This can be expressed as the product of the molar mass of water, the relative humidity, and the saturated vapor concentration [39] :

$M_v\left(\partial C_v\right)/\partial t+M_{v}u\cdot \nabla C_v+\nabla \cdot g=G$ (9)

$g=-M_{v}D{C}_v$ (10)

$C_v=\varphi C_{sat}$ (11)

With:

$M_v$: the molar mass of water vapor (kg/mol)

Φ: the relative humidity (dimensionless)

$C_{sat}$: the saturated vapor concentration (mol/m³)

D: the diffusion coefficient of vapor in air (m²/s)

$u$: the air velocity field (m/s)

$G$: the source (or sink) of moisture (kg/(m³∙s))

For conditions of higher vapor concentration, the variation in moisture content is expressed through the transport of the mass fraction of vapor, as follows [35] :

${\rho }_g\left(\partial {\omega }_v\right)/\partial t+{\rho }_{g}u\cdot \nabla {\omega }_v+\nabla \cdot g_w=G$ (12)

$g_w=-{\rho }_{g}D\nabla {\omega }_v$ (13)

${\omega }_v=\left(M_v{C}_v\right)/{\rho }_g$ (14)

where ${\rho }_g$ is the density of moist air.

The transport of vapor concentration occurs through convection and diffusion in the moist air.

In this study, the interface “Moisture transport in air” is applied to domain 3, which represents the air contained within the solar still. The initial value of the relative humidity was set to 0.5, corresponding to the initial average ambient air condition.

The initial concentration of liquid water on the evaporation (wet) air-water surface was defined as 182.35 mol/m²; this value represents the saturation of liquid water at the initial temperature of the system. Then, the evaporation factor (K) was determined by applying Stefan flow modeling resulting from evaporation from the water surface (Equation (15)). The calculation was performed under constant ambient conditions: at low temperature (T₀ = 20˚C), the vapor concentration for ϕ = 1 is relatively low (C_sat ≅ 1 mol/m³), which resulted in a moderate vapor concentration gradient and a low Stefan velocity. Therefore, the effect on the vapor concentration can be neglected.

$K=\left(1610.7\times {10}^{7.5\cdot \left(\left(T-273.15\right)/\left(-37.85+T\right)\right)}\right)/\left(R\cdot T\right)\cdot \left(2.6\times {10}^{-5}\times 1000\right)/18$ (15)

With R (J/mol∙K) is the universal gas constant, and T is the temperature.

Step 5: Mesh Creation

The mesh is the division of the model’s geometry into small cells or elements, allowing for precise numerical resolution of the physical equations associated with the model. A standard triangular mesh is used in this study. This choice was motivated by the need to minimize the computation time and memory requirements.

Figure 8 illustrates the mesh of the structure of our model, which consists of 3253 domain elements and 585 boundary elements.

Step 6: Launching the simulation

This final step involved executing the numerical model using COMSOL’s computation algorithms to produce the results and visualizations of the studied phenomena. Before launching the simulation, it was important to consider the following simplifying assumptions.

The numerical simulation was conducted for 12 h, from 6 AM to 6 PM, with a time step of one hour.

Experimental tests were conducted to validate the results of the numerical simulations and to verify the reliability of the model. The experiments were carried out on-site in a neighborhood of Manazary Ilafy, 2.9 km from Ambatobe in the city of Antananarivo, the capital of Madagascar, during the summer and winter seasons, within the time frame of 9 AM to 4 PM.

The collected data, including the air temperature inside the still, temperature of the inner surface of the glass cover, and production of distilled water, were compared with the numerical results.

This comparison allowed for the assessment of the model’s accuracy and confirmed its ability to predict the actual behavior of the solar still.

To evaluate the reliability of the accuracy of our predictive model, we chose to use statistical indicators such as the Mean Squared Error (MSE) and Coefficient of Determination (R²).

The Mean Squared Error (MSE) is a standard statistical measure used to evaluate the performance of a predictive model. It represents the average of the squares of the differences between the observed values and the values predicted by the model [40] .

The MSE was always positive. The closer the values predicted by the model are to the observed values, the smaller the differences, and the closer the MSE will be to zero. A low MSE indicates that the model performs well and accurately predicts values [40] .

The coefficient of determination R² represents the proportion of variance in the dependent variable that is predictable from the independent variables [41] .

The R² value can be negative, indicating a poorly performing regression, or equal to zero, meaning that the model does not explain the variability in the data. Positive R² values range between 0 and 1, with 1 representing a perfect prediction. A high R² indicates that the model performs well and effectively explains the variability in the data effectively [41] .

Figure 9 illustrates the variation in solar radiation used for the numerical simulation. Figure 10 presents the temperature profiles obtained for the two studied models at the end of the simulation. These temperature profiles are essential for understanding the thermal behavior of stills under different design configurations.

Regarding the liquid water concentration, the curves in Figure 11 illustrate the liquid water concentrations on the wet surfaces inside the normal concrete still and those inside the acrylic plastic still, specifically on the air-water surface and internal surface of the glass cover.

The liquid water concentration (mol/m²) on the air-water surface refers to the amount of evaporated water, whereas the concentration of water on the internal surface of the glass cover represents the amount of condensed water. Table 8 presents the data related to Figure 11 .

Table 8 shows the estimated evaporated water and water productivity of the two stills for a given irradiance of 1200 W/m². At the end of the day, at 6 PM, for 2109.95 mL/m² of evaporated water, the normal concrete still has a distilled water production of 1383.93 mL/m². In contrast, for 4455.53 mL/m² of evaporated water, the acrylic plastic still produces 2925.98 mL/m². It can also be noted from Table 8 that the amount of evaporated water is significantly greater than the amount of condensed water over time. This suggests that the stills have evaporation potential. However, based on the observed values, it appears that this potential is not fully utilized within stills. These results clearly show that acrylic plastic still offers a higher yield than concrete, both in terms of evaporation and distilled water production.

The graphs in Figure 12 present comparisons between the simulation results and experimental data for the air and glazing temperatures inside the stills on October 23, 2023.

They show that for the same irradiance data, the temperatures obtained from the simulation during the summer days of October 23, 2023, show an increase compared to the experimental data.

The graphs in Figure 13 present comparisons between the simulation results and the experimental data for the air and glazing temperatures inside the stills on June 7, 2024. They show that for the winter day of June 7, 2024, it can be observed that the temperature data obtained from the model remain high and show a significant discrepancy compared to the experimental data.

Using the data provided in Figure 12 and Figure 13 , the reliability of the model can be determined by calculating the Mean Squared Error (MSE) and Coefficient of Determination (R²). Table 9 summarizes the calculated values of the MSE and R².

The validation of the model’s reliability for predicting temperatures inside the “Pano Rano” solar stills revealed the following.

1) For normal concrete still, the MSE values range from 71.53 to 123.42 during summer and winter days, and the R² values range from −1.34 to −0.25, indicating low reliability of the model.

2) For the molten plastic still, the MSE values are very high compared to those of normal concrete, and the R² values are relatively low (−10.06 to −2.86), indicating very low reliability during summer and winter days.

The graphs in Figure 14 present comparisons between the simulation results and experimental data for the cumulative productivities of the stills on October 23, 2023.

It can be observed that, during the day of October 23, 2023.

The graphs in Figure 15 present comparisons between the simulation results and experimental data for the cumulative productivities of the stills on June 7, 2024. They show that for June 7, 2024, for normal concrete still, the productivity values from the numerical model are very low compared to those from the experimental model. In contrast, for molten plastic still, the trend is reversed: the productivity of the numerical model exceeds that of the experimental model.

The results of the numerical simulation ( Figure 14 and Figure 15 ) show that the water productivity of the “Pano Rano” still is strongly influenced by the choice of basin material. These results indicate that, in terms of efficiency and performance, the acrylic plastic solar still “Pano Rano” surpasses normal concrete owing to its thermal properties and low density. This observation suggests that the type of material, whether normal concrete or acrylic plastic, significantly impacts the performance of stills owing to their distinct thermal and mechanical properties, affecting heat transfer, evaporation, and condensation. This is consistent with the findings of the study by Rai Khare et al. (2017) [23] , which showed that using the right materials for the still basin can optimize its productivity.

However, even though the simulation predictions indicate that the productivity of the molten plastic solar still is better than that of normal concrete, the experimental results show the opposite. This divergence can be explained by the incidents that occurred during the experiment. The increase in the internal temperature of the still caused cracks ( Figure 16 ), reducing its efficiency by allowing heat losses, thereby decreasing its water productivity. These results highlight the importance of material durability in the design of solar stills, indicating that theoretical performance does not guarantee practical performance, particularly under extreme conditions.

Figure 14 and Figure 15 also show that for the experimental trials, the productivity of solar stills is higher in summer than in winter. This can be explained by the fact that the productivity of distilled water from solar stills is strongly influenced by climatic conditions, as demonstrated by Panchal and Patel [13] . Furthermore, research based on the theoretical model by Hinai et al. [42] has shown that an increase in the ambient temperature from 23˚C to 33˚C can improve the yield of solar stills by 8.2%.

Table 10 shows the results of determining the reliability of the predictive model by calculating the Mean Squared Error (MSE) and R² of the productivities obtained from the experimental and numerical models of the solar stills during the summer and winter.

During the summer, the normal concrete still shows high MSE values and R² values ranging from −0.01 to 0.40, indicating low reliability of the model. In contrast, for the melted plastic still, the MSE values were low, and the R² values were high (from 0.57 to 0.95), demonstrating better reliability.

In winter, the model becomes more reliable for normal concrete with R2 values between 0.20 and 0.79, while for melted plastic, the MSE values increase significantly, and the R2 values become negative, indicating very low reliability.

Evaluation of the model’s performance over the two periods revealed marked variability. The most accurate results were obtained during the summer period, particularly on October 23, 2023, with an MSE of 16.24 and an R² of 0.95, indicating very high model accuracy. Conversely, some measurements, especially in winter, showed high MSE values and negative R² values, reflecting insufficient performance. The low reliability of the numerical model and less accurate predictions were attributed to simplifying the assumptions. The high MSE and low R² values illustrate the limitations of these assumptions, suggesting that the model is not sufficiently robust to capture the complexity of real conditions.