Fuzzy logic-based prediction and parametric optimizing using particle swarm optimization for performance improvement in pyramid solar still

All living things need energy to function. As a result of higher energy use, daily demand is on the rise. Fossil fuels, such as natural resources, have met human energy needs for many years. However, their use has caused significant environmental damage, leading to issues like global warming and the melting of ice in the polar regions. Renewable energy resources are expected to impact future energy demand substantially. Renewable energy sources are environmentally friendly and do not produce pollution like fossil fuels (Yuvaperiyasamy et al. 2023a; Shivhare et al. 2024). Based on data from the World Health Organization, it is estimated that there will be a 56% deficit in supply relative to demand by 2025. The issue of water scarcity is complex and has wide-ranging effects, such as malnutrition, degradation of ecosystems, desertification, and potential risks to global peace. A large portion of the Earth's surface is surrounded by water; however, the amount of drinkable water available is limited. The remaining 98% is deemed unsuitable due to its high salt content (Ghandourah et al. 2022; Somashekar et al. 2023; Khan et al. 2024). Earth contains an estimated 150 million cubic kilometers of water springs. Water consumption from the existing sources will be halted due to the presence of contaminants like fragments, organic scum, and elevated levels of total dissolved solids (TDS), resulting in multiple human infections (Yuvaperiyasamy et al. 2024a). Tian et al. (2024a) explored the effects of light and microbes on polyurethane plastics (PU-PS) degradation. Their results indicated that microorganisms have a more pronounced impact on PU-PS degradation than light. Tian et al. (2024b) delved into the breakdown of microplastics (MPs) derived from coated controlled-release fertilizers (CRF) in soil. They observed an enhancement in the metabolism of amino acids and polymers, which acted as a protective measure against stress induced by MPs. Guo et al. (2024) highlight the complex nature of the environmental impact of MPs, which is influenced by their characteristics and the changing dynamics of freshwater ecosystems. The fluorometric and hydrodynamic data evaluation demonstrated the presence of unique mixing zones above the canopy, with canopy physical characteristics playing a significant role in their formation (Stride et al. 2023). Noori et al. (2022) underscores the importance of reducing the levels in Tehran's water sources to ensure the well-being of the vast population of over 13 million people who depend on the Tehran Public Water (TPW) supply. Therefore, it is crucial to purify the contaminated water. Efforts to purify polluted water and protect clean water sources are crucial and require exploring innovative techniques. However, additional evaluation is needed for alternative desalination methods in terms of their feasibility and economic viability (AI-Mezeini et al. 2023). Kateshia & Lakhera (2022) Utilizing stearic acid as the phase change material (PCM) enabled the development of a meticulous mathematical model that showcases the direct correlation between the amount of stearic acid and the volume of distillate yielded in the solar still. Additionally, integrating a PCM layer may enhance the efficiency of solar distillation systems. These elements can discharge the thermal energy stored during daylight hours at night through sensible heat, latent heat, or a combination of the two (Ravishankara et al. 2013; Afolabi et al. 2023). Arunkumar et al. (2013) The study involved using commercially available paraffin wax and a solar hemispherical basin connected to a concentrator. This research improved thermal energy storage, resulting in a significant increase of 26% in daily production. Abdullah et al. (2023a, 2023b) The study aimed to evaluate the impact of incorporating a 10 mm thick PCM layer at the bottom of the solar still basin on its performance. The investigation revealed a significant 20% increase in output due to utilizing PCM. PCMs can store and release considerable thermal energy as heat because of their intrinsic latent heat characteristic (Asbik et al. 2016). A solar still was created employing paraffin wax, resulting in a nighttime production that was 400% more than regular solar still. Increasing PCM charging and discharging rates may be possible using highly conductive nanoparticles (Yousef et al. 2019). Incorporating nanoparticles into the base materials has significantly improved the efficiency of single basin solar still, leading to increased production of distilled water (Sahota & Tiwari 2016; Shalaby et al. 2022). Chaichan & Kazem (2018) researched the effects of incorporating Al₂O₃ nanoparticles into paraffin wax and discovered notable improvements in properties, including heat conductivity and homogeneous mixture stability. Modi et al. (2023) The study examined how different nanomaterials affect the condensing surface of solar stills to assess their impact on efficiency. Additionally, the study compared the outcomes of drop-wise and film-wise condensation on solar still production. The primary goal of the research was to explore how Nano-silicon material could enhance condensing surfaces. Researchers found that adding nanoparticles to the condensation process changed its behavior significantly. A remarkable advancement in the distillate was achieved by introducing CuO and Al₂O₃ nanoparticles into both active and passive solar still systems. The observed enhancements ranged from 60 to 90% (Naveenkumar et al. 2020). Assessed three distinct altered single-slope solar stills, each featuring a unique heating system and design. The study's findings indicate that the external condenser utilizing silver nanofluid demonstrated the highest output of 7,760 cc/m²/day. The research found that solar systems with added condensers had an efficiency improvement of 26.30% (Alenezi & Alabaiadly 2023). Passive and active solar stills underwent monthly and annual performance evaluations in diverse Indian climates. Maximizing the output of solar stills was made possible by changing the water depth and positioning the condensing cover according to the latitude of the specific area (Singh & Tiwari 2004). An investigation was conducted to analyze how ambient temperature and solar radiation influenced the performance of tubular and triangular solar stills, employing mathematical simulations and experimental procedures. The findings revealed that the tubular solar still exhibited a substantial 20% advantage over the triangular solar still. However, further research is necessary to simulate solar desalination stills effectively and evaluate the evaporation rate (Abdullah et al. 2024).

Existing literature indicates that only a limited number of experiments have delved into the effect of integrating nanoparticles with paraffin wax on the yield of the pyramid solar still (PSS). At the same time, very few studies have concentrated on optimizing the performance of solar stills. The primary goal of this research was to assess the potential application of paraffin wax infused with silver nanoparticles as a PCM in PSS and to optimize process parameters for increased productivity. The input process parameters considered in this study are solar intensity ranging from 350 to 950 W/m², nanoparticle concentration ranging from 0.5 to 1.5%, and water depth ranging from 4 to 8 cm and the corresponding performance parameters are productivity and temperatures of the distilled water and glass. This study introduces a new methodology that combines an artificial intelligence-driven fuzzy logic interface with particle swarm optimization.

Figure SI1 shows the procedure for preparing the nanocomposite. The black plate absorbs heat through radiation during daylight hours. Consequently, the heat will be transferred to the paraffin/silver nanoparticle. This investigation explored the implementation of silver nanoparticles (Ag) in molten paraffin at a temperature of 45 °C. The investigation focused on the effects of different weight proportions (0.5, 1, and 1.5%) of silver nanoparticles distributed in the paraffin medium. Using these nanoparticles as PCMs was intended to increase solar still's efficiency. The nanofluid was mixed for 2 h using intermittent 95% power to avoid overheating. In the daytime, mainly after 2 pm, paraffin undergoes a melting process where its temperature remains constant until the melting is finished. Paraffin begins to cool after 7 pm and maintains a constant temperature until it solidifies. It then cools back to the surrounding temperature. In the absence of paraffin, the water temperature increases during the daytime. After 7 pm, the water temperature increases due to the presence of PCM. To maximize the operational efficiency of the solar still during nighttime, a layer comprising paraffin/silver nanoparticles has been integrated beneath the basin (Sathyamurthy 2023). Silver nanoparticles are used in solar stills because of their excellent thermal conductivity and plasmonic properties, facilitating efficient heat transfer and improved light absorption. Figure SI2(a) presents the silver nanoparticle size and particle morphology. Similarly, figure SI2(b) shows the morphology of the paraffin wax. The morphology influences the process efficiency, and hence, it is studied.

In engineering, the Taguchi quality control technique emphasizes research and design to create dependable and efficient products (Chandra et al. 2022). The Taguchi technique efficiently identifies the optimal number of trials required within the permissible range of parameters and levels (Shunmugasundaram et al. 2021). This study examines the variation of three input parameters at three levels using a L₉ (3^3) orthogonal design. The factors and corresponding values are displayed in Table SI3.

Here, A_i signifies the ith selection that fulfills the jth specification, and x_ij represents the corresponding numerical value of that ith choice. Implementing the TOPSIS methodology involves various stages.

So, the vector's unit length for every attribute is identical.

The factors linked to benefit and cost criteria are crucial in identifying the most optimal choice. The alternatives A* and A⁻ represent the most favorable and unfavorable options in a specific sequence (Sharma 2020).

If A_i is equal to A*, C_i* is set to 1; if A_i is equal to A⁻, C_i* is set to 0. As Ci* approaches 1, A_i is converging toward A*.

It is now feasible to sort a set of choices in descending C_i* order according to their selection.

Using fuzzy logic, fuzzy rule-based systems, and fuzzy expert systems is widespread in applying logical rules to build correlations between input variables and outputs. The categorizing process involves four interconnected modules that are closely intertwined. The system has several key components: the rule framer, fuzzifier, implication, and output processor (Sathish Kumar et al. 2023). The equation y = f quantitatively expresses the reaction (x). After establishing the criteria for linking input variables to prediction variables, the fuzzy network can assess the network's response to slight modifications. Using the fuzzy network's membership function (MF), rules can be derived, allowing for informed decisions by established regulations (Palanikumar & Rajasekaran 2017; Choudhury & Chandrasekaran 2023). A set of IF–THEN rules provides correct translation from input to output. It is possible to use fuzzy logic to find output solutions that are opaque, imprecise, or unclear. Using suitable IF–THEN rules in the FIS reasoning process can decrease uncertainties in input data. Furthermore, the precision of the prediction model could be enhanced by incorporating precise data and rule bases into the FIS framework (Azmi et al. 2011; Zhang et al. 2023). The functional components of FIS are shown in Figure SI4, whereby inputs are transformed into separate outputs using a rule basis. The optimal number of membership functions (MF) and their respective values allocated to the set depend primarily on the intended response. Commonly used techniques in fuzzy systems include Sugeno and Mamdani's implication methods.

Figure SI5 illustrates the triangular MF used to measure productivity and relative closeness (RC) with varying numbers of MFs. The low, medium, and high triangle MFs are used to represent the input variables. Five triangle MFs (very low (VL), low, medium, high, and very high (VH)) represent the output variables.

Machine learning methods can analyze large datasets efficiently and uncover patterns and insights that are not immediately apparent to humans. They often require significant computational resources and large amounts of labeled data and can be challenging to interpret and understand.

In the interval [0,1], let and stand for two uniformly distributed random values, and Δt = 1 for the time increment. For time-discrete iterative systems with unit time increments, Δt is defined as 1 and holds significance. Furthermore, the variable x represents the most suitable solution for particle i, considering its search history up to iteration t. At the same time, g* denotes the optimal solution for the entire population at that specific iteration (Najjar et al. 2023). Furthermore, it is common for the learning parameters and to be assigned values within the range of [0,2]. An investigation was performed to assess the significance of these parameter values and their possible influence on the algorithm's stability. It is not surprising that there are numerous variants (Gholami & Melenka 2023).

Various input control parameters were utilized during the experiments, such as solar radiation (ranging from 350 to 950 W/m²), water depth (ranging from 4 to 8 cm), and nanoparticle concentration (ranging from 0.5 to 1.5%). Corresponding output responses include productivity values ranging from 1.367 to 2.050 kg/m², glass temperatures spanning from 35.08 to 47.86 °C, and basin water temperatures ranging from 52.49 to 63.86 °C.

Table 1 displays the input and output response of PSS. The results indicate a positive correlation between solar intensity, nanoparticle concentration, productivity, and basin temperature. The basin's depth significantly influences the water's temperature and productivity. Evaporation rises when the water depth decreases, increasing temperature and productivity. By keeping the water at its minimum depth, the SS can effectively improve the production of the distillate (Abdullah et al. 2023a, 2023b). By incorporating silver nanoparticles into the PCM, there has been a substantial enhancement in heat transfer efficiency and an increase in water productivity in areas with relatively low solar intensity. The rate at which the PSS system produces drinkable water depends on the temperature difference (T_w − T_g) between the water in the basin and the glass cover, affecting the condensation process. (Prasad et al. 2019). Elevating the temperature differential (T_w − T_g) between the water surface and glass cover in a PSS typically results in greater potable water output. This is primarily due to the expanded heat gradient, enabling faster evaporation and condensation processes. The quantity of distilled water produced varies depending on the temperature of the water in the basin and the glass container. Lowering the temperature of the glass cover can be accomplished through modifications like adjusting the water level and incorporating nanoparticles with PCM (Bekele et al. 2023).

Table 1

Input and output response of PSS

Exp.No .	Input control parameters .			Output responses (⁠⁠) .
	SI (W/m2) .	WD (cm) .	NSp (%) .	P (kg/m2) .	Tg (°C) .	Tw (°C) .
1	350	4	0.5	1.367	35.08	53.52
2	350	6	1.0	1.497	36.25	59.89
3	350	8	1.5	1.681	47.86	52.49
4	650	4	1.0	1.958	37.21	57.60
5	650	6	1.5	1.732	38.50	60.26
6	650	8	0.5	1.623	35.56	58.45
7	950	4	1.5	1.972	41.04	61.25
8	950	6	0.5	2.050	36.83	58.10
9	950	8	1.0	1.751	37.12	63.86

Exp.No .	Input control parameters .			Output responses (⁠⁠) .
	SI (W/m2) .	WD (cm) .	NSp (%) .	P (kg/m2) .	Tg (°C) .	Tw (°C) .
1	350	4	0.5	1.367	35.08	53.52
2	350	6	1.0	1.497	36.25	59.89
3	350	8	1.5	1.681	47.86	52.49
4	650	4	1.0	1.958	37.21	57.60
5	650	6	1.5	1.732	38.50	60.26
6	650	8	0.5	1.623	35.56	58.45
7	950	4	1.5	1.972	41.04	61.25
8	950	6	0.5	2.050	36.83	58.10
9	950	8	1.0	1.751	37.12	63.86

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The TOPSIS method is used to optimize PSS responses with multiple objectives. Initially, it is necessary to transform the outputs into a normalized sequence, where all values are adjusted to fall within the range of 0 to 1. Table 2 displays the normalized values for productivity, basin water temperature, and glass. Subsequently, after the normalization process, weights are assigned to variables such as productivity, glass temperature, and basin water. It is important to note that all responses in this analysis are given equal weightage. Once the weighted normalization is completed, the positive ideal solutions are determined using Equation (5), as presented in Table 3.

Table 2

Normalization of the output responses

Exp. No .	.			Normalization .
	P (kg/m2) .	Tg (°C) .	Tw (°C) .	P (kg/m2) .	Tg (°C) .	Tw (°C) .
1	1.8687	1,230.6064	2,864.3904	0.260	0.303	0.305
2	2.2410	1,314.0625	3,586.8121	0.285	0.313	0.341
3	2.8258	2,290.5796	2,755.2001	0.320	0.414	0.299
4	3.8338	1,384.5841	3,317.7600	0.373	0.322	0.328
5	2.9981	1,482.2500	3,631.2676	0.330	0.333	0.343
6	2.6341	1,264.5136	3,416.4025	0.309	0.307	0.333
7	3.8888	1,684.2816	3,751.5625	0.376	0.355	0.349
8	4.2033	1,356.4489	3,375.6100	0.391	0.318	0.331
9	3.0660	1,377.8944	4,078.0996	0.334	0.321	0.364

Exp. No .	.			Normalization .
	P (kg/m2) .	Tg (°C) .	Tw (°C) .	P (kg/m2) .	Tg (°C) .	Tw (°C) .
1	1.8687	1,230.6064	2,864.3904	0.260	0.303	0.305
2	2.2410	1,314.0625	3,586.8121	0.285	0.313	0.341
3	2.8258	2,290.5796	2,755.2001	0.320	0.414	0.299
4	3.8338	1,384.5841	3,317.7600	0.373	0.322	0.328
5	2.9981	1,482.2500	3,631.2676	0.330	0.333	0.343
6	2.6341	1,264.5136	3,416.4025	0.309	0.307	0.333
7	3.8888	1,684.2816	3,751.5625	0.376	0.355	0.349
8	4.2033	1,356.4489	3,375.6100	0.391	0.318	0.331
9	3.0660	1,377.8944	4,078.0996	0.334	0.321	0.364

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The response table is generated to identify the optimal conditions, considering the RC values related to each level, as presented in Table SI4.

According to the ANOVA table, the mathematical model established could be better as there is fuzziness in the obtained data. Hence, fuzzy logic is applied to improve the distinctness of the available data outputs. The fuzzy inference module and MF utilized in this research are depicted in Figure SI5. It comprises three inputs (solar intensity, water depth, and % nanoparticle concentration) and three outputs: productivity, glass temperature, and basin water temperature. All three input values (low, medium, and high) are considered when evaluating triangular MFs. The triangular MF is commonly utilized and is, therefore, the focus of this study. The inputs consist of three MFs, while the outputs consist of nine MFs, aiming to enhance the accuracy of predictions. The triangular MFs shown in Figure SI5 comprise nine subsets, which are very small (VS), small (S), VL, low (L), medium (M), low (L), VL, high (H), and VH. The rules were created by analyzing the input and output data and assigning values to MFs. The variations in output and input were documented using a set of guidelines. The rule editor in Figure SI6 utilizes IF–THEN rules (Ambigai & Prabhu 2019, 2021; Selvarajan et al. 2023) to implement expert system knowledge. According to the rules, the predicted results for the given input values are productivity, glass temperature, and basin water temperature.

The PSO algorithm maximizes productivity and basin water temperature while minimizing glass temperature. The parametric conditions are

• 350 < solar intensity (W/m²) < 950

• 4 < water depth (cm) < 8

• 0.5 < Ag nanoparticle (1%) < 1.5

The source code for PSO was written using MATLAB R2018a. The program's execution was contingent upon meeting the termination conditions of having 50 search agents and 100 iterations.

With the obtained optimized condition, a confirmation experiment was performed for TOPSIS-fuzzy and PSO ideal conditions, as tabulated in Table 5.

The confirmation table compares the outcomes of three optimization methods: TOPSIS, fuzzy logic, and PSO. Each approach was assessed using input parameters, including solar intensity, water depth, and nanoparticle concentration. This evaluation aimed to determine the productivity and the water and glass temperatures. The TOPSIS method determined the optimal parameters to be a solar intensity of 950 W/m², a water depth of 4 cm, and a concentration of 1% nanoparticles. As a result, the corresponding output values were a basin water temperature of 59.16 °C, a glass temperature of 35.96 °C, and a productivity of 2.112 kg/m².

In contrast, the application of fuzzy logic produced slightly varied outcomes. The water depth increased to 6 cm, productivity reached 2.225 kg/m², and the temperatures recorded were 59.96 °C for water and 34.52 °C for glass. The PSO algorithm yielded results that include a water depth of 5.85 cm, a productivity of 2.321 kg/m², and temperatures of 60.10 °C for water and 34.21 °C for glass. The percentage of variation between TOPSIS and fuzzy methods is 5.07% for productivity, 1.33% for basin water temperature, and 4% for glass temperature. The differences in productivity between TOPSIS and PSO are 9%, the basin water temperature is 1.56%, and the glass temperature is 4.86%. The percentage of variation between fuzzy and PSO is 4.13% for productivity, 0.23% for basin water temperature, and 0.89% for glass temperature. The confirmation table found that results obtained from the PSO are similar to those obtained from the fuzzy.

This research presents a novel application of a fuzzy logic system to enhance the performance data analysis for PSS. By carefully selecting appropriate MFs and formulating if–then rules, the research provides a method for generating clear and concise outputs. Incorporating silver nanoparticles into the PCM and using fuzzy logic and PSO to optimize conditions is an innovative approach that advances the field of solar desalination. The outcomes of this research endeavor include:

• 1. The study demonstrates a positive correlation between solar intensity, productivity, and basin water temperature. The highest level of productivity is observed when the water depth is increased from 4 to 6 cm. However, after reaching this point, productivity starts to decline. The temperature of glass decreases with an increase in water depth. The heat transfer rate and water productivity can be enhanced by incorporating silver nanoparticles into PCM in low solar intensity conditions. The temperature differential between the glass cover and the water in the basin (Tw − Tg) impacts the efficiency of producing clean water and condensation rate in PSS. An increase in the temperature of the basin water, with a decrease in the temperature of the glass, results in a more significant amount of distilled water. It clearly shows that adding Ag nanoparticles to the absorber basin of PSS will enhance productivity up to a water depth of 6 cm. Hence, the optimal conditions are the solar intensity of 950 W/m², water depth of 6 cm, and Ag nanoparticle of 1%.

• 2. The optimal values attained with TOPSIS analysis are solar intensity of 950 W/m², water depth of 4 cm, and Ag nanoparticle of 1%. A substantial interaction exists among the input factors, and the selected level values and optimal values show a considerable interaction effect. The R² value obtained before the incorporation of fuzzy logic is 88.55% with an adjusted R² value of 54.19%; the contribution of solar intensity is higher at 78.79%, followed by water depth by 7.17% and Ag nanoparticle by 2.58%. The error % is higher (11.45%).

• 3. A fuzzy system is considered to have three triangular MFs as inputs and five triangular MFs as outputs. The R² value obtained for fuzzy-RC is 96.24% with an adjusted R² value of 84.97%. The contribution of solar intensity is higher at 81.68%, followed by water depth at 10.21% and Ag nanoparticle at 4.35%. The error % is 3.76, which falls under the 95% confidence interval. The fuzziness in the system is reduced, and a crisp output is obtained, which is helpful for interpretation with the adoption of fuzzy logic. The optimal condition obtained is solar intensity of 950 W/m², water depth of 6 cm, and Ag nanoparticle of 1%.

• 4. The optimal conditions obtained by PSO are solar intensity of 950 W/m², water depth of 5.85 cm, and Ag nanoparticle of 1%, which is nearly similar to the optimal conditions obtained from the fuzzy analysis.

Future studies explore using other nanomaterials and their impact on PSS efficiency, which could further enhance water productivity. Combining fuzzy logic with other optimization techniques, such as genetic algorithms, could improve the accuracy and applicability of the model.

No funding was received for this research work.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.