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Article

Statistical Model for the Sizing of a Prototype Solar Still Applicable to Remote Islands

1
International Program on Energy Engineering, National Cheng Kung University, Tainan 70101, Taiwan
2
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan
*
Author to whom correspondence should be addressed.
Water 2022, 14(21), 3510; https://doi.org/10.3390/w14213510
Received: 12 September 2022 / Revised: 25 October 2022 / Accepted: 28 October 2022 / Published: 2 November 2022
(This article belongs to the Section Wastewater Treatment and Reuse)

Abstract

:
The topography and location of many remote islands limit the available freshwater resources present for use by the inhabitants. However, the abundance of solar and seawater resources and small population size makes them ideal candidates for solar still application. A prototype solar still was designed and fabricated for use on such applications; however, for its implementation, a statistical model was developed to assess its productive performance at a pilot location, Dongji Islet. Experiments were conducted to collect data and construct a multivariable regression model by means of the jack-knife procedure and best subsets technique. The model was then used to size the solar still for implementation by applying the TMY data of Dongji Islet. The daily total global solar radiation, average ambient temperature, and extent of cloud cover were found to be the most suitable predictor variables for the model based on their correlation to the productivity of the protype solar still and their p-value. The model predicted a maximum daily yield of 5.88 L/day in July and a minimum of 1.97 L/day in December. In relation to the annual predicted yield, the length of the solar still can be increased by 88.6% in order to satisfy the daily minimum requirement of 7.5 L per day per person.

1. Introduction

Freshwater scarcity is increasingly becoming a global issue especially in recent times due partly to the changes in global weather patterns and the increase in demand resulting from continual growth in population size and industrial development. This is also compounded by the energy crisis as fossil fuel reserves are being depleted at an increasingly fast rate which limits the technologies that can be used to generate affordable freshwater.
Solar stills have been deemed one possible solution to resolve both the water and energy crisis, for small-scale applications, as it uses a renewable energy form (solar energy) to desalinate saline water. A more intensive review on the current progress on solar still techniques, including passive and active solar stills, is referred to in our previous paper [1]. To this end a prototype solar still was designed and fabricated in a previous study [1] to satisfy the minimum freshwater requirement of 7.5 L per day per person as recommended by the World Health Organization (WHO) [2]. This prototype was intended to supply freshwater to remote islands due to some of its pertinent features such as the simplicity of its design, its durability, its ability to operate without the need for electrical power, and its low cost and maintenance. These features are necessary for application on remote islands as the use of more sophisticated technologies, such as reverse osmosis system, were deemed unsuitable for some remote areas as they generally require frequent maintenance, manufacturer support and trained technicians, and have relatively high capital and operating costs [3]. Furthermore, remote islands are generally characterized by small land area, low population size, abundant seawater and solar resource, few skilled workers, and are generally secluded, making them not easily accessible and thus better suited for solar still application.
Despite the suitability of solar stills for remote islands, they inherently have low productivity. In light of this, various studies have been conducted to identify factors that influence their performance with the aim of making them more efficient but also cost effective. Researchers have identified three main categories of factors, namely meteorological, design, and operational factors [4,5,6,7,8]. Consideration was given to the design and operational factors during the designing and fabrication stages of the protype solar still used in this study. On the other hand, it was necessary to consider the meteorological factors for the implementation stage of the solar still as this would be a deciding factor in selecting a suitable location based on the level of freshwater demand.
The modelling of a prototype to assess its performance for implementation is a common technique used by product developers and engineers alike. Such models vary in accuracy, complexity, costs, time for development, technology, and technical expertise. Common approaches taken by researchers to model the productive performance of a solar still include thermal [9,10,11,12,13,14], intelligent [15,16,17,18], and statistical modelling [19,20,21,22,23,24,25,26,27,28,29]. For the purpose of this study, the statistical modelling approach was adopted. To estimate the productivity of a solar still, statistical models used by previous researchers utilized, in some instances, meteorological factors only [19,28], operational factors only [20], or a combination of meteorological, design, and operational factors [21,22,23,24,26,27,29] as predictor variables as outlined in Table 1. Additionally, a variation of linear, non-linear, and exponential regression models have been based on the stepwise technique. The regression models were, however, specific to the solar still for which it was developed and not adoptable to other stills. Moreover, further use and evaluation of the regression models beyond the experimental sites were not carried out. To this end, typical meteorological year (TMY) data have been applied to a developed statistical model for the purpose of predicting the productivity of the solar still used in the present study. TMY data sets are a compilation of one year of meteorological data representing the hourly weather information of a specific geographical site based on long-term records [30]. The TMY data sets are intended to reflect the normal weather conditions of a location by concatenating 12 observational months. An algorithm is used to select the months from different years, based on the Sandia method [31].
The purpose of this study is to develop a multivariable regression model using the jack-knife and best subsets technique for a prototype solar still that can be implemented beyond the experimental site. A small remote island, Dongji Islet, located in the Taiwan Strait at 23.3° N and 119.7° E, with an approximate population size of 20 permanent residents and a land area of 1.54 km2 was used as the pilot location for demonstration [32]. The regression model will then be used to estimate the productivity of the prototype solar still based on the TMY data of the Islet as it possesses its own weather station and as TMY data is an accurate long-term representation of the weather condition at a given location. Furthermore, the sizing of the solar still with reference to the freshwater demand of 7.5 L per day per person will be performed in accordance with the projected productivity. However, due to the limited land area and the hazards that disposal of the concentrated brine residue may have on the environment, the brine residue will be repurposed to make crude solar sea salt. The payback period of the prototype solar still will also be determined based on its estimated performance.

2. Solar Still Set-Up and Experimental Methods

2.1. Experimental Setup and Data Collection

A prototype solar still was designed and fabricated according to [1]. The protype was used to collect experimental data between the period of 9 February to 26 June 2022. A total of 13 experimental trials were conducted over the period. The experimental setup used is shown in Figure 1. Figure 2 shows the schematic diagram of the system along with the electronic components used to control the water level, as well as the thermometers used to record the temperature at various positions within the solar still.
The experiments were carried out on the grounds of the National Cheng Kung University (NCKU), Guiren Campus, which is located in Tainan City (22.9° N, 120.3° E), Taiwan. The procedure that was used is as follows:
  • The solar still was positioned in a north-south orientation such that the inclined glass covers were directed east and west while the heat pipe evacuated tube collectors (ETC) were inclined at 23° to the horizontal facing the south.
  • The feedwater storage tub was filled with 120 L of water at least one day ahead of the experiment and the topside glass panes, side wall glass panes, fins, basin, and the solar water heater were also cleaned ahead of the experiment.
  • Cotton wicks were prepared by cutting a roll of cotton yarn into 145 strands that were 0.002 m in diameter and 0.50 m long and were then placed in the feedwater to soak overnight.
  • On the first day of the experiment at approximately 5:30 am, the glass cover was removed and two sets of fins were installed in the basin of the solar still, following which an electronic water pump was used to fill the basin with feedwater to the maximum water level (0.027 m for 60% recovery ratio, 0.040 m for 70% recovery, and 0.077 m for 80% recovery). The high-water level switch (Figure 2) controls the maximum water level by deactivating the water pump and electronic valve A simultaneously. The cotton wicks were then added to the larger rectangular fins (Figure 3).
  • The glass cover was then repositioned on the basin and the gaps between the basin and the frame of the glass cover were sealed with foam material to prevent vapor from escaping.
  • Hourly measurements of weather parameters such as total global solar radiation (Rad), ambient temperature (Ta), ambient relative humidity (RHa), and windspeed (Ws) were performed using equipment already installed at the test site such as the Eppley precision spectral pyranometers (Model PSP), Young Co. anemometer (Model 05103L), and Dwyer humidity-temperature transmitter (Model 657-1). Data for the fraction of cloud cover (CCr) for Tainan City was obtained from the Central Weather Bureau [33].
  • The data acquisition modules along with the LabVIEW engineering software were used to make hourly measurements of the basin water temperature (TB), inner glass temperature, vapor temperature, and humidity of the solar still.
  • The volume of distillate produced were measured at 8 am (overnight) and 5 pm (daytime) daily until the volume of the feedwater in the basin reaches the minimum level (approximately 1 cm above the absorber plate) which was controlled by the low water level switch (Figure 2).
  • The low-level water switch activates electronic valve A which allows the residue water to flow into the brine residue storage tub. The volume of the residue water was then measured after which it was placed in an open area to evaporate (Figure 4a) and the mass of the crude salt that remained was measured by gravimetric analysis (drying and reweighing until two consecutive masses were within 1 g of each other and recorded).
  • Following the collection of the residue water, the wicks and fins were then removed and cleaned.
  • The procedure was then repeated for the next experiment.
It must be noted that due to time constraints, a portion of the residue water was boiled off in some cases to estimate the concentration of the residue water and crude salt produced. Furthermore, two of the thirteen experiments carried out were based on a 60% and 80% recovery ratio while the remaining experiments were done using a 70% recovery ratio.

2.2. Uncertainty Analysis

The measurement of physical quantities is susceptible to random errors, systematic errors, or a combination of both. As a result, a measurand cannot be stated with absolute certainty. Errors in measurement may arise from the environment, the instrument being used, the methodology employed, or the limitations of the observer. The external errors associated with the measuring instruments used are shown in Table 2.
The internal uncertainty was done according to the method used by [34]. The following equations were used:
x ¯ = 1 n i = 1 n x i
σ = 1 n 1 i = 1 n ( x i x ¯ ) 2
U ( x ) = σ n
where x ¯ represents the mean of the measured values, n represents the number of measurements, x i represents the ith measured value, σ represents the standard deviation, and U ( x ) represents the standard uncertainty.

2.3. Data Analysis

Multivariable regression analysis was performed on the data collected in order to formulate an empirical model to predict the yield of the solar still at 70% recovery ratio operational mode according to correlated weather parameters.
Outliers within the data sets based on the 70% recovery ratio were first identified by constructing box plots using the statistical software Minitab 19 [35]. Data points that were beyond 1.5 of the interquartile range (IQR) value, represented by the whiskers of the box plots, were deemed outliers and thus removed from the data sets. A Pearson’s correlation [36] matrix was then plotted to measure the extent of correlation between the weather parameters and the productivity of the prototype solar still. The jack-knife technique was repeatedly applied to the data sets after which the best subset method was carried out [37,38]. The model was then measured for accuracy based on the selected statistical indicators: standard error of the regression (S), Durbin-Watson statistic ( d ), variance inflation factor (VIF), Mallows Cp, probability value (p-value), coefficient of determination (R2), adjusted-R squared ( R a 2 ), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). These parameters were calculated using the following equations [17,39,40]:
R 2 = 1 i = 1 n ( Y i Y ˆ i ) 2 i = 1 n ( Y i Y ¯ i ) 2
R a 2 = 1 [ ( 1 R 2 ) ( n 1 ) n k 1 ]
MAE = 1 n i = 1 n | Y i Y ˆ i |
MSE = 1 n i = 1 n ( Y i Y ˆ i ) 2
RMSE = MSE
MAPE = 1 n i = 1 n | Y i Y ˆ i Y i | × 100
S = 1 R a 2   ×   1 n 1 i = 1 n ( Y i Y ˆ i ) 2
SE   = 1 n 2 i = 1 n ( Y i Y ˆ i ) 2 i = 1 n ( X i X ¯ i ) 2
Cp = ( SSE k MSE all _ k ) n + 2 ( k + 1 )
SSE = i = 1 n ( Y i Y ˆ i ) 2
VIF j = 1 1 R j 2
d = i = 2 n ( e i e i 1 ) 2 i = 1 n e i 2
e i   =   Y i Y ˆ i
e i 1 =   Y i 1 Y ˆ i 1
where k represents the number of variables, j represents the jth predictor variable, i represents the ith observation, e i represents the ith error term, e i 1 represents the ( i 1 )th error term, Y represents the observed values, X i represents the value of the ith predictor variable, X ¯ represents the mean of the predictor variable, Y ˆ represents the predicted values, and Y ¯ represents the mean of the observed values.
The values of selected weather parameters corresponding to the typical meteorological year (TMY) of Dongji Islet were applied to the statistical model to estimate the daily and yearly productivity of the prototype solar still based on the weather condition of Dongji Islet [33,41]. In relation to the World Health Organization’s (WHO) recommended minimum freshwater requirement of 7.5 L per day per person for drinking and food preparation [2], the size of the solar still necessary to meet this demand was determined. The estimated productivity of the protype solar still on Dongji Islet, combined with manufacturing and maintenance costs, were used to perform an economic analysis of the solar still.

3. Results and Discussion

3.1. Summary of Data Collected from Experimental Trials at 70% Recovery Ratio

A total of thirteen experimental trials were conducted for the purpose of this study. Furthermore, eleven of the thirteen experimental trials were based on a 70% recovery ratio (Table 2), while the other two, trials 12 and 13, were based on an 80% and a 60% recovery ratio (Table 3), respectively. The eleven experimental trials were then further subdivided into ten training data sets and one testing data set (trail 9).
Table 3 shows that the average daily Rad has a significant effect on the yield of potable water generated by the prototype solar still and duration of the experimental trials. Trial 1 with the lowest average daily yield and longest duration is observed to have the lowest average daily Rad, while the opposite is true for trial 13. Moreover, trials 3, 6, 10, and 12 all have similar duration and average yield, which can be attributed to the near identical amount of average daily Rad received.

3.2. Effect of Recovery Ratio on the Performance of the Solar Still

The recovery ratio (RR) of a desalination system is defined as the volume of distillate (VD) that results from a unit volume of feedwater (VF) expressed as a percentage or decimal [42,43]. The recovery ratio may be expressed as:
RR = V D V F × 100
The prototype solar still was operated at three different recovery ratios, 60%, 70%, and 80%. Table 4 summarizes the effect of changing the recovery ratio on the performance of the solar still. Owing to the design of the solar still, a minimum water level of approximately 1 cm above the absorber plate must be maintained in the basin for heat transfer and to prevent dry spots from appearing on the absorber plate and the ETCs during operation. As a result, different volumes of feedwater must be used to achieve different recovery ratios as shown in Table 4.
In relation to the total volume of distillate produced over the testing period, at 80% recovery ratio, the solar still recorded the highest productivity despite having the longest duration to reach the minimum water level and similar weather conditions compared with the other recovery ratios. On the other hand, at 60% recovery ratio, the opposite effect is observed. This effect can be explained by the distance of the basin water surface in relation to the inner condensing glass cover. At the 80% recovery ratio, the basin water surface is relatively closer to the glass cover compared with the 60% recovery ratio, since a larger volume of water is required to achieve a higher recovery ratio. Studies have shown that the size of the fluid layer between the water surface and the glass cover affects the level of thermal resistance there is for convective and evaporative heat transfer, i.e., less fluid layer results in lower thermal resistance [44,45,46,47]. Therefore, the shorter the average distance between the water surface and the transparent cover, the better the heat transfer which results in improved productivity since condensation can occur sooner. Additionally, the overnight distillate production at 80% recovery ratio is significantly higher than that of the other recovery ratios. The greater overnight yield resulted from the large volume of water acting as a thermal energy storage medium which was able to store more of the thermal energy captured during the daytime and reuse it at night in the absence of solar radiation.
A comparison of the average rate of change of productivity for the different recovery ratios shows that there is a decrease of 30.9% and 59.2% when the 60% recovery ratio is compared with the 70% and 80%, respectively. This is accompanied by corresponding longer operational periods of 1.5 and 2.7 more when the 60% recovery ratio is compared with the 70% and 80%, respectively. These observations resulted from the higher energy demand needed for evaporation of the more concentrated brine residue [48]. The reduction in the rate of productivity with increased recovery ratio may have also been influenced by increased vapor leakage of the solar still over the longer periods of operation under higher recovery ratios.
Although the 70% recovery ratio did not generate the highest productivity, it was chosen over the other recovery ratios for normal operation on the basis of energy requirement, frequency of feedwater refilling operations, and long-term effects of scaling due to high brine residue concentration associated with higher recovery ratios.

3.3. Effect of Recovery Ratio on the Concentration of the Brine Residue

Apart from affecting the performance of a solar still, changing the recovery ratio also impacts the concentration of the brine residue that remains from the desalination process. Table 5 shows the concentration of the brine residue at different ratios. As is evident from Table 5, the concentrations of the brine residue increase drastically from 93,900 mg/L to 220,650 mg/L as the recovery ratio increases from 60% to 80%, respectively. A 10% increase in the recovery ratio results in a 22.3% increase in the concentration of the brine residue, while a 20% increase results in a 135% increase in concentration. The non-linear relationship between the variation of the recovery ratio and change in concentration of the brine residue may be due to the increased requirement in the volume of feedwater, as shown in Table 5, needed to achieve a higher recovery ratio based on the design of the prototype solar still.
Given that such highly concentrated brine residue is harmful to the environment and can thus be a challenge for disposal, it was repurposed to make crude solar sea salt (Figure 4) for food preservation and industrial use. According to Table 5, an average of 1.7 kg of sea salt can be generated per 14.8 L of residue volume at a recovery ratio of 70%.

3.4. Modelling of the Daily Distillate Yield

Multivariable linear regression was adopted to develop the statistical model used to estimate the yield of the prototype solar still using the TMY data of Dongji Islet. Multivariable linear regression expresses a response or dependent variable ( y ) in terms of two or more dissimilar predictor or independent variables ( x ). A linear regression equation is generally expressed as:
y = β 0 + β 1 x 1 + β 2 x 2 + + β k x k + ε
where β 0 represents a constant term and β k represent the coefficient of the kth predictor variable and ε represents a random error term [49].

3.4.1. Identification of Outliers

The box plots shown in Figure 5 illustrate the preliminary outliers that were detected within the various data sets. The horizontal line within the boxes represents the median value, the horizontal lines at the end of the boxes represent the quartiles (upper and lower), and the length of the whiskers is 1.5 times the value of the IQR. On surveying Figure 3, there were at most two outliers within a single data set. The data sets belonging to the yield, global radiation, and ambient relative humidity registered two outliers each represented by the shaded dots in the graphs, while the data sets for cloud cover and ambient temperature registered one outlier each. The low number of outliers detected in the data sets, based on the 1.5 IQR rule, suggests that the observations are statistically uniform despite the random nature of the weather.
Besides identifying outliers, the boxplots are also useful in analyzing the variability among observed values. Overall, the observations for ambient temperature and ambient relative humidity displayed the least spread in datapoints over the experimental trials compared with the other variables. Amongst the 10 trials, trial number 5 (T5) most frequently displayed the greatest variation in the observations for the variables in Figure 3, while trial number 3 (T3) most frequently displayed the smallest variation. This implies that the weather condition for trial number 5 was the least stable while that for trial number 3 was the most stable over the testing period.

3.4.2. Predictor Variables and Pearson’s Correlation Matrix

The Pearson’s correlation matrix shown in Figure 6 was used to identify the most significant or correlated predictor variables with regards to the productivity or yield of the protype solar still. Taking yield as the response variable, Figure 4 shows that Rad is the most correlated predictor variable since it has the highest correlation coefficient (r), while Ws is the least correlated. Rad highly correlated to the yield because of the direct relationship between absorbed radiative energy and evaporation, while Ws is least correlated due to the unknown directional nature of its effect on the solar still. This is evident as the cooling effect of forced air movement across the glass cover of a solar still resulted in a positive effect on the productivity of a solar still in some studies and a negative effect in others [50,51,52].
Since Ws was the least correlated variable it was omitted as a feasible predictor variable and the variables Rad, Temp, RHa, and CCr were used.

3.4.3. Model Building

The first stage of training/building the model involved adopting the jack-knife technique [37]. The regression model was fitted and data points with standard absolute residual values of at least two were removed. The model was refitted using the remaining observations until there was no data point with a standard absolute residual greater than or equal to two. Figure 7 shows the standardized absolute residuals of the fitted regression model before and after the jack-knife technique was performed. After the repeated application of this technique, the total number of data points were reduced by approximately 47%. Additionally, attention was also paid to predictor variables with a p-value greater than 0.05, which indicates that the variable is statistically insignificant in accounting for the variation in the response variable, i.e., yield. Hence, RHa with a p-value of 0.618 as shown in Table 6 was omitted from the model.
Table 6 also shows significant reductions in the standard error (SE) of the coefficients as well as the p-values for Temp and CCr. These reductions indicate how closely the model fits the observed data. On the other hand, Rad and Temp show increases in the variable inflation factor (VIF). The VIF is generally used to measure the correlation between two or more predictor variables (multicollinearity). Although the existence of multicollinearity impacts the estimated values of the regression coefficients and the sum of squares for the estimates of the model, it does not affect the predictive ability of the model [39]. Despite the increases, the VIF values lie within the range 2–5 which is indicative of an acceptable level of multicollinearity [53].
Another statistic used to assess the quality of the model was the Durbin–Watson statistic (d). It is often used to measure the extent of correlation between the residuals in a linear regression model (autocorrelation). For the purpose of this study, the existence of autocorrelation would violate one of the four assumptions of a linear regression model which would negatively impact the validity of the predictor variables [39,49,54]. The initial value of d for the model at the beginning of the jack-knife regression technique was 1.517, while the final value after the technique was performed was 2.399. Since the Durbin–Watson statistic is within the range 1.5–2.5, the extent of autocorrelation is not significant [55].
To substantiate the regression model that resulted from the jack-knife technique, a best subset selection [38] was also performed. Table 7 shows the results of the best subset selection. The Mallows Cp-statistic shown in Table 7 is used to assess the extent of bias in the predicted values based [39]. Severe bias exists if the Cp value is significantly higher than k + 1. Table 7 confirms that the regression model with Rad, Temp, and CCr as the predictor variables is the most acceptable based on the high R2 and R a 2 values and low Mallows Cp and S values. Both R a 2 and R2 are used to measure the extent to which variation in a response variable can be accounted for by one or more predictor variables. However, the use of R a 2 is preferred over R2 for multivariable regression models as the value of R2 increases with increasing number of predictor variables without regard for the significance of the variable to the model [49].
From the values in Table 6, the final regression model was expressed as:
yield = 1.073 + 0.31015   Rad   + 0.04074   Temp   0.0741   CCr
where the yield is measured in liters (L), Rad represents the total daily global solar radiation measured in units of MJ/m2, Temp represents the average daily ambient temperature measured in °C, RHa represents the average daily ambient relative humidity measured as a percentage (%), Ws represents the average daily windspeed measure in m/s, and CCr represents the cloud cover which is given a value ranging from 0 to 10, where 0 indicates a clear sky and 10 indicates that the sky is overcast.

3.4.4. Model Evaluation

Apart from the R2, R a 2 , and S values, the accuracy of the model was also evaluated using the statistical measures mean absolute error (MAE), root mean squared error (RMSE), and mean absolute percentage error (MAPE). Table 8 shows the values of the different metrics used to evaluate the accuracy of the regression model. In consideration of the relatively low values of MAE, RMSE, and MAPE shown in Table 8, the regression model can be considered a good fit for the observations. Moreover, Feria-Díaz et al. developed a performance rating scale based on MAPE values whereby the model is rated as excellent (0% ≤ MAPE ≤ 10%), good (10% < MAPE ≤ 20%), fair (20% ≤ MAPE< 50%), or inaccurate (50% ≤ MAPE) [56]. Since the MAPE value of the model developed in this study lies within the range 0% to 10%, the model can be rated as excellent. The small variations between the training and testing values shown in Table 8 imply that the model has relatively consistent performance.

3.5. Application of Typical Meteorological Year (TMY) Data to the Model

In order to predict the performance of the prototype solar still on Dongji Islet, the TMY data for the Islet was applied to the model. The TMY was constructed from 12 typical meteorological months (TMM), which were selected from different years as displayed in Table 9. Using data obtained from Dongji weather station over the period 2004–2018, the TMY for Dongji was determined by Hsieh [41] in accordance with the TMY3 method [31] and the full weight for global radiation. Table 9 depicts the monthly average TMM values along with the corresponding predicted yield of the protype solar still for Dongji Islet. The highest daily yield is estimated at 5.88 L in July (summer season) while the lowest is 1.97 L in December (winter season). This trend is more clearly shown in Figure 9 and is largely influenced by the weather pattern on Dongji Islet since May, June, July, and August are the summer months which are usually hot. The winter months, including November, December, January, and February are generally cold. The general trend shown in Figure 8 indicates that both the predicted yield (green color) and total global solar radiation (orange color) have minimum values during the winter period which then rise to a maximum during the summer period, with July being the optimum month. The standard error bars of Figure 8 also reflect some variations in the daily standard errors which correspond to the predicted yield and total daily solar radiation. The daily standard error for the predicted yield has a range of 0.228–0.504 L, while that for the total daily solar radiation spans from 0.05–1.56 MJ. In spite of the standard errors, the approximate variation of predicted productivity in response to the seasonal weather pattern also validates the reliability of the model.
The regression model was also used to determine annual production estimates of potable water yield and the byproduct, crude sea salt, from the operation of the prototype solar still as shown in Table 10.

3.6. Sizing of the Solar Still Prototype

Dongji Islet is populated by approximately 20 permanent residents who obtain freshwater by means of periodical delivery by the coastguard as well as the collection of rainwater. According to the World Health Organization (WHO), a minimum of 7.5 L of fresh water per day per person is needed for survival, of which 2 L is for drinking and 5.5 L for food preparation [2]. On an annual basis, this would require 730 L of water at the level of 2 L per day per person and 2737.5 L at the level of 7.5 L per day per person. From Table 10, a single unit of the prototype solar still is capable of supplying 200% of the required 2 L per day per person and 53% of the required 7.5 L per day per person. This implies that the prototype solar still can provide a family of at most two people with potable water at the level of 2 L per day per person.
To fulfill the annual requirement of 2737.5 L, several units of the protype solar still can be used, a rainwater catchment can be installed, or the size of a single unit can be increased to meet the daily demand. However, in order to maintain the thermal performance of the protype solar still, only the length should be increased and not the width as depicted in Figure 9. This is necessary since the direction of heat transfer within the basin is along the fins, parallel to the width. As a result, changing the length would maintain the thermal performance of the prototype and the validity of the statistical model. The new dimensions of the prototype solar still that would be necessary to supply the required 2737.5 L of water annually is depicted in Table 11. To satisfy this minimum requirement, the basin length, area, and number of ETC collectors should be increased by 88.6%, 88.5%, and 91%, respectively.

3.7. Payback Period

The payback period (PP) of the protype solar still on Dongji Islet was determined by considering factors such as the remoteness of Dongji Islet, the capital cost ( C capital ), the annual cash flow ( ACF ), the estimated annual potable water yield (M), and crude sea salt production ( M S a l t ). Additionally, the projected selling price of the distillate ( SP w ) and crude sea salt ( SP s a l t ) along with their anticipated market penetration rate ( MPR ) were also taken into account. These parameters were determined using the following formulas:
PP = C capital ACF
ACF = [ ( M × SP w ) + ( M S a l t × SP s a l t ) ] × MPR
The market price of bottled water (less than 1 L), NTD 18, for Dongji Islet (estimated based on a prince range of NTD 13–23 and the distance from the neighboring town of Magong in the Penghu Archipelago), crude sea salt (NTD 23 per kg), and an estimated market penetration rate of 60% was applied in Equation (19). The capital cost ( C capital ) was evaluated using the formula:
C capital = C equipment + C installation
where C equipment represents the cost of fabrication as previously determined by [1] and C installation represents the cost of installation on Dongji Islet, which is estimated at 15% of C equipment based on the remoteness of the Islet. The estimated values resulting from Equations (18)–(20) are shown in Table 12. The prototype solar still has an estimated payback period of 6.1 years based on its installation on Dongji Islet.

4. Conclusions

The application of multivariable linear regression by means of the jack-knife and best subsets technique based on TMY data has been proven to be a viable approach in modelling the productivity of a prototype solar still. TMY data is considered to be a reliable representation of the weather pattern of Dongji Islet since it is based on the collection of weather data over a long period of time. The multivariable linear regression model, Equation (17), was developed with R-squared (R2), adjusted R-squared ( R a 2 ), MAE, RMSE, and MAPE values of 99.5%, 99.4%, 0.144, 0.167, and 9.71%, respectively. This study also showed that the meteorological parameters of daily total global solar radiation, ambient temperature, and extent of cloud cover were the most appropriate variables for predicting the productivity of the prototype solar still.
The suitability of the prototype solar still for application on remote islands is related to its characteristic features such as the simplicity of its design, its durability, its ability to operate without the need for electrical power, and its low capital cost and maintenance. A single unit of the prototype solar still is estimated to supply 2 L of potable water per day per person to a family of at most two people. On the other hand, in order to meet the recommended minimum daily requirement of 7.5 L of water per day per person according to the WHO, the length of the prototype solar still can be increased by approximately 88.6% with a proportional increase of 91% in the number of ETCs.
The installation of the prototype solar still on Dongji Islet has an estimated payback period of 6.1 years in direct relation to the volume of potable water generated and the mass of crude sea salt that can be repurposed from the brine residue.

Author Contributions

Conceptualization, A.S. and K.-C.C.; Methodology, K.-C.C.; Validation, A.S. and K.-C.C.; Formal Analysis, A.S.; Investigation, A.S. and K.-C.C.; Data Curation, A.S.; Original Draft Preparation, A.S.; Writing-Review and Editing, K.-C.C.; Supervision, K.-C.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not Applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

SymbolDescription
β 0 Constant regression term
β k Coefficient of the kth predictor variable
ε Random error of regression model
σ Standard deviation, Equation (2)
ACFAnnual cash flow (NTD), Equation (21)
CCrCloud cover
C capital Capital cost (NTD), Equation (21)
C equipment Cost of fabrication (NTD)
C installation Cost of installation (NTD)
C P Mallows test statistic, Equation (12)
d Durbin-Watson statistic, Equation (15)
e i The ith error term, Equation (16)
e i 1 The ( i 1 )th error term, Equation (17)
ETCEvacuated tube collector
iith observation
IQRInterquartile range
j jth predictor variable
k Number of variables
MAnnual freshwater yield (L)
MAEMean absolute error, Equation (6)
MAPEMean absolute percentage error, Equation (9)
MPRMarket penetration rate (%)
M S a l t Annual solar salt production (kg)
MSEMean square error, Equation (7)
n Number of observations
NTDNew Taiwanese dollar
PPPayback period, Equation (21)
p-valueProbability value
rCorrelation coefficient
R2Coefficient of determination, Equation (4)
R a 2 Adjusted R-squared, Equation (6)
RadDaily total global solar radiation
RHaAmbient relative humidity
RMSERoot mean square error, Equation (8)
RRRecovery ratio (%), Equation (18)
SStandard error of the regression, Equation (10)
SEStandard error, Equation (11)
SP s a l t Selling price of crude solar sea salt (NTD)
SP w Selling price of distillate (NTD)
SSESum of squares error, Equation (13)
TaAmbient temperature (°C)
TBBasin water temperature (°C)
TempDaily average ambient temperature (°C)
TDSTotal dissolved solids (mg/L)
TMMTypical meteorological month
TMYTypical meteorological year
T-valueTest statistic
U ( x ) Standard uncertainty, Equation (3)
USDUnited States dollar
VDVolume of distillate (L)
VFVolume of feedwater (L)
VIFVariance inflation factor, Equation (11)
WHOWorld Health Organization
WsWindspeed (m/s)
x Predictor or independent variable
x ¯ Mean of measured values, Equation (1)
x i ith measured value, Equation (1)
X ¯ Mean of the predictor variable
X i The value of the ith predictor variable
y Response or dependent variable, Equation (19)
Y Observed value
Y ˆ Predicted value
Y ¯ Mean value

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Figure 1. Experimental setup of the solar still installed in a north-south orientation: (a) northern side, (b) southern side.
Figure 1. Experimental setup of the solar still installed in a north-south orientation: (a) northern side, (b) southern side.
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Figure 2. Schematic diagram showing the electronic components used for water level control and data capture.
Figure 2. Schematic diagram showing the electronic components used for water level control and data capture.
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Figure 3. Aerial view of the fins and wicks installed in the basin.
Figure 3. Aerial view of the fins and wicks installed in the basin.
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Figure 4. (a) Production of crude solar sea salt crystals from the evaporation of the brine residue. (b) Samples collected from different experimental trials.
Figure 4. (a) Production of crude solar sea salt crystals from the evaporation of the brine residue. (b) Samples collected from different experimental trials.
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Figure 5. Identification of outliers within the Guiren experimental data sets of: (a) yield; (b) solar radiation; (c) cloud cover; (d) ambient temperature; (e) ambient relative humidity (NB: the shaded dots represent the outliers).
Figure 5. Identification of outliers within the Guiren experimental data sets of: (a) yield; (b) solar radiation; (c) cloud cover; (d) ambient temperature; (e) ambient relative humidity (NB: the shaded dots represent the outliers).
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Figure 6. Matrix plot of Pearson’s correlation showing the extent of correlation between the variables yield, cloud cover (CCr), ambient relative humidity (RHa), wind speed (Ws), ambient temperature (Temp), and daily total global solar radiation (Rad).
Figure 6. Matrix plot of Pearson’s correlation showing the extent of correlation between the variables yield, cloud cover (CCr), ambient relative humidity (RHa), wind speed (Ws), ambient temperature (Temp), and daily total global solar radiation (Rad).
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Figure 7. Standardized absolute residuals: (a) before; (b) after the jack-knife technique.
Figure 7. Standardized absolute residuals: (a) before; (b) after the jack-knife technique.
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Figure 8. Variation in the monthly average predicted yield and TMY global solar radiation for Dongji Islet.
Figure 8. Variation in the monthly average predicted yield and TMY global solar radiation for Dongji Islet.
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Figure 9. Recommended modification of the solar still’s basin dimension.
Figure 9. Recommended modification of the solar still’s basin dimension.
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Table 1. Predictor variables used in previous regression models.
Table 1. Predictor variables used in previous regression models.
Meteorological FactorsDesign FactorsOperational FactorsReferences
ambient temperature, sky temperature, global radiation, and wind velocity--[19]
daily solar radiation, average ambient air temperature--[28]
--heat storage material, basin water depth, basin cover thickness and external mirrors[20]
percentage of daylight, ambient temperature, solar radiation, windspeedglass cover thicknessfeedwater salinity, initial water depth[21]
relative humidity, windspeed, solar radiation, ambient temperature-feedwater temperature, feedwater flow rate and feedwater TDS[22]
-inclination anglefeedwater temperature, salt concentration, depth[23]
solar radiation intensity, ambient temperature-temperature difference between feedwater and inner condensing cover[24]
-inclination anglefeedwater temperature[26]
-quantity of stone used as energy storing medium, area of the double glazing usedfeedwater level[27]
solar radiation-feedwater temperature[29]
Table 2. The uncertainty, range, make, and model of the measuring instruments used.
Table 2. The uncertainty, range, make, and model of the measuring instruments used.
InstrumentMake/ModelRangeUncertainty (%)
Resistance thermocoupleThermoway/PT1000–100 °C±0.47%
Spectral pyranometersKipp & Zonen/CMP110–1400 W/ m 2 ±1.41%
AnemometerVector Instruments/A100L20–77 m/s±1.16%
Humidity-temperature transmitterDwyer/657-10–100%
0–100 °C
±1.73%
±0.58%
BeakerYeasten0–500 mL±2%
Top-pan balanceShimadzu/EB-4300D0–4300 g±2.5%
Table 3. Experimental data collected based on a 70% recovery ratio.
Table 3. Experimental data collected based on a 70% recovery ratio.
TrialDuration (Days)Average Daily Yield (L/day)Average Daily Rad (MJ/m2)Average Daily Ta (°C)Average Daily CCrAverage Daily Ws (m/s)Average Daily RHa (%)
1172.55211.7317.287.221.7582.57
294.87918.2721.332.642.1070.73
376.20521.3723.272.891.9263.16
4123.63113.9323.438.221.9977.98
594.77817.9020.645.322.2174.97
676.10121.5225.323.911.9165.76
7104.29116.1025.367.761.8180.21
876.12621.7028.807.431.4976.44
9104.27015.4528.008.361.5979.62
1076.10021.8129.697.731.3471.94
1167.11723.9229.966.981.6868.19
Table 4. Comparison of performance parameters of the solar still based on recovery ratio.
Table 4. Comparison of performance parameters of the solar still based on recovery ratio.
ParametersRecovery Ratio (%)
607080
Volume of feedwater (L)4661100
Total volume of distillate (L)Overnight4.229 (15.3%)7.279 (16.9%)20.661 (25.8%)
Daytime23.436 (84.7%)35.690 (83.1%)59.534 (74.2%)
Total27.665 (100%)43.001 (100%)80.195 (100%)
Duration (days)6916
Avg. productivity (L/day)4.614.785.01
Avg. rate of change of productivity (L/day2)0.7680.531 (−30.9%)0.313 (−59.2%)
Avg. daily total insolation (MJ/m2)17.117.918.6
Avg. daily Ta (°C)26.520.625.7
Avg. daily RHa (%)75.775.076.5
Avg. daily TB (°C)40.435.239.8
Table 5. Comparison of the brine residue concentrations at different recovery ratios.
Table 5. Comparison of the brine residue concentrations at different recovery ratios.
Recovery Ratio (%)Mass of Salt (g)Volume of Water Sample (L)Concentration of Brine Residue (mg/L)Change in Concentration (%)
60187.82.00 b93, 900-
701, 700.0 a14.8 a114, 86522.3
80441.32.00 b220, 650135.0
Note(s): a Averaged value; b Sample of water collected and boiled off.
Table 6. Summary of the model statistics for the constant term and different predictor variables used.
Table 6. Summary of the model statistics for the constant term and different predictor variables used.
TermCoefficientSE of Coefficientp-ValueVIF
InitialFinalInitialFinalInitialFinalInitialFinal
Constant−1.786−1.0730.9260.140.0570.00000--
Rad0.3340.310150.01770.007750.0000.000004.414.92
Temp0.02070.040740.01950.009050.2910.000042.863.13
CCr0.0015−0.07410.02790.01090.9580.000002.572.35
RHa0.0020.00280.01020.00550.8440.618002.362.84
Note(s): SE—standard error; p-value—probability value; VIF—variable inflation factor.
Table 7. Model statistics based on best subset selection.
Table 7. Model statistics based on best subset selection.
Number of Variables (k)R2 (%) R a 2 (%) Mallows CpSRadTempRHaWsCCr
199.099.041.90.23036
159.358.53887.91.4768
299.399.219.20.20031
299.199.133.10.21900
399.599.42.50.17350
399.399.219.90.20033
499.599.44.10.17445
499.599.44.40.17504
599.599.46.00.17602
Note(s): S—standard error of the regression. ✓—represents variable(s) used in the model.
Table 8. Metric used for cross-validation and evaluation of the accuracy of the model.
Table 8. Metric used for cross-validation and evaluation of the accuracy of the model.
Data Set UsedMAERMSEMAPE (%)R2 (%) R a 2 (%)
Training0.1440.1679.70999.4799.44
Testing0.1780.2464.59198.3997.58
Table 9. Monthly average values of the three predictor variables and predicted yield for each TMM.
Table 9. Monthly average values of the three predictor variables and predicted yield for each TMM.
MonthTMY YearMonthly Average TMM ValuesDaily Average Predicted Yield (L)
Rad (MJ/m2)Temp (°C)CCr
January20139.1617.057.021.99
February201110.4817.286.482.53
March201512.2720.756.553.10
April201415.6723.265.304.34
May200718.5825.974.055.45
June200517.7627.097.125.06
July201520.0328.285.545.88
August201617.9828.365.365.26
September201317.0327.353.495.07
October201714.7026.854.204.27
November200910.2523.086.082.72
December20178.4319.586.391.97
Table 10. Predicted annual yield of potable water and crude sea salt production.
Table 10. Predicted annual yield of potable water and crude sea salt production.
LocationAvg. YieldAvg. Days per BatchAvg. Batches per YearAnnual Yield (L)a Annual Mass of Crude Salt (kg)
L/DayL/M2 Day
Dongji Island3.9784.85112.529.31456.949.8
Note(s): a Based on an average of 1.7 kg per trial or batch of brine residue as shown in Table 5.
Table 11. Scaled solar still specifications.
Table 11. Scaled solar still specifications.
Locationa Basin Length Requirementb Basin Size Requirementc Evacuated Tube Requirement
Value (m)% IncreaseValue (m2)% IncreaseAmount% Increase
Dongji Islet3.3088.61.9588.52391
Note(s): Based on the present: a length of 1.748 m; b width of 0.5919 m; c amount of 12 evacuated tubes.
Table 12. Estimated payback period.
Table 12. Estimated payback period.
Location C c a p i t a l (NTD) ACF (NTD)PP (Years)
Dongji Islet100, 567.5 (USD 3, 419.3)16, 421.76 (USD 558.3)6.1
Note(s): NTD 1 = USD 0.034.
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Samuel, A.; Chang, K.-C. Statistical Model for the Sizing of a Prototype Solar Still Applicable to Remote Islands. Water 2022, 14, 3510. https://doi.org/10.3390/w14213510

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Samuel A, Chang K-C. Statistical Model for the Sizing of a Prototype Solar Still Applicable to Remote Islands. Water. 2022; 14(21):3510. https://doi.org/10.3390/w14213510

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Samuel, Alinford, and Keh-Chin Chang. 2022. "Statistical Model for the Sizing of a Prototype Solar Still Applicable to Remote Islands" Water 14, no. 21: 3510. https://doi.org/10.3390/w14213510

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