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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}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

- 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 (T
_{a}), 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 (T
_{B}), 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.

#### 2.2. Uncertainty Analysis

#### 2.3. Data Analysis

^{2}), adjusted-R squared (${\mathrm{R}}_{\mathrm{a}}^{2}$), 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]:

## 3. Results and Discussion

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

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

_{D}) that results from a unit volume of feedwater (V

_{F}) expressed as a percentage or decimal [42,43]. The recovery ratio may be expressed as:

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

#### 3.4. Modelling of the Daily Distillate Yield

#### 3.4.1. Identification of Outliers

#### 3.4.2. Predictor Variables and Pearson’s Correlation Matrix

#### 3.4.3. Model Building

^{2}and ${\mathrm{R}}_{\mathrm{a}}^{2}$ values and low Mallows Cp and S values. Both ${\mathrm{R}}_{\mathrm{a}}^{2}$ and R

^{2}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 ${\mathrm{R}}_{\mathrm{a}}^{2}$ is preferred over R

^{2}for multivariable regression models as the value of R

^{2}increases with increasing number of predictor variables without regard for the significance of the variable to the model [49].

^{2}, 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

^{2}, ${\mathrm{R}}_{\mathrm{a}}^{2}$, 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

#### 3.6. Sizing of the Solar Still Prototype

#### 3.7. Payback Period

## 4. Conclusions

^{2}), adjusted R-squared (${\mathrm{R}}_{a}^{2}$), 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.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Description |

${\beta}_{0}$ | Constant regression term |

${\beta}_{\mathrm{k}}$ | Coefficient of the kth predictor variable |

$\epsilon $ | Random error of regression model |

$\sigma $ | Standard deviation, Equation (2) |

ACF | Annual cash flow (NTD), Equation (21) |

CCr | Cloud cover |

${\mathrm{C}}_{\mathrm{capital}}$ | Capital cost (NTD), Equation (21) |

${\mathrm{C}}_{\mathrm{equipment}}$ | Cost of fabrication (NTD) |

${\mathrm{C}}_{\mathrm{installation}}$ | Cost of installation (NTD) |

${\mathrm{C}}_{\mathrm{P}}$ | Mallows test statistic, Equation (12) |

$\mathrm{d}$ | Durbin-Watson statistic, Equation (15) |

${\mathrm{e}}_{\mathrm{i}}$ | The ith error term, Equation (16) |

${\mathrm{e}}_{\mathrm{i}-1}$ | The ($\mathrm{i}-1$)th error term, Equation (17) |

ETC | Evacuated tube collector |

i | ith observation |

IQR | Interquartile range |

$\mathrm{j}$ | jth predictor variable |

$\mathrm{k}$ | Number of variables |

M | Annual freshwater yield (L) |

MAE | Mean absolute error, Equation (6) |

MAPE | Mean absolute percentage error, Equation (9) |

MPR | Market penetration rate (%) |

${\mathrm{M}}_{Salt}$ | Annual solar salt production (kg) |

MSE | Mean square error, Equation (7) |

$\mathrm{n}$ | Number of observations |

NTD | New Taiwanese dollar |

PP | Payback period, Equation (21) |

p-value | Probability value |

r | Correlation coefficient |

R^{2} | Coefficient of determination, Equation (4) |

${\mathrm{R}}_{\mathrm{a}}^{2}$ | Adjusted R-squared, Equation (6) |

Rad | Daily total global solar radiation |

RHa | Ambient relative humidity |

RMSE | Root mean square error, Equation (8) |

RR | Recovery ratio (%), Equation (18) |

S | Standard error of the regression, Equation (10) |

SE | Standard error, Equation (11) |

${\mathrm{SP}}_{salt}$ | Selling price of crude solar sea salt (NTD) |

${\mathrm{SP}}_{w}$ | Selling price of distillate (NTD) |

SSE | Sum of squares error, Equation (13) |

T_{a} | Ambient temperature (°C) |

T_{B} | Basin water temperature (°C) |

Temp | Daily average ambient temperature (°C) |

TDS | Total dissolved solids (mg/L) |

TMM | Typical meteorological month |

TMY | Typical meteorological year |

T-value | Test statistic |

$\mathrm{U}\left(x\right)$ | Standard uncertainty, Equation (3) |

USD | United States dollar |

V_{D} | Volume of distillate (L) |

V_{F} | Volume of feedwater (L) |

VIF | Variance inflation factor, Equation (11) |

WHO | World Health Organization |

Ws | Windspeed (m/s) |

$x$ | Predictor or independent variable |

$\overline{x}$ | Mean of measured values, Equation (1) |

${x}_{\mathrm{i}}$ | ith measured value, Equation (1) |

$\overline{\mathrm{X}}$ | Mean of the predictor variable |

${\mathrm{X}}_{\mathrm{i}}$ | The value of the ith predictor variable |

$y$ | Response or dependent variable, Equation (19) |

$\mathrm{Y}$ | Observed value |

$\stackrel{\u02c6}{\mathrm{Y}}$ | Predicted value |

$\overline{\mathrm{Y}}$ | 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 2.**Schematic diagram showing the electronic components used for water level control and data capture.

**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 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 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 8.**Variation in the monthly average predicted yield and TMY global solar radiation for Dongji Islet.

Meteorological Factors | Design Factors | Operational Factors | References |
---|---|---|---|

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, windspeed | glass cover thickness | feedwater salinity, initial water depth | [21] |

relative humidity, windspeed, solar radiation, ambient temperature | - | feedwater temperature, feedwater flow rate and feedwater TDS | [22] |

- | inclination angle | feedwater temperature, salt concentration, depth | [23] |

solar radiation intensity, ambient temperature | - | temperature difference between feedwater and inner condensing cover | [24] |

- | inclination angle | feedwater temperature | [26] |

- | quantity of stone used as energy storing medium, area of the double glazing used | feedwater level | [27] |

solar radiation | - | feedwater temperature | [29] |

Instrument | Make/Model | Range | Uncertainty (%) |
---|---|---|---|

Resistance thermocouple | Thermoway/PT100 | 0–100 °C | ±0.47% |

Spectral pyranometers | Kipp & Zonen/CMP11 | 0–1400 W/${\mathrm{m}}^{2}$ | ±1.41% |

Anemometer | Vector Instruments/A100L2 | 0–77 m/s | ±1.16% |

Humidity-temperature transmitter | Dwyer/657-1 | 0–100% 0–100 °C | ±1.73% ±0.58% |

Beaker | Yeasten | 0–500 mL | ±2% |

Top-pan balance | Shimadzu/EB-4300D | 0–4300 g | ±2.5% |

Trial | Duration (Days) | Average Daily Yield (L/day) | Average Daily Rad (MJ/m^{2}) | Average Daily T_{a} (°C) | Average Daily CCr | Average Daily Ws (m/s) | Average Daily RHa (%) |
---|---|---|---|---|---|---|---|

1 | 17 | 2.552 | 11.73 | 17.28 | 7.22 | 1.75 | 82.57 |

2 | 9 | 4.879 | 18.27 | 21.33 | 2.64 | 2.10 | 70.73 |

3 | 7 | 6.205 | 21.37 | 23.27 | 2.89 | 1.92 | 63.16 |

4 | 12 | 3.631 | 13.93 | 23.43 | 8.22 | 1.99 | 77.98 |

5 | 9 | 4.778 | 17.90 | 20.64 | 5.32 | 2.21 | 74.97 |

6 | 7 | 6.101 | 21.52 | 25.32 | 3.91 | 1.91 | 65.76 |

7 | 10 | 4.291 | 16.10 | 25.36 | 7.76 | 1.81 | 80.21 |

8 | 7 | 6.126 | 21.70 | 28.80 | 7.43 | 1.49 | 76.44 |

9 | 10 | 4.270 | 15.45 | 28.00 | 8.36 | 1.59 | 79.62 |

10 | 7 | 6.100 | 21.81 | 29.69 | 7.73 | 1.34 | 71.94 |

11 | 6 | 7.117 | 23.92 | 29.96 | 6.98 | 1.68 | 68.19 |

Parameters | Recovery Ratio (%) | |||
---|---|---|---|---|

60 | 70 | 80 | ||

Volume of feedwater (L) | 46 | 61 | 100 | |

Total volume of distillate (L) | Overnight | 4.229 (15.3%) | 7.279 (16.9%) | 20.661 (25.8%) |

Daytime | 23.436 (84.7%) | 35.690 (83.1%) | 59.534 (74.2%) | |

Total | 27.665 (100%) | 43.001 (100%) | 80.195 (100%) | |

Duration (days) | 6 | 9 | 16 | |

Avg. productivity (L/day) | 4.61 | 4.78 | 5.01 | |

Avg. rate of change of productivity (L/day^{2}) | 0.768 | 0.531 (−30.9%) | 0.313 (−59.2%) | |

Avg. daily total insolation (MJ/m^{2}) | 17.1 | 17.9 | 18.6 | |

Avg. daily T_{a} (°C) | 26.5 | 20.6 | 25.7 | |

Avg. daily RHa (%) | 75.7 | 75.0 | 76.5 | |

Avg. daily T_{B} (°C) | 40.4 | 35.2 | 39.8 |

Recovery Ratio (%) | Mass of Salt (g) | Volume of Water Sample (L) | Concentration of Brine Residue (mg/L) | Change in Concentration (%) |
---|---|---|---|---|

60 | 187.8 | 2.00 ^{b} | 93, 900 | - |

70 | 1, 700.0 ^{a} | 14.8 ^{a} | 114, 865 | 22.3 |

80 | 441.3 | 2.00 ^{b} | 220, 650 | 135.0 |

^{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.

Term | Coefficient | SE of Coefficient | p-Value | VIF | ||||
---|---|---|---|---|---|---|---|---|

Initial | Final | Initial | Final | Initial | Final | Initial | Final | |

Constant | −1.786 | −1.073 | 0.926 | 0.14 | 0.057 | 0.00000 | - | - |

Rad | 0.334 | 0.31015 | 0.0177 | 0.00775 | 0.000 | 0.00000 | 4.41 | 4.92 |

Temp | 0.0207 | 0.04074 | 0.0195 | 0.00905 | 0.291 | 0.00004 | 2.86 | 3.13 |

CCr | 0.0015 | −0.0741 | 0.0279 | 0.0109 | 0.958 | 0.00000 | 2.57 | 2.35 |

RHa | 0.002 | 0.0028 | 0.0102 | 0.0055 | 0.844 | 0.61800 | 2.36 | 2.84 |

Number of Variables (k) | R^{2} (%) | ${\mathbf{R}}_{\mathit{a}}^{2}$ (%) | Mallows Cp | S | Rad | Temp | RHa | Ws | CCr |
---|---|---|---|---|---|---|---|---|---|

1 | 99.0 | 99.0 | 41.9 | 0.23036 | ✓ | ||||

1 | 59.3 | 58.5 | 3887.9 | 1.4768 | ✓ | ||||

2 | 99.3 | 99.2 | 19.2 | 0.20031 | ✓ | ✓ | |||

2 | 99.1 | 99.1 | 33.1 | 0.21900 | ✓ | ✓ | |||

3 | 99.5 | 99.4 | 2.5 | 0.17350 | ✓ | ✓ | ✓ | ||

3 | 99.3 | 99.2 | 19.9 | 0.20033 | ✓ | ✓ | ✓ | ||

4 | 99.5 | 99.4 | 4.1 | 0.17445 | ✓ | ✓ | ✓ | ✓ | |

4 | 99.5 | 99.4 | 4.4 | 0.17504 | ✓ | ✓ | ✓ | ✓ | |

5 | 99.5 | 99.4 | 6.0 | 0.17602 | ✓ | ✓ | ✓ | ✓ | ✓ |

Data Set Used | MAE | RMSE | MAPE (%) | R^{2} (%) | ${\mathbf{R}}_{\mathit{a}}^{2}$ (%) |
---|---|---|---|---|---|

Training | 0.144 | 0.167 | 9.709 | 99.47 | 99.44 |

Testing | 0.178 | 0.246 | 4.591 | 98.39 | 97.58 |

Month | TMY Year | Monthly Average TMM Values | Daily Average Predicted Yield (L) | ||
---|---|---|---|---|---|

Rad (MJ/m^{2}) | Temp (°C) | CCr | |||

January | 2013 | 9.16 | 17.05 | 7.02 | 1.99 |

February | 2011 | 10.48 | 17.28 | 6.48 | 2.53 |

March | 2015 | 12.27 | 20.75 | 6.55 | 3.10 |

April | 2014 | 15.67 | 23.26 | 5.30 | 4.34 |

May | 2007 | 18.58 | 25.97 | 4.05 | 5.45 |

June | 2005 | 17.76 | 27.09 | 7.12 | 5.06 |

July | 2015 | 20.03 | 28.28 | 5.54 | 5.88 |

August | 2016 | 17.98 | 28.36 | 5.36 | 5.26 |

September | 2013 | 17.03 | 27.35 | 3.49 | 5.07 |

October | 2017 | 14.70 | 26.85 | 4.20 | 4.27 |

November | 2009 | 10.25 | 23.08 | 6.08 | 2.72 |

December | 2017 | 8.43 | 19.58 | 6.39 | 1.97 |

Location | Avg. Yield | Avg. Days per Batch | Avg. Batches per Year | Annual Yield (L) | ^{a} Annual Mass of Crude Salt (kg) | |
---|---|---|---|---|---|---|

L/Day | L/M^{2} Day | |||||

Dongji Island | 3.978 | 4.851 | 12.5 | 29.3 | 1456.9 | 49.8 |

^{a}Based on an average of 1.7 kg per trial or batch of brine residue as shown in Table 5.

Location | ^{a} Basin Length Requirement | ^{b} Basin Size Requirement | ^{c} Evacuated Tube Requirement | |||
---|---|---|---|---|---|---|

Value (m) | % Increase | Value (m^{2}) | % Increase | Amount | % Increase | |

Dongji Islet | 3.30 | 88.6 | 1.95 | 88.5 | 23 | 91 |

^{a}length of 1.748 m;

^{b}width of 0.5919 m;

^{c}amount of 12 evacuated tubes.

Location | ${\mathbf{C}}_{\mathbf{c}\mathbf{a}\mathbf{p}\mathbf{i}\mathbf{t}\mathbf{a}\mathbf{l}}$ (NTD) | ACF (NTD) | PP (Years) |
---|---|---|---|

Dongji Islet | 100, 567.5 (USD 3, 419.3) | 16, 421.76 (USD 558.3) | 6.1 |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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

**Chicago/Turabian Style**

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