Figure 1 on the following page shows a spatial representation of the solar distiller to be studied and the different “views” (directions and orientations of observation) through which the distillation sections are observed.

Figure 2(a) , Figure 3(a) and Figure 3(b) show respectively: the diagram of the “view” of a distiller cross-section in a horizontal plane, the diagram of “view 1” in the plane of symmetry plane along axis 1 ↔ 2 (or axis North-South) and the diagram of “view 4” in the plane of symmetry along axis 3 ↔ 4 (or axis East-West) of the distiller ( Figure 1 ).

The distiller we are describing is a solar distiller with two vertical planes of symmetry. The intersection of the first plane of symmetry with a absorbeur’s horizontal plane forms the axis marked “1 ↔ 2” (or axis North-South) and the intersection of the second plane with this horizontal plane forms the axis marked

“3 ↔ 4” (or axis East-West) ( Figures 1-3 ).

The distiller has five compartments ( Figure 1 and Figure 2 ): the central compartment covered by three glass panes V₅, V₆ and V₇, topped by the vertical reflector panel facing away from the sun and reflecting the sun’s rays above the distiller towards the entrance to the central compartment, at the bottom of which is the central absorber; compartments 1 and 2, covered respectively by glass panes V₁ and V₂, which can each be covered by one removable plywood cover; at the bottom of compartments 1 and 2 are tanks 1 and 2 respectively, which can contain the solution to be distilled; compartments 3 and 4, each communicating with compartments 1 and 2 via slots F_1-3 and F_1-4, F_2-3 and F_2-4, these compartments are covered respectively by glass panes V₃ and V₄, each of which is covered by one fixed plywood cover.

It should be noted that the absorber block “tank 1 + absorber + tank 2” is made from the same stainless steel sheet, which is painted with a matt black paint on top.

All the distiller’s compartments are covered by ordinary glass, all the same thickness of 5 mm.

The thickness of the glass is not too small so that it is more resistant (glass does not break under its own weight or during handling), and not too large so that, when exposed to the sun, it absorbs less solar radiation.

These glass panes have two important features:

Depending on requirements, the glass is used to transmit solar radiation or to condense water vapour.

1) Glass V₁ and glass V₂

Glass 1 (_V1) and glass 2 (_V2) cover Compartment 1 and Compartment 2 respectively.

In principle, they play two roles at the same time: transmitting solar radiation, thereby promoting the greenhouse effect, and acting as a condensation surface for the steam produced.

However, they can also be covered by a plywood lid.

In this case, they act as a condensation surface for the water vapour produced.

These two glass panes have the same dimensions:

length: 153 cm

width: 29 cm

thickness: 5 mm

2) Glass V₃ and glass V₄

Glass 3 (_V3) and glass 4 (_V4) cover compartment 3 and compartment 4 respectively, each one is covered by a plywood lid.

They act as a condensation surface for the part of the steam produced in compartments 1 and 2 which is diverted through the four slots in compartments 1 and 2 to compartments 3 and 4.

These two glass panes have the same dimensions:

length: 105 cm

width:34.5 cm

thickness:5 mm

3) Glass V₅, V₆ and glass V₇

Glass 5 (_V5), Glass 6 (_V6) and Glass 7 (_V7) cover the central absorber compartment respectively from top to bottom.

Their role is to transmit solar radiation to the central absorber and promote the greenhouse effect in this compartment.

These two glass panes have the same dimensions:

length: 144.5 cm

width: 44 cm

thickness: 5 mm

The central absorber and the two tanks are made from the same 2 mm thick stainless steel sheet, the top surface of which is painted matt black.

1) Central absorber

The central absorber is an empty, parallelepiped-shaped container, the inside surface of which is painted matt black to maximize absorption of solar radiation transmitted through the glass panes of this compartment.

Its dimensions are ( Figure 4 ):

length: 149.5 cm

width: 44 cm

hauteur: 4 cm

For this distiller, the ratio between the width l and length L of the central absorber is:

$l/L=0.3045$

2) Tanks

Tank 1 and Tank 2 are parallelepiped-shaped container whose inner surfaces

are painted matt black to optimize absorption of solar radiation.

These two tanks receive the solution to be distilled and heat it for evaporation.

These two trays are located at the base of Compartment 1 and Compartment 2 respectively, and are attached to the central absorber.

These two bins have the same dimensions, which are ( Figure 4 ):

length: 149.5 cm

width: 22.5 cm

thickness: 4 cm

The reflector panel is made of rectangular plywood and has one side covered with reflective plastic sheeting.

This panel has two roles depending on experimental requirements:

Its dimensions are:

length: 145 cm

width: 45 cm

thickness: 1.5 cm

Plywood lids cover the windows to prevent solar radiation from reaching them (to prevent solar radiation from passing through them or heating them up by absorbing part of this radiation).

They allow natural air circulation between the lids and the glass panes they cover (natural convection cooling glass panes)

This can lead to condensation of water vapour under the inner surface of the glass panes.

1) Cover 1 and cover 2

The cover 1 and cover 2 are rectangular, movable and may or may not cover glass panes

V₁ and V₂ respectively, as required. These covers have the same dimensions:

length: 163 cm

width: 39 cm

thickness: 1.5 cm

2) Cover 3 and cover 4

The cover 3 and cover 4 are rectangular, fixed and always cover glass panes V₃ and V₄ respectively.

These covers have the same dimensions:

lengthr: 115 cm

width: 35.5 cm

thickness: 1.5 cm

Thermal insulation minimizes heat loss by thermal conduction to the outside of the distiller.

For the distiller, the thermal insulators used are:

plywood and expanded polystyrene.

The arrangement and dimensions of these two insulators depend on the position of the distiller’s outer walls:

The central absorber is heated by solar radiation transmitted through the glass panes, it is accentuated by the greenhouse effect.

Since the central absorber and the two tanks are made from the same stainless steel sheet, heat is transferred from the central absorber to the two tanks by conduction within the sheet.

In addition to the heat transferred by conduction to the tanks, heat is also transferred by solar radiation transmitted through each pane of glass in compartments 1 and 2, to the solution to be distilled and to tanks 1 and 2 ( Figure 7 ).

As a result, water evaporates from the solution to be distilled in tanks 1 and 2, whose vapors rise by natural convection respectively, towards glass panes V₁ and V₂ respectively.

Some of the vapors from compartments 1 and 2 condense under the glass panes V₁ and V₂ into droplets which are collected by the respective gutters of compartments 1 and 2 to form, at their outlets, the distillates (also known as condensates) from compartments 1 and 2.

The other part of the steam from compartment 1 is diverted through slots F₁₋₃ and F₁₋₄ to compartments 3 and 4, and the other part of the steam from compartment 2 is diverted diverted through slots F₂₋₃ and F₂₋₄ into compartments 3 and 4.

These vapors from compartments 1 and 2 into compartments 3 and 4 condense under their respective panes V₃ and V₄ to form distillates from compartments compartments 3 and 4.

Figure 8 show several instruments.

Figure 8(a) : the “EPPLEY PsP” pyranometer, used to measure irradiance received on a horizontal surface, with a flat solar sensor inside that can receive solar radiation at a solid angle of 2π steradians, the pyranometer delivers a

millivolt voltage, proportional to the irradiance received $E=\frac{V}{k}$ where E is the

irradiance received by the pyranometer in W/m², V is the potential difference delivered by the pyranometer in mV and k = 10.41 × 10⁻³ mV·m²/W is the pyranometer’s proportionality coefficient.

Figure 8(b) : the “PICO” brand voltage recorder with a U.S.B. cable for connection to a computer.

Figure 8(c) : the computer to which the voltage recorder is connected.

It should be noted that irradiance is recorded every minute.

Figure 9(a) shows the experimental set-up on the roof of the research building.

This Figure 9(a) including: solar distiller, pyranometer, thermocouples and recording unit.

Figure 10 shows the “SF-400C” electronic balance with digital display used to measure the masses (in grams to an accuracy of 0.01 g) of distillates collected 2 hours 30 minutes after the start of the experiments, then every 1 hour until the end of the experiment.

The daily solar irradiation $I$ is the solar energy received per unit area over the entire duration $\Delta t=t_f-t_i$ of the day’s experiment, from the initial instant $t_i$ (the 1^st measurement) to the final instant $t_f$ (the last measurement) with ∆_t = 10 h30 min = 10.5 h, it is given by the formula:

$I={\displaystyle {\int }_{t_i}^{t_f}E\left(t\right)\text{d}t}$ (1)

where $I$ is in Wh/m², $E\left(t\right)$ is irradiance at time $t$ in W/m².

Productivity $P_r$ in kg/m² is the total mass of water collected per unit area of the water tanks [2] :

$P_r=\frac{m_d}{2\times S_1}$(2)

where $m_d$ is the total mass of water collected from compartments 1, 2, 3 and 4 in kg and $S_1$ is the surface area of one of the tanks in m².

The declination $\delta$ is given by the formula proposed by Vincent Bourdin:

$\delta =0.38+23.26\times \sin \left(\frac{360\times n_0}{365.24}-1.395\right)+0.375\times \sin \left(\frac{2\times 360\times n_0}{365.24}-1.47\right)$ (3)

where $n_0$ is the number of the day of the year, counted from January 1, 2013 to December 31, 2023 ( $n_0=1$ on January 1, 2013) and $\delta$ in degrees of arc [3] .

At the location under consideration, we have:

$H_{slv}=H_{ml}-\frac{E_T}{60}$

where $H_{slv}$ is local true solar time, $H_{ml}$ is local mean solar time and $E_T$ is the equation of time in minute also proposed by Vincent Bourdin:

$\begin{array}{c}{E}_T=7.36\times \sin \left(\frac{360\times n_0}{365.242}-0.071\right)+9.92\times \sin \left(\frac{2\times 360\times n_0}{365.42}+0.357\right)\\ +0.305\times \sin \left(\frac{3\times 360\times n_0}{365.42}+0.256\right)\end{array}$ (4)

where $n_0$ is the number of the day of the year, counted from January 1, 2013 to December 31, 2023 ( $n_0=1$ on January 1, 2013) [3] .

We have the following relationships:

$\{\begin{array}{l}{H}_l=T.U+\Delta H\\ H_{ml}=T.U+\frac{\lambda }{15}\end{array}$

where $H_l$ is local time, $T.U$ is universal time, $\Delta H$ is the time zone offset and $\lambda$ is the longitude of the location.

We can write that with formulas (2.5.4) and (2.5.4):

$H_{svl}-\frac{E_T}{60}=H_l-\Delta H-\frac{E_T}{60}$(5)

Since $H_{svl}=\frac{W}{15}+12$, where $w$ is the hour angle in degrees of arc, we finally find:

$w=15\times H_l+\lambda -15\times \Delta H-\frac{E_t}{4}-180$(6)

Instead of experimenting, we have the following data:

Longitude $\lambda =-3.98833333333˚$, latitude $L=+5.344722222222˚$, time zone offset $\Delta H=0$ hour, so:

$w=15\times H_l-186.98833333333-\frac{E_T}{4}$(7)

Figure 11(a) and Figure 11(b) show the three-dimensional local Cartesian reference frame (O, X, Y, Z) linked to the central compartment with the sun’s height $h$ and azimuth $a$, then in two dimensions on the central absorber.

Consider the vectors $i,j,k$ which are the respective unitary director vectors of these axes. In this reference frame, the coordinates of the vector $u\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$ which

is the unitary directing vector of the line passing through the origin O of the reference frame, through the center of the sun and towards the sun, are such that:

$\{\begin{array}{l}x=\cos h\times \cos a\\ y=-\cos h\times \sin a\\ z=\sin h\end{array}$(8)

or:

$\{\begin{array}{l}x=\cos \delta \times \sin L\times \cos W-\cos L\times \sin \delta \\ y=-\cos \delta \times \sin W\\ z=\cos \delta \times \cos L\times \cos W+\sin L\times \sin \delta \end{array}$(9)

Figure 12(a) and Figure 12(b) show vector $u_{xz}$ which is the projection of vector $u$ in the (OXZ) plane, and vector $u_{yz}$ that of vector $u$ in the (OYZ) plane. We define the following angle measures:

$\{\begin{array}{l}{\alpha }_x=Mes\left(\widehat{i,u_{xz}}\right)\text{et}{\alpha }_x\in \left[0^{\circ };{180}^{\circ }\right]\\ {\alpha }_y=Mes\left(\widehat{i,u_{yz}}\right)\text{et}{\alpha }_y\in \left[0^{\circ };{180}^{\circ }\right]\end{array}$(10)

In addition

$\{\begin{array}{l}\cos {\alpha }_x=\frac{x}{\sqrt{x^2+z^2}}\\ \sin {\alpha }_x=\frac{z}{\sqrt{x^2+z^2}}\end{array}$ et $\{\begin{array}{l}\cos {\alpha }_y=\frac{y}{\sqrt{y^2+z^2}}\\ \sin {\alpha }_y=\frac{z}{\sqrt{y^2+z^2}}\end{array}$(11)

The angular deviation between the direction of the sun and the vertical plane passing through the East-West axis is characterized by the angle measure defined by: ${\alpha }_{xz}=Mes\left(u_{xz},\hat{k}\right)$ for ${\alpha }_x\ge 0$, during the day ( Figure 10 ). We have:

${\alpha }_{xz}=\frac{\pi }{2}-{\alpha }_x$(12)

${\alpha }_{xz}=90-{\alpha }_x$(13)

We can say that:

The Figure 13 shows the angle ${\alpha }_{xz}=Mes\left(u_{xz},\hat{k}\right)$ between the direction of the sun and the vertical plane passing through the East-West axis in the (OXZ) plane.

So, to characterize the proximity of the sun’s path (the apparent trajectory of the sun in the sky) to the vertical plane passing through the East-West axis, over the duration of the measurements, we defined the absolute mean angle $\overline{{\alpha }_{ax}}$ as the average absolute value of ${\alpha }_{xz}$, relative to the entire measurement time, for values of ${\alpha }_x\ge 0$.

We have:

$\overline{{\alpha }_{ax}}=\left|average\left({\alpha }_{xz}\right)\right|=\{\begin{array}{c}\left|average\left(\frac{\pi }{2}-{\alpha }_x\right)\right|\\ ou\\ \left|average\left(90-{\alpha }_x\right)\right|\end{array}$ (14)

The program for calculating the absolute mean angle $\overline{{\alpha }_{ax}}$ is based on the algorithmic flowchart shown respectively in Figures 14-17 .

The Student’s t.test for a two-tailed distribution of samples measuring daily irradiation matched to samples measuring productivity of size 11, gives a p-value equal to

1.2539 × 10⁻⁷ < 0.01.

This Student’s t.test shows that there is a highly significant correlation between productivity and daily irradiation.

The coefficient of determination of productivity as a function of daily irradiation is R² = 0.9289, close to 1.

This sends to study the linear correlation between productivity and daily irradiation. Table 1 was used to construct Figure 18 , showing the scatterplot of productivity as a function of daily irradiation.

In Figure 18 , the arrangement of the scatterplot of productivity as a function of irradiation allows a linear fit with the regression line:

$y=1.2599\times x-1.9743$(15)

This correlation is strong because the correlation coefficient r = 0.9638 is close to 1.

Productivity therefore depends on daily irradiation and increases with daily irradiation.

This is because solar irradiation is the main source of energy used to heat and vaporise the water needed for distillation.

The Student’s t.test for a two-tailed distribution of samples measuring the absolute mean angle matched to samples measuring productivity of size 11, gives the p-value equal to

0.0296 < 0.05.

This Student’s t.test shows that there is a significant correlation between productivity and absolute mean angle.

The coefficient of determination for productivity as a function of absolute mean angle is

R² = 0.6333, greater than 0.5.

Let’s look at the linear correlation between absolute mean angle and productivity.

Table 1 was used to construct Figure 19 .

In Figure 19 , the arrangement of the scatter plot of productivity as a function of the absolute mean angle $\overline{{\alpha }_{ax}}$, allows a linear fit with the regression line:

$y=-0.0651\times x+4.4676$(16)

The correlation coefficient is r = −0.7958.

Productivity therefore depends on the absolute mean angle; when this angle decreases, productivity increases.

The productivity of the distiller depend on the absolute mean angle $\overline{{\alpha }_{ax}}$, that’s to say the proximity of the sun’s path to the vertical plane passing through the East-West axis.

If we look at Figure 20 we can see that, because of the “masking effects” due to the geometry of the central compartment and compartments 1 and 2, the closer the sun’s path is to the vertical plane passing through the East-West axis (the absolute mean angle $\overline{{\alpha }_{ax}}$ is small), the larger the surfaces (on the central absorber and the tanks) directly illuminated by the sun’s rays and the closer the solar rays are to the normal to these surfaces, so the conversion of this incident solar radiation into heat is more efficient, resulting in an increase in productivity.

If we follow the same reasoning, we understand that moving the sun’s path away from the vertical plane passing through the East-West axis (the absolute mean angle $\overline{{\alpha }_{ax}}$ is large) will cause the decrease in productivity.

We have seen that the productivity of the distiller depend on both the daily irradiation and the absolute mean angle $\overline{{\alpha }_{ax}}$ over the duration of the measurements.

This explains, on the bar graphs in Figure 21 , showing productivity as a function of daily irradiation, the decrease in productivity on 1 March 2017 and 16 February 2018, compared with the 12 April 2017 ( Table 1 ), despite the fact that the irradiation levels on irradiations on 1 March 2017 and 16 February 2018 were higher than those on 12 April 2017. In fact, in Table 1 , the absolute mean angles for 1 March 2017 and 16 February 2018 are higher than that of 12 April 2017.

The coefficient of multiple determination of productivity as a function of daily irradiation and absolute mean angle is R² = 0.979404577 is close to 1.

This justifies the use of multiple linear correlation formulas linking productivity to daily irradiation and absolute mean angle, to predict or estimate distiller productivity [4] [5] .

Let’s call X₁, X₂ and Y the respective physical quantities: daily irradiation over the duration of the measurements in kWh/m², the absolute mean angle $\overline{{\alpha }_{ax}}$ in degrees of arc over the duration of the measurements (with ${\alpha }_x\ge 0$) and productivity in kg/m²;

$r_{X_{1}Y}$, $r_{X_{2}Y}$ and $r_{X_1{X}_2}$ the correlation coefficients between the respective quantities $Y$ and $X_1$, $Y$ and $X_2$, $X_1$ and $X_2$, ${\sigma }_{X_1}$, ${\sigma }_{X_2}$ and ${\sigma }_Y$ the standard deviations of the respective quantities $X_1$, $X_2$ and $Y$.

The multiple correlation coefficient $R_{Y,X_1{X}_2}$ of productivity as a function of daily irradiation and the absolute mean angle $\overline{{\alpha }_{ax}}$, is given by the formula:

$R_{Y,X_1{X}_2}=\sqrt{\frac{{\left(r_{X_{1}Y}\right)}^2+{\left(r_{X_{2}Y}\right)}^2-2\times {\left(r_{X_{1}Y}\right)}^2\times {\left(r_{X_{2}Y}\right)}^2\times {\left(r_{X_1{X}_2}\right)}^2}{1-{\left(r_{X_1{X}_2}\right)}^2}}$ (17)

The linear equation that gives productivity as a function of irradiation and the absolute mean angle $\overline{{\alpha }_{ax}}$ is:

$Y=A_{X_{1}Y,X_2}\times X_1+A_{X_{2}Y,X_1}\times X_2+B_{Y,X_1{X}_2}$(18)

The partial correlation coefficient $A_{X_{1}Y,X_2}$ of Y in X₁ when X₂ is constant is such that:

$A_{X_{1}Y,X_2}=\frac{{\sigma }_Y}{{\sigma }_{X_1}}\times \frac{r_{X_{1}Y}-r_{X_{2}Y}\times r_{X_1{X}_2}}{1-{\left(r_{X_1{X}_2}\right)}^2}$ (19)

The partial correlation coefficient $A_{X_{2}Y,X_1}$ of Y in X₂ when X₁ is constant is such that:

$A_{X_{1}Y,X_2}=\frac{{\sigma }_Y}{{\sigma }_{X_2}}\times \frac{r_{X_{2}Y}-r_{X_{1}Y}\times r_{X_1{X}_2}}{1-{\left(r_{X_1{X}_2}\right)}^2}$ (20)

Calculations using the values in Table 1 give:

$R_{Y,X_1{X}_2}=0.989648714$(21)

This means that productivity is strongly linearly correlated with both irradiation and the mean absolute angle. Using the data in Table 1 , we obtain the equation giving productivity:

$Y=1.006697395\times X_1-0.024139062\times X_2-0.585536498$(22)

We find the following average errors on productivity:

The indices “mes” and “cal” refer to the measured and calculated values respectively.

The error on the calculation of the productivity with the formula (15) is such that:

$\{\begin{array}{l}\overline{\Delta Y_1}=0.127065461\text{kg}/{\text{m}}^2\\ \text{with a frame of error}\\ 0\text{kg}/{\text{m}}^2\le \overline{\Delta Y_1}\le 0.351545991\text{kg}/{\text{m}}^2\end{array}$ (23)

The error on the calculation of the productivity with the formula (16) is such that:

$\{\begin{array}{l}\overline{\Delta Y_2}=0.304384035\text{kg}/{\text{m}}^2\\ \text{with a frame of error}\\ 0\text{kg}/{\text{m}}^2\le \overline{\Delta Y_2}\le 0.628516033\text{kg}/{\text{m}}^2\end{array}$ (24)

The error on the calculation of the productivity with the formula (22) is such that:

$\{\begin{array}{l}\overline{\Delta Y}=0.070499419\text{kg}/{\text{m}}^2\\ \text{with a frame of error}\\ 0\text{kg}/{\text{m}}^2\le \overline{\Delta Y}\le 0.162544132\text{kg}/{\text{m}}^2\end{array}$ (25)

Relations (23), (24) and (25) have allowed to write:

$\{\begin{array}{l}\overline{\Delta Y}<\overline{\Delta Y_1}\\ \overline{\Delta Y}<\overline{\Delta Y_2}\end{array}$(26)

From the relation (26), we can say that the multiple correlation formula (22) gives a better approximation than the simple correlation formulas (15) and (16).