Theoretical Analysis of the Effect of a Bottom Reflector on a Vertical Multiple-Effect Diffusion Solar Still Coupled with a Basin-Type Still

Abstract

The effect of a flat plate bottom reflector on a vertical multiple-effect diffusion solar still coupled with a basin-type still is analyzed theoretically at 30°N latitude. The still has a right-angled triangular cross-section consisting of a horizontal basin liner, a sloping double glass cover, and the vertical multiple-effect diffusion still (multiple-effect section) at the vertical rear wall. A geometrical model was constructed to calculate the amount of solar radiation reflected by the reflector and absorbed on both the basin liner and the first partition of the multiple-effect section. The reflected radiation from the reflector can be a thermal input to the multiple-effect section both in cases where the reflected radiation is absorbed on the basin liner and on the first partition. It is predicted that the overall daily productivity of the still can be increased about 16%, 13%, and 29% on the spring and autumn equinoxes, the winter solstice, and the summer solstice, respectively, by adding the external reflector and setting the reflector inclination to the proper angle according to seasons.

1. Introduction

A solar still can be used to get sanitary fresh water from saline water using only solar energy. Various types of solar stills have been presented and studied as reviewed by Tiwari et al. [1], Kaushal and Varun [2], Sampathkumar et al. [3], and Kabeel and El-Agouz [4]. Among them, multiple-effect diffusion solar stills, which consist of a series of closely spaced parallel partitions in contact with saline-soaked wicks, have great potential because of their high productivity and simplicity [5–23].

In the multiple-effect diffusion still, solar thermal energy absorbed on the first partition is used to evaporate the saline water in the wick of the first partition. Water vapor diffuses through the humid air layer between partitions and condenses on the second partition. The latent heat of condensation causes further evaporation from the wick of the second partition. In this manner, the evaporation and condensation process is repeated, and the solar energy is recycled on all partitions in the still to increase distillate productivity.

In a multiple-effect diffusion still, the size of diffusion gaps between partitions is a very important parameter. Smaller diffusion gaps can increase distillate productivity considerably, mainly because the smaller gaps reduce thermal resistance through each single gap as well as throughout the whole still. Accordingly, vertical multiple-effect diffusion stills have an advantage over inclined ones, since the deformation of partition plates due to gravity can be reduced, and then the diffusion gaps can be smaller for the vertical stills than for the inclined ones [10, 17–20, 22, 23]. If the partition plates are deformed at small diffusion gaps, contact between the evaporating wick and facing condensing surface is liable to be occurred, which leads to the contamination of distilled water with saline water and/or the reabsorption of distilled water by the wick.

Tanaka et al. [17–19] theoretically and experimentally studied a vertical multiple-effect diffusion solar still coupled with a basin-type still shown in Figure 1. The still is a single slope basin-type still with a double glass cover and a vertical multiple-effect diffusion still (or multiple-effect section) which is attached to the vertical rear wall of the basin-type still. Water in the basin liner is heated by solar energy and evaporated. Water vapor from basin liner condenses on the inner glass cover and the vertical rear wall, which is the first partition of the multiple-effect section. Latent heat of condensation on the inner glass cover is just released to the surroundings through the double glass cover, while the latent heat of condensation on the rear wall can be used as a heat source for the multiple-effect section. Therefore, the latent heat of condensation as well as solar energy directly absorbed on the rear wall is utilized repeatedly in the multiple-effect section as mentioned above.

Figure 1:

Schematic diagram of a vertical multiple-effect diffusion still coupled with a basin-type still using a flat plate bottom reflector.

The vertical multiple-effect diffusion still coupled with the basin-type still has very simple structure; that is, the still can be constructed by adding the outer glass cover and some partitions to the conventional single slope basin-type still, which is the simplest solar still. However, the still is highly productive, and Tanaka et al. [19] reported that a still with 5 mm diffusion gaps between 11 partitions in the multiple-effect section produces 19.0–24.0 kg/m² day of distillate per unit base area at 20.7–22.4 MJ/m² day solar radiation on the glass cover in outdoor experiments, which is in good agreement with theoretical predictions calculated from the heat and mass transfer model developed by Tanaka et al. [17].

Recently, many studies on the effect of flat plate external reflectors on basin-type stills have been reported [24–37], although few reports had been presented before the year 2000 [38–40]. The results showed that an external reflector can increase the solar radiation absorbed on the basin liner as well as the distillate productivity of the still if the reflector angle is set properly. Among them, Tanaka [34] theoretically studied a basin-type still with a flat plate external bottom reflector extending from the front wall of the still and found that the increase in the still's daily productivity obtained by adding an external bottom reflector and setting its inclination to the optimum angle is predicted to be about 21%, 23%, and 12% on the spring equinox and summer and winter solstice days.

A flat plate bottom reflector can be applied to the vertical multiple-effect diffusion still coupled with a basin-type still, but in this case, the reflected radiation not only on the basin liner but also on the rear wall (the first partition of the multiple-effect section) should be taken into consideration, since the reflected radiation on the rear wall can be used directly as thermal input to the multiple-effect section. To analyze the external bottom reflector on the basin-type still, Tanaka [34] presented a geometrical model by introducing the mirror-symmetric plane of a basin liner to the external reflector and determined the amount of radiations reflected from the external reflector and then absorbed on the basin liner by calculating the direct radiation that goes through the external reflector and is then absorbed on the mirror-symmetric plane. However, the reflected radiation on the rear wall cannot be calculated with this model.

Therefore, in this paper, a new geometrical model is presented to calculate the amount of solar radiation reflected from the external bottom reflector and then absorbed on the basin liner as well as on the rear wall and determine the effect of the external reflector on the distillate productivity of a vertical multiple-effect diffusion still coupled with basin-type still at 30°N latitude.

2. Vertical Multiple-Effect Diffusion Solar Still Coupled with Basin-Type Still Using Flat Plate External Bottom Reflector

The proposed still is shown in Figure 1. The still has a right-angled triangular cross-section consisting of a horizontal basin liner, a sloping double glass cover facing due south, and a vertical multiple-effect diffusion still (multiple-effect section) at the vertical rear wall. The basin liner and the vertical rear wall (the first partition of the multiple-effect section) are assumed to be painted black. The bottom and side walls are insulated. The two inner side walls are assumed to be covered with highly reflective materials. The multiple-effect section consists of a series of closely spaced vertical parallel partitions in contact with saline-soaked wicks. Saline water is fed to the wicks constantly. A flat plate external bottom reflector is attached to the front wall of the still.

Direct and diffuse solar radiation as well as reflected solar radiation from the external reflector transmits through the double glass cover and is absorbed on the basin liner (b) and the first partition (p1). Water vapor from the basin liner condenses on the inner glass cover (gi) and the first partition (p1) to become distilled water. The latent heat of condensation and the solar radiation directly absorbed on the first partition is utilized repeatedly in the multiple-effect section.

The double glass cover maintains a high temperature on the inner glass cover. This causes an increase in the ratio of condensate on the first partition p1 to that on the inner glass cover gi and a reduction in radiative and convective heat loss through the glass cover.

Tanaka et al. [18] showed that the average daily productivity of the still for four typical days (the spring and autumn equinoxes and the summer and winter solstices) was predicted to be maximum when the glass cover inclination is about 40° to 45°. Therefore, the inclination of the glass cover θ _s is determined as 40° in this paper. The inclination angle of the reflector θ _m is changed in the calculation to find the optimum angle of θ _m .

The basin liner and the external reflector are determined to be the same size at 1 m² (1 m × 1 m). So the area of each glass plate of the double glass cover and each partition plate in the multiple-effect section would be about 1.31 m² and 0.84 m².

3. Theoretical Analysis

3.1. Reflected Radiation from External Reflector Absorbed on the Basin Liner and the First Partition

To simplify the calculations, it is assumed that the same amount of solar radiation obstructed by one side wall hits the other inner side wall since both inner side walls are assumed to be covered with highly reflective materials. This assumption would be valid if the still's width is sufficient in relation to the length of the basin liner. The height of the front wall and thickness of the double glass cover are assumed to be negligible in calculating the absorption of solar radiation.

The reflected radiation from the external reflector cannot all be absorbed on the basin liner and the first partition, and part of or all of the reflected radiations would escape to the surroundings without hitting the basin liner and the first partition. Figure 2 shows a schematic diagram of the shadow of the reflector as well as a projection of the reflected sunrays from the reflector on a horizontal surface caused by direct solar radiation. ABCD is the basin liner, ABFE is the double glass cover, CDEF is the first partition, and ABGH is the external reflector. l _s and l _m are the length of the basin liner and reflector, and θ _s and θ _m are the angle of the glass cover and reflector from the horizontal. w is the width of both the still and the reflector. φ and ϕ are the azimuth and altitude angles of the sun. The glass cover is facing due south.

Figure 2:

The shadow of the still and reflector and the reflected projection of the reflector on a horizontal surface caused by direct radiation. l₁ = l _m {sinθ _m tan (π – ω₃ – 2θ _m ) – cosθ _m }, l₂ = l₁tanω₁, l₃ = l₁tanω₂, l₄ = l _m sinθ _m cosφ/tanϕ, l₅ = l _m sinθ _m sin|φ|/tanϕ, l₆ = l _m cosθ _m – l₄, l₇ = l _s tanθ _s cosφ′/tanϕ′, l₈ = l _s tanθ _s sin|φ′|/tanϕ′, tanω₁ = l₅/l₆, tanω₂ = l₈/(l₇ + l _s ), tanφ′ = l₂/(l₁ + l _m cosθ _m ), and tanϕ′ = l _m sinθ _m cosφ′/(l₁ + l _m cosθ _m ).

The shadow of the reflector and the projection of the reflected sunrays from the reflector caused by direct radiation are shown as ABG′H′ and ABG ″ H ″, respectively. Assuming that radiation with the same azimuth (φ′) and altitude angles (ϕ′) as the reflected sunrays from the reflector hits the glass cover, the shadow of the glass cover would be an area shown as ABF ″ E ″. The amount of reflected radiation from the reflector absorbed on the basin liner and the first partition can be determined as the overlapping area of the reflected projection of the reflector (ABG ″ H ″) and the shadow of the glass cover (ABF ″ E ″) shown as trapezoid ABKH ″. The residue shown as triangle BG ″ K would escape to the ground. It follows that the portion of the reflected radiation absorbed on either the basin liner or the first partition can be determined as area ABIJ or JIKH ″, respectively. Since the reflected radiation from the reflector is condensed or diluted (i.e., the length of the shadow (l₆) and reflected projection (l₁) of the reflector are not equal), the intensity of the reflected radiation from the reflector on a horizontal surface should be determined as G _dr × l₆/l₁. Here, G _dr is direct solar radiation on a horizontal surface. Therefore, the solar radiation reflected from the reflector and absorbed on the basin liner, Q_{re, b}, or the first partition, Q_{re, p1}, can be expressed as

$Q_{re,b}=G_{dr}\frac{l_{\mathrm{6}}}{l_{\mathrm{1}}}{\left\{{\tau }_g\left({\beta }^{\prime }\right)\right\}}^{\mathrm{2}}{\rho }_m{\alpha }_b\times s_b,$

(1)

$s_b=l_s\left\{w-\mathrm{0.5}{l}_s(\text{tan}{\omega }_{\mathrm{1}}-\text{tan}{\omega }_{\mathrm{2}})\right\},$

(2)

$Q_{re,p\mathrm{1}}=G_{dr}\frac{l_{\mathrm{6}}}{l_{\mathrm{1}}}{\left\{{\tau }_g\left({\beta }^{\prime }\right)\right\}}^{\mathrm{2}}{\rho }_m{\alpha }_{p\mathrm{1}}\times s_{p\mathrm{1}},$

(3)

$s_{p\mathrm{1}}=\left(l_{\mathrm{1}}-l_s\right)\left[w-\mathrm{0.5}\left\{l_{\mathrm{2}}-l_{\mathrm{3}}+l_s\left(\text{tan}{\omega }_{\mathrm{1}}-\text{tan}{\omega }_{\mathrm{2}}\right)\right\}\right],$

(4)

where τ _g is the transmittance of each glass cover, β′ is the incident angle of the reflected sunlight from the reflector to the glass cover, ρ _m is the reflectance of the external reflector, α _b and α_p1 are the absorptance of the basin liner and the first partition, and s _b and s_p1 are the area of ABIJ and JIKH ″.

To calculate (1) to (4), the length of the reflected projection of the reflector, l₁, has to be determined. The side view of the external reflector is shown in Figure 3. Since the incident angle and reflection angle of the sunrays for the reflector can be expressed as π/2 – ω₃ – θ _m , the length of the reflected projection of the reflector, l₁, can be determined with the angle ω₃ (= tan⁻¹(cosφ/tanϕ)) as

$l_{\mathrm{1}}=l_m\left\{\text{sin}{\theta }_m\text{tan}(\pi -{\omega }_{\mathrm{3}}-\mathrm{2}{\theta }_m)-\text{cos}{\theta }_m\right\}.$

(5)

Therefore, Q_{re, b} and Q_{re, p1} can be calculated with the lengths l₁ to l₈ and angles ω₁ to ω₃ shown in Figures 2 and 3.

Figure 3:

Side view of the external reflector. l₁ = l _m {sinθ _m tan (π – ω₃ – 2θ _m ) – cosθ _m } and tanω₃ = l₄/l _m sinθ _m = (l _m sinθ _m cosφ/tanϕ)/(l _m sinθ _m ) = cosφ/tanϕ.

When Q_{re, b} and Q_{re, p1} are calculated, there are some exceptions as follows.

1. When length l₂ is longer than the still's width w as shown in Figure 4, s _b and s_p1 should be determined as follows:case i (ω₂ < ω₄):

   $\begin{array}{c}{s}_b=\frac{\mathrm{0.5}{w}^{\mathrm{2}}\text{sin}\left(\pi /\mathrm{2}-{\omega }_{\mathrm{1}}\right)\text{sin}\left(\pi /\mathrm{2}+{\omega }_{\mathrm{2}}\right)}{\text{sin}\left({\omega }_{\mathrm{1}}-{\omega }_{\mathrm{2}}\right)},\\ s_{p\mathrm{1}}=\mathrm{0}\end{array}$

   (6)

   case ii (ω₄ ≤ ω₂ < ω₅):

   $\begin{array}{c}{s}_b=\mathrm{0.5}\left\{l_{s}w+l_s^{\mathrm{2}}\left(\text{tan}{\omega }_{\mathrm{2}}-\text{tan}{\omega }_{\mathrm{4}}\right)\right\},\\ s_{p\mathrm{1}}=(\mathrm{0.5}{\left\{l_s\left(\text{tan}{\omega }_{\mathrm{2}}-\text{tan}{\omega }_{\mathrm{4}}\right)\right\}}^{\mathrm{2}}\text{sin}\left(\frac{\pi }{\mathrm{2}}-{\omega }_{\mathrm{1}}\right)\hspace{1em}\hspace{1em}\times \text{sin}\left(\frac{\pi }{\mathrm{2}}+{\omega }_{\mathrm{2}}\right))\times {\left(\text{sin}\left({\omega }_{\mathrm{1}}-{\omega }_{\mathrm{2}}\right)\right)}^{-1},\end{array}$

   (7)

   case iii (ω₂ ≥ ω₅):

   $\begin{array}{c}{s}_b=\mathrm{0.5}\left\{l_{s}w+l_s^{\mathrm{2}}\left(\text{tan}{\omega }_{\mathrm{2}}-\text{tan}{\omega }_{\mathrm{4}}\right)\right\},\\ s_{p\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}{\left\{l_s\left(\text{tan}{\omega }_{\mathrm{5}}-\text{tan}{\omega }_{\mathrm{4}}\right)\right\}}^{\mathrm{2}}\hspace{1em}\times \frac{\text{sin}\left(\pi /\mathrm{2}-{\omega }_{\mathrm{1}}\right)\text{sin}\left(\pi /\mathrm{2}+{\omega }_{\mathrm{5}}\right)}{\text{sin}({\omega }_{\mathrm{1}}-{\omega }_{\mathrm{5}})}\hspace{1em}+\frac{\mathrm{1}}{\mathrm{2}}\left(l_1^{\mathrm{2}}-l_s^{\mathrm{2}}\right)\left(\text{tan}{\omega }_{\mathrm{2}}-\text{tan}{\omega }_{\mathrm{5}}\right).\end{array}$

   (8)

2. When length l₁ is shorter than l _s , the reflected radiation from the reflector does not hit the first partition. So s_p1 should be zero, and s _b should be determined as follows:

   $\begin{array}{c}\left(\text{a}\right)l_{\mathrm{2}}\le w\text{:}\\ \hspace{1em}{s}_b=l_{\mathrm{1}}\left\{w-\mathrm{0.5}\left(l_{\mathrm{2}}-l_{\mathrm{3}}\right)\right\},\\ \left(\text{b}\right)l_{\mathrm{2}}>w\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\omega }_{\mathrm{2}}<{\omega }_{\mathrm{5}}\text{:}\\ \hspace{1em}{s}_b=\frac{\mathrm{0.5}{w}^{\mathrm{2}}\text{sin}\left(\pi /\mathrm{2}-{\omega }_{\mathrm{1}}\right)\text{sin}\left(\pi /\mathrm{2}+{\omega }_{\mathrm{2}}\right)}{\text{sin}\left({\omega }_{\mathrm{1}}-{\omega }_{\mathrm{2}}\right)},\\ \left(\text{c}\right)l_{\mathrm{2}}>w\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\omega }_{\mathrm{2}}\ge {\omega }_{\mathrm{5}}\text{:}\\ \hspace{1em}{s}_b=\mathrm{0.5}\left\{l_{\mathrm{1}}w+l_{\mathrm{1}}^{\mathrm{2}}\left(\text{tan}{\omega }_{\mathrm{2}}-\text{tan}{\omega }_{\mathrm{5}}\right)\right\}.\end{array}$

   (9)

3. When the reflection angle π/2 – ω₃ – θ _m is larger than the reflector inclination θ _m (π/2 – ω₃ – θ _m > θ _m ), all of the reflected radiations will escape to the surroundings. So both Q_{re, b} and Q_{re, p1} will be zero.

4. During the months of April to August, when the sun moves north, that is, the absolute value of the azimuth angle of the sun, |φ|, is larger than π/2 in the morning and evening, Q_{re, b} and Q_{re, p1} should be calculated by determining lengths l₁ and l₆ as

   $\begin{array}{c}{l}_{\mathrm{1}}=l_m\left\{\text{sin}{\theta }_m\text{tan}\left(\pi +{\omega }_{\mathrm{3}}-\mathrm{2}{\theta }_m\right)-\text{cos}{\theta }_m\right\}\\ l_{\mathrm{6}}=l_m\text{cos}{\theta }_m+l_{\mathrm{4}}.\end{array}$

   (10)

5. When the altitude angle of the sun is very small and the reflector would shade a part of the basin liner in the early morning and late evening, Q_{re, b} and Q_{re, p1} should be zero.

Figure 4:

Reflected projection and shadow of the glass cover when l₂ > w. l₉ = (l₂ – w)/tanω₁, l₁₀ = l₉ – (l₁ – l _s ), l₁₁ = l₁₀tanω₁, tanω₄ = l₁₁/l _s , and tanω₅ = (l₂ – w)/l₁.

3.2. Absorption of Direct and Diffuse Radiation on the Basin Liner and the First Partition

Direct solar radiation absorbed on the basin liner, Q_{dr, b}, can be determined as

$Q_{dr,b}=G_{dr}{\left\{{\tau }_g\left(\beta \right)\right\}}^{\mathrm{2}}{\alpha }_b\times l_{s}w,$

(11)

where β is the incident angle of the sunrays to the glass cover. Direct solar radiation absorbed on the first partition, Q_{dr, p1}, can be determined by assuming that the azimuth and altitude angles of the reflected sunrays, φ′ and ϕ′, in Figure 2 would be those of the direct radiation, φ and ϕ, and considering the shadow area of the first partition (DCF ″ E ″). This can be expressed as

$Q_{dr,p\mathrm{1}}=G_{dr}{\left\{{\tau }_g\left(\beta \right)\right\}}^{\mathrm{2}}{\alpha }_{p\mathrm{1}}w{l}_{p\mathrm{1}}\frac{\text{cos}\phi }{\text{tan}\varphi },$

(12)

where l_p1 is the height of the first partition.

When the sun moves north during the months of April to August, Q_{dr, p1} would be zero, and Q_{dr, b} should be determined taking into consideration the shadow of the first partition on the basin liner as

$\begin{array}{c}{Q}_{dr,b}=G_{dr}{\left\{{\tau }_g\left(\beta \right)\right\}}^{\mathrm{2}}{\alpha }_b\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left\{l_{s}w-l_{p\mathrm{1}}\frac{\text{cos}\phi }{\text{tan}\varphi }\left(w-\mathrm{0.5}{l}_{p\mathrm{1}}\frac{\text{sin}\left|\phi \right|}{\text{tan}\varphi }\right)\right\}.\end{array}$

(13)

Diffuse solar radiation absorbed on the basin liner and the first partition, Q_{df, b} and Q_{df, p1}, can be determined assuming that diffuse radiation comes uniformly from all directions in the sky dome and can be expressed as

$\begin{array}{c}{Q}_{df,b}=G_{df}{\left({\tau }_g\right)}_{df,b}{\alpha }_b\times l_{s}w,\\ Q_{df,p\mathrm{1}}=G_{df}{\left({\tau }_g\right)}_{df,p\mathrm{1}}{\alpha }_{p\mathrm{1}}\times l_{p\mathrm{1}}w,\end{array}$

(14)

where G _df is the diffuse solar radiation on a horizontal surface and (τ _g )_{df, b} and (τ _g )_{df, p1} are functions of the inclination of the glass cover, θ _s , and is calculated by integrating the transmittance of the double glass cover for diffuse radiation from all directions in the sky dome. This can be expressed as

$\begin{array}{c}{\left({\tau }_g\right)}_{df,b}=\mathrm{0.484}-\mathrm{2}.\mathrm{82}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}\times {\theta }_s+\mathrm{1}.\mathrm{22}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}\times {\theta }_s^{\mathrm{2}},\hspace{0.17em}\hspace{0.17em}{\theta }_s\left[{}^{\text{o}}\right],\\ {\left({\tau }_g\right)}_{df,p\mathrm{1}}=\mathrm{0.253}+\mathrm{3}.\mathrm{88}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{1}{\mathrm{0}}^{-\mathrm{3}}\times {\theta }_s-\mathrm{4.47}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}\times {\theta }_s^{\mathrm{2}},\hspace{0.17em}\hspace{0.17em}{\theta }_s\left[{}^{\text{o}}\right].\end{array}$

(15)

3.3. Absorption of Solar Radiation on the Glass Cover

All of the reflected or direct solar radiations which go through the double glass cover should be absorbed on the basin liner and the first partition. So the amount of the solar radiation absorbed on each glass cover can be determined by adding the area of (1) and (3) for reflected radiation and (11) and (12) for direct radiation. Diffuse solar radiation absorbed on each glass cover can be determined with the same assumption for Q_{df, b} and Q_{df, p1}.

3.4. Effect of Shadow of External Reflector

When the altitude angle of the sun ϕ is very small, the external reflector will shade a part of the basin liner. In the calculations, this effect is taken into account and was described in a previous paper in detail [34].

A longer reflector length l _m and/or a greater reflector inclination θ _m would cast a shadow on even the first partition p1. But for our calculations, a shadow on the first partition was not observed with the numerical conditions employed in this paper.

3.5. Heat and Mass Transfer in the Still

Heat and mass transfer in the still was described in a previous paper in detail [17]. The equations for the solar radiation absorbed on the basin liner, the first partition and each glass cover, the energy balance equations for all components of the still, and the equations of properties were solved together to find the temperatures, heat and mass transfer rates, and the distillate production rates on the inner glass cover and each partition throughout the day. Temperatures of each component of the still were set to be equal to the ambient air temperature at just before sunrise for the initial values. The weather and design conditions are listed in Table 1.

Table 1:

Design and weather conditions.

l s = l m = w = 1 m, θ s = 40°

Number of partitions = 10

Diffusion gaps between partitions = 5 mm

Base area = 1.05 m2

Thickness of glass plate and partition plate = 3 mm and 0.5 mm

Air gap between double glass cover = 10 mm

Depth of basin water = 10 mm

Thickness and thermal conductivity of insulation = 50 mm and 0.04 W/mK

α b = αp1 = 0.9, α g i = α g o = 0.08, and ρ m = 0.8

τ g (β) = 2.642 cos β – 2.163 cos2 β – 0.320 cos3 β + 0.719 cos4 β [17]

Feed rate of saline water to wicks: twice as large as the steady-state evaporation rate from wicks which have been calculated on the assumption that solar radiation will stay constant at the maximum of the day.

G dr , G df : Bouguer's and Berlage's equations [41] with solar constant of 1370 W/m2 and transmittance of atmosphere of 0.7 at 30°N latitude

T a = 25°C, 33°C, 30°C, and 20°C on spring equinox, summer solstice, autumn equinox, and winter solstice

Ambient wind velocity = 1 m/s

4. Results and Discussions

Figure 5 shows the variation of the daily amount of solar radiation absorbed on the basin liner, Q_{sun, b}, and on the first partition, Q_sun,p1, with reflector inclination θ _m on three typical days (spring equinox and summer and winter solstice days) at 30°N latitude. Q_{sun, b} and Q_sun,p1 are the sum of the direct, diffuse, and reflected radiation absorbed on the basin liner and the first partition, respectively. Q_{sun, b} is greatest in summer and smallest in winter, while Q_sun,p1 is greatest in winter and smallest in summer, and the reflector inclination which can increase Q_{sun, b} and Q_sun,p1 is smallest in winter and greatest in summer. This is because the altitude angle of the sun becomes nearly vertical in summer and is lowest in winter. Solar radiation absorbed on both the basin liner and the first partition can be increased by setting the reflector inclination to the proper angle according to the seasons on these three days.

Figure 5:

Daily solar radiation absorbed on the basin liner, Q_sun,b, and the first partition, Q_sun,p1, varying with reflector inclination θ _m on the spring equinox and the summer and winter solstice days at 30°N.

On each day, both Q_{sun, b} and Q_sun,p1 start to increase at a certain reflector inclination, and Q_sun,p1 has a sharp peak, while Q_{sun, b} has a gentle peak. The reason for this is as follows: all of the reflected sunrays from the external reflector escape to the surroundings during θ _m < π/2 – ω₃ – θ _m (see Figure 3), and after θ _m becomes more than the angle π/2 – ω₃ – θ _m by increasing the reflector inclination, reflected sunrays start to hit the first partition and the basin liner. A greater reflector inclination θ _m results in a shorter length of l₁ (shown in Figure 2), so Q_{re, p1} rapidly decreases to zero, but Q_{re, b} increases for a while and decreases to zero with an increase in reflector inclination θ _m . Further, when θ _m is larger than about 35° or 60° on the winter solstice or spring equinox, the external reflector shades the basin liner. So Q_{sun, b} is less than that θ _m = 0° which is equivalent to the still without a reflector.

The time variation of the overall distillate production rate (Distillate), that is, the sum of distillate production rates on all condensing surfaces (gi and p1 to p10), global solar radiation on 1 m² horizontal surface (Global), and absorption of direct, diffuse, and reflected solar radiation on the basin liner (Q_{dr, b}, Q_{df, b}, and Q_{re, b}) and the first partition (Q_{dr, p1}, Q_{df, p1}, and Q_{re, p1}) on a spring equinox day at θ _m = 35° is shown in Figure 6. At θ _m = 35°, the reflected sunrays can be absorbed on both the basin liner and the first partition (Figure 5). In this paper, the distillate production rate as well as the daily amount of distillate is defined as those per unit base area of the still. The overall distillate production rate peaks about 30 min later than that of the global solar radiation because of the heat capacity of the still. Absorption of direct solar radiation (Q_{dr, b} and Q_{dr, p1}) is dominant, and the amounts of diffuse solar radiation (Q_{df, b} and Q_{df, p1}) are very low on both the basin liner and the first partition.

Figure 6:

Time variation of overall distillate production rate (Distillate), global solar radiation on a horizontal surface (Global), and absorption of direct (Q _dr ), diffuse (Q _df ), and reflected (Q _re ) solar radiation on the basin liner and the first partition on a spring equinox day at θ _m = 35°.

Figure 7 shows the temperature distribution in the still at the peak of the overall distillate production rate and the daily amount of distillate on each condensing surface on a spring equinox day at θ _m = 35°. Overall daily productivity (Total) is shown in 1/5 scale. The temperature is highest at the basin liner (b) and decreases as it passes through the double glass cover and the multiple-effect section to the ambient air (a). The daily amount of distillate from gi is greater than that on p1 since the temperature differences between the basin liner and the inner glass cover gi (T _b – T _gi ) and the first partition p1 (T _b – T_p1) are almost the same, but the area of the inner glass cover is larger than that of the first partition. The daily amount of distillate on the second partition p2 is greater than that on p1 since the latent heat of condensation in addition to the solar radiation absorbed on the first partition is utilized to evaporate the saline water from the wick of the first partition. The daily amount of distillate decreases progressively from p2 to the last partition p10, since the thermal energy is consumed to heat up saline water fed to each wick, and the mean temperature of evaporating and condensing surfaces of each distilling cell decreases progressively to the outer partition.

Figure 7:

Temperature distribution at the peak of the overall distillate production rate and the daily amount of distillate in the still on a spring equinox day at θ _m = 35°.

To evaluate whether it is better to absorb the reflected sunrays on the basin liner or on the first partition, the variation in the daily amount of solar radiation absorbed on the basin liner Q_{sun, b} and the first partition Q_sun,p1, the daily amount of distillate on the inner glass cover, m_{d, gi}, and the first partition, m_{d, p1}, with reflector inclination θ _m on a spring equinox day is shown in Figure 8. Both the absorption of solar radiation, Q_{sun, b} and Q_sun,p1, are shown in 1/10 scale. The temperature difference between the basin liner and the inner glass cover or the first partition (T _b – T _gi or T _b – T_p1) at the peak of overall distillate production rate is also shown in Figure 8. Reflected sunrays from the external reflector start to hit both the basin liner and the first partition at about θ _m = 30°, and Q_sun,p1 (or Q_{re, p1}) has a sharp peak at about θ _m = 31°, and Q_{sun, b} (or Q_{re, b}) has a gentle peat at around θ _m = 40°.

Figure 8:

Variation of daily solar radiation absorbed on the basin liner Q_sun,b and the first partition Q_sun,p1, daily amount of distillate on the inner glass cover m_{d, gi} and the first partition m_{d, p1}, and temperature difference between basin water and inner glass cover, T _b – T _gi , and the first partition, T _b – T_p1, with reflector inclination θ _m on a spring equinox day.

When Q_sun,p1 peaks (θ _m = about 31°), the daily amount of distillate on the inner glass cover m_{d, gi} increases, while that of the first partition m_{d, p1} slightly decreases. This is because the greater Q_sun,p1 causes a higher temperature on p1 (or less temperature difference T _b – T_p1) and a higher ratio of condensation of water vapor from the basin water on the inner glass cover to that on the first partition. In a range of θ _m = 31° to 40°, Q_sun,p1 decreases but Q_sun,b increases, and this causes an increase in difference of the temperature difference between T _b – T_p1 and T _b – T _gi , since the absorption of solar energy on the first partition decreases, while that on the inner glass stays almost the same. Therefore, the daily amount of distillate on p1, m_{d, p1}, greatly increases, while that on gi, m_{d, gi}, slightly decreases with an increase in reflector inclination θ _m in this range. When θ _m is larger than 40°, Q_sun,p1 remains constant (Q_{re, p1} is zero), and Q_sun,b decreases with an increase in θ _m . This causes a decrease in difference of the temperature differences between T _b – T_p1 and T _b – T _gi , and m_{d, p1} decreases faster than m_{d, gi} with an increase in θ _m .

Figure 9 shows the daily amount of distillate on condensing surfaces when Q_{re, p1} is max (θ _m = 31°) and Q_{re, b} is max (θ _m = 40°) on a spring equinox day. The results for a still without a reflector (NRS) are also shown. When Q_{re, p1} is max, the daily amount of distillate on gi and p1 is almost the same with those of NRS, but the daily amount of distillate from p2 to p10 are greater than for NRS. This shows that the reflected radiation directly absorbed on the first partition p1 can be used adequately as a heat source for the multiple-effect section without escaping to the surroundings through the double glass cover. When Q_{re, b} is max, the daily amount of distillate on gi is almost the same, while that on the first partition p1 is considerably greater than for NRS, and the daily amount of distillate from p2 to p10 would also be greater than for NRS. This indicates that the reflected radiation absorbed on the basin liner can be transferred to the multiple-effect section as a form of latent heat of condensation. As a result, the reflected radiation from the external reflector can be used in the multiple-effect section adequately both in cases where the reflected radiation is absorbed on the basin liner or on the first partition.

Figure 9:

The daily amount of distillate on condensing surfaces when Q_{re, p1} is max (θ _m = 31°) and Q_{re, b} is max (θ _m = 40°) on a spring equinox day. NRS shows the still without the external reflector.

Figure 10 shows variations of (a) overall daily productivity of the still and (b) the increase ratio with reflector inclination θ _m on four typical days (spring and autumn equinoxes and summer and winter solstice days) at 30°N. The increase ratio is the value comparing a still without an external reflector and defined as

$\text{increase}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{ratio}=\left(\text{overall}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{daily}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{productivity}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{of}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{a}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{still}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{with}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{a}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{reflector}\right)/(\text{overall}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{daily}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{productivity}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{of}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{a}\text{still}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{without}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{a}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{reflector}).$

(16)

Figure 10:

Variation of (a) overall daily productivity of the still and (b) increase ratio with reflector inclination on four days (spring and autumn equinoxes and summer and winter solstice days) at 30°N.

Daily global solar radiation on a horizontal surface is about 23.5, 22.6, 30.4, and 12.5 MJ/m² day on spring and autumn equinoxes and summer and winter solstice days, respectively. As mentioned above, reflected radiation from the external reflector can be used adequately as a heat source for the multiple-effect section both in cases where the reflected radiation is absorbed on the basin liner or on the first partition. Therefore, the overall daily productivity and increase ratio have gentle peaks around Q_sun,b, and Q_sun,p1 (or Q_{re, b} and Q_{re, p1}) have peaks, except for on the winter solstice. Only on the winter solstice, the altitude angle of the sun is very low, and Q_sun,p1 is greater than Q_sun,b (Figure 5), so the overall daily productivity and increase ratio have a sharp peak when Q_sun,p1 has maximum value (θ _m = 20°).

The overall daily productivity of the still can be increased by using a flat plate bottom reflector and setting the inclination of the reflector to the proper angle according to the seasons. The reflector inclination should be set to be about 30° to 40° on the spring and autumn equinoxes, 50° to 60° on the summer solstice, and around 20° on the winter solstice. Increase ratio would be 1.16, 1.13, and 1.29 on the spring and autumn equinoxes, the winter solstice, and summer solstice, respectively, when the glass cover inclination is set at 40° at 30°N.

5. Conclusions

The effect of an external flat plate bottom reflector on a vertical multiple-effect diffusion solar still coupled with a basin-type still was analyzed theoretically when the glass cover inclination is 40° from a horizontal at 30°N latitude. The results of this work are summarized as follows.

1. A geometrical model to calculate the amount of solar radiation reflected by an external reflector and then absorbed on the basin liner as well as the vertical rear wall (the first partition of the multiple-effect section) was constructed.

2. Solar radiation absorbed on both the basin liner and the first partition can be increased by adding the external reflector and setting its inclination to the proper angle according to seasons.

3. The reflected radiation from the reflector can be utilized adequately both in cases where the reflected radiation is absorbed on the basin liner and on the first partition.

4. The reflector inclination should be set to be around 30° to 40° on the spring and autumn equinoxes, 50° to 60° on the summer solstice, and 20° on the winter solstice, respectively.

5. The external reflector can increase the overall daily productivity of the still about 16%, 13%, and 29% on the spring and autumn equinoxes, the winter solstice, and the summer solstice, respectively.