1. Introduction
Probably no one needs convincing how important drinking water and free access to it is for humanity. Drinking impure and saline water can lead to various medical problems, including cholera, polio [
1], increased risk of hypertension, as well as skin and diarrheal diseases [
2]. The World Health Organization estimates that by 2025, 50% of the population will be living in water-stressed areas [
1]. This is the result of an increasing population, the uneven distribution of water resources, and the fact that approximately 97% of the Earth’s water resources are saline water [
3]. Therefore, water desalination is necessary and the number of water desalination plants has been growing since the 1960s to reach the number of 15,906 in 2018 [
4].
Existing water desalination plants are based mostly on reverse osmosis (RO) and multi-stage flash (MSF) distillation. Generally, RO and MSF desalination plants are large-scale and high-capacity units that require large capital expenditure. Nevertheless, building large-scale desalination plants or delivering water by pipelines or trucks in some areas may be uneconomical and challenging. Therefore, the use of solar stills to desalinate water in such cases appears to be a reasonable alternative.
Passive SS are devices of simple construction with a lack of moving parts and additional energy sources, and they do not require significant capital expenditure [
5]. The water obtained in the SS is free of salt and other non-volatile compounds [
6]. However, the main drawback of SS is low productivity, usually less than 5 L/m
2/day [
7]. Therefore, there is a lot of effort spent on improving the productivity of SS.
The main driving force behind the water evaporation process, and hence SS productivity, is the temperature difference between water and cover ∆
T [
8], and the water temperature,
Tw [
9]. Tsilingiris [
10] reported that as the ∆
T increases from 5 to 10 °C, the productivity increases by a factor of 2.6, while for a fixed ∆
T, an increase of the SS temperature by 20 °C leads to productivity improvement by a factor of 3.2. Therefore, the methods of increasing the productivity of SS can be divided into two groups: increasing the temperature of the water and decreasing the temperature of the cover [
11]. The first way includes, for example: implementing the internal [
12] or external [
13] reflectors to increase the incident solar irradiance on the water and absorber, or coating the absorber with TiO
2 nanoparticles to increase its absorptivity [
14]. On the other hand, decreasing the temperature of the cover can be realized by using the flow of cold water over the cover [
15], applying radiative cooling [
16], or using an additional external condenser [
17].
Based on the above reflections, it appears that desalinating water in SS would be more effective, in terms of productivity, if it was conducted at night when the temperature of the ambient air is low. Therefore, thermal energy storage materials have been proposed to be used in SS to store the excess heat during the day and release it during the night [
18]. Among the group of thermal-energy storage materials, phase change materials (PCM) are of great interest for use in SS [
19,
20]. PCM are materials that undergo reversible phase transitions, most commonly solid-liquid, during which the PCM stores thermal energy in the form of latent heat.
Yousef and Hassan [
21] reported that using paraffin wax as PCM can increase the SS daily productivity by 9.5%. Chaichan and Kazem [
22] investigated experimentally the performance of SS without PCM (paraffin wax), with pure paraffin wax, and with paraffin wax doped with Al
2O
3 nanoparticles. It was found that the daily productivity of SS with pure paraffin wax and with paraffin wax doped with Al
2O
3 was greater than the productivity of SS without PCM, by 14.3%, and 60.5%, respectively. A similar finding was reported by Kabeel et al. [
23], who used pure paraffin wax and paraffin wax with different mass fractions of graphite nanoparticles in the SS. The results showed that the productivity of SS with pure paraffin wax, and with paraffin wax doped with graphite nanoparticles (20% mass fraction) increased by 62.6%, and 94.5%, respectively, compared to the SS without paraffin wax. Furthermore, it was concluded that the productivity of SS increased with the increasing mass fraction of graphite nanoparticles in the paraffin wax. The addition of high thermal conductive nanoparticles into the PCM increases the thermal-conductivity of the latter and intensifies the heat transfer, leading to faster melting and solidification [
24].
In the works cited in the above paragraph, the researchers investigated only one PCM-to-water mass ratio. However, the PCM-to-water mass ratio in SS is an issue that has been investigated by several researchers. Shalaby et al. [
25] tested the SS without, and with 18 kg of paraffin wax as PCM. The PCM-to-water mass ratio was changed by using 25, and 35 kg of water. The daily productivity of SS without PCM and with 25 kg of water was 3.36 kg/m
2/day, while the productivity of the SS with PCM and 25, and 35 kg of water was 3.76, and 2.22 kg/m
2/day, respectively. A similar experimental and theoretical investigation, using the Dunkle model, was conducted by Sonker et al. [
26], who used 1.3 kg of lauric acid, stearic acid, and paraffin wax as PCM. The depth of water in the SS was changed from 1 to 5 cm. It was reported that the productivity of the SS with paraffin wax was greater than when the lauric and stearic acids were used. Additionally, the daily productivity decreased with the decreasing PCM-to-water mass ratio.
As can be seen, one way to investigate the PCM-to-water mass ratio is by changing the mass of water, and, in such cases, the productivity of SS increases with the increasing PCM-to-water mass ratio. On the other hand, Mousa et al. [
27] used tricosane as PCM in their research, and the PCM-to-water mass ratio was 0.17, 0.24, and 0.51. The PCM-to-water mass ratio was set by varying the mass of the PCM while keeping the mass of water constant. The results showed that the daily productivity of the SS was 627, 630, 545, and 550 mL for the SS with 0 (without PCM), 0.17, 0.35, and 0.51 PCM-to-water mass ratios, respectively. Thus, only in one case (0.17 mass ratio) was the application of PCM advantageous, in terms of improving productivity by 0.5%. The authors justified this result by the fact that as a large amount of PCM is present in the SS, a lot of energy is used to heat and melt the PCM. Consequently, the temperature of the water, as well as the productivity, decreases. On the other hand, Al-Harahsheh et al. [
28] investigated the SS coupled with solar collector, cover cooling, and different masses of PCM. The maximum productivity improvement was obtained for the maximum investigated PCM-to-water mass ratio of 0.38, and the improvement was 19.0%, compared to the SS without PCM.
Mousa and Gujarathi [
29] investigated theoretically, using the Dunkle model, the effects of PCM mass on the productivity of the SS. As the mass of the PCM increased, more heat from the solar radiation was used to raise the PCM temperature and melt it, at the expense of reducing the energy input to the water and increasing the water temperature. As a result, the productivity of the SS decreased with an increasing mass of the PCM. Kateshia and Lakhera [
30] investigated the influence of both the water mass and PCM mass on the performance of SS. It was reported that the productivity increased with decreasing water mass and increasing PCM mass.
Another interesting finding was reported by Tabrizi et al. [
31] and Sarhaddi et al. [
32], who concluded that using the PCM in SS is more beneficial during cloudy than sunny days. Tabrizi et al. [
31] reported an increase of 61.9% in productivity when the PCM was used on a cloudy day, but a decrease of 5.6% was observed after applying the PCM on a sunny day. This finding was confirmed further in the theoretical analysis carried out by Sarhaddi et al. [
32]. They observed a productivity improvement of 28.6% after applying the PCM during a cloudy day and a productivity reduction of 6.0% after applying the PCM during a sunny day.
Based on the literature review presented above, it can be concluded that:
However, there are still some knowledge gaps that need to be filled:
The conditions under which the PCM improve and under which they reduce the productivity of SS, have not yet been established;
The optimal PCM-to-water mass ratio remains unclear;
The water evaporation rate in the function of Tw and ∆T has not been dealt with in depth.
Therefore, this paper aims at filling the abovementioned research gaps by:
Investigating the impact of the PCM-to-water mass ratio on the performance of the passive SS, depending on the operating conditions;
Determining the conditions that must be met for PCM to improve the productivity of SS;
Establishing the evaporation rate of water in the function of Tw and ∆T in the entire range of operating temperatures of the SS.
This paper presents the results of the experimental and theoretical studies on the performance of passive single-slope SS with two different PCM, a variable PCM-to-water mass ratio, and in four operating conditions. Moreover, a comprehensive analysis of the impact of Tw and ∆T on the water-evaporation rate is conducted. Additionally, the following aspects make this paper original, compared to the previously published papers:
The experiments were conducted in laboratory conditions, which allowed the minimization of the influence of atmospheric conditions on the measurement results;
The heat- and mass-transfer phenomena occurring in the SS were described by a mathematical model that has not been widely used so far;
The PCM were placed in flexible plastic bags, which, according to the best of the authors’ knowledge, has not been done before in the field of solar stills;
This is the first time that the SS has been investigated in Poland.
3. Mathematical Modeling
A mathematical model of the SS was established, to predict the performance of the SS without and with PCM. Thus, the energy-balance equations for the absorber, water, inner and outer glass-cover, as well as PCM in the case of SS with PCM, were created and solved using MATLAB software, version 9.8 (R2020a, the MathWorks, Inc., Natick, MA, USA). The mathematical model based on the energy-balance equations for individual elements of the SS is a commonly used method of modeling for this type of system, and has been used by numerous researchers [
37,
38,
39]. To simplify the model, the following assumptions were made [
34]:
There was no temperature gradient in the water;
The mass of water was constant, as the feedwater tank supplied the freshwater to the SS;
The thermophysical properties of water and humid air were variable in the function of temperature, while the thermophysical properties of the absorber and glass cover were assumed to be constant;
The specific heat capacity of the insulating material was neglected;
Heat conduction from the absorber to the side walls above the free surface of the water was neglected;
The SS was assumed to be a lumped system, and worked in quasi-steady-state conditions with a 60 s time step, ∆t;
The glass cover was divided into two parts, inner and outer, with equal masses, and the temperature gradient between them was taken into account;
The water and inner-glass-cover surfaces were assumed to be parallel and the view factors between them were assumed to be 1;
The humid air in the SS did not affect or participate in the radiation heat transfer;
The relative humidity of the humid air in the SS was 100%, and its total pressure, pt, was 101,300 Pa;
There was no temperature gradient and no natural convection in the PCM;
The density and thermal conductivity of the PCM were assumed to be constant.
3.1. Solar Still without PCM
The energy-balance equations for the components, i.e., the absorber, water, inner, and outer glass-cover of the SS without PCM, can be written as [
34]:
where
a,
A,
cp,
e,
h,
m,
,
Q,
r,
t,
T, and
λ are the ratio coefficient, surface area (m
2), specific heat capacity (J/(kg·K)), thickness (m), heat transfer coefficient (W/(m
2·K)), mass (kg), mass flow rate (kg/s), heat transfer rate (W), heat of vaporization (J/kg), time (s), temperature (°C), and thermal conductivity (W/(m·K)), respectively. The subscripts
air,
b,
c,
g,
gi,
go,
h,
loss,
r,
sky, and
w stand for ambient air, absorber, convective, glass cover, inner glass-cover, outer glass-cover, electric heater, loss to ambient, radiative, sky, and water, respectively. The heat transfer rate from the electric heaters,
Qh, changed every hour, as shown in
Figure 3, while the temperature of sky,
Tsky, was assumed to be equal to the
Tair because the experiments were conducted under laboratory conditions. The heat transfer coefficients were calculated as [
40,
41,
42,
43,
44]:
where
Gr,
Lc,
Nu,
Ra,
v,
ε, and θ are the Grashof number, characteristic length (m), Nusselt number, Rayleigh number, wind velocity (m/s), emissivity, and inclination angle of the cover (°), respectively. The subscripts
ins and
f stand for insulation and humid air (air–water-vapor mixture in the SS), respectively. The Nusselt number for the water,
Nuw, was calculated as follows [
41,
42]:
where the Rayleigh number for water
Raw is:
where
g,
Pr,
β,
μ, and
ρ are the gravitational acceleration (m/s
2), Prandtl number, coefficient of thermal expansion (1/K), dynamic viscosity (Pa∙s), and density (kg/m
3), respectively. The thermophysical properties of water were taken from [
45]. The Rayleigh number,
Raf, for the humid air in the SS (Equation (8)) was calculated as [
41,
42]:
The properties of the humid air were calculated from the correlations given by Tsilingiris [
46,
47].
The total evaporation rate of the water in the SS is calculated using the Dunkle model in the majority of papers published in the open literature. However, in this work, the Chilton–Colburn analogy was used to calculate the evaporation rate of water, and it was assumed that the water vapor is an ideal gas [
41]:
where
Dwa,
Mw,
p, and
R are the binary diffusion coefficient of water vapor in air (m
2/s), molar mass of water vapor (kg/mol), partial water-vapor-pressure (Pa), and universal gas constant (J/(mol·K)), respectively. The
Dwa,
pw, and
pgi can be calculated as presented in Equations (15), (16), and (17), respectively [
40,
41]:
However, the mass flow rate of water vapor condensed on the cover, , is less than the total evaporation rate, , due to the following reasons:
Water vapor condenses partially on the side walls of the SS;
Some of the water droplets do not flow down the cover;
Slight vapor-leakage from the SS can occur;
Some of the water vapor is used to raise and maintain the 100% relative humidity inside the SS.
Therefore, and were assumed to be 75% and 25% of the , respectively.
In the field of solar stills, the most widely used model for calculating the convective and evaporative heat-transfer-coefficients, and thus the productivity of the SS, is the Dunkle model. However, that model has some limitations, which include [
40]:
The characteristic length between water and cover is not taken into account;
The equivalent temperature difference between water and cover is 17 °C;
The thermophysical properties of humid air are calculated for the water at the temperature of 50 °C.
Therefore, in this paper, another relationship (see Equations (8) and (13)) was used to calculate the convective heat-transfer-coefficient. In addition, the Chilton–Colburn analogy was used to calculate the evaporation rate of water. This allowed the above-mentioned limitations of the Dunkle model to be overcome.
3.2. Solar Still with PCM
The energy-balance equations for the absorber, water, inner and outer glass-cover, and PCM can be written as follows:
where
cPCM is the effective heat capacity of the PCM, and can be calculated as [
48]:
Subscripts avg, endset, l, onset, and s stand for average, endset, liquid, onset, and solid, respectively, and L is the latentheat of fusion (J/kg). Furthermore, the convective heat- transfer-coefficient, hc,w-PCM was calculated similarly to hc,b-w,(Equations (6), (11) and (12)), but with the other characteristic length and changing Tb with TPCM.
The productivity
V (mL/m
2) of the SS in the time step:
The equations presented in
Section 3.1 and
Section 3.2 were implemented into the MATLAB software (version 9.8, R2020a) and Equations (1)–(4) and (18)–(22) were solved using the
ode45 function. The calculated temperatures of the absorber, water, inner and outer cover, and PCM were used to calculate the appropriate heat-transfer-coefficients in the next time-step. The measured power of the electric heaters, ambient air temperature, initial temperatures of the absorber, water, inner and outer glass-cover, and PCM were taken as the main input parameters during modeling in MATLAB. The values of the rest of the inputs for the mathematical model are summarized in
Table 3. It is worth noting that some of the input parameters depended on the mass of the PCM, and these parameters are summarized in
Table 4.
5. Conclusions
This paper presents the results of the experimental and theoretical investigation of the effects of using PCM on the performance of the passive single-slope solar still. The experimental data were used to validate the mathematical model describing the heat and mass transfer inside the SS. The ordinary differential equations from the model were solved numerically, and the approximate solutions of the temperatures were obtained. Then, through the mathematical modeling, a detailed relationship between the water-evaporation rate, Tw, and ∆T was determined. Furthermore, the influence of the PCM-to-water mass ratio on the productivity of SS, depending on the operating conditions, was established. These novel findings fill a knowledge gap representing the lack of explanation as to why PCM does not improve the productivity of SS in certain cases.
When the initial temperatures of the absorber, water, cover, and PCM are the same, and they are at least 15 °C lower than the maximum temperature of the water, the productivity of the SS with PCM is lower by up to 10.8%, compared to the SS without PCM. However, when the initial temperatures of the water and PCM are 50 °C, the PCM can improve the productivity of the SS by up to 2.4%. Furthermore, if the PCM were heated up outside of the SS and put into the SS when the temperature of the water starts decreasing, it could improve productivity by up to 47.1%. In the case of a lack of thermal insulation of the SS, the PCM can take over the role of an insulator and improve productivity, by up to 1.1%.
This paper undoubtedly provides a valuable new contribution to the field of solar stills.