Theoretical Analysis of a New Design of a Concentration Based Solar Distiller

. This paper includes a theoretical study of energy balance for all parts of new design of solar concentration distiller using a parabolic concentrator with a half-cylinder basin. Our goal is to analyze the thermal efficiency of the new device to use in the Morocco's Rabat-Sale-Kenitra region. The methodology concentrates on solving the thermal collector's energy balance equations whose components are the glass cover, the brackish water and half-cylinder absorber. Numerical resolution of the energy balance equations was performed using a MATLAB code based on the method of 4th order Runge-Kutta. The results show a good theoretical performance of the new device


Introduction
In the world, water forms about 97 % in the oceans, it is also stored as ice in polar places for about 2%, and just of 1 % is fresh water that is required for animals and plants and the needs of human life [1,2].The world's need for potable water continues to rise day by day because of the increasing growth in the population, industry and agriculture.Nowadays, the scarcity problem threatens even regions that are considered as water-rich.The several issues associated in the world with the lack of fresh and clean water are well documented: 1.2 billion people complain about of lack of access to clean drinking water and more than millions die each year from diseases caused by human excreta or contaminated water [3].
Face to the fresh water shortage, the solar distillation use can be found a promising alternative for turning salty or brackish water into fresh water, by using solar energy which is widely and sufficiently available on Earth [2].
Chouchi et al. [4] showed that the solar energy thermal conversion using solar concentrators allowed salt water to be boiled at high temperatures at pressures equal or higher than atmospheric pressures.
Khalifa et al. [5] performed an experimental analysis on new designs of solar stills of the basin type and designed multiple single and double solar stills.They conducted the tests to determine the impact on results of such changes, such as productivity and efficiency.Some improvements including preheating of feeding water via a solar heater and the use internal and external condensers for condensing vapors along with preheating water for feeding.The results revealed advancements in solar stills production and performance.
El Gharbi et al. [6] conducted a comparative analysis between linear Fresnel reflector and collector's parabolic trough.They found that the linear Fresnel reflector's thermal efficiency and optical quality is lower due to a E3S Web of Conferences 234, 00003 (2021) https://doi.org/10.1051/e3sconf/202123400003ICIES 2020 ICIES'2020 higher cosine factor and incidence angle effect, but it also has lower costs.
Auti [7] manufactured a solar thermal system consisting of a manual tracking parabolic concentrator, a black receiver and the water vapor passes to the main receiver which is a finned system used for condensation and distilled water collects into a container.
Arunkumar et al. [8] compered a compound parabolic concentrator-tubular solar still(CPC-TSS), a compound conical concentrator (CCC) and a singleslope solar still.The daily yield rate is measured per square meter of CPC-TSS, CCC solar still and single slope solar still absorber area are 6,100,18,000 and 2,100, respectively.The CCC solar still has been found to produce the maximum yield.
Bouzaid et al. [9] analyzed both experimentally and theoretically the thermal performances and productivity of the stepped solar still new pattern.The results show that there is a good agreement between the experimental measures and the calculation which confirms the mathematical model.The new cascade solar still daily production can achieve important values that can exceed up to 7 kg/m2 when compared with other solar still results.
Al-harahsheh et al. [10] performed the water desalination experimental study using a solar still has phase change material (PCM) and linked to a solar collector.They also studied the impact on the fresh water amount generated by the cooling water flow rate, hot water circulation flow rate and basin water level.The results show that the desalinated water output rate was proportional to the increase in ambient temperature and the flow rate of hot water circulation.During the nighttime period the selected PCM operated well to provide energy without any change in thermal behavior.In addition, it was noticed that rising the water level in the basin leads to decrease the productivity.
Lachhab et al. [11] manufactured a solar cogeneration system that consists of a concentrated solar system (CSP) and photovoltaic (PV).They conducted the theoretical and experimental study of heat transfer on system in the (Rabat-sale-kenitra) region.They have developed a calculation code.The findings of the comparison between the experimental data and the numerical solution relating to the different thermal collect temperature are recommended.
The parabolic tough concentrators are the most generally utilized linear concentrator for solar thermodynamic conversion, practically in domestic and industrial areas where operating temperatures between 80 °C and 160 °C are needed.The parabolic concentrator is the most encourage technologies to replace nonrenewable energies (nuclear energy and fossil fuels), practically in the industrial sector, like hybrid systems power plants, desalination, air condition, irrigation, or refrigeration [12][13][14][15] .

New design of solar concentration description
In the present study, a new design for a solar parabolic concentrator used for water desalination is considered.It consists of two parts that are described below.

The parabolic concentrator assembly
The parabolic trough concentrator has a half-cylinder basin absorber of solar distiller made of copper with an effective selective layer installed in the parabola's focal line.The limited surface that has a high solar radiation absorption coefficient.
The reflector is designed to operate the focal line (f) is 1 m away the vertex (V), the device aperture (W) is 2.25 m.The equation of the designed system will be: The designed dimension, aperture distance is 3 m and the maximum height at the end of the parabola is 0.47 m.
The angle of rim φ defined in terms of the ratio of the width of aperture to the focal length and is such as Where f and W the focal length and the aperture width of the system, respectively.The rim angle of the system is 58.72 o .
The reflector assembly consists of two parts.A parabolic base made of several pieces of steel flat bar must be supported by steel hollow bar (tube).The other part, the reflector consists of several segments or pieces of mirrors.

The half-cylinder basin solar distiller
The part of the half-cylinder of diameter 0.3 m, designed separately and connected to the basin, constitutes of the solar distiller's bottom part of thickness 0.004 m.
The entire volume of half-cylinder absorber servers as storage of water.
The top solar distiller surface is a glass cover of (0.36 m x 3 m) area, 0.3 m height and 0.002 m thickness.
The top cover places a uniform resting double slope of 30° on the grooves.The segments of fresh-water collection have a length of 3 m and a width of 0.030 m.
The center line of half-cylinder basin mounts at the parabolic concentrator's focal line.
The system concentration ratio is the ratio of the aperture area of the parabolic trough concentrator to the half-cylinder receiver area.The system concentration ratio is 4.774.
A diagram view of the new design of solar concentration is shown in Figure 1.-In half-cylinder basin (absorber surface) which contains brackish water inside the basin through a glass cover.
-In the parabolic concentrator where the radiation will be reflected at the focal line where the absorber surface and solar radiation concentration in the absorbent surface leads to heat the brackish water and rises the temperature.
The brackish water evaporates once heated and the water vapor rises until it touches the inner glass cover surface.Afterwards, the water vapor condenses on the glass cover from the internal surface and becomes fresh water stored and turns into fresh water collected in the segments located on each side of the system.

Mathematical Modeling of Heat Transfer
The modeling principle is based on an energy balance for the essential elements of the device will be applied to three places: brackish water, glass cover and halfcylinder basin.
Figure 2 shows transversal view of the parabolic concentrator with solar distiller with the heat transfer paths throughout the distiller containing three components (glass cover, brackish water, half -cylinder basin).
The equations of the energy balance are used for determining the temperatures of the elements of the device.
In these equations Tg, Tw and Tb denote the glass temperature, the brackish water temperature and the halfcylinder basin temperature, respectively.
The equations of the energy balance are written based on the following assumptions: 1-The parabola form is symmetrical.2-The absorber shadow effect on the parabola is negligible.3-The physical properties are constant for different materials.4-The heat capacity of the sides and bottom walls is neglected.
5-Constant temperature gradient with water depth and glass cover thickness.6-Dry air and water vapor are known to be ideal gases.7-No vapor leakage occurs inside the device.8-Condensation occurs only the internal glasscover surface.9-Water level is constant in the half-cylinder basin.10-Losses from the segments and walls sides are neglected.

The energy balance for the glass cover
The energy balance equations of the glass cover are the following: (3) Where Qin is the energy entering and gained by the glass.
Where Qout is the energy that the glass losses.
Where Qe,w-g, Qc,w-g and Qr,w-g are the heat transfer fluxes from the brackish water and the glass cover through evaporation, convection and radiation, respectively.
Qc,g-a represents the heat transfer flux from the glass cover and the ambient air through convection.
Qr,g-sky represents the heat transfer flux from the glass cover to the sky through radiation.αgAgI(t) is portion of the solar incident radiation is absorbed by the glass cover.
The heat transfer flux between the brackish water and the glass cover caused by the convection is calculated as follows: (6)

ICIES'2020
Where hc,w-g is the heat transfer coefficient by convection.It is given by [9,[16][17][18][19] (7) Where Pg and Pw indicate the pressures of saturated water vapor at the temperatures of the glass cover and water respectively, which are given as follows: ( 8) The heat transfer flux between the brackish water and the glass cover caused by the evaporation is calculated as follows: (10) Where he,w-g is the heat transfer coefficient caused by the evaporation.It is given by [9,[16][17][18][19] and using the following expression: The radiative heat transfer flux between the glass cover and water is calculated by the following expression: (12) Where hr,w-g is the radiative heat transfer coefficient of water to glass cover using Stefan -Boltzmann law and from [20].(13) Where σ is the constant of Stefan Boltzmann σ =5.67x10 -8 (w/m 2 .k).Where ɛeff is the effective diffusivity given by: (14) Where ɛw=0.96 and ɛg=0.88 are the water and the glass emissivity, respectively.
After, replacing formulas ( 7) in ( 6), ( 11) in ( 10) and ( 14) and ( 13) in (12) and substituting in equation ( 4), the equation becomes as follows: (15) The convective heat transfer between the glass cover and the ambient air.(16) Where hc,g-a is the convection heat transfer coefficient can be determined by the suggested equation by Johnson et al. [21]: (17) Where Vwd is the wind speed average in meter per second.
The radiative heat losses from the glass cover and the ambient air can be expressed as follows: (18) The radiation heat transfer coefficient (hr,g-sky( is given by: (19) Where the temperature of sky (Tsky) given by [20][21][22] (20) After, replacing formulas ( 17) in ( 16) and ( 19) in (18) and by substituting in equation ( 5), the equation becomes as follows: (21) The heat transfer equation at the glass cover is more complicated.The glass cover temperature depends on a large number of variables and different parameters.A simplified expression, using ( 15) and ( 21) by substituting in equation ( 3), could be obtained as follows: ICIES'2020 (22) Dividing equation ( 22) by the quantity (CpgMg) leads to the differential equation in the following form:

The energy balance for the brackish water
The energy balance equations of the brackish water are the following: α wτgAwI(t) is fraction of the solar incident radiation is absorbed by the brackish water.
Where Qc,b-w is the energy loss through convection, from the half-cylinder basin and the brackish water.

(27)
Where hc,b-w is the convection heat transfer coefficient that can be determined using the correlation of Nusslet Number for horizontal cylinder [23]: Where Gr, Pr and Nu are the Grashof, Prandtl and Nusselt numbers given by: The values of the constants for the Nusslet number equation can be estimated from the Table 1.
Where Kw and Lc are the water heat conductivity and the characteristic length of the absorber.After, replacing formula (28) in (27) and by substituting in equation ( 25), the equation becomes as follows: (32) Replacing formulas ( 7) in ( 6), ( 11) in ( 10) and ( 14) and ( 13) in (12) and by substituting in equation (26) gives the expression of the heat losses as follows: (33) Equations ( 32) and (33) are gathered in equation (24) to obtain the energy balance in the brackish water: ICIES'2020

Numerical resolution
The system of the three ODE equation ( 23), ( 35) and (42) is solved using a simulation code based on the 4th order Runge-Kutta method under MATLAB® to find Tg, Tw and Tb.Through the MATLAB code, the productivity of water and the system efficiency were determined by using equations ( 43), (46).The design and physical operating parameters used in theoretical model are illustrated in Table 2.It is first necessary to define the heat without solar irradiation conditions.In our case, Morocco's Rabat-Sale-Kenitra that has a high interesting geographical position is considered.The solar radiation data were obtained from measurements of the European satellite PVGIS [25][26][27] with metrological parameters on the zone of the location 34°01'53.34''North, 6°50'39.72''West, and elevation is 139 m.The optimal inclination angle is 31 degrees [11].
It is observed from Figure 3 that the incident solar radiation rises continuously from 7:22 in the morning and reaches a maximum amount of 860 W/m 2 at 12.22 and after that it starts to decrement with time until 17.52.
Additionally, the variations of ambient temperature and the wind speed with time are taken into account.
The maximal wind speed of 4.85 m/s is recorded at 15.22 to 15.52 and the maximal ambient temperature of 33 °C is recorded at 12.22.

Results and discussion
The theoretical study of new design of a concentration based solar distiller was carried out to evaluate the performance and the developed design.As for the figures shown below, all the parameters in the graphs show the same tendency.They gradually increase from morning and reache the maximum values at t = 12:22 and then decrease until sunset.The present paper proposes the study of a solar desalination device that consists of a concentrated solar system by parabolic trough concentrator and a halfcylinder basin.

The glass cover, brackish water and halfcylinder basin temperatures
The goal of the new design of this system is to obtain high throughput of distilled water.
In this work, we focused on the theoretical heat transfer study on the solar distiller.
The study was based on the energy balance of the components (the glass cover-the brackish water-the absorber).
A group of differential equations is used to determine different elements temperatures.The system of the obtained energy balance equations was solved using a computer simulation code based on the 4th order Runge-Kutta method with programming under MATLAB® .This study was carried out according with the meteorology of the region of the (Rabat-sale-kenitra) of Morocco for test.
The obtained results show that the solar distiller performance and efficiency with the solar concentrator is better than the solar distiller without the solar concentrator.As well, these results encourage the development and improvement our research to increase the heat rate based on the solar collection system.This research through good results is very useful for desalination by obtaining pure distilled water suitable for drinking.And that the device that will be manufactured can be used in areas that suffer from a shortage of drinking water.And also the use of a desalination system via solar concentrators does not affect the environment because the energy source for it is the sun and it does not emit any pollution, noise, etc.
All these conclusions encourage us to develop and improve our research in desalination of non-drinking water via solar energy.

ICIES' 2020 Fig. 1 .
Fig. 1.A diagram view of new design of device

Table 1 .
Estimation of constants a and n for the Nusslet number equation[23]

Fig. 3 .
Fig. 3.Estimated by PVGIS of the daily solar radiation, wind speed and ambient temperature for region of Rabat in Morocco.

Figure 4
Figure4illustrates the evolution of the glass cover, brackish water and half-cylinder basin temperatures for the solar concentration distiller.All the parameters show the same tendency, they begin to increase until the achievement of peaks assessed at 12.22.The maximum value of the glass cover, brackish water and half-cylinder basin temperatures is 83.77°C, 93.09 °C and 107.4 °C, respectively.After this instant, they start to fell down until the sunset.

Table 2 .
Physical properties used in theoretical calculation