Desalination 354 (2014) 1–8

Contents lists available at ScienceDirect

Desalination

j ourna l homepage: www.e lsev ie r .com/ locate /desa l

Modeling an active solar still for sea water desalinationprocess optimization

Osman Ali Hamadou, Khamlichi Abdellatif ⁎Department of Physics, Faculty of Sciences at Tetouan, Tetouan 93030, Morocco

H I G H L I G H T S

• Comprehensive numerical modeling of an active solar still was performed.• All the design parameters with those of the heat transfer fluid were included in the model.• Validation of the modeling was assessed by comparison to an exact 2-D model of a humid-air cavity.• Adequacy of the correlations used for the heat transfer coefficients was verified.• Extensive parametric studies enabled to determine the relative influence of the key factors.

⁎ Corresponding author. Tel.: +212 600769960.E-mail address: khamlichi7@yahoo.es (K. Abdellatif).

http://dx.doi.org/10.1016/j.desal.2014.09.0190011-9164/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 3 August 2014Received in revised form 12 September 2014Accepted 13 September 2014Available online xxxx

Keywords:Active solar stillDesalinationSolar energyHeat transferOptimization

Active solar stills can provide enhanceddistillate productivity, as salinewater is circulated andput in contactwithan additional heat source which supplies extra heat to the system. In this work, a single slope solar still having atransparent glass cover is considered to be heated at its bottom by a circulating heat transfer fluid. The rate of thisflow and the temperature at the inlet of the still are assumed to be controlled such that they are adjusted to thedesired values. A modeling based on relevant correlations giving the heat transfer coefficients and the vaporiza-tion heatflux as function of Rayleigh numberwas derived. This takes the formof a set of highly coupled nonlinearordinary differential equations in terms of time-dependent temperatures of the still components. The obtainedmodel enabled to take into account the effects of heat transfer fluid rate, inlet temperature, sea water rate,basin depth, ambient temperature, wind speed and relative humidity of ambient air. Extensive parametricstudies were performed and optimization of the rate and yield of distilled water was discussed.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Water is scarce inmany areas across theworld. As demand onwateris increasing, supply of water constitutes a critical problem to deal withby using innovative concepts.

Literature review indicates that approximately 99% of availablewater on Earth is salty, brackish or frozen and that only 1% is freshand set to consumption as potable water. Sea water desalination canhelp addressing the water-shortage problem. The separation of saltsfrom sea water requires however a large amount of energy. Therefore,it is beneficial to employ renewable energy sources for desalination.Solar energywhich is in abundance, particularly in dry areas, can be uti-lized for this purpose in order to produce potable water with reducedenergy cost [1].

Solar thermal desalination comprises direct and indirect processes[2]. Direct processes are such that all parts are integrated into one

system whereas indirect processes are those for which heat is comingfrom a separate solar collecting system such as solar collectors or solarponds. The most common solar thermal arrangements that are used inpractice for desalination are solar stills and solar ponds. Solar stillswere recognized to be more efficient for small production scale wherethe freshwater demand does not exceed 100 m3 per day [3].

Natural convection takes place in solar stills because of the buoyantforce caused bydensity variation due to the temperature and concentra-tion gradients. Various numerical approaches were considered in orderto study evaporation and condensation in a humid air-filled cavity,which constitutes the physical representation of solar stills.

The first family of approaches use coupled heat and mass transferequations for natural convection taking place in the cavity with wallsurface condensation and evaporation. Talukdar et al. [4] have per-formed 3-D computational fluid dynamics simulations for convectiveheat and mass transfer between water surface and humid air in a hori-zontal rectangular duct. They have concluded that the numerical resultscompare well for most of the observed experimental data. Sun et al. [5]have analyzed the case where the cavity walls are subjected to uniform

swd

L

Distillate output

Heat transfer fluid

Sea water level

Insulation layer

Hl

rH

gr

Brine Sea water Copper heating plate

Fig. 1. Schematic diagram of an active single-slope solar still.

2 O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

time-dependent temperatures. The authors noted that various transientflow structures occur during condensation and evaporation processes.The thickness distributions of the water films condensed at the wallwere discussed as function of the aspect ratio. By using a control volumebased finite element method, Chouikh et al. [6] have analyzed the two-dimensional natural convection flow resulting from the combinedbuoyancy effects of thermal and mass diffusion in an inclined cavity.They showed that the performance of the solar distiller can be enhancedwhile a single cell develops and rotates in a sense that allows enoughtime for the vapor to cool down.

Rahbar and Esfahani [7] have investigated the natural convectioneffect in a 2-D single slope solar still. A numerical model basedon SIMPLEC (Semi Implicit Method for Pressure Linked EquationsCorrected) was used for the solution of mass, momentum, energy andconcentration equations. A new correlation for estimating the convec-tive heat transfer coefficient has been derived as function of the aspectratio and Rayleigh number. Alvarado-Juárez et al. [8] have performeda numerical study of conjugate heat and mass transfer in a solar stilldevice having the form of an inclined cavity. Double-diffusive naturalconvection and surface thermal radiation were considered through asteady state 2-D approximation of the equations which were solvedby the finite volume method. The authors noted strong effect on theNusselt number which results from the surface thermal radiation as itincreases the velocity near the walls.

The previousmodels based on rigorousmathematical formulation ofthe problem provide thorough understanding of the complex physicalphenomena occurring in the humid air filled cavity. These modelsshow for instance the existence ofmultiple cells depending on the inter-vening parameters. They can thus be used to enhance the design of solarstills in terms of aspect ratio, inclination angle and Rayleigh number [8].Their use in practice is effective however only within the framework ofsteady-state approximation to the problem where time variations areneglected. The possibility of employing them for transient regimes in astep-by-step analysis type is highly time consuming. Besides, this re-quires using the same precision level models for the heat transferfluid, ambient air and heat transfer occurring in the absorption plateand glass cover domains. Even with a 2-D approximation of the prob-lem, huge numerical difficulties arise then.

Table 1Thermo-physical properties of the system.

Density(kg m−3)

Thermal conductivity(W m−1 K−1)

Heat capacity(J kg−1 K−1)

Volume(m3)

Sear water 1027 0.6038 3976 VariableGlass 2500 1.2 720 0.025Copper 8933 420 385 0.005Heat transfer fluid 768.6 0.1663 2515.6 0.5

In the second family of approaches to modeling of solar stills, use ismade directly of proper correlations between the Nusselt number andRayleigh number. This enables to obtain the convective heat transfer co-efficient as a function of fluid properties, still geometry, flow character-istics and operating temperature. Various experimental models havebeen proposed to evaluate the internal heat transfer coefficients.These include for instance Jacob's model [9], Dunkle's model [10],Corcione's model [11] and Chilton–Colburn model [12]. Enhanced cor-relations giving the evaporation/condensation heat transfer coefficientcan also be obtained by means of the first family of approaches [11].Conservation equations ofmass,momentum, energy, and concentrationcan be solved enabling to perform extensive parametric simulations.Using the simulation data and the least square method through a loga-rithmic multiple regression procedure, Rahbar and Esfahani [7] havefitted a dimensionless equation that gives the Nusselt number as func-tion of the aspect ratio and Rayleigh number. The authors have shownthat in the investigated range of parameters, the obtained results areclose to those given by the Chilton–Colburn model or by Dunkle'smodel.

Using such correlations for the heat transfer coefficient enables ef-fective computation of the temperatures during the transient regimewhile avoiding solution of bulky coupled time-dependent nonlinearpartial differential equations.

Considering passive solar stills which use directly solar radiation toproduce distillate water, various parametric studies and optimizationmethods have been considered by using the heat transfer correlationbased modeling [13–16]. The objective is to increase the performanceof solar stills. But, these systems have yet low productivity which ex-plains their rather limited commercial utility. Currently availablestate-of-the-art single-effect solar stills have an efficiency not exceeding40% [3]. The main reason is the heat loss which is quite large for theseinstallations [17–19]. As a main step toward improving solar still pro-ductivity, optimizing the system input and geometric configuration isof crucial importance. Design problems encountered with passivesolar stills dealwith brine depth, vapor tightness of the enclosure, distil-late leakage,materials for thermal insulation, and cover slope and shape[20]. One can add to this list of design parameters, the aspect ratio andthe work Rayleigh number in the solar still cavity [7,8].

Productivity in terms of distillate yield can be increased by pro-viding extra supply of heat to the sea water through an exchangewith a heat transfer fluid heated previously in a solar collector sys-tem. It is then of interest modeling and optimizing the active solarstill having this configuration. This is really important in practice inorder to achieve control of the heat transfer fluid rate and the inputtemperature as in practice the ambient temperature and solar inten-sity are continuously changing.

In this work, modeling of an active solar still with a circulating heattransfer fluid that supplies heat to the still bottom is performed. Threecorrelations giving explicitly the intervening heat transfer coefficients

0 5000 10000 15000290

300

310

320

330

340

350

360

Time (s)

Tem

pera

ture

(K

)

TgiTswTcTh

Fig. 2. Time variation of temperatures for the different still components.

3O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

are used. The energy balance equations for the different components ofthe active solar still are derived. A set of coupled nonlinear ordinary dif-ferential equations is obtained as a function of the temperatures in eachstill component, aswell asmaterial and geometry parameters of the stilland the heat transfer fluid. This system is solved numerically by using aprogram developed under Matlab software package. Calculations arethen carried out to assess the instantaneous productivity and the yieldof the still as a function of the saline water rate, the heat transfer fluidrate, still basin depth, ambient temperature, wind speed and relativehumidity. Optimization of this system design is finally discussed.

2. Modeling of the system

The solar still is a basin enclosing saline water. It is covered by atransparent envelop constitutedwith glass or plastic material. The prin-ciple of operation of these devices is evaporation and condensation. The

Table 2The stationary distilled water ratem

�

ve;∞ and the yield Γ as function of the considered com-bination of factors levels.

With solar radiation Without solarradiation

Dh (m3/h) Dsw (m3/h) dsw (m) m�

ev;∞ (kg/h) Γ (kg) m�

ev;∞ (kg/h) Γ (kg)

1.4 0.01 0.02 8.22 192 7.43 1771.4 0.01 0.04 8.22 155 7.43 1431.4 0.01 0.06 8.21 131 7.5 1211.4 0.015 0.02 7.92 186 7.19 1721.4 0.015 0.04 7.92 151 7.19 1391.4 0.015 0.06 7.92 127 7.27 1181.4 0.02 0.02 7.65 182 6.96 1681.4 0.02 0.04 7.65 147 6.98 1361.4 0.02 0.06 7.65 124 6.96 1152 0.01 0.02 8.27 197 7.61 1842 0.01 0.04 8.27 159 7.65 1492 0.01 0.06 8.26 143 7.61 1272 0.015 0.02 8.0 192 7.39 1802 0.015 0.04 8.04 155 7.41 1462 0.015 0.06 8.0 140 7.39 1242 0.02 0.02 7.75 187 7.18 1762 0.02 0.04 7.79 152 7.19 1432 0.02 0.06 7.75 129 7.18 1222.8 0.01 0.02 8.31 210 7.73 1902.8 0.01 0.04 8.31 170 7.74 1532.8 0.01 0.06 8.3 147 7.73 1392.8 0.015 0.02 8.05 197 7.53 1862.8 0.015 0.04 8.06 166 7.73 1502.8 0.015 0.06 8.05 143 7.53 1362.8 0.02 0.02 7.82 192 7.33 1822.8 0.02 0.04 7.84 163 7.33 1472.8 0.02 0.06 7.82 140 7.33 133

solar still cover traps solar energy within the enclosure, which heatsthen the saline water. This causes evaporation and condensation onthe inner surface of the sloped transparent cover. The resultant con-densedwater on the underside of the roof cover runs down into troughsin which the purified distillate is collected.

A simple design of an active solar still is shown in Fig. 1. The still hasthe formof a basinwith a certain depth of salinewater and a glass trans-parent cover that let solar radiation to enter while blocking the longwavelengths radiation emitted by the interior surfaces of the still. Thetransparent sloped cover provides a cool surface for condensation ofwater vapor andmakes easy flow of thewater droplets into the conden-sate trough. The basin liner of the still is blackened on the interior sur-face to maximize absorption of solar radiation. It contains also anembedded copper plate which enables fast heating of the sea water bymeans of a heat transfer fluid that is circulating beneath it. The bottomand the lateral sides of the still are insulated on the exterior surface tominimize heat losses. The aspect ratio of this single-slope solar still isdefined as a = (Hℓ + Hr)/(2L). The width of the still is linked to thearea of saline water surface by the equation w = Asw/L.

Basic assumptions in the modeling of the desalination system takeaccount of negligible temperature stratification within the evaporatorbasin. Temperatures are supposed to be uniform within each still com-ponent, while they are time-dependent. The governing equations ex-press conservation of saline water mass and that of the contained saltsas well as energy balance of the system which is a consequence of thefirst law of thermodynamics. Only pure water contained in salinewater is assumed to be evaporated and the still is considered to befree from any vapor leakage.

The masse balance writes [21]

m�

sw ¼ m�

ev þm�

b ð1ÞDenoting xsw the concentration of salt in the feed flow and xb that in

the brine, salt balance writes [21]

m�

swxsw ¼ m�

bxb ð2Þ

Themaximum salt content in the brine is determined by the solubil-ity of the salts. This is very important in practice in order to avoid saltprecipitation leading to problems of encrustation and blockage. Theconcentration factor η is defined as the ratio of brine concentration tofeed concentration

η ¼ xbxsw

ð3Þ

This parameter is usually used to fix a limit threshold to not exceedduring the evaporation process. Using Eqs. (2) and (3), Eq. (1) simplifiesto

m�

b ¼ 1ηm�

sw ð4Þ

m�

ev ¼η−1η

m�

sw ð5Þ

Eq. (5) defines the yield as it gives, for the stationary conditions, therate of distillate as a function of the feed rate. The distillate yield or re-covery rate is then defined as

φ ¼ η−1η

ð6Þ

As an example, for Mediterranean sea xsw = 35 mg l−1 and xb =60 mg l−1 [21], so η = 1.7 and φ = 0.4. This means that only 40% of

0 5000 10000 150000

1

2

3

4

5

6

7

8

9

Time (s)

Dis

tille

d w

ater

rat

e (k

g/h)

Fig. 3. Time variation of distilled water rate.

4 O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

saline water can be transformed to distillate water in order to avoidscaling.

Using Eq. (1), the energy balance yields the following set ofequations that hold for each system component. They are associatedrespectively to outer glass, inner glass, sea water, copper plate andheat transfer fluid.

ρgVgcp;g2Ag

dTge

dt¼ λg

egTgi−Tge

� �−hcge am Tge−Tamb

� �−hrge sky Tge−Tsky

� �ð7Þ

ρgVgcp;g2Ag

dTgi

dt¼ α0

gI tð Þ þ hcsw gi þ hrsw gi

� �Tsw−Tgi

� �

þm�

evLvAg

−λg

egTgi−Tge

� �ð8Þ

ρcVccp;cAc

dTc

dt¼ α0

cI tð Þ þ hch c Th−Tcð Þ−hcc sw Tc−Tswð Þ ð9Þ

ρhVhcp;hAh

dTh

dt¼ 2ρhDhcp; f

AhTh;in−Th

� �−hch c Th−Tcð Þ−Uloss Th−Tambð Þ

ð10Þ

ρswVswcp;swAsw

dTsw

dt¼ α0

swI tð Þ þ hcc sw Tc−Tswð Þ þ ρswDsw

Asw

cp;swTsw;in−cp;bTb;out

� �−m

�

ev

AswhL−cp;bTb;out

� �− hcsw gi þ hrsw gi

� �Tsw−Tgi

� � ð11Þ

Table 3Quadratic polynomials fitting the results given in Table 2.

Equation of the correlation

With solar radiation Γ ¼ 1:14þ 0:1731Dh−0:1315Ds

þ0:2381 Dswdsw−0:01738 D2h þ

m�

ev;∞ ¼ 1:0338þ 0:08294Dh−0þ1:707� 10−4Dhdsw þ 0:01805

Without solar radiation Γ ¼ 1:1834þ 0:1988Dh−0:1298þ0:3665Dhdsw þ 0:2105Dswdsw−m�

ev;∞ ¼ 0:9024þ 0:2036Dh−0:0−0:09165Dhdsw−0:4522Dswdsw

The heat transfer coefficient between the exterior glass and ambienttemperature depends on the wind velocity; taking Mc Adams correla-tion [22] this coefficient is approximated by

hcge amb ¼5:621þ 1151:2v

Tambif vb4:88 m s−1

604:29v

Tamb

� �0:78if 4:88≤vb30:48 m s−1

8>><>>: ð12Þ

The radiation heat transfer coefficient between the exterior coverglass and the sky is of the form

hrge sky ¼ εvσ T2ge þ T2

sky

� �Tge þ Tsky

� �ð13Þ

with the sky temperature given by [23]

Tsky ¼ Tamb 0:74þ 0:006 θð Þ0:25 ð14Þ

in which θ is the dew point temperature given by [24]

θ ¼ 237:3

17:27−ln ξþ 17:27Tamb−4061Tamb−35:85

ln ξþ 17:27Tamb−4061Tamb−35:85

� �ð15Þ

where ξ is the relative humidity.To express the heat transfer coefficient between salinewater and the

glass cover bymeans of an explicit correlation, Dunkle'smodel is used inthe following. It should be noted that two versions of this model are en-countered in the literature. The first form of Dunkle'smodel as analyzedby Tsilingiris [12] gives large overestimation of the heat transfer coeffi-cient. Whereas, the refined second form gives under some conditionsresults that are close enough to those provided by Chilton–Colburnmodel [12], which are on their turn very close to those of a complete2-D modeling for steady-state conditions [7].

Using the second form of Dunkle's model, the heat transfer coeffi-cient between saline water and the glass cover writes [10]

hcsw gi ¼ 0:884 Tsw−Tgi þpsw−pgi� �

Tsw

2:689� 105−psw

0@

1A

13

ð16Þ

with

p� ¼ exp 25:317−5144T �

� �ð17Þ

where (·) stands either for (sw) or (gi).The radiation heat transfer coefficient between the saline water and

the cover glass is

hrge sky ¼ εeffσ T2sw þ T2

gi

� �Tsw þ Tgi

� �ð18Þ

R2 (%)

w−4:3dsw−0:04264DhDsw þ 0:1919Dhdsw0:0255 D2

sw þ 7:322d2sw

99.4

:1868Dsw þ 0:0869dsw þ 0:04052DhDsw

Dswdsw−0:0539D2h þ 0:02011D2

sw−0:3895d2sw

99.8

Dsw−4:955dsw þ 0:01261DhDsw

0:07232D2h þ 0:014D2

sw þ 9:08d2sw

99.7

7273Dsw þ 0:3115dsw þ 0:04253DhDsw

−0:0901D2h−0:04479D2

sw−0:7267d2sw

97.6

5O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

with

εeff ¼1εsw

þ 1εg

−1

!−1

ð19Þ

The vaporization heat flux appearing in the second half of Eq. (8) isgiven by [10]

m�

evhL ¼ hevAsw Tsw−Tgi

� �ð20Þ

in which the latent evaporation heat is given by [10]

hL ¼ 3146−2:36Tsw ð21Þ

and the heat transfer coefficient by evaporation is given in [10] underthe following form

hev ¼ 0:016273 hcvsw gipsw−pgiTsw−Tgi

ð22Þ

with hcvsw _ gi evaluated according to Eqs. (16) and (17).It should be noticed that for the evaporative heat transfer coefficient

to be accurately estimated by the correlations defined by Eqs. (16) and(22), the following conditions should be satisfied [7]: the aspect ratio2.5 ≤ a ≤ 5.5, the inclination angle 10° ≤ ψ ≤ 30° and the Rayleighnumber 5 × 106 ≤ Ra ≤ 5 × 107. If these conditions are not fulfilledthen evaluation of the evaporative heat transfer coefficient could bedone either experimentally of by using a refined 2-D modeling of theproblem such as those considered in [7,8,11].

Flow velocities of the heat transfer fluid and seawater are always low;thus quasi-steady convective heat transfer can be assumed to take place,on one side, between the basin liner copper plate and the heat transferfluid and, on the other side, between this plate and sea water.

Dh: 6%Dsw: 1%

dsw: 92%

Other: < 1%

Without solar radiation

Yield

Dh:6%Dsw: 3%

dsw: 91%

Other: < 1%W

ith solar radiation

Fig. 4. Influence of factors on the yield dist

To express the associated heat transfer coefficients, the Nusseltnumber is assumed to be obtained through a general correlation ofthe form [25]

Nu ¼ c Rað Þn ¼ c Gr Prð Þn ð23Þ

where Ra is the Rayleigh number, Gr the Grashof number and Pr thePrandtl number, and c and n are constants which depend on the rangeof Gr number. The Grashof number is given by

Gr ¼ βgρ2L3ΔTμ2 ð24Þ

and the Prandtl number is

Pr ¼ μcpλ

ð25Þ

For a horizontal heating plate, the correlation giving the Nusseltnumber writes [25]

Nu ¼1 if Grb105

0:5 Rað Þ0:25 if 105bGrb2� 107

0:15 Rað Þ0:33 if GrN2� 107

8><>: ð26Þ

Knowing the Nusselt number, the heat transfer coefficient betweenthe basin liner copper plate and sea water can be calculated as

hcvc sw ¼ Nuc swλsw

Lð27Þ

Dh: 36%

Dsw: 61%

Other: < 1%dsw: 3%

Stationary distilled water rate

Dh: 6%

Dsw: 93%

dsw: < 1%Other: < 1%

illate and the stationary distillate rate.

23

45

6

0.40.6

0.81

140

160

180

200

220

Wind speed (m/s)Relative humidity

Yie

ld (

kg)

23

45

6

0.40.6

0.816

7

8

9

Wind speed (m/s)Relative humidity

Sta

tiona

ry d

istil

led

wat

er ra

te (k

g/h)

Fig. 5. Effects of wind speed and relative humidity on distillate yield and stationary distilled water rate.

6 O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

while that between the heat transfer fluid and the copper plate is

hcvh c ¼Nuh cλh

Lð28Þ

The heat loss coefficient can be approximated by the following for-mula

Uloss ¼λis

eisð29Þ

As the basin liner has a large length, the temperature Tb,out can beassumed to be equal to that of the copper plate Tc.

3. Results and discussion

The mathematical model as resulting from Eqs. (7) to (28) is time-dependent. The solution of the temperature Eqs. (7) to (11) is calculatedby means of a Matlab program that we have developed based on thecommand ode15s. This command is very fast as compared to the com-mand ode45 which is slow for our system with a mass matrix.

Under constant environmental conditions, the system goes to asteady state within a finite time duration of the order of 1 h. To beable to assess more visibly the influence of the key factors interveningin the problem, constant environmental conditions are imposed. Thesimulation is performed with constant solar radiationI(t) = I0 =700 W m−2, constant wind velocity v, constant relative humidityξ and constant ambient temperature fixed at the value Tamb =298.15 K. Also, the input heat transfer fluid temperature is fixed in thefollowing at the value Tsw,in = 258.15 K.

Knowing the temperatures, Eq. (21) gives then the mass rate of theevaporated water as

m�

ev ¼hevAsw Tsw−Tgi

� �hL

ð30Þ

The yield is defined as the amount of produced distillatewater over afixed period of time. Taking the time interval [0, tc], the distillate yield iscalculated by

Γ ¼Z tc

0m�

ev tð Þdt ð31Þ

The distillate mass rate is a function of a lot of parameters. However,in the following focus is on variations of m

�

ve as function of the salinewater depth dsw = Vsw/Asw, the volume rates of saline water Dsw andheat transfer fluid Dh. Comparison between the two situations wheresolar radiation is present (case with solar radiation) and where it isabsent (case without solar radiation) is considered.

The thermo-physical properties of the system used in simulation aregiven in Table 1. The insulation layer material is assumed to have ther-mal conductivity λis = 0.04 W m−1 K−1 and thickness eis = 0.05 m.The length of the solar still is L = 1 m, and the heights are Hℓ =0.05 m and Hr = 0.627 m. The areas of heat exchange are fixed atAc = Ah = Asw= 5 m2. That of the glass cover is calculated for an incli-nation angle of 20°; Ag = 5.321 m2. The emissivities for both glass andsaline water are fixed at εg = εsw = 0.9.

The viscosities are chosen to be μh = 8.12 × 10−4 N s m−2 andμsw = 9.64 × 10−4 N s m−2. The coefficients of volumetric thermalexpansion are respectively βh=9.45 × 10−4K−1 and βsw= 2.4276 ×10−4K−1. The other constants are: g = 9.81 m s−2 and xsw =40 mg l−1.

From the geometry parameters one can evaluate the aspect ratio ofthe solar still: a = 2.953. By using the common humid air propertiessuch as those given in [26] one can calculate the Rayleigh number asso-ciated to heat transfer between sea water and the inner face of glasscover. The obtained value is Ra = 3.776 × 107. All the conditions arethen met in order to use the Dunkle's model correlations. The obtainedheat transfer coefficient evaluated by Eq. (16) was in fact within a mar-gin of 2 % when compared to that evaluated according to the 2-D exactmodeling considered in [7].

Under the previous conditions, a design of experiment table L27wasconsidered in order to assess the influence of the key factors dsw, Dsw

andDh on the distillate yield Γ and the stationary value of distilledwater rate production m

�

ve;∞.Fig. 2 gives, in the case with solar radiation with I(t) = I0 =

700 W m−2, the calculated temperatures as function of time for the fol-lowing set of parameters: v=5 m s−1 and ξ= 0.75. The factors valuesused for this simulation are: dsw = 0.02 m, Dsw = 0.01 m3/h andDh =1.4 m3/h, which corresponds to line 1 of Table 2 The total time of simu-lation is fixed at tc = 1.8 × 104s, which is found to be largely enough toget a stationary regime for distilled water production.

Fig. 3 shows the instantaneous value of distilled water production.The stationary regime is attained after only a period of 5000 s which isalmost 1 h and 23 min.

Table 2 summarizes the obtained results in terms of the consideredcombination of factors. Columns 4 and 5 give the results for the activestill where the effects of solar radiation and heat transfer fluid areadded (case with solar radiation). Columns 6 and 7 give the results inthe absence of any solar radiation intensity (case without solarradiation).

Using the results of Table 2, a quadratic polynomial regression can beperformed to give explicit representation of the system performanceoutputs as function of the considered factors over their associatedranges. Table 3 gives the obtained correlations. These achieved high ad-equacy as all the R2 values exceed 97%. In fact, even linear correlationscan be used for Γ and m

�

ev;∞ as the calculated R2 values exceeded 95%in all cases.

7O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

Analysis of variance was performed on the data given in Table 2.Fig. 4 gives the influence of factors on the results for the yield and thestationary distilled water rate, for the two cases with solar radiationand without solar radiation.

To analyze the effect of environmental conditions that influencecooling of the glass cover, a parametric study is considered in terms ofthe relative humidity ξ and wind speed v. Fig. 5 gives variations of theyield distillate and the stationary distillate water rate as function ofthese parameters.

Figs. 2 and 3 show that a stationary regime is reached after a timeduration of about 5000 s. When this regime is attained, temperaturesas well as the production rate of distilled water will remain constant ifthe exterior and environmental conditions are kept invariable. Produc-tion of distilled water will then be proportional to time. In reality solarradiation intensity, ambient temperature, wind speed and relativehumidity are constantly changing. The system will thus follow thesevariations with some delay as the response time to these perturbationsis not short. About 1 h of time will in fact be needed in order that thesystem reaches a new static state with all the dynamic effects vanishingafter being subject to someperturbation. This is why during simulationsthe environmental conditions were taken constant. The objective is toassess more visibly the influence of the key controlled factors: dsw, Dsw

and Dh on the active still performance.Fig. 4 gives the influence of these factors on the yield Γ and the sta-

tionary distilled water ratem�

ev;∞. In the investigated ranges of parame-ters, the water depth is found to be the main factor that controlsvariations of the yield. It is followed by the heat transfer fluid rate.Whereas for the output m

�

ev;∞ , the saline water rate is found to be themajor factor. It is followed by the heat transfer fluid rate.

Fig. 5 shows that wind speed has a major effect on both Γ andm�

ev;∞,while humidity appears to have a negligible influence on the results.Solar stills have to be preferably installed in windy sites or sufficientlyelevated from the ground surface in order to profit at a maximumfrom wind ventilation.

FromTable 2 one can see thatwhen increasing the heat transfer fluidrate, solar radiation has only small effect on the produced distilledwater. The effect of solar radiation is found here to be small becausethe amount of heat supplied by the heat transfer fluid was taken to belarge. Table 2 shows also that the relation between distilled water pro-duction and the heat transfer fluid rate is not linear as doubling this rateincreases the yield by only 9%.

The model presented in this work can be used to conduct an eco-nomic optimisation of the active solar still by taking into account thecost of energy associated to the heat transfer fluid. The optimum rateand input temperature of the heat transfer fluid can be determined si-multaneously with the still parameters dsw and Dsw.

It was assumed here that sea water desalination produces salt-freewater in one stage; this is known to be too energy consuming. Themultiple-effect nature should be considered in order to minimize theenergy requirement. The basic model introduced in this work can thenbe adapted to deal with optimization of a multiple-stage plant.

4. Conclusions

Modeling of an active solar still was performed in this work. Quasi-steady heat transfer assumption was used to express the complexenergy exchange taking place in the system. A set of time-dependentnonlinear ordinary differential equationswhich unknowns are the tem-peratures in the still components was obtained. Closure of the problemwas performed by using standard effective correlations to express theintervening heat transfer coefficients. The obtained explicit equationswere then solved by means of a program that has been developedunder Matlab environment. Extensive parametric studies wereperformed after that. They showed that, in the investigated ranges ofparameters, the water depth controls predominantly the yield of dis-tilled water. It is followed by the heat transfer fluid rate. Results of

simulation have shown also that wind speed has a major effect on dis-tilled water production and that it is more convenient to set the desali-nation installation in windy sites or at adequate elevation from theground surface.

It was found besides that the yield is not proportional to the heattransfer fluid rate and thus a cost optimisation has to be performed inorder to determine the optimal design parameters of the active solarstill. The model presented herein can be used usefully for this purpose.

NomenclatureAc Area of copper plate (m2)Ah Area of heat transfer fluid (m2)Ag Area of glass cover (m2)Asw Area of saline water (m2)a Aspect ratioc Constant in the Nusselt correlationcp,b Heat capacity of brine (J kg−1 K−1)cp,c Heat capacity of copper (J kg−1 K−1)cp,g Heat capacity of glass (J kg−1 K−1)cp,h Heat capacity of heat transfer fluid (J kg−1 K−1)cp,sw Heat capacity of saline water (J kg−1 K−1)Dh Rate of volume change for heat transfer fluid (m3 s−1)Dsw Rate of volume change for sea water (m3 s−1)dsw Saline water depth (m)eg Thickness of glass cover (m)eis Thickness of insulation material (m)I(t) Solar intensity (W m−2)I0 Constant solar intensity (W m−2)Hℓ Height of the left side of the still (m)Hr Height of the right side of the still (m)H Mean height of the still (m)hL Latent heat of vaporization (kJ kg−1)Gr Grashof numberg Gravity acceleration (m s−2)L Length of the basin liner in the direction of flow (m)m�

sw Mass rate of saline water (kg m−3)m�

ev Produced mass rate of vapor (kg m−3)m�

ev;∞ Limit stationary value of produced mass rate of vapor(kg m−3)

m�

b Mass rate of brine (kg m−3)Nu Nusselt numbern Exponent in the Nusselt correlationPr Prandtl number

pgi Partial pressure of thewater at the interior cover glass surface(Pa)

psw Partial pressure at the sea water temperature surface (Pa)Ra Rayleigh numberTamb Ambient temperature (K)Tb,out Output temperature of the brine (K)Tc Copper plate temperature (K)Th Temperature of heat transfer fluid (K)Th,in Inlet temperature of heat transfer fluid (K)Tge Temperature at the exterior of cover glass (K)Tgi Temperature at the interior of cover glass (K)Tsw Temperature of sea water (K)Tsw,in Inlet temperature of sea water (K)Tsky Sky temperature (K)t Time (s)tc Time duration for calculation of the yield (s)Uloss Loss factor per unit surface (W m−2 K−1)Vc Volume of copper plate (m3)Vh Volume of heat transfer rate (m3)Vg Volume of glass cover (m3)Vsw Volume of sea water (m3)v Wind speed (m s−1)w Width of the solar still (m)

8 O.A. Hamadou, K. Abdellatif / Desalination 354 (2014) 1–8

xsw Concentration of salts in feeding sea water (mg l−1)xb Concentration of salts in brine (mg l−1)αc′ Fraction of solar energy absorbed by the basin liner material,

dimensionlessαg′ Fraction of solar energy absorbed glass cover, dimensionlessαsw′ Fraction of solar energy absorbed by saline water,

dimensionlessβ Coefficient of volumetric thermal expansion (K−1)βh Coefficient of volumetric thermal expansion for the heat

transfer fluid (K−1)βsw Coefficient of volumetric thermal expansion for sea water

(K−1)εeff Effective emissivity, dimensionlessεg Emissivity of cover glass, dimensionlessεsw Emissivity of saline water, dimensionlessφ Feed recovery rateη Feed concentration factorΓ Yield (kg)λ Thermal conductivity (W m−1 K−1)λg Thermal conductivity of glass (W m−1 K−1)λis Thermal conductivity of insulation material (W m−1 K−1)λc Thermal conductivity of copper (W m−1 K−1)λh Thermal conductivity of heat transfer fluid (W m−1 K−1)λsw Thermal conductivity of saline water (W m−1 K−1)μ Viscosity (N s m−2)μh Viscosity of heat transfer fluid (N s m−2)μsw Viscosity of saline water (N s m−2)ρ Density (kg m−3)ρc Copper density (kg m−3)ρh Heat transfer fluid density (kg m−3)ρg Glass density (kg m−3)ρsw Saline water density (kg m−3)ψ Inclination angle (°)σ Stefan–Boltzmann's constant (5.6697 × 10−8 W m−2 K−4)θ Dew point temperature (K)ξ Relative humidityΔT Temperature difference in heat transfer by natural convection

(K)

References

[1] R. Dev, G.N. Tiwari, Characteristic equation of a passive solar still, Desalination 245(2009) 246–265.

[2] H.M. Qiblawey, F. Banat, Solar thermal desalination technologies, Desalination 220(2008) 633–644.

[3] L.G. Rodriguez, Seawater desalination driven by renewable energies: a review,Desalination 143 (2002) 103–113.

[4] P. Talukdar, C.R. Iskra, C.J. Simonson, Combined heat and mass transferfor laminar flow of moist air in a 3D rectangular duct: CFD simulation andvalidation with experimental data, Int. J. Heat Mass Transfer 51 (2008)3091–3102.

[5] H. Sun, G. Lauriat, X. Nicolas, Natural convection and wall condensation or evapora-tion in humid air-filled cavities subjected to wall temperature variations, Int. J.Therm. Sci. 50 (2011) 663–679.

[6] R. Chouikh, L.B. Snoussi, A. Guizani, Numerical study of heat andmass transfer in in-clined glazing cavity: application to a solar distillation cell, Renew. Energy 32 (2007)1511–1524.

[7] N. Rahbar, J.A. Esfahani, Estimation of convective heat transfer coefficient in asingle-slope solar still: a numerical study, Desalin. Water Treat. 50 (2012)387–396.

[8] R. Alvarado-Juárez, G. Álvarez, J. Xamán, I. Hernández-López, Numerical study ofconjugate heat and mass transfer in a solar still device, Desalination 325 (2013)84–94.

[9] Y.A. Cengel, Heat and Mass Transfer: A Practical Approach, third ed. McGraw-Hill,Singapore, 2006.

[10] R.V. Dunkle, Solar water distillation; the roof type still and a multiple effect diffusionstill, International developments in heat transfer ASME, Proceedings of internationalheat transfer part V, University of Colorado, 1961, p. 895.

[11] M. Corcione, Effects of the thermal boundary conditions at the sidewalls uponnatural convection in rectangular enclosures heated from below and cooled fromabove, Int. J. Therm. Sci. 42 (2003) 199–208.

[12] P.T. Tsilingiris, Modeling heat and mass transport phenomena at higher tempera-tures in solar distillation systems—the Chilton–Colburn analogy, Sol. Energy 84(2010) 308–317.

[13] H.E.S. Fath, Solar distillation: a promising alternative for water provision with freeenergy, simple technology and a clean environment, Desalination 116 (1998)45–56.

[14] S.K. Shukla, V.P.S. Sorayan, Thermal modeling of solar stills: an experimental valida-tion, Renew. Energy 30 (2005) 683–699.

[15] F. Agyenim, N. Hewitta, Ph. Eames, M. Smytha, A review of materials, heat transferand phase change problem formulation for latent heat thermal energy storagesystems (LHTESS), Renew. Sust. Energ. Rev. 14 (2010) 615–628.

[16] A. Ahsan, Kh.M. Shafiul Islam, T. Fukuhara, A.H. Ghazali, Experimental study onevaporation, condensation and production of a new tubular solar still, Desalination260 (2010) 172–179.

[17] B.A. Akash, M.S. Mohsen, O. Osta, Y. Elayan, Experimental evaluation of a single-basin solar still using different absorbing materials, Renew. Energy 14 (1998)307–310.

[18] A.S. Nafey, M. Abdelkader, A. Abdelmotalip, A.A.Mabrouk, Enhancement of solar stillproductivity using floating perforated black plate, Energy Convers. Manag. 43(2002) 937–946.

[19] S.K. Shukla, V.P.S. Sorayan, Thermal modeling of solar stills: an experimental valida-tion, Renew. Energy Fuel Energy Abstr. 30 (2005) 683–699.

[20] J.A. Eibling, S.G. Talbert, Solar stills for community use—digest of technology, Sol.Energy 13 (1971) 263–276.

[21] J. Kucera, Desalination; Water from Water, Scrivener Publishing LLC, Salem,Massachusetts, 2014.

[22] W.H. McAdams, Heat Transmission, third ed McGraw-Hill Kogakusha, Tokyo, Japan,1954.

[23] E. Clark, P. Berhal, Radiative cooling: resources and applications, Proceedings of thePassive-Cooling Workshop, Amherst, MA, 1980.

[24] F. Murray, On the computation of saturation vapor pressure, J. Appl. Meteorol. 6(1967) 203–204.

[25] M.A.S. Malik, G.N. Tiwari, A. Kumar, M.S. Sodha, Solar Distillation, Pergamon PressLtd, UK, 1982.

[26] V.K. Dwivedi, G.N. Tiwari, Comparison of internal heat transfer coefficients inpassive solar stills by different thermal models: an experimental validation,Desalination 246 (2009) 304–318.