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Current Environmental Engineering, 2014, 1, 73-81
73
Modelling of Solar Evaporation Assisted by Floating Light-Absorbing
Porous Materials
Bahman A. Horri*,1, Meng N. Chong1, Xiao D. Chen2,3 and Huanting Wang3
1
Chemical Engineering Discipline, Monash University Malaysia, Bandar Sunway 46150, Malaysia
2
Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen
University, Xiamen 361005, China
3
Department of Chemical Engineering, Monash University, Clayton 3800, Australia
Abstract: Solar energy is one of the promising renewable energy sources for desalination of saline and brackish waters.
The solar evaporation rate could be enhanced by placing light-absorbing agents on the surface or bottom of the solar
ponds. So far, different materials such as various dyes, blackened wet jute cloth, black plastic bubble sheets, black rubber,
floating porous plates, etc. were used to induce the solar evaporation rate, but the evaporation results showed that the
water evaporation enhancement using those materials was quite limited. We have recently reported the use of solar lightabsorbing carbon-Fe3O4 particles and achieved a 230% increase in solar evaporation rate. This paper focuses on
mathematical modeling of the solar evaporation process assisted by this kind of floating light-absorbing material. The
proposed model was used to predict the evaporation rate of the experimental tests and results showed an acceptable
compatibility between the experimental and calculated evaporation rates by an error lower than 13%.
Keywords: Evaporation, evaporation enhancement, modeling, solar, solar light-absorbing.
1. INTRODUCTION
Drinking water is a basic requirement for humans to be
survived. Although, water is one of the most abundant
resources on earth, covering three-fourths of the planet’s
surface, but the most portion of the earth’s water (about
97%) is characterized as the salted water [1]. According to
the World Health Organization (WHO), the permissible limit
of salinity in water is 500 ppm and for special cases up to
1000 ppm while most of the water available on earth has the
salinity up to 10,000 ppm whereas seawater normally has
salinity in the range of 35,000-45,000 ppm in the form of
total dissolved salts [2]. In Australia, fresh surface water is
rare and decreasing especially in the arid interior and
northern coastal areas. Also, the ground-water is often
brackish and contains high levels of fluorides and nitrate
with a total salt concentration of 1500-5000 mg.L-1 [3, 4].
Water desalination has become an increasingly important
source of fresh water in many parts of world. A number of
desalination technologies such as nanofiltration and reverse
osmosis membrane processes, electrodialysis, vapour
compression, multistage flash distillation, multiple-effect
distillation and solar distillation have been developed [1, 5].
But many water purification processes are not suitable for
most of remote areas because of technical or economical
barriers such as lack of electricity and technical
infrastructure, long transportation distance, and high cost of
fuel [6-8]. Cost-benefit analysis of different techniques for
*Address correspondence to this author at the Chemical Engineering
Discipline, Monash University Malaysia, Bandar Sunway 46150, Malaysia;
Tel: +60 3 5514 4420; Fax: +60 3 5514 6207;
E-mail: [email protected]
2212-7186/14 $58.00+.00
supplying potable water to such remote locations has shown
that using the renewable energy sources (e.g. solar-thermal
energy, solar- photovoltaic energy, wind energy, and
geothermal energy) in those areas is an optimal strategy to
provide the fresh water [1, 8-11].
Solar energy is one of the most attractive applications of
renewable energies for salted-water desalination [9, 12]. Sun
is a given-free, non-polluting, and virtually inexhaustible
source of energy and most arid and semi-arid regions have
an abundance of sunshine during a day that can be harnessed
by solar collection systems [6, 13]. Solar still is an example
of conventional collection systems that uses the greenhouse
effect to evaporate the salted-water. The maximum thermal
efficiency of a solar still is around 35% with a daily water
production of 3-4 L.m-2 [1]. Several investigations including
material variation, shape modification, and design parameter
alteration have been carried out to increase the overall
performance of the solar stills [1, 11]. Enhancing the
evaporation rate by adding light absorbing material was
reported in some literature [14-15]. In an investigation,
various dyes were used to darken the water to increase its
solar radiation absorbtivity that could increase the
productivity of a deep basin solar still by 29% [14]. In
another attempt, the production rate of still was improved by
immersing charcoal pieces in the water to reduce the thermal
inertia of the still [15]. Covering the water surface with the
blackened wet jute cloth performed as the solar wicks was
investigated by some researchers and resulted to increase the
still efficiency by 4% [16-18]. Covering the bottom of the
solar ponds with the black plastic bubble sheets was also
tested, but the evaporation rate increased only by 10% [19].
Black rubber as the light absorbing material was used, which
© 2014 Bentham Science Publishers
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Current Environmental Engineering, 2014, Vol. 1, No. 2
increased the daily water productivity of a single slopped
solar still by 38% [20, 21]. Employing floating porous plates
in the water basins is the subject of another research and
could speed up the evaporation rate of the conventional open
basins by 20% [22]. In a newer work, a floating perforated
black aluminum plate was employed to enhance film
evaporation in a soar still that caused to increase the
productivity of the still by 15% [23].
Further enhancement in water evaporation rate is
required to improve the thermal efficiency and overall
performance of the solar evaporation systems. In our
previous paper, we described a new strategy for solar
evaporation enhancement using light-absorbing, floating and
magnetic carbon-Fe3O4 particles that could improve the rate
of salt water evaporation by a factor of 2.3 [24]. The present
paper focuses on mathematical modeling of the solar
evaporation process assisted by the synthesized floating
light-absorbing material. The aim of this work is to
investigate the parameters influencing the evaporation rate
and the performance of the solar evaporation process and
also to set the operational process condition for both solid
and liquid phases to get the highest possible evaporation rate.
The models and results obtained in this study could be used
to optimize the performance and also to design the solar
evaporation systems using floating light-absorbing materials
for different applications such as solar stills, solar ponds,
solar-thermal storage systems, and combined renewable
energy systems.
2. EXPERIMENTAL AND MODELLING METHODS
2.1. Experimental Method
The experimental procedure for the synthesis and
characterization of the floating light-absorbing material
(carbon-Fe3O4 composite particles with average particle size
of 500 nm) and also the experimental results of evaporation
tests were presented in our previous paper [24]. Briefly, the
floating light-absorbing carbon-Fe3O4 particles were
synthesized by modifying a two-step polymerization of
furfuryl alcohol (FA) dispersed with a small amount of
Fe3O4 nanoparticles [25].
The salt water evaporation experiments were conducted
at room temperature and a relative humidity of 50%. A
10 mL beaker with an inner diameter of 2.2 cm filled with
5 g of 3.5% NaCl aqueous solution (salt water) was placed
under a sunlight source. A sunlight simulator (CHF-XM500,
20 A) was used as the light source and a radiation meter (FZA) was used to measure the light intensity. The radiation
intensity was varied from 430-1355 Wm−2. The amount of
water evaporated was determined by monitoring the weight
change. The water surface and bottom temperatures of 5 g
and 10 g of 3.5% salt water were measured using two
thermocouples, respectively (Type K, Amprobe 38XR-A).
2.2. Mathematical Formulation
The geometry of the evaporation system consisting of the
hydrophobic floating light-absorbing material (porous-solid
phase) on top and salt water (liquid phase) underneath is
Horri et al.
schematically illustrated in Fig. (1). The total solar irradiance
(solar power per unit area, Eir) received by the surface of the
porous material, is balanced by different heat losses due to
radiation, conduction, convection, and evaporation
phenomena as follows:
Fig. (1). Schematic diagram of the analysed physical evaporation
system.
2.2.1. Radiation Heat Loss
The upper surface of the light-absorbing material can lose
part of the absorbed heat to its surrounding atmosphere
through the radiation mechanism if its surface temperature is
higher than ambient temperature. Although, the lightabsorbing material could be assumed as a black body, but a
more accurate relationship for the radiation heat loss can be
obtained by assuming the atmosphere and the lightabsorbing material as two parallel gray surfaces for
exchanging radiation to each other. Consequently, the
radiation heat flux lost from the upper surface of the lightabsorbing material could be expressed as [26]:
𝑄! =
! !!! !!!!
! !
! !!
!! !!
(1)
2.2.2. Convection Heat Loss
The absorbed energy is lost by the convection
mechanism from both upper and lower surfaces of the lightabsorbing material. At the start of radiation process or lack
of solar irradiance source (during night), it is assumed that
both liquid and porous-solid phases have a same temperature
equals to the wet-bulb temperature (saturation temperature at
the gas-liquid interface). The saturation temperature is
essentially lower than the ambient temperature that means
the light-absorbing material initially receives the required
heat for evaporating its underneath water from the
surrounding atmosphere (mainly through convection
mechanism). Gradually, by absorbing the solar irradiance,
the temperature of the porous-solid phase exceeds the
ambient temperature that causes the absorbed heat to be
transferred to both ambient (by the upper surface) and liquid
phase (by the lower surface) through the convection
Transport Phenomena and Fluid Mechanics
Current Environmental Engineering, 2014, Vol. 1, No. 2
75
mechanism. The flux of convective heat loss from the upper
surface could be formulated as:
the lower surface of the light-absorbing material through the
convection mechanism that could be formulated as:
𝑄!" = ℎ! 𝑇! − 𝑇!
𝑄!" = ℎ!! 𝑇! − 𝑇!"
(2)
The convective heat transfer coefficient between the
static surrounding air (no wind) and the porous-solid surface
(ℎ! ), could be calculated using the local Nusselt’s number
correlation (for natural-convection) developed by Pop and
Cheng [27]:
𝑁𝑢! =
!! .!
!!
!
!
= 0.413𝑅𝑎!
!"#$ !! !!!
! ! !!
!! .!
!!
𝑃𝑒! =
(5)
(6)
!!
In equation 4 and 6, 𝛼! is the overall thermal diffusivity
of the fluid saturating (flowing in) pores of the lightabsorbing material which is defined as [29]:
𝛼! =
!!
!!!
(10)
𝑅𝑎! =
where, 𝑃𝑒! has the following definition:
!! .!
!!
!
= 0.65𝑅𝑎!!
(4)
!
= 0.886𝑃𝑒!!
!!" .!
𝑁𝑢! =
where, 𝑅𝑎! is the Darcy modified Rayleigh’s number that
can be calculated at l (half length of the porous-solid
material) using the following equation:
For the case of forced-convection between the lightabsorbing material and the upper surrounding atmosphere
(i.e. existence of wind), the obtained value of Peclet’s
number has the main role to determine the convective heat
transfer coefficient. For this case, Nield and Bejan
correlation could be used as follows [28]:
𝑁𝑢! =
In equation 9, the convective heat transfer coefficient of
salt water (ℎ!" ) could be obtained using Kimura et al.
correlation as [30]:
(3)
where, the local Rayleigh’s number (𝑅𝑎! ) is:
𝑅𝑎! =
(9)
(7)
!
!"#$ !! !!!"
!!" !!
(11)
2.2.3. Evaporation Heat Loss
The total energy transferred to the salt water by
convective mechanism (𝑄!" ) can be either directly used for
evaporation or increasing the liquid phase temperature. The
heat flux used for water evaporation (𝑄!"# ) could be derived
from the heat and mass balance equations as follows:
𝑄!"# = 𝑁! ∆𝐻!" (12)
where, 𝑁! is the water evaporation flux from the surface of
liquid phase into the pores of the light-absorbing material.
By assuming the void fraction of the light-absorbing material
to be saturated by water vapor at the saturation temperature,
the mass transfer flux is controlled by the external condition
that could be calculated by the Newton’s law of mass
transfer as:
𝑁! = 𝐾! (𝑌!"# − 𝑌! )
(13)
where, 𝑘! is the overall thermal conductivity of the poroussolid (light-absorbing) material that could be calculated
using the following relationship [29]:
where, 𝐾! is the mass transfer coefficient. For the case of
laminar regime, 𝐾! could be obtained by Schlichting
correlation as [31]:
𝑘! = 1 − 𝜙 𝑘! + 𝜙𝑘!
𝑆ℎ =
(8)
The local heat transfer coefficients obtained from
equations 3 and 5 were numerically integrated over the total
length of the solid material subjected to the convective heat
transfer to calculate the average values of the convective heat
transfer coefficients.
By applying an energy balance on the surface of the
system, the net energy absorbed by the light-absorbing
material (𝑄!"# ) can be obtained by subtracting the total
energy loss (summation of radiation and convective energy
losses at the upper surface) from the total irradiance. The
absorbed net-energy is used to induce the water evaporation
rate through different manners such as increasing
temperature of both porous-solid and salt water and also
providing a direct source of the latent heat required for water
evaporation. By starting the light-absorbing process, a part of
the absorbed net-energy (𝑄! ) is used to increase the overall
temperature of the porous-solid material (both solid and gas
phases). By increasing the temperature difference between
the porous-solid and liquid phases, another part of the
absorbed net-energy (𝑄!" ) is transferred into the salt water by
!! !!! !!
!! !!"
= 0.664 𝑅𝑒 !.! 𝑆𝑐 !.!!
(14)
Smolsky and Sergeyev dimensionless correlation could also
be used to calculate the mass transfer coefficient (𝐾! ) at
turbulent regime as the following [32]:
𝑆ℎ =
!! !!! !!
!! !!"
= 0.094 𝑅𝑒 !.! 𝑆𝑐 !.!! 𝐺𝑢 !.!
(15)
2.3. Governing Equations
The salt water evaporation process using the hydrophobic
floating light-absorbing material on the surface could be
modeled using the transient energy conservation over the
solid and liquid phases. The porous-solid was assumed to be
in contact with the liquid phase, and the solid phase could be
treated as the lumped geometry with a uniform temperature
due to its very low Biot number (𝐵𝑖 ≤ 0.0083). For the
static liquid phase receiving heat from the top layer, the onedimensional heat conduction could be applied as the
prevailing heat transfer mechanism in the salt water. This is
especially applicable for a big solar pond as the length and
width of the salt water container are much longer than its
depth that means one-dimensional analysis could be applied
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Current Environmental Engineering, 2014, Vol. 1, No. 2
Horri et al.
for the liquid phase. The heat transferred into the fluid
flowing in the pores of the porous-solid could be neglected
due to its low heat transfer coefficient and small thickness of
the solid layer. In addition, it could be assumed that the air
condition (temperature, pressure, wind speed and relative
humidity), the solar irradiance and the liquid-phase level to
be constant during the evaporation process.
The governing equation could be presented by writing a
heat rate balance over the solid-liquid system. By assuming a
uniform distribution of porosity in the porous structure of the
light-absorbing material, the cross-sectional surface area for
the void-space and solid phases are 𝜙. 𝐴! , and 1 − 𝜙 . 𝐴! ,
respectively. Accordingly, the governing equation could be
written as:
𝐸!" − 𝑄! − 𝑄!" − 𝑄!" . 1 − 𝜙 . 𝐴! − 𝑄!"# . 𝜙. 𝐴! =
!"
𝑚. 𝐶!
(16)
! !"
Equation (16) could be rewritten by substituting the
equivalent equation obtained for each term as follows:
𝐸!" −
𝑁!
!
! !!! !!!!
! !
! !!
!! !!
!!!
− ℎ! 𝑇! − 𝑇! − ℎ!" 𝑇! − 𝑇!" −
∆𝐻!" −
!! .!!! .!! !!!
!!!
!"
=0
(17)
Equation (17) is the governing equation for the lightabsorbing material in the transient period. The first term in
equation (17) represents the solar irradiance energy absorbed
through the upper surface of the light-absorbing material.
The second and the third terms in the governing
equations stand for the heat transferred through the uppersurface of the porous-solid material by radiation and
convective mechanisms, respectively. The temperature status
of both porous-solid material and ambient air (𝑇! and 𝑇! )
determines the sign of the second and the third terms in
equation (17). For water evaporation occurring either at night
(without solar irradiance source) or at the initial step of
absorbing solar energy, the porous-solid temperature is
below (or up to equals) the ambient temperature (𝑇! ≤ 𝑇! ),
and consequently the second and the third terms of equation
(17) would be negative. This means the required heat for
water evaporating should be essentially supplied from the
surrounding atmosphere through radiation mechanism
(negligible) and convection mechanism (prevailing).
Gradually, by absorbing the solar radiation energy, the
temperature of the porous-solid material would then be
increased. For the case that the porous-solid temperature is
higher than the ambient temperature (𝑇! > 𝑇! ), both the
second the third terms in equation (17) would be positive
that means losing the absorbed solar energy to the
surrounding atmosphere by the radiation and convection
mechanisms. In general, the importance of the radiation term
depends on the magnitude of temperature difference between
the solid material and the surrounding air. Usually, for lower
temperature differences, the radiation term is negligible.
The last three terms in equation (17) present the energy
transferred into the salt water by convection mechanism, the
energy used for water evaporation, and the transient
temperature differential change in the porous-solid phase.
The initial condition for equations (17) can be expressed
as:
!"#
@ 𝜃 = 0: 𝑇! = 𝑇!"
(18)
The above condition means the evaporation process starts
at the uniform saturation temperature (wet-bulb temperature
of the liquid phase) for both phases.
The governing equation for the liquid phase in the
transient period and on the domain of 0 ≤ 𝑧 ≤ 𝑙!" was
derived as:
𝛼!" .
!! !!"
!! !
−
!!!"
!"
−
!
!
!! .!!" .!!!" !!"
+
!!
!!
𝐿𝑛
!!
!!
+
!!
!! !!
!!
=0
(19)
The first and second terms in equation (19) represent the
one-dimensional conductive heat transfer term and the
transient temperature differential change inside the liquid
phase, respectively. The last term in equation (19) represents
the heat lost through side walls of the salt water container.
That is obvious for the case of using a container covered by
insulation or a big solar pond; the last term of equation (19)
should be neglected. The initial and boundary conditions for
equation (19) can be expressed as:
!"#
@ 𝜃 = 0, 𝑎𝑛𝑦 𝑍: 𝑇!" = 𝑇!"
@ 𝜃 > 0, 𝑧 = 0: −𝐾!"
@ 𝜃 > 0, 𝑧 = 𝑙!" : !!!"
!!!"
!"
!"
= ℎ! 1 − 𝜙 (𝑇! − 𝑇!" )
=0
(20)
(21)
(22)
Equation (20) represents the initial condition of heat
transfer by assuming the saturation temperature (wet-bulb
temperature) for the liquid phase at the start of evaporation
process. The first boundary condition (equation 21)
represents the convective heat transfer mechanism in the
solid-liquid interface (lower surface of solid material). The
second boundary condition (equation 22) represents
assuming adiabatic condition at the bottom of the geometry
that can be explained by using an insulation layer over the
bottom of the container.
2.4. Method of Solution
Both differential equations (17) and (19) were
simultaneously solved using numerical methods for the total
range of the transitional period. The parabolic PDE (equation
19) was solved using the finite difference method. For the
non-homogeneous linear ODE (equation 17), the RungeKutta method was used to obtain the solution. The
computations were repeated until all calculated variables
satisfied the convergence criteria. A fixed time step-size
(Δ𝜃) was selected according to the applied finite difference
formulation as:
Δ𝜃 ≤
!!
!! !"
(23)
where, the depth step-size (Δ𝑧) was selected to be 1 mm (for
shallow salt water containers). For deep containers (e.g. 1 m
or more in depth), bigger depth step-size (around 1 cm)
could be selected. Sharqawy et al. polynomials were used to
determine the physical properties of the liquid phase
including the density, specific-heat capacity, thermal
conductivity, dynamic and kinematics viscosity, saturation
Transport Phenomena and Fluid Mechanics
partial pressure and latent-heat of vaporization [33]. The
physical properties of the porous-solid material were
calculated for the mixture using the porosity and the
properties of the pure phases [34].
3. RESULTS AND DISCUSSIONS
The evaporation experimental results including the
liquid-phase temperature and also the evaporation rate
measurements were used to test the validity of the
mathematical modelling analysis. The physical properties
and parameters used in the experimental measurements and
also calculation of the modelling results have been
summarized in Table 1. The measured surface/bottom
temperature of the liquid phase and the calculated (predicted)
temperatures by the model have been represented in Fig. (2).
As shown in this figure, the model could predict the
temperature behaviour of the liquid phase with an acceptable
consistency. It seems there is a gap between the predicted
and experimental temperatures. The gap at the beginning
period (the first half an hour) could be explained by the
assumption expressed in equation 20 to solve the governing
equation. In this equation, it is assumed the starting
temperature of the liquid phase is the wet-bulb temperature
of the water in the container. This assumption is almost valid
for a real solar evaporation case (such as a solar pond or a
lake illustrated in Fig. (1) with insulated bottom and sidewalls), but for a small beaker applied in the experimental
measurements, the liquid-phase temperature is almost similar
to the ambient temperature which is higher than the wet-bulb
temperature. Therefore, as it is seen in Fig. (2), the predicted
temperatures are smaller than the measured temperatures in
the beginning period of the evaporation process. After this
period, results show higher values for the predicted
temperatures compared to the experimental temperatures.
This could be explained by assuming the insulation condition
Current Environmental Engineering, 2014, Vol. 1, No. 2
77
for the bottom of the evaporation container (the second
boundary condition, equation 22). It should be noted that, for
a real solar evaporation case with a high surfaces area and a
little depth, both side-wall and bottom heat-losses could be
approximately negligible and this means, the liquid phase
could achieve higher average temperature assisting the
overall evaporation rate that has been accordingly predicted
by the model.
To quantify the amount of evaporation enhancement
resulted by using the synthesized floating light-absorbing
material, the evaporation tests were carried out with and
without the absorbing materials. Fig. (3) has compared the
experimental and the predicted cumulative water evaporation
for both cases (with and without using the floating solid
material). This figure shows enough consistency between the
calculated and the experimental evaporation results for both
cases. But the overall predicted evaporation rates are
somewhat greater than the experimental ones. At the start of
evaporation process the experimental evaporation rates are
greater than the calculated rates. This could be due to the
long time-intervals (half an hour) used between each weightloss measurement and also the forward derivation applied to
calculate the experimental evaporation rates. In addition, the
water evaporation rate in the experimental tests was reduced
at the end of the process due to the increased salinity of the
salt water. Another reason could be the application of the
insulation boundary condition at the bottom of the container
while for the real case; there is a small heat loss from the
bottom of the salt water beaker.
To compare the evaporation results with and without the
light-absorbing material, the modified Penman's correlation
was used to calculate the free evaporation rate from the
liquid phase surface [35]. As the modified Penman's
correlation was developed for the steady-state condition, the
Fig. (2). Experimental and predicted surface/bottom temperature of the liquid phase.
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Current Environmental Engineering, 2014, Vol. 1, No. 2
Table 1.
Horri et al.
Summary of physical data used for calculation of the salt water evaporation process.
Air Phase (Ambient)
Solar Light-Absorbing Material (Porous-Solid Phase)
Property
Quantity
Temperature, 𝑇! (°C)
25
Pressure, 𝑃!
Property
Emissivity, 𝜖
Wind speed, 𝑈!
(m/s)
Relative humidity, 𝐻! (%)
2
Solar irradiance, 𝐸!" (W/m )
Daily solar shining
(h)
Applied duration of solar shining, (h)
0.85
(-)
Porosity, 𝜙
101.3
(kPa)
Quantity
40
(-)
Salt Water (Liquid Phase)
Property
Salinity, 𝑆!"
(Wt. %)
Height, 𝑙!"
(cm)
Quantity
3.5
2
0
Skeletal density, 𝜌! (kg/m3)
1.43
-
-
50
Particles surface area, 𝑆! (m2/g)
429
-
-
120
-
-
0.25
-
-
710
-
-
0.5
-
-
1355
2
Thermal conductivity, 𝑘! (W/m .°C)
Thickness, 𝐿!
10
6
(mm)
Specific heat, 𝐶𝑝!
(J/kg.°C)
Average particle Size, 𝑑! (µ)
actual amount of water evaporated was slightly lower than
the predicted one. On the other hand, the evaporation process
in the experimental tests is started by the salt water available
at the ambient temperature, meaning that it has an unsteady
state period (transitional) with lower evaporation rate before
reaching its highest rate at the steady-state condition.
The experimental and calculated values of the average
evaporation rate using different amounts of the lightabsorbing material are shown in Table 2. The calculated
values show that the application of light-absorbing material
enhances the water evaporation rate by approximate factors
of 2.3, 2 and 1.8, which corresponds to the weight of solid
particles of 0.045, 0.023 and 0.015 g, respectively. This
trend is in good agreement with the experimental result. In
every case, the calculated evaporation rate shows higher
value compared to the experimental results. The governing
equations (17 and 19) could be applied for a real solar pond
that physically has a vast surface area (semi-infinite) and a
small height (shallow). Also, the real salt water pond is more
likely to have insulated side-walls, that means the last term
in equation (19) could be neglected.
CONCLUSION
The solar evaporation process enhanced by the
hydrophobic floating light-absorbing porous material was
mathematically modeled. Thermal energy losses through the
radiation, conduction and evaporation mechanisms were
Fig. (3). Experimental and predicted cumulative water evaporation flux with and without the solar light-absorbing solid.
Transport Phenomena and Fluid Mechanics
Table 2.
Current Environmental Engineering, 2014, Vol. 1, No. 2
79
Experimental and calculated values of evaporation rate for different solid loading.
Weight of Light-Absorbing Material Loaded on the Salt Water Surface (g)
Average Evaporation Rate
0
0.015
0.023
0.045
Experimental (mm/hr)
0.9655
1.6122
1.8227
2.2059
Calculated (mm/hr)
1.0107
1.8253
1.9910
2.3643
Error (%)
+4.67
+13.22
+9.23
+7.18
mathematically expressed. The required heat transfer
coefficients and other physical properties were calculated
using appropriate dimensionless groups. The governing
equations were derived by conducting the transient energy
conservation over the solid and liquid phases. The solid
phase was treated as the lumped geometry with a uniform
temperature while the one-dimensional heat conduction
mechanism was applied in the liquid phase. The validity of
the proposed models was tested by comparing the
experimental and calculated results in terms of temperature
and cumulative evaporation rate. There was a gap between
the predicted and experimental results which was due to
applying the insulation condition at the bottom of the salt
water container. The modelling result showed that by using
0.045, 0.023 and 0.015 g of the light-absorbing material, the
evaporation rate can be enhanced by approximate factors of
2.3, 2 and 1.8, respectively.
SYMBOLS
𝐴! : Cross-sectional surface area of heat transfer for the lightabsorbing material (m2).
𝐵𝑖: Biot Number (
!! .!!
!!
, dimensionless).
𝑘! : Thermal conductivity of the fluid saturating pores of the
light-absorbing material (W/m.K).
𝑘! : Overall thermal conductivity of the light-absorbing
material (W/m.K).
𝑘! : Thermal conductivity of the (solid phase of) lightabsorbing material (W/m.K).
𝐾! : Overall mass transfer coefficient (Kg dry air/m2.s).
𝐾!" : Thermal conductivity of salt water (W/m ºC).
𝑙 : Half length of the porous-solid material subjected to
convective heat transfer (m).
𝑙! : Characteristic length of light-absorbing material (m).
𝑙! : Total length of light-absorbing material subjected to
convective heat transfer (m).
𝐿! , 𝐿!" : Thickness of the light-absorbing material, and salt
water, respectively (m).
𝑙!" : Depth (thickness) of salt water phase (m).
𝑚. 𝑐!
(J/ºC).
!
: Overall heat capacity of light-absorbing material
𝑁! : Water evaporation flux of salt water (Kg/m2.s).
𝐶𝑝! : Specific heat capacity of fluid saturating the pores of
the light-absorbing material (J/Kg ºC).
𝑁𝑢! : Nusselt’s number at half length of light-absorbing
material (dimensionless).
𝐶𝑝! : Overall specific heat capacity of the light-absorbing
material (J/Kg ºC).
𝑁𝑢! : Local Nusselt’s number (dimensionless).
𝐶𝑝!" : Specific heat capacity of salt water (J/Kg ºC).
𝑃! : Total pressure of the fluid saturating pores of the lightabsorbing material (Pa).
𝑑! : Mean pore-diameter of the light-absorbing material (m).
𝑃𝑒! : Local Pecltet’s number (dimensionless).
𝐷!" : Diffusion coefficient for water vapour in air (m2/s).
𝑄!" , 𝑄!" : Flux of convective heat loss from the lower surface,
and upper surface, respectively (W/m2).
𝐸!" : Total solar irradiance received by the light-absorbing
material (W/m2).
𝑔: Gravitational acceleration (m2/s).
𝐺𝑢: Gukhman’s number (
!"#
!! !!!!
!!
, dimensionless).
ℎ! : Convective heat transfer coefficient of air (W/m2 ºC).
ℎ! : Overall convective heat transfer coefficient (W/m2 ºC).
ℎ!" : Convective heat transfer coefficient of salt water (W/m2
ºC).
𝐾: Permeability of the light-absorbing material (m2).
𝑘! : Thermal conductivity of salt water container (W/m.K).
𝑄!"# : Heat flux spent for water evaporation (W/m2).
𝑄!"# : Net energy absorbed by the light-absorbing material
(W/m2).
𝑄! : Radiation heat loss from the upper surface of the lightabsorbing material (W/m2).
𝑄! : Heat flux spent for increasing the temperature of the
light-absorbing material (W/m2).
𝑄!"!! , 𝑄!"!! : Inlet and outlet heat fluxes for an elemental
heat transfer analysis in salt water, respectively (W/m2).
𝑟! , 𝑟! : Inside, and outside radius of the salt water container,
respectively (m).
𝑅: Ideal gas constant, (8.314 J/gmol.K).
80
Current Environmental Engineering, 2014, Vol. 1, No. 2
Horri et al.
𝑅𝑎! : Local Rayleigh’s number (dimensionless).
CONFLICT OF INTEREST
𝑅𝑎! : Darcy modified Rayleigh’s number (dimensionless).
The authors confirm that this article content has no conflict
of interest.
𝑅𝑒: Reynolds’s number, (
𝑆𝑐: Schmidt’s number, (
!! !! !!
!!
!!
!!"
𝑆ℎ: Sherwood’s number, (
, dimensionless).
ACKNOWLEDGEMENTS
, dimensionless).
!! !!! !!
!! !!"
This work was supported by the Australian Research
Council. The authors thank the Australian Research Council
for a future fellowship (FT100100192).
, dimensionless).
𝑇! : Ambient (wet air) temperature (ºC).
𝑇! : Average temperature of gas phase saturating (flowing in)
pores of light-absorbing material (ºC)
𝑇! : Temperature of the light-absorbing material (ºC).
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𝑇!" : Temperature of salt water (ºC).
!"#
: Salt water saturation temperature (ºC).
𝑇!!
[3]
𝑈! : Velocity of ambient air (wind speed, m/s).
[4]
𝑢! : Velocity of gas phase flowing in pores of light-absorbing
material (m/s).
[5]
𝑥: Coordinate along the horizontal plate (m).
[6]
𝑌! : Absolute humidity of ambient air (Kg water/Kg dry air).
[7]
𝑌!"# : Saturated absolute humidity at the gas-liquid interface
(Kg water/Kg dry air).
[8]
𝑧, 𝑑𝑧: Vertical distance from the surface of salt water, and
unit of length in differential analysis, respectively (m).
[9]
[10]
𝑍: Vertical coordinate (m).
[11]
GREEK SYMBOLS
𝛼! : Overall thermal diffusivity of the fluid saturating pores
of the light-absorbing material (m2/s).
[12]
[13]
2
𝛼!" : Thermal diffusivity of salt water (m /s)
𝛽: Coefficient of thermal expansion for fluid (K-1).
[14]
𝛿: Stefan-Boltzmann constant (5.67*10-8 W/m2.K4).
[15]
∆𝐻!" : Latent heat of water vaporization for salt water (J/Kg).
[16]
𝜖! , 𝜖! : Radiation emissivity coefficient of ambient air and the
light-absorbing material (dimensionless)
[17]
𝜃: Time (S).
[18]
𝜇! : Viscosity of the fluid flowing in pores of the light
absorbing material (Pa.s)
[19]
𝜈! , 𝜈! , 𝜈!" : Kinematic viscosity of (wet) air, the fluid flowing
in pores of the light absorbing material, and salt water phase,
respectively (m2/s).
[20]
[21]
𝜌! : Density of the fluid saturating (flowing in) pores of lightabsorbing material (Kg/m3).
[22]
𝜌! : Overall density of light-absorbing material (Kg/m3).
[23]
3
𝜌!! : Density of salt water (Kg/m ).
𝜙 : Open porosity
(dimensionless)
of
the
light-absorbing
material
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Revised: October 19, 2014
Accepted: October 20, 2014