Numerical temperature and concentration distributions in an insulated salinity gradient solar pond
 Ridha Boudhiaf^{1}Email author
Received: 14 July 2014
Accepted: 23 April 2015
Published: 28 July 2015
Abstract
In this paper, the temperature and concentration distributions in an insulated salinity gradient solar pond are studied numerically in transient regime. The dimensionless governing equations are solved by finitevolume method using SIMPLER algorithm with HYBRID scheme. The pond is filled with a mixture of salt and water to form three zones: upper convective zone (UCZ), nonconvective zone (NCZ) and lower convective zone (LCZ). The vertical walls and the bottom of the pond are thermally insulated. The bottom of the pond is black painted. The simulation of the insulated salinity gradient solar pond shows the existence of salt diffusion from the bottom to the free surface. This numerical study shows also that the buoyancy ratio increases and decreases the temperature and concentration in the LCZ and in the UCZ, respectively.
Keywords
Background
Salinity gradient solar ponds have been widely studied experimentally, analytically, and numerically due to their excellent collection of solar radiation and storage of thermal energy. Many research works have been developed and published (Sodah et al. 1981; Joshi and Kishore 1984; Singh et al. 1994; Sezai and Tasdemiroglu 1995) to control the construction techniques of solar ponds, and different numerical models have been developed to analyze the performance of salinity gradient solar ponds. Ouni et al. (1998) studied the transient behavior of a onedimensional salinity gradient solar pond in the southern part of Tunisia. In this research work, the temperature is constant in the UCZ and in the LCZ. Kurt et al. (2000) studied experimentally, analytically, and numerically the thermal behavior of a onedimensional salinity gradient solar pond in transient regime. They showed that the experimental, numerical, and analytical temperatures are constant in the UCZ and in the LCZ. Angeli and Leonardi (2004) studied numerically the salt diffusion in a solar pond using a onedimensional mathematical model. Ould Dah et al. (2005) studied experimentally the transient evolution of the temperature and concentration profiles in a solar pond with high salinity NaCl solution. Mansour et al. (2004) studied numerically the transient evolution of the temperature and concentration profiles in a twodimensional salinity gradient solar pond. Kurt et al. (2006) studied experimentally and numerically the performance of a salinity gradient solar pond by using the sodium carbonate salt to create a salinity gradient in the solar pond. They studied the temperature and concentration profiles in the solar pond by developing a onedimensional mathematical model in transient regime. Hammami et al. (2007) studied numerically the transient evolution of the temperature and concentration profiles in a stratified enclosure with a buoyancy ratio equal to 1,000. In their research work, the internal heating of the fluid due to the absorption of solar radiation is not considered into account. In the research work of Ould Dah et al. (2010), a onedimensional numerical model was developed to simulate the transient evolution of the temperature and concentration profiles in a mini solar pond. Karakilcik et al. (2006) studied experimentally and numerically the temperature distributions in an insulated solar pond during daytimes and nighttimes. Sakhrieh and AlSalaymeh (2013) studied experimentally and numerically the temperature distribution in an insulated solar pond under Jordanian climate conditions. In their research work, the temperature is constant in the UCZ and in the LCZ. ElSebaii et al. (2011) presented a review study of the history of the solar ponds. They found that the temperature and concentration are almost constant in both the UCZ and the LCZ. Suarez et al. (2010) studied numerically the evolution of the temperature and concentration profiles with time in a salinity gradient solar pond with stable stratified layers.
After reviewing the literature, the anterior numerical and analytical models studied the transient evolution of the temperature and concentration profiles in a onedimensional salinity gradient solar pond. The equations of heat transfer by conduction and mass diffusion are solved by a finitedifference method in transient regime. In these models, the authors neglected the convective movements and assumed that the temperature is constant in the upper and lower convective zones.
In very recent works, the authors have studied the hydrodynamic, heat, and mass transfer behaviors of a twodimensional salinity gradient solar pond in transient regime (Boudhiaf et al. 2012; Boudhiaf and Baccar 2014). The results have shown, in particular, that the internal Rayleigh number, the buoyancy ratio, and the aspect ratio have an important effect, respectively, on the thermal performance of the pond, on the stability, and on the distribution of temperature and velocity fields in the salinity gradient solar pond. In the present paper, we are interested to study numerically the temporal evolution of the temperature and concentration distributions in a twodimensional insulated salinity gradient solar pond under the influence of buoyancy ratio and for an aspect ratio equal to three. The resolution of continuity, momentum, energy, and mass transfer equations is conducted using finitevolume technique discretization in transient regime.
Mathematical modeling and numerical method
Mathematical modeling
 a.
The velocity, temperature, and concentration variation along the ydirection is considered small enough so that it is negligible. Therefore, the velocity, temperature, and concentration distributions within the pond are twodimensional.
 b.
Both the vertical and the bottom walls of the pond are well insulated and impermeable. The free surface of the pond is subjected to heat losses by convection, evaporation, and radiation.
 c.
The incident solar radiation upon the free surface of the pond is a supposed constant and has an average value in transient regime.
 d.
The pond’s bottom is black painted in order to raise the absorption of solar radiation by salty water layers. The solar radiation that reaches the bottom of the pond is entirely absorbed by the fluid at this depth.
 e.
The mixture of salt and water is assumed to be incompressible and Newtonian.
 f.
The fluid properties are assumed independent of temperature and salt concentration, except the density that varies according to Boussinesq approximation:
The process of hydrodynamic, heat, and mass transfer in the salinity gradient solar pond can be described by a set of differential equations. To simplify the study, before the numerical calculation, the governing equations are nondimensionalized with the following dimensionless parameters:
X = x/H; Z = z/H; U = u/(α/H); W = w/(α/H); τ = t/(H ^{2}/α); P = p/(ρ _{0} α ^{2}/H ^{2})
θ = (T − T _{a})/ΔT; φ = (C − C _{min})/ΔC; Ra_{T} = gβ _{T}ΔTH ^{3}/(αν); Ra_{I} = gβ _{T} q _{0} H ^{4}/(λ _{w} αν)
Ra_{S} = gβ _{C}ΔCH ^{3}/(αν); N = β _{C}ΔC/(β _{T}ΔT); Pr = ν/α; Le = α/D
where H, α/H, H ^{2}/α, ρ _{0} α ^{2}/H ^{2}, ΔT, and ΔC are used as characteristic scales for length, velocity, time, pressure, temperature, and concentration, respectively.
Therefore, the resulting continuity, momentum, energy, and mass transfer equations can be written in dimensionless form as follows:
With the following initial and boundary conditions:
At initial time, the salty water is considered in rest condition and the pond has an ambient temperature.
τ = 0: U = W = 0, θ = 0, and P = 0

The LCZ is saturated in salt (φ = 1).

In the NCZ, the concentration of salt increases with the depth as follows: φ = (Z _{NCZ} − Z)/(Z _{NCZ} − Z _{LCZ}).

The UCZ is a zone without salt (φ = 0).
Concerning the boundary conditions, the vertical walls of the pond are thermally insulated and impermeable. At the bottom of the pond, we impose a zero mass flux and we assume that the heat flux is equal to the solar radiation reaching this depth. At the free surface of the pond, the corresponding boundary condition of velocities is: ∂U/∂Z = 0 and W = 0.
We consider only half of the pond for reasons of symmetry. At the symmetrical vertical plane, the corresponding boundary conditions are: U = 0, ∂W/∂X = 0, ∂θ/∂X = 0, and ∂φ/∂X = 0.
Heat losses through the top surface of the pond are detailed in Appendix.
Numerical method
The governing Equations (2 to 6) are discretized using the finitevolume method of Patankar (1980). In addition, the hybrid scheme interpolation is employed to discretize spatially the governing equations. The Alterning Directions Implicit (ADI) method is used to integrate temporally the discretized equations over a time step. The SIMPLER (SemiImplicit Method for PressureLinked Equation Revised) algorithm (Patankar 1980) is employed to handle the coupling of pressurevelocity. A dimensionless time step 10^{−8} was found to be sufficient for producing accurate results at reasonable computed time.
In order to ensure the numerical results independency with respect to the number of nodes employed in the process of discretization, four uniform grid patterns spacing in OX and in OZ have been tested. The results obtained from this research, which were published in Boudhiaf and Baccar (2014), have shown that the uniform grid of 100 × 100 appears relatively suitable for the present work. In order to ensure the accuracy of numerical results, we validated the numerical code specifically developed for the present work in three ways. The results obtained from this study, which were published in Boudhiaf et al. (2012) and Boudhiaf and Baccar (2014) and hence are not repeated here for the sake of space, have shown a good comparison. The study of the grid independence, the validation of the numerical code specifically developed for the present research, and the algorithm of the numerical method have been detailed elsewhere (Boudhiaf et al. 2012; Boudhiaf and Baccar 2014; Boudhiaf 2013).
Results and discussion
In this work, we will present numerical results to study the importance of the effect of buoyancy ratio on the temporal evolution of temperature and concentration distributions in the three zones constituting the salinity gradient solar pond. The Prandtl and Schmidt numbers are kept constant (Pr = 6 and Sc = 1000), which correspond to the average characteristics of salty water (Hammami et al. 2007; Boudhiaf et al. 2012; Boudhiaf and Baccar 2014). The results are generated for fixed values of thermal Rayleigh number, internal Rayleigh number, and aspect ratio (Ra_{T} = 10^{7}, Ra_{I} = 1.4 × 10^{8}, and A = 3).
Temporal evolution of temperature distribution
Temporal evolution of concentration distribution
Conclusions
We have numerically studied the temperature and concentration distributions within a twodimensional insulated salinity gradient solar pond under the effect of the buoyancy ratio, in transient regime. The dimensionless equations obtained for the model have been solved numerically by employing the finitevolume method.
From the results obtained by this numerical study, we have concluded that the buoyancy ratio has a very important effect on the average temperature of UCZ, NCZ, and LCZ.
In addition, the buoyancy ratio has an important influence on the temporal evolution of average salt concentration in the UCZ, NCZ, and LCZ constituting the salinity gradient solar pond.
We have proved the existence of a salt diffusion from the lower convective zone to the upper convective zone. This diffusion of salt is generally due to the molecular diffusion.
We have also concluded that the buoyancy ratio is important to reduce the diffusion of salt from the bottom to the free surface of the pond and to increase the temperature in the lower convective zone.
Nomenclature
A Aspect ratio, (A = L H ^{−1}),
C Concentration of solution, (kg m^{−3}),
ΔC Difference of concentration, (ΔC = C _{max} − C _{min}),
C _{pa} Specific heat of air, (kJ kg^{−1}°C^{−1}),
C _{p} Specific heat of solution, (kJ kg^{−1}°C^{−1}),
D Diffusion coefficient, (m^{2} s^{−1}),
H Height, (m),
h _{c} Convective heat transfer coefficient, (W m^{−2}°C^{−1}),
L Length, (m),
LCZ Lower convective zone,
L _{v} Latent heat of water evaporation, (J kg^{−1}),
Le Lewis number,
N buoyancy ratio,
NCZ Nonconvective zone,
p Pressure, (Pa),
P Dimensionless pressure,
Pr Prandtl number,
P _{s} Vapor pressure of water at the pond’s surface, (Pa),
P _{v }Partial pressure of water vapor in the air, (Pa),
P _{atm} Atmospheric pressure, (Pa),
Q _{c} Heat losses due to convection, (W m^{−2}),
Q _{e} Heat losses due to evaporation, (W m^{−2}),
Q _{r} Heat losses due to radiation, (W m^{−2}),
q _{0 }Solar radiation penetrating the top surface of the pond, (W m^{−2}),
q(z) Solar radiation absorbed at depth z, (W m^{−2}),
Ra_{T} Thermal Rayleigh number,
Ra_{I} Internal Rayleigh number,
R _{h} Relative humidity,
Ra_{S} Solutal Rayleigh number,
Sc Schmidt number,
t Time, (s),
ΔT Difference of temperature, (ΔT = T _{max} − T _{min}),
T Temperature, (°C),
T _{max} Maximum temperature, (°C),
T _{min} Minimum temperature, (°C),
T _{s} Temperature at the top surface of the pond, (°C),
T _{a} Ambient temperature, (°C),
T _{sky} Sky temperature, (°C),
V Wind average velocity, (m s^{−1}),
u, w Velocity components, (m s^{−1}),
U, W Dimensionless velocity components,
UCZ Upper convective zone,
x, z Cartesian coordinates, (m),
X, Z Dimensionless Cartesian coordinates,
Z _{LCZ} Dimensionless depth of the solar pond measured at the LCZNCZ boundary,
Z _{NCZ} Dimensionless depth of the solar pond measured at the NCZUCZ boundary,
Greek symbols
α Thermal diffusivity, (m^{2} s^{−1}),
β _{T} Thermal expansion coefficient, (K^{−1}),
β _{C} Concentration expansion coefficient, (m^{3} kg^{−1}),
λ _{w} Thermal conductivity of water, (W m^{−1} K^{−1}),
ν Cinematic viscosity, (m^{2} s^{−1}),
μ Dynamic viscosity, (kg m^{−1} s^{−1}),
ρ Density, (kg m^{−3}),
φ Dimensionless concentration,
θ Dimensionless temperature,
τ Dimensionless time,
ε Salty water extinction coefficient, (m^{−1}),
ε _{w} Water emissivity,
σ Constant of StefanBoltzmann, (W m^{−2} K^{−4}),
Ф Dimensionless absorption coefficient, (Ф = εH),
Subscripts
a Ambient,
b Bottom,
w Water,
l Local,
max Maximum value,
min Minimum value.
Declarations
Acknowledgements
The author wishes to acknowledge Professor Mounir Baccar, the director of Computational Fluid Dynamics and Transfer Phenomena (National School of Engineers of Sfax, University of Sfax, Tunisia), for helpful suggestions and fruitful discussions.
Authors’ Affiliations
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