International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt A fully coupled, transient double-diffusive convective model for salt-gradient solar ponds Francisco Suárez a,*, Scott W. Tyler b, Amy E. Childress c a Graduate Program of Hydrologic Sciences, University of Nevada, Reno 1664 N. Virginia St. MS 175, Reno NV 89557, USA Department of Geological Sciences and Engineering, University of Nevada, Reno 1664 N. Virginia St. MS 175, Reno NV 89557, USA c Department of Civil and Environmental Engineering, University of Nevada, Reno 1664 N. Virginia St. MS 258, Reno NV 89557, USA b a r t i c l e i n f o Article history: Received 26 June 2009 Available online 27 January 2010 Keywords: Solar pond Convection Double-diffusive convection Stability Radiation absorption Transient model a b s t r a c t A fully coupled two-dimensional, numerical model that evaluates, for the first time, the effects of doublediffusive convection in the thermal performance and stability of a salt-gradient solar pond is presented. The inclusion of circulation in the upper and lower convective zone clearly shows that erosion of the nonconvective zone occurs. Model results show that in a two-week period, the temperature in the bottom of the solar pond increased from 20 °C to approximately 52 °C and, even though the insulating layer is being eroded by double-diffusive convection, the solar pond remained stable. Results from previous models that neglect the effect of double-diffusive convection are shown to over-estimate the temperatures in the bottom of the solar pond. Incorporation of convective mixing is shown to have profound impacts on the overall stability of a solar pond, and demonstrates the need to actively manage the mixing and heat transfer to maintain stability and an insulating non-convective zone. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction A salt-gradient solar pond (termed ‘‘solar pond” for this work) is a large-scale solar energy collection system which absorbs solar radiation and stores it as thermal energy for a long period of time [1]. A solar pond consists of three thermally distinct layers (Fig. 1): the upper convective zone, the non-convective zone, and the lower convective zone. The upper convective zone is a relatively thin layer of cooler, less salty water. The non-convective zone has gradients in temperature and salinity and acts as a critical insulator for the thermal storage zone, or lower convective zone. The lower convective zone is a layer of high salinity brine where temperatures are the highest. The solar radiation that penetrates the pond’s upper layers and reaches the lower convective zone heats the highly concentrated brine. The heated brine will not rise beyond the lower convective zone because the effect of salinity on density is greater than the effect of temperature. The stored thermal energy can only escape back to the atmosphere from the lower convective zone by conduction, which makes the stability and thickness of the non-convective zone a critical operating parameter for efficient solar pond operation. Because the brine has a relatively low thermal conductivity, the heat losses by conduction are * Corresponding author. Tel.: +1 775 784 4986; fax: +1 775 784 1953. E-mail addresses: suarezf@unr.nevada.edu (F. Suárez), styler@unr.edu (S.W. Tyler), amyec@unr.edu (A.E. Childress). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.01.017 relatively small. The hot brine in the storage zone, which can reach temperatures greater than 90 °C, may then be used directly for heating, thermal desalination, or for other low-temperature thermal applications [2–5]. Previous studies on solar ponds include experimental, analytical, and numerical investigations carried out to understand the thermal behavior under different operating conditions [6–11]. However, almost all of the previous studies were performed using a one-dimensional thermal analysis without consideration of the dynamics of fluid layers. Kurt et al. [7] modeled the non-convective zone as a series of flat layers, including both heat and mass transfer between layers. The upper and lower convective zone were each modeled as a single, homogenous layer. The thicknesses of each zone were assumed fixed, implying that salt diffusion is negligible or controlled (this is reasonable only when freshwater is added to the upper convective zone and highly saline brine is added to the lower convective zone). This one-dimensional model was compared to experimental data from a prototype pond at Istanbul Technical University. The model predicted the shape of the temperature profile; however, the calculated temperature differed from the experimental data by as much as 8 °C. Atkinson and Harleman [8] developed a one-dimensional wind-mixed model for large-scale solar ponds. Using a turbulent entrainment model, they were able to predict the upper convective zone thickness. They showed that wind mixing is a major problem in large-scale solar ponds, and that management of wind effects via floating grids or F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 1719 Nomenclature cy cp d ea esa esw fi g k t y z A As B C D/Dt DDC Fep GS GT L Lc LCZ NCZ P PP Patm Pr Q RaS RaT RH S Si Sc T Tav Twv UCZ ~ U ~ V wind drag coefficient [–] specific heat at constant pressure [J kg C1] distance [m] vapor pressure [N m2] saturation vapor pressure [N m2] saturated vapor pressure at the water surface [N m2] fraction of energy in the ith bandwidth [–] gravitational acceleration [m s2] thermal conductivity [W m1 C1] time [s] height [m] depth [m] empirical coefficient (Eq. (28)) [–] surface albedo [–] empirical coefficient (Eq. (28)) [–] cloud fraction [–] substantial (or material) time derivative double-diffusive convection apparent salt flux [kg s1] salt-gradient [kg m4] temperature gradient [°C m1] latent heat of vaporization [J kg1] characteristic length [m] lower convective zone non-convective zone pressure [N m2] precipitation [m s1] atmospheric pressure [N m2] Prandtl number [–] heat flux [W m2] solutal Rayleigh number [–] thermal Rayleigh number [–] relative humidity [%] salinity [%, w/w] normalized spectral distribution [–] Schmidt number [–] temperature [°C or K] virtual temperature of air [K] virtual temperature of water surface [K] upper convective zone wind velocity [m s1] velocity field [m s1] other structures is important. This was the first work that allowed for a transient upper convective zone; however, the effect of solute (salt) transport on the entrainment process was neglected in their analysis. Mansour et al. [12] studied the transient heat and mass transfer and stability within a solar pond using a two-dimensional model over a 46-week period. Using the density stability ratio, which is used for static stability, it was found that there are two critical zones: one immediately beneath the water surface, and the other near the bottom of the pond. However, the fluid motion caused by buoyancy within the solar pond was neglected. For this reason, the temperature profile did not show the existence of a well-mixed upper or lower convective zone (i.e., conduction was the dominant process in these zones). Thus, even though some of the instabilities were predicted, mixing in these zones and potential erosion of the non-convective zone were not. While previous work on stability of solar ponds has been conducted, the coupling and potential mixing of both heat and salt transport has not been included, likely due to the complexity of coupling these transport phenomenons. Mixing, driven by coupled Greek symbols as solar altitude angle [rad] solutal expansion coefficient [%1] bS thermal expansion coefficient [°C1] bT attenuation length [m] di ea atmospheric emissivity [–] gi composite attenuation coefficient [m1] jS solutal diffusivity [m2 s1] jT thermal diffusivity [m2 s1] l dynamic viscosity [N s m2] m kinematic viscosity [m2 s1] h refracted angle of the beam light [rad] q density [kg m3] r Stefan-Boltzman constant [=5.67 108 W m2 K4] ~ s wind shear stress [N m2] rate of internal heat generation [W m3] Uh D difference r gradient r divergence r2 Laplace operator Subscripts/superscripts 0 reference state a air bottom bottom of the pond e evaporative heat forced forced convection free free convection i incident l long-wave radiation max maximum min minimum net net heat flux r reflected s sensible heat sed sediment swr short-wave radiation w water y height z depth processes, is known as double-diffusive convection (DDC), in which convective motion is driven by buoyancy where two components with different diffusivities exist simultaneously and make opposing contributions to the vertical density gradient [13]. In solar ponds, as in the ocean, the components that are diffusing simultaneously are heat and salt. Several researchers have studied the stability of a thermohaline horizontal layer with linear gradients, such as in the non-convective zone, using linear perturbation theory [14–16]. Results obtained from these investigations have provided information concerning the onset of instabilities inside a solar pond as well as possible unstable or stable states. However, because of the assumption of infinitesimal perturbations or the boundary conditions used, these theoretical findings cannot be extrapolated to real situations where the solar pond is subject to large and drastic perturbations as a consequence of meteorological conditions. Only few researchers have taken into account the double-diffusive phenomena beyond the onset of convection within a solar pond in two dimensions [17,18]. Hammami et al. [17] studied the transient natural convection in an enclosure with a vertical solute gradient. This research used the Navier-Stokes, energy, and 1720 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 i Qswr r Qswr Ql Qe 0 Qswr UCZ z Fluid Qs Qswr NCZ Co Te nc m pe en ra tra tu tio re n Useful heat LCZ Fig. 1. Stratification and heat balance in a salt-gradient solar pond. UCZ is upper convective zone, NCZ is non-convective zone, and LCZ is lower convective zone. mass conservation equations to evaluate the performance of a solar pond during the night. It took into account fully coupled momentum, heat, and mass transfer but did not include the internal heating of the fluid due to solar radiation absorption. Giestas et al. [18] developed a two-dimensional numerical model for solar ponds using the same equations as those used by Hammami et al. [17]. They validated their model using experimental data and, even though there were some three-dimensional effects present in the experiments, which were not included in the model, the twodimensional simulations were found to capture the main features of the circulation and the heat and mass transfer rates of their experiments [18]. However, the internal heating of the fluid due to the absorption of solar radiation was not included. Whereas previous studies provide insight into individual components of solar pond behavior for long-term operation, investigation of critical factors in solar pond operation or maintenance over days to weeks, such as the effects of solar radiation absorption, salt-gradient adjustment, or wind on the stability of the internal zones has not been investigated. In particular, the timing and nature of the erosion of the non-convective zone has not been adequately modeled nor have the transient behavior of the interfaces been simulated in great detail. In the current investigation, a fully coupled two-dimensional numerical model is developed to predict the transient performance of a solar pond during energy collection and storage. The main objective for developing this model is to represent short- and medium-term operation, e.g., maturation or maintenance, of a solar pond by allowing the interfaces between layers to dynamically adjust. The mass conservation for water and species, momentum, and energy equations are coupled with the equation for fluid density. The effect of wind shear on the surface of the solar pond is also taken into account as is a complete energy balance at the pond surface, including internal heating by solar radiation absorption. 2. Materials and methods 2.1. Model description Mass conservation for water and species (i.e., continuity and convective-diffusive equations), momentum (i.e., Navier-Stokes), and energy conservation equations are coupled with the density of the fluid, assumed to be linear with temperature and salinity. These equations are given by [19]: @q VÞ ¼ 0 þ r ðq~ @t @ VSÞ ¼ r ðqjS rSÞ ðqSÞ þ r ðq~ @t @ ~ V~ VÞ ¼ rP þ lr2 ~ V þ q~ ðqVÞ þ r ðq~ g @t DT DP ¼ r ðkrTÞ þ bT T þ Uh Dt Dt qðS; TÞ ¼ q0 ½1 bT ðT T 0 Þ þ bS ðS S0 Þ qcp ð4Þ ð5Þ 3 where q0 = 1088.6 kg m , T0 = 60 °C, and S0 = 15%, w/w are the reference density, temperature, and salinity, respectively; bT = 5.24 104 °C1 and bS = 6.82 103%1 are the assumed thermal and solutal expansion coefficients, respectively, and correspond to a sodium chloride (NaCl) solution [1]; and all the other parameters are defined in the nomenclature section. The fluid properties (e.g., viscosity, thermal conductivity), as a function of temperature and salinity, have been estimated using thermo-physical properties previously reported [20]. 2.1.1. Boundary conditions The net heat flux across the water surface (Fig. 1), Q 0net , includes net long-wave radiation emission, Ql, evaporation, Qe, and sensible heat flux, Qs: Q 0net ¼ Q l þ Q e þ Q s ð6Þ and is controlled by meteorological forcing. The parameterization of the surface heat fluxes is presented in Appendix A. For this study, latent and sensible heat formulations appropriate for typical solar pond environments (warm and sunny environments, with variable wind speed and humidity) and requiring typically collected meteorological data have been implemented. The net heat flux across the surface does not include the penetrating solar short-wave radiation flux, rather, this heat flux is distributed along the fluid column. At the pond surface, the wind causes a shear stress, ~ s , to act on the surface of the water that can be represented as [21]: ~ y jU ~y ~ s ¼ qa cy jU ð7Þ ~y is the wind velocity at a height of y. Here we assume a where U wind drag coefficient, cy, derived using a logarithmic wind profile (Charnock formulation) [22]. Although no salt actually crosses the water surface, evaporation and precipitation can cause an apparent salt flux, Fep, which can be estimated by [23]: F ep ¼ S½PP q Q e =L ð8Þ The bottom of the pond was considered impermeable and adiabatic (i.e., fluid, salt, and heat fluxes equal zero), and the no-slip condition was applied. Also, it was assumed that any solar radiation that reaches the bottom is absorbed by the fluid at this deepest depth. ð1Þ ð2Þ ð3Þ 2.1.2. Internal heating due to solar radiation absorption The short-wave solar radiation that penetrates the air–water interface, Q 0swr , is treated separately from the heat flux across the water surface because it causes internal heating of the fluid. It is attenuated with the distance, d, traveled by the light beam within 1721 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 the fluid, and is wavelength-specific [1]. To represent this heat flux at a depth z within the fluid, Q zswr , various relationships have been proposed [3,24,25]. For low-turbidity seawater, Hull et al. [1] recommend the Rabl-Nielsen simplified four-term series fit [3], which was obtained by discretizing the full spectrum into four bandwidths with different extinction coefficients: 4 X i¼1 4 X d z Si exp Si exp ¼ Q 0swr di di cosðhÞ i¼1 ð9Þ The values of Si and di for low-turbidity seawater are presented elsewhere [6]. In the model developed here, the internal heating of the fluid column is treated as a volumetric heat source, Uh, in the energy equation: Uh ðzÞ ¼ @Q zswr @z ð10Þ 2.2. Numerical methods and validation Fluent 6.3 [26], a commercial computational fluid dynamics code, using a finite-volume discretization, was used to solve the governing equations. The pressure-based, segregated solver was selected and spatial discretization on cell-center and second order upwind weighting were used. The gradients and derivatives were evaluated using the Green-Gauss cell-based method; the SIMPLE (semi-implicit method for the pressure linked equations) algorithm was used for the pressure-velocity coupling, and because the fluid is buoyancy-driven, the PRESTO! (pressure staggering option) scheme was selected to interpolate the pressure. To ensure independency of numerical results with respect to spatial discretization, several grids were tested. Cells as large as 0.02 0.02 m were found to be appropriate for the problem under study. To guarantee convergence of the solution, the values of residuals of the governing equations were monitored. A converged solution 2.2.1. Validation of momentum and heat transport in a shallow lake Mirror Lake is a small eutrophic lake located on the University of Connecticut campus. Meteorological data from this study site, collected at 30-min intervals, were obtained as described by Branco and Torgersen [27]. Temperature data at different depths within the lake and in the sediment beneath the lake bottom were available at 5-min interval. A chain of thermistors with a precision of ±0.01 °C (and a variability amongst thermistors of ±0.02 °C) were used to obtain these measurements [27,30]. The simulations started at 12:00 am on March 19, 2002 and finished at 12:00 am on March 24, 2002. Fig. 2(a) shows the meteorological data for this time period. At the initial time, the temperature profile inside the lake was uniform and equal to 6.29 ± 0.03 °C. This value and zero fluid velocity were used as initial conditions. The heat flux formulation (Eq. (6)) and the wind Incident solar radiation Air temperature 800 10 600 5 400 0 200 -5 0 0 1 2 3 4 10 -10 5 Incident solar radiation [W m-2] (b) Model evaluation 15 Air temperature [°C] Incident solar radiation [W m-2] (a) Model validation 1000 1200 36 1000 30 800 24 600 18 400 12 200 6 0 Air temperature [°C] Q zswr ¼ Q 0swr was achieved with residuals as small as 104 for the continuity, momentum, and species equations, and 106 for the energy equation. In addition, the local Reynolds number (based on the cell length) was monitored to check the flow regime of the simulations. As the simulations showed the local Reynolds number to be less than 400 in the entire domain, a laminar viscous model was used [19]. Because only limited experimental data for solar ponds are available, the model was validated in two ways. First, the momentum and heat transport, including the internal heating of the water column due to solar absorption, were validated against limnological data from a shallow lake. Mirror Lake in Storrs, Connecticut USA was chosen because it: (i) shows diel stratification or mixing, and (ii) has readily available weather data (Fig. 2) [27]. In the second validation, double-diffusive convection without consideration of internal heating of the fluid was validated by comparing the results of a steady-state simulation with experimental and numerical results published by Han and Kuehn [28,29]. 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 100 5 50 80 4 40 3 30 2 20 1 10 60 4 40 2 20 0 0 0 1 2 3 4 Days Starting on March 19, 2002 at 12:00 am 5 0 -1 Relative humidity [%] 6 Wind speed [m s ] Wind speed 8 Relative humidity [%] Wind speed [m s-1] Relative humidity 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Days Starting on August 7, 2002 at 12:00 am Fig. 2. Mirror Lake meteorological data (30-min intervals) utilized in the simulations. (a) Input data used to validate momentum and heat transport, including the internal heating of the water column. (b) Input data used to evaluate the transient performance of the solar pond under summer conditions. 1722 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 shear stress (Eq. (7)) were used as boundary conditions at the lake surface. A time step of 10 s and a constant-level water surface of 1.2 m were assumed. A lake section of 1.0 m width (in the horizontal direction) was simulated. The sides of this section were modeled using the symmetry boundary condition. The no-slip condition was used as the boundary condition at the bottom of the lake. The solar radiation that penetrates the air–water interface was estimated using the incident solar radiation and the water surface albedo (see Appendix A). Because the bottom boundary conditions for heat are uncertain, three different scenarios were considered: (i) any solar radiation that reaches the lake bottom was absorbed by the fluid and no heat exchange occurs between the sediment and the lake (i.e., insulated bottom scenario). Hereafter, this will be referred to as thermal boundary condition 1 (TBC1); (ii) in addition to the solar radiation that reaches the lake bottom, the temperature difference between the bottom of the lake and the sediment temperature at 0.05 m below it were used to estimate an additional heat flux, which was calculated using a sediment thermal conductivity of 1.5 W m1 °C1 [31] (TBC2); and (iii) in addition to the solar radiation that reaches the lake bottom, the sediment temperatures at 0.05 and 0.15 m below the bottom of the lake were used to estimate a heat flux that was assumed to occur at the lake bottom (TBC3). The short-wave radiation flux that causes the internal heating of the water column was represented by [27]: Q zswr ¼ Q 0swr 7 X fi expðgi zÞ ð11Þ the simulation) data were run to analyze the influence of time averaging on the output. The initial thicknesses of the upper, non-convective, and lower convective zones were chosen to be 0.1, 0.4, and 0.5 m, respectively. Initial salinities were 6% and 26% in the upper and lower convective zones, respectively. The initial salinity distribution in the non-convective zone was varied linearly. A constant temperature of 20 °C and zero fluid velocity were used as initial conditions. These initial conditions are representative of typical solar ponds at construction. Evolution of the pond dynamics was simulated using a time step of 0.5 s. A uniform grid with cells of 0.02 0.02 m was used. For simplicity, a constant-level water surface was assumed as was zero flux of salts across the top of the pond. To compare the thermal behavior when double-diffusive convection is considered with that of previous models, which neglected salt transport in the evolution of the interfaces, simulations with a constant salinity profile and fixed zone thickness were also conducted (i.e., neglecting or controlling salt diffusion). The upper and lower convective zones were modeled assuming instantaneous mixing of these zones [7]. Salinities of 6% and 26% were used in the upper and lower convective zones, respectively. Salinity varied linearly in the non-convective zone, i.e., the same salt profile used as the initial condition when double-diffusive convection is included. In contrast to the fully coupled simulation, the heat transfer in the non-convective zone was modeled using the energy equation for pure conduction and internal heat generation [19]: i¼1 where the values of fi and gi for pure water are presented elsewhere [27]. 2.2.2. Validation of double-diffusive convection in an enclosed cavity The second validation step was derived from the results of Han and Kuehn [28,29], who experimentally and numerically studied double-diffusive convection in a vertical rectangular cavity of aspect ratio 4:1 with temperature and concentration gradients imposed in the horizontal direction. Both aiding and opposing buoyant conditions, occurring when the temperature and buoyant forces act in the same or opposite direction, respectively, were studied. For the numerical model, a two-dimensional steady-state flow within the cavity was considered. A non-uniform grid spacing in the horizontal direction was used. In the vertical direction, a uniform grid spacing was used to find the possible multiple convective cells in the flow structure. The typical numbers of cells used in their simulations were 34 and 130 in the horizontal and vertical directions, respectively. The simulation presented here was designed for opposing buoyant conditions, with the following parameters: thermal and solutal Rayleigh numbers of RaT = 3.2 106 and RaS = 2.4 107, respectively; Prandtl number, Pr = 8; and Schmidt number, Sc = 2000. Opposing buoyant conditions were tested by fixing Tmin and Smin at the left wall and Tmax and Smax at the right wall. The top and bottom walls were impermeable and adiabatic. The grid used was uniform with 100 and 200 cells in the horizontal and vertical directions, respectively. qcp @T @ @T ¼ k þ Uh ðzÞ @t @z @z A constant temperature of 20 °C in the entire domain was used as the initial condition. In addition, two surface boundary conditions were compared: (i) use of the heat flux formulation presented in Eq. (6), and (ii) use of the air temperature as the temperature of the water surface. The second boundary condition was tested because it has been used in several previous models [1]. 2.3.2. Stability and interface motion A solar pond cannot operate without an internally stable nonconvective zone. Elementary buoyancy considerations yielded a minimum requirement for static stability: density must be at least uniform or increasing downward to prevent gravitational overturn. This static stability criterion is not adequate to insure dynamic stability because of the potential energy available from the destabilizing temperature distribution that occurs inside a solar pond [1,13,32]. When taking into account the large difference between heat and salt diffusivities (a double-diffusive system), the salinity gradient in the non-convective zone must be larger than that required to satisfy the elementary static buoyancy criterion alone. Dynamic marginal stability may be defined as [15]: RaT ¼ m þ jS jS jS 27 4 RaS þ 1 þ 1þ p m þ jT jT m 4 ð13Þ The Rayleigh numbers are the ratios between buoyancy driving force (due to thermal or solutal expansion or contraction) and viscous drag force, and are defined by: 2.3. Model application RaT ¼ 2.3.1. Transient behavior of a solar pond After the validation of the model using the two situations presented above, the transient behavior of a simulated solar pond was evaluated. Simulations utilized a 1-m deep 1-m wide pond with NaCl as the solute. The solar pond was subject to the meteorological conditions recorded for Mirror Lake during the summer of 2002 (starting at 12:00 am on August 7) (Fig. 2 (b)). Simulations using 30-min, daily-averaged, and averaged (constant throughout ð12Þ gbT DTL3c mjT RaS ¼ gbS DSL3c mjT ð14Þ The dynamic marginal stability criterion was obtained for a layer of fluid with constant gradients of temperature and salinity, as well as defined boundary conditions within the fluid. Nevertheless in a real solar pond, linear profiles of temperature and salinity are rarely present [33]. In addition, the boundary conditions of the non-convective zone are difficult to define as the interfaces between the non-convective zone and the upper and lower zones are in constant F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 motion for time-varying conditions [1,34]. Empirically, it has been found that for an ideal simple case in which the circulation of a mixed zone (e.g., the lower convective zone) is due to a constant convective heat transfer and where all other parameters (e.g., level of the interfaces, temperature, salinity) have reached steady-state, the boundaries of internal convective zones are well described by a power law [34,35]: GS ¼ 28G0:63 T ð15Þ where GS (°C m1) and GT (kg m4) are the salt and temperature gradients close to each interface, respectively. This empirical relationship is not a stability threshold, but instead reflects the possible temperature and salt gradients that can coexist at the zone boundaries in a stationary state; and is termed ‘‘boundary equilibrium”. Eq. (15) provides a basis to predict what will occur when either the temperature or salt gradients are altered. For salt gradients greater than those predicted by Eq. (15), gradient growth will occur (i.e., growth of the non-convective zone) and the nearby convective zone will reduce its thickness to accommodate this change. On the other hand, for salt gradients less than those predicted by Eq. (15), gradient erosion will occur (i.e., erosion of the non-convective zone) and the nearby convective zone will expand. Interface velocities equal to or less than 0.14 cm day1 are typically considered as equilibrium conditions [34]. Eqs. (13) and (14) were combined to describe the dynamic marginal stability of the non-convective zone in terms of temperature and salt gradients. Stability and interface motion within the numerical solar pond simulations were studied by comparing different simulated temperature and salt gradients with the internal stability and boundary equilibrium lines, respectively. These gradients were evaluated within the boundaries of the non-convective zone using z = 0.02 m. 3. Results and discussion 3.1. Model validation 3.1.1. Validation of momentum and heat transport in a shallow lake The validation of the momentum and heat transfer components of the model, including the internal heating of the water column due to solar radiation absorption, was performed by predicting the temperatures from the Mirror Lake data set. The corresponding meteorological data for this time period is presented in Fig. 2(a). The measured and modeled temperatures at different depths within Mirror Lake are presented in Fig. 3. In general, the trend of the measured temperatures is well represented by the model, with a small underestimation of the temperatures. For depths of 0.10 and 0.35 m, the measured and modeled temperatures agree fairly well throughout the simulation period. When it is assumed that the lake bottom is insulated and that the solar radiation that reaches the lake bottom is absorbed by the fluid at this depth, i.e., TBC1, the differences between the measured and modeled temperatures are less than 2 °C during the entire simulation. For depths greater than 0.35 m, the model did not adequately represent the observed temperature peak of the third day. At this time, the largest temperature differences between the experimental and modeled data are as much as 5 °C, but for the other times the temperature differences are always less than 2 °C and typically less than 1 °C. Because greater differences between the measured and modeled data occur deeper in the lake, and at the bottom where the thermal boundary conditions are uncertain, the thermal interaction between the sediment beneath the lake and the lake bottom was also considered. As a first approach, the sediment temperatures beneath the lake were used to estimate the magnitude of the heat flux that occurs at the bottom. This heat flux had the same 1723 order of magnitude as that of the solar radiation at the lake bottom (10-20 W m2). In this case, it is important to consider the thermal interaction between the sediment and the lake, as well as the solar radiation that reaches this point. When the temperature at 0.05 m below the lake bottom was used to estimate an additional heat flux at the sediment-water interface, i.e., TBC2, the measured and modeled temperatures are better represented during the first four days of simulations, but in the fifth day the modeled temperatures using TBC2 are lower than those simulated using TBC1. If the sediment-water heat flux is estimated using the sediment temperatures between 0.05 and 0.15 m beneath the lake bottom, i.e., TBC3, the measured and modeled temperatures agrees well at all depths. A very good agreement occurs at the final time of simulation. The representation of the peak of the third day at deeper depths is slightly improved when TBC2 and TBC3 are used, but remain poorly correlated for this time period. The agreement with the experimental data presented here appears satisfactory considering that both experimental and modeling uncertainties exist. For example, the meteorological variables were not measured in the lake but a few km from it, and a rain event of 1.6 cm during the second day of simulation was neglected which may have produced both surface changes as well as runoff. In addition, water column turbidity may have varied and produced significant changes in the values of the extinction coefficients. This validation also demonstrates that this model is not limited only to thermohaline systems; therefore, it can be used in natural water bodies where circulation is driven by density gradients created by the diffusion of only one component (e.g., temperature). 3.1.2. Validation of double-diffusive convection in an enclosed cavity As the model reasonably represented the momentum and heat transport within a natural water body, where the effect of solute concentration on water circulation is not important, the next step in the validation was to include both the effect of temperature and solute concentration on fluid density. Fig. 4 shows the velocity vectors, stream function, dimensionless temperature (i.e., ½T T min =½T max T min , and dimensionless concentration (i.e., ½S Smin =½Smax Smin ) obtained in the simulations to validate the double-diffusive convective phenomenon. The temperature and concentration fields, as well as the flow pattern are similar to theoretical and experimental results obtained by Han and Kuehn [28,29]. A three-cell structure is obtained inside the vertical cavity. The fluid in these cells moves in the counterclockwise direction and is driven primarily by thermal buoyancy. The fluid inside the thin solutal boundary layers, i.e., in the region near the vertical walls, moves against the bulk fluid motion and does not appear to affect the overall fluid motion in the center of the cavity. As Han and Kuehn [29] point out, this type of flow pattern has not been observed in flows where only one component is diffusing. This complex flow structure occurs because the buoyant forces due to thermal and solutal gradients act in opposing directions on each vertical wall. 3.2. Model application 3.2.1. Transient behavior of a solar pond Based upon the validation results of the various model components, the model was next used to predict the transient behavior of a solar pond. Fig. 5 shows the evolution of the temperature, density, and salinity profiles using 30-min interval meteorological data recorded at Mirror Lake during the summer of 2002 (Fig. 2(b)). After 13 days of simulation, the temperature in the upper convective zone always remained below 28 °C, and was influenced by the meteorological conditions. In this zone, the density increased from 1040 to 1080 kg m3 and the salinity from 6% to nearly 12%. The decrease in temperature at the surface is mainly due to evaporation, which 1724 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 Modeled - TBC1 Experimental 10 Temperature [°C] 9 8 7 7 6 6 5 5 4 4 3 3 2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10 Temperature [°C] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2.5 3.0 3.5 4.0 4.5 5.0 10 Depth: 0.55 m 8 7 7 6 6 5 5 4 4 3 3 2 0.0 Depth: 0.75 m 9 8 2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10 9 Depth: 0.35 m 9 8 9 Modeled - TBC3 10 Depth: 0.10 m 2 0.0 Temperature [°C] Modeled - TBC2 0.0 0.5 1.0 1.5 10 Depth: 1.10 m 9 8 8 7 7 6 6 5 5 4 4 3 3 2 0.0 Depth: 1.17 m 2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time [d] 0.0 0.5 1.0 1.5 2.0 Time [d] Fig. 3. Measured and modeled temperatures at different depths in Mirror Lake using the fully coupled numerical model. Simulations started at 12:00 am on March 19, 2002. TBC1 is when an insulated lake bottom is considered, TBC2 is when the sediment temperature at 0.05 m below the bottom was used to estimate a sediment-water heat flux, and TBC3 is when the sediment temperatures at 0.05 and 0.15 m below the bottom were used to estimate a sediment-water heat flux. cools the pond. The temperature, density, and salinity show a uniform profile in the upper convective zone, consistent with a mixed zone, due to natural convection or wind shear at the surface. The temperature in the lower convective zone increased to greater than 50 °C, while density and salinity slowly decreased in this zone, from approximately 1197–1164 kg m3 and 26–24%, respectively. The increase in the temperature at the bottom of the lower convective zone is a result of the assumption that any solar radiation that reaches this point is absorbed by the fluid at this depth. In addition, the density profile (Fig. 5(b)) clearly shows that both the upper and lower convective zone thicknesses increase with time. As a result, the nonconvective zone is progressively eroded by convective mixing above and below. This thickness reduction reduces the efficiency of the pond, as higher conductive heat transfer occurs through the nonconvective zone. While an increase in the thickness of the lower convective zone will provide more heat storage, this occurs at the expense of the heat losses that occur in the other zones. The temperature field and velocity vectors for different times of simulation are presented in Fig. 6 (see also Supplementary Video 1 of the Electronic Annex in the online version of this article). At 6.5 days (noon) (Fig. 6(a)), in the upper convective zone, no natural convection cells are observed. Solar radiation absorption warms the fluid that is at shallow depths and helps maintain a stable configuration in almost the entire upper convective zone. An unstable zone does occur in the surface boundary layer due to the heat fluxes across the surface and the wind shear stress, driving the relatively weak and shallow circulation that is observed at the surface of the pond. The non-convective zone is a stable zone where the temperature increases from approximately 23–42 °C. No fluid motion is observed in this zone; hence, global circulation is suppressed by the non-convective zone. In the lower convective zone, the temperature in the bottom boundary layer is higher due to solar radiation absorption. Irregular convective cells are observed in this zone. These cells are larger than those observed in the upper convective zone, although the maximum velocity is smaller. At 7.0 days (midnight) (Fig. 6(b)), the temperature profile behave in the same fashion as that at 6.5 days, however lower temperatures are observed due to night-time cooling (see also Supplementary Fig. 1). In the upper convective zone, larger cells than those found at 6.5 days (previous noon) are observed due to the 1725 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 Present model 1.00 0.94 0.89 0.83 0.78 0.72 0.67 0.61 0.56 0.50 0.44 0.39 0.33 0.28 0.22 0.17 0.11 0.06 0.00 Han and Kuehn model [29] V Vmax ψ ψmax T − Tmin Tmax − Tmin S − Smin Smax − Smin Fig. 4. Comparison of the velocity vectors, stream functions (w), temperature, and concentration (in dimensionless form) between the present model and the model presented by Han and Kuehn [29]. Aspect ratio = 4:1 (vertical:horizontal). heat exchange across the surface and the heat transferred from the fluid of the non-convective zone. In the lower convective zone, irregular convective cells are observed. These are larger than those observed at 6.5 days. At 12.5 days (noon) (Fig. 6 (c)), larger convective cells than those observed at 6.5 or 7.0 days occur in the lower convective zone. In addition, the temperatures across the entire pond are greater than those observed at 6.5 days. At 13.0 days (midnight) (Fig. 6 (d)), the flow pattern observed is similar to the flow pattern observed at 7.0 days (midnight), but again larger convective cells are observed in both the upper and lower convective zone. At this time (13.0 days), the shape of the cells is more defined than before, with wide and short cells in the upper zone and more squared cells in the lower zone. The minimum temperature at the surface is lower than that observed at the previous noon and is 1726 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 0.0 0.1 0.2 Time Depth [m] 0.3 0.4 0.5 0.6 0.7 0.8 (a) 0.9 1.0 15 20 25 30 35 40 45 50 55 60 Temperature [ºC] 0.0 Time 0.1 0.2 Depth [m] 0.3 0.4 0.5 0.6 Time 0.7 0.8 (b) 0.9 1.0 1040 1060 1080 1100 1120 1140 1160 1180 1200 Density [kg/m3 ] 0.0 Time 0.1 0.2 Depth [m] 0.3 0.4 0.0 days 1.5 days 3.5 days 5.5 days 7.5 days 9.5 days 11.5 days 13.0 days 0.5 0.6 0.7 0.8 0.9 Time (c) 1.0 5 10 15 20 25 30 Salinity [%] Fig. 5. Transient evolution of the (a) temperature, (b) density, and (c) salinity profiles. similar to the minimum temperature observed at 7.0 days, suggesting that the temperature at the surface is primarily influenced by meteorological conditions. The maximum velocity was found to occur at the surface of the solar pond for the entire simulation and was always less than 2.0 102 m s1, resulting in a local Reynolds number that was always less than 400. The simulated velocities are in agreement with velocities measured in the upper and lower convective zones of solar ponds reported in the literature [1]. These are on the order of 1.0 102 m s1, with corresponding Reynolds numbers that are far from the transition to turbulent flow. As the maximum velocity inside the pond was observed at the surface, wind and heat surface exchange are critical factors that should be taken into account for design, operation, and maintenance of solar ponds. It is important to note that even though the non-convective zone thickness de- creased, it still suppresses convection and does not allow complete overturn of the solar pond. Additionally, the convective cells in Fig. 6 were generally found to be irregular-shaped, suggesting that changes in meteorological conditions did not give enough time for the cells to develop a regular shape, such as that observed in steady-state experiments [28]. This emphasizes the importance of having a transient model to represent the hydrodynamics within solar ponds driven by real climate forcing and for transient situations such as heat extraction or fresh water addition. Fig. 7 shows the final temperature profiles at 13 days for simulations using different temporal averaging of the input data, i.e., 30-min, daily-averaged, and averaged, different thermal boundary conditions at the surface, i.e., using the heat flux formulation presented in this work and using air temperature, and with or without the inclusion of double-diffusive convection, i.e., dynamic or static internal boundaries. Fig. 7(a) shows a comparison of constant and variable fluid properties (e.g., viscosity, thermal conductivity as a function of temperature and salinity) for the 30-min interval data, considering double-diffusive convection, and using the heat flux formulation described in Eq. (6) and Appendix A. For constant fluid properties, the temperature in the lower convective zone is 51.9 °C, while for variable fluid properties it is 53.0 °C. Fig. 7(a) also shows a comparison of constant fluid properties for the different averages of meteorological data. For the daily-averaged and averaged data, the final temperatures in the lower convective zone are 53.9 and 54.4 °C, respectively. Fig. 7(b) shows the difference in the temperature profile when the air temperature is used as the surface boundary condition, considering double-diffusive convection, and for constant fluid properties. In addition, the temperature profile obtained using the heat flux formulation is shown. For all the different meteorological averaged data and using air temperature, the lower convective zone temperature is overpredicted when compared to the temperature profile estimated using the heat flux formulation. The averaged data predicts a temperature of 56.4 °C in the lower convective zone. Fig. 7(c) shows the difference in the temperature profile with or without the inclusion of double-diffusive convection and using the heat flux formulation. For 30-min interval data, and when double-diffusive convection is not considered, a temperature of 55.8 °C is observed in the lower convective zone. When air temperature is used as the surface boundary condition, and for 30-min interval data, as is shown in Fig. 7(d), the estimated temperature in the lower convective zone is 58.0 °C. For averaged data, the estimated temperature in this zone is 58.2 °C; therefore, simplified models may over-estimate efficiencies for the collected and stored heat. The inclusion of double-diffusive convection inside the solar pond has an important effect on the erosion of the non-convective zone. This erosion is augmented by convective cells that circulate in the lower and upper convective zones. If double-diffusive convective effects were negligible, the erosion of the non-convective zone will be much slower and mainly due to salt diffusion. 3.2.2. Stability and interface motion Fig. 8(a) shows the dynamic marginal stability line (solid line) calculated for 15% NaCl at 60 °C, which predicts the onset of double-diffusive convection for infinitesimal perturbations [15]. Theoretically, points falling below this line represent unstable states and points falling above this line represent stable states. Fig. 8(a) also shows the boundary equilibrium (dashed line) calculated using Eq. (15). Points on the dashed line suggest stationary interfaces. For points below the boundary equilibrium, gradient erosion occurs and for points above it, gradient growth occurs [1]. As the simulations showed that the depths of the non-convective zone boundaries were not fixed in time (described below), the positions of the salt and temperature gradients plotted in this stability diagram were not fixed, i.e., for 1 hr of simulation, the upper boundary 1727 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 (a) 6.5 days (noon) (b) 7.0 days (midnight) 0.0 Tmin = 21.7 °C Depth [m] 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 Tmin = 20.4 °C 0.1 20 25 30 0.9 Tmax = 47.8 °C Tmax = 43.0 °C 35 1.0 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 40 (c) 12.5 days (noon) (d) 13.0 days (midnight) 0.0 0.0 Tmin = 23.8 °C Depth [m] 0.1 Tmin = 20.8 °C 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 Tmax = 57.2 °C 45 Temperature [°C] 0.0 50 55 60 Tmax = 52.0 °C 1.0 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 x position [m] 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x position [m] Depth [m] Depth [m] Fig. 6. Temperature field and velocity vectors inside the solar pond for different simulation times. The maximum velocity never exceeded 2 102 m s1 and the local Reynolds never exceeded 400, suggesting laminar flow throughout the simulation. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Heat flux formulation Considering double-diffusive convection (a) Constant properties, 30-min Variable properties, 30-min Constant properties, daily-averaged Constant properties, averaged 15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 25 30 35 40 45 50 55 60 (c) DDC, 30-min No DDC, 30-min No DDC, daily-averaged No DDC, average 20 25 30 35 40 45 Temperature [°C] 50 55 60 Air temperature at the surface Considering double-diffusive convection (b) 30-min, HFF 30-min Daily-averaged Average 15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Heat flux formulation 15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 25 30 35 40 45 50 55 60 55 60 Air temperature at the surface (d) DDC, 30-min, HFF DDC, 30-min No DDC, 30-min No DDC, daily-averaged No DDC, averaged 15 20 25 30 35 40 45 Temperature [°C] 50 Fig. 7. Temperature profile inside the solar pond at the end of the simulations (13 days) using 30-min, daily-averaged, and 13-day average intervals for the meteorological data. (a) Considering double-diffusive convection (DDC) and the heat flux formulation (HFF) as top boundary condition with constant and variable fluid properties. (b) Considering double-diffusive convection and air temperature as top boundary condition. The results using the heat flux formulation with 30-min interval data are also plotted. (c) Without considering double-diffusive convection and the heat flux formulation as top boundary condition. The results considering double-diffusive convection with 30-min interval data are also plotted. (d) Without considering double-diffusive convection and air temperature as top boundary condition. Results considering doublediffusive convection with 30-min interval data are also plotted. 1728 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 of the non-convective zone (i.e., the UCZ–NCZ interface) was located at approximately 0.12 m depth, but at 13 days this interface was located at 0.22 m depth. Two situations were considered, using (i) 30-min interval input data, and (ii) averaged input data. For these situations, three different times of simulation were plotted: 1.0 day (midnight), 6.5 days (noon), and 13 days (midnight). All the gradients plotted on the stability diagram (Fig. 8(a)) fall above the dynamic marginal stability line, showing that the temperature and salt gradients in the non-convective zone are in the theoretically stable zone. Even though these points correspond to a stable state, there are some instabilities in the interfaces of each zone. In fact, the general trend of the gradients plotted in the stability diagram is to move towards an unstable state because of the non-convective zone erosion. In addition, as is shown in Fig. 6, the fluid velocities become non-zero at the initial internal boundaries of the non-convective zone, confirming that the upper and lower convective zones are increasing their thicknesses. This issue must be taken into account for maintenance purposes, and the present fully coupled model, as contrary of previous models, is able to assess this. At 1 day of simulation, the gradients at the lower convective zone interface for 30-min input data fall on the boundary equilibrium (Fig. 8(a)), suggesting a stationary interface, which can be supported by the results presented in Fig. 8(b). For the aver- 1.0 day 6.5 days 13.0 days 30-min data - void symbols Average data - filled symbols UCZ - squares LCZ - circles 3.0 2.6 y ilit tab s al n Bo rgi c mi ma na 2.2 Log GS ium libr qui ye ar und Dy 1.8 1.4 1.0 (a) Theoretically unstable 0.6 0.6 1.0 1.4 1.8 Log GT 2.2 2.6 3.0 0.0 0.1 0.2 UCZ-NCZ interface Depth [m] 0.3 0.4 0.5 LCZ-NCZ interface 0.6 0.7 30-min Averaged 0.8 0.9 (b) age meteorological data, these gradients fall just above the boundary equilibrium, suggesting that the configuration may be close to a stationary state. For 6.5 and 13 days, all the gradients fall below the boundary equilibrium line, confirming erosion of the non-convective zone, which is the general trend observed in the simulations. The transient evolution of the interfaces is presented in Fig. 8(b) for (i) 30-min interval input data, and (ii) averaged input data. Defining the boundaries of the upper and lower convective zones as the zones where the difference in dimensionless density (i.e., ½q qmin =½qmax qmin ) differs by less than 0.25% for different elevations, the interfaces of the zones within the solar pond were found for each time of simulation. In both situations, the UCZ– NCZ interface moves downwards and the lower boundary of the non-convective zone (i.e., the LCZ–NCZ interface) moves upwards. Thus, both convective zones increase in size by eroding the nonconvective zone. When 30-min interval data is used, the upper boundary moves rapidly at the beginning of the simulation and then becomes slower. On the other hand, the lower interface always moves more slowly. If averaged data are used, the changes in thickness of the upper and lower zones are slower than those observed for 30-min interval data. At the end of the simulations, the positions of the interfaces are similar when predicted by both types of input data. In the 13 days of simulation, the upper and lower convective zone thicknesses increase from 0.10 to 0.22 m and from 0.50 to 0.56 m, respectively. As a result of the increase in the thicknesses, the non-convective zone decreases from 0.40 to 0.22 m at the end of the simulation. The main cause of the non-convective zone erosion seems to be the fast growth in the thickness of the upper zone. This growth is explained by wind effects and penetrative convection as a consequence of the surface cooling at night, i.e., rapid changes that occur due to meteorological conditions. This emphasizes how critical is to take into account wind and heat exchange that occur at the surface of the pond. A larger salt-gradient could be one solution to halt the upper zone growth. This analysis clearly shows that the non-convective zone continually erodes. Previously published numerical models usually assume that the distinct layers of solar ponds remain in stable equilibrium and have constant thicknesses [6,7,11]. In the simulations presented here, the non-convective zone remains stable throughout the simulation, but the simulation is capable of allowing the internal interfaces to be in constant motion, and thereby correctly modeling the physics of the erosion of the non-convective zone. This is in agreement with previous experimental works that point out that the equilibrium positions of the boundaries can be substantially changed without losing the internal stability of the non-convective zone [35,36]. To our best knowledge, this is the first numerical model that represents the interface motion without losing the internal stability of the non-convective zone. This work also shows that the change in the positions of the interfaces always occurs naturally and could be exacerbated by solar pond management, such as the modification of the salt profile by liquid injection. Finally, it should be noted that if long-term simulations without maintenance are carried out, salt diffusion tends to create a uniform salt-gradient and thus, the pond overturns and loses its capacity to collect and store heat. Therefore, zone and gradient management must occur to maintain solar pond performance. 1.0 0 1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 Time [d] Fig. 8. (a) Internal stability and boundary equilibrium as a function of the temperature and salt gradients (GT and GS). Eq. (13), calculated for 15% NaCl at 60 °C, was used to calculate the dynamic marginal stability. Eq. (15) was used to calculate the boundary equilibrium. (b) Interface motion within the solar pond using 30-min and average data. 4. Conclusions A fully coupled, transient double-diffusive convective model was developed. This model was able to predict the transient performance of a natural water body as well as the performance of a solar pond. Model validation, without fitting any parameters, 1729 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 showed good agreement between measured and modeled temperatures within a small-shallow lake. In addition, it showed good agreement when simulating double-diffusive convection in a rectangular cavity under opposing buoyant conditions. When used to simulate realistic solar pond conditions, results showed that in a two-week period, for a 1.0 m deep pond under summer conditions and without heat extraction, the temperature of the lower convective zone increased from 20 to approximately 52 °C, confirming this promising technology for renewable energy, especially for low-temperature applications. Different assumptions were tested to estimate the effect of considering double-diffusive convection on pond performance. It was shown that when double-diffusive convection is not considered, the temperature obtained in the lower convective zone is overpredicted, thus the efficiency for the collected and stored heat may be overestimated, especially in models that use air temperature as the surface boundary condition in conjunction with monthly-averaged data. The model showed that, when double-diffusive convection is included, the thicknesses of the upper and lower convective zones increase by approximately 100% and 20%, respectively, contrary to previous simplified analyses that required these interfaces to remain static. A stability analysis revealed that the temperature and salt gradients were always large enough to suppress convection within the non-convective zone, although there is constant motion of the internal interfaces that result in erosion of the non-convective zone, which was predicted by correctly representing the doublediffusive phenomena within the solar pond. The convective mixing that occurs in both the upper and lower convective zones increases the erosion rate in the non-convective zone. In addition, the reduction of the non-convective zone and the increase of the upper convective zone results in higher conductive heat losses to the atmosphere. The results presented here can be used to determine appropriate operation and maintenance parameters of a solar pond. In particular, the model shows the importance of the internal interface migration, and can be used to predict pond turnover and overall thermal efficiency. By incorporating coupled momentum, heat and mass transport, the dynamic nature of the interface can now be modeled for a variety of meteorological conditions. In addition, management of the non-convective zone thickness via injection of fresh or saline water into the convective layers, or through heat extraction, can be accurately simulated and assessed. This would lead to larger efficiencies for the collected and stored heat. in which Ta is expressed in K, and ea can be expressed as [37]: ea 8 > 0:84 ð0:1 9:973 106 ea Þð1 CÞ þ 3:491 105 ea > > > > < 0:0 < C 6 0:6 ¼ > 0:87 ð0:175 29:92 106 ea Þð1 CÞ þ 2:693 105 ea > > > > : 0:6 < C 6 1:0 ð18Þ where ea can be estimated by: ea ¼ RH esa 4157 esa ¼ 2:1718 1010 exp T a 33:91 ð19Þ ð20Þ To represent Q w l the water surface can be treated as a black body emitter, therefore [38]: 4 Qw l ¼ 0:972rT w ð21Þ in which Tw is expressed in K. Negative heat fluxes mean that the heat is leaving the body of water. A.2. Evaporative heat flux (Qe) The evaporative or latent heat flux across the air–water interface, which includes forced convection due to the horizontal air movement across the water surface and free convection under an unstable atmosphere (produced by the density difference between the water surface and air temperature), can be described by [39]: Q e ¼ ðQ 2forced þ Q 2free Þ1=2 ð22Þ where the wind-forced and free convection are given by: Q forced ¼ 3:1 102 U 2 ðesw ea Þ ( Q free ¼ ð23Þ 2:7 102 ðT wv T av Þ1=3 ðesw ea Þ T wv > T av 0 T w v 6 T av ð24Þ in which U2 is the wind speed measured at 2 m height. The saturated vapor pressure at the water surface, esw, is estimated using Eq. (20) with Ta = Tw; and the virtual temperatures of the water surface (Twv) and air (Tav) are estimated using [31]: T wv ¼ Tw ð1 0:378 esw P1 atm Þ T av ¼ Ta ð1 0:378 ea P1 atm Þ ð25Þ Acknowledgments The authors wish to thank the Department of Energy for funding Grant No. DE-FG02-05ER64143. The authors greatly appreciate the help of Thomas Torgersen for providing the validation data and for his thoughtful suggestions. We also wish to recognize the valuable input from Clay Cooper and Jazmín Aravena. A.3. Sensible heat flux (Qs) The sensible heat flux, which is driven by temperature difference between the water surface and the air, has been described using [40]: Q s ¼ 1:5701 U 2 ðT w T a Þ Appendix A. Heat flux formulation A.4. Solar short-wave radiation A.1. Long-wave radiation flux (Ql) The long-wave radiation flux across the air–water interface is estimated using the long-wave radiation from the atmosphere to the water free surface, Q al , and the emission of long-wave radiation from the water free surface to the atmosphere, Q w l . Q l ¼ Q al þ Q w l ð16Þ where Q al can be represented by [37]: Q al ¼ 0:97ea rT 4a ð26Þ ð17Þ The incident solar radiation that reaches the water surface, Q iswr , is reflected to the air or transmitted to the water. The short-wave radiation that penetrates the air–water interface, Q 0swr , is given by: Q 0swr ¼ Q iswr ð1 As Þ ð27Þ where the surface albedo, As, is a function of the zenith angle, water conditions, and cloud cover. A mean annual value between 0.06 and 0.08 is usually used in water quality or thermal models [37]. In this investigation, a simple empirical relationship that takes into account the cloud fraction was used: 1730 F. Suárez et al. / International Journal of Heat and Mass Transfer 53 (2010) 1718–1730 As ¼ A aBs ð28Þ where the constants A and B are presented in the work of Henderson-Sellers [37]. A.5. Heat flux at the bottom of the solar pond It is assumed that any solar radiation that reaches the bottom, , is absorbed by the fluid at this depth. Eqs. (9) and (11) Q bottom swr are used to estimate this heat flux. In addition, the heat loss to the sediment is described using Fourier’s Law; therefore, the heat flux at the bottom of the solar pond, Qbottom, is described by: Q bottom ¼ Q bottom ðksed rTÞ swr ð29Þ 1 where T (°C m ) is the temperature gradient between the bottom of the solar pond and the sediment at an arbitrary position. If it is assumed that the bottom is adiabatic, then the heat flux at the bottom of the solar pond corresponds to the short-wave radiation that ). reaches this position (i.e., Q bottom ¼ Q bottom swr Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2010.01.017. References [1] J.R. Hull, C.E. Nielsen, P. Golding, Salinity-Gradient Solar Ponds, CRC Press, Florida, 1989 [Chapters 1, 3 and 4]. [2] G. Caruso, A. Naviglio, A desalination plant using solar heat as a heat supply, not affecting the environment with chemicals, Desalination 122 (2–3) (1999) 225–234. [3] A. 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