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Transient natural convection in an enclosure with vertical solutal gradients
Solar Energy 81 (2007) 476–487 www.elsevier.com/locate/solener
Transient natural convection in an enclosure with vertical solutal gradients Moez Hammami *, Mohamed Mseddi, Mounir Baccar Unite´ de Dynamique des Fluides Nume´rique et Phe´nome`nes de Transfert, De´partement de Ge´nie Me´canique, Ecole Nationale d’Inge´nieurs de Sfax, Route de Sokra, B.P. W 3038 Sfax, Tunisia Received 12 August 2005; received in revised form 4 August 2006; accepted 13 August 2006 Available online 26 September 2006 Communicated by: Associate Editor Aliakbar Akbarzardeh
Abstract Transient natural convection in an enclosure with vertical solutal gradients has been studied in this paper. Transfers in a rectangular cavity configuration translating hydrodynamic and thermal phenomena are numerically predicted by means of computational fluid dynamics (CFD) in transient regime. The objective of this numerical study is to give a fine knowledge of the hydrodynamic and thermal characteristics during energy storage in an enclosure filled with water stratified by downward salinity gradient. The enclosure is divided into three zones with different salinity level such as salt gradient pond (SGP). Water is heated by a heating device at the bottom of the cavity. The Navier–Stokes, energy and mass equations are discretized using finite-volume method, and a two-dimensional analysis of the hydrodynamic and thermal behaviors generated in transient regime in the cavity are performed. The mathematical modelling has allowed the prediction of the storage performances by developing parametrical study in view to search the convective heat transfer coefficient at the bottom of the enclosure. Velocity vector fields show the presence of recirculation zones caused only in the lower region and permit to explain the increase of the temperature in the lower convective zone (LCZ). This study shows also the importance of the salinity in the preservation of the high temperature in the bottom of the cavity, and the important reduction of the phenomenon of thermal transfer across the non-convective zone (NCZ). 2006 Elsevier Ltd. All rights reserved. Keywords: Storage pond; Salt gradient; Natural convection; Coupled heat and mass transfers; Finite-volume method
1. Introduction A solar pond is a large area solar collector using a simple technology that uses water and salt. It is an artificially constructed water pool in which a significant increase of temperature is caused in the lower region by preventing convective currents with a gradient of salinity. Hence, pond can be coupled with a thermal power station and used for *
Corresponding author. Tel.: +216 74 274 088x551; fax: +216 74 275 595. E-mail addresses: [email protected] (M. Hammami), [email protected] (M. Mseddi), [email protected] (M. Baccar). 0038-092X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2006.08.004
storage of a huge amount of energy as in nuclear reactor plant. Also, ponds present the advantage to have a long time heat storage performances throughout months, seasons or even a year. The importance of the use of this technology appears especially in the applications which need a large quantity of hot water (80–90 C), such as textile processing and food industries, production of hot air for industrial usage such as drying agricultural and chemical products, the heater of agricultural spaces, desalination of water, . . . The solar pond is made up of a cavity filled with water (Fig. 1) and subdivided into three salt water layers in which the salinity and hence the density are different. The lower layer, called lower convective zone ‘LCZ’ or storage
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Nomenclature C concentration, kg m3 DC concentration gap, kg m3 Cp specific heat, J kg1 K1 D diffusivity, m2 s1 g gravitational acceleration, m s2 H height of the cavity, m HLCZ height of the lower convective zone (LCZ), m h heatffi transfer at the bottom, W m2 K1 poverall ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ gbT h0 HLCZ characteristic velocity, m s1 L length of the cavity, m P pressure, Pa T temperature, K T0 ambient temperature, K Ts serpentine temperature, K TLCZ lower convective zone (LCZ) average temperature, K t time, s V horizontal velocity component, m s1 W vertical velocity component, m s1 X horizontal coordinate, m Z vertical coordinate, m Greek symbols q fluid density, kg m3 l dynamic fluid viscosity, Pa s bT thermal expansion coefficient, K1
convective zone is a convective layer of almost saturated salt water. The second is formed by several fine layers in which the salinity and the temperature decrease linearly from the bottom to the top (Fig. 2). In this condition, the convective phenomena are cancelled, and that is why it is called non-convective zone (NCZ). The least thick
bm mass expansion coefficient, m3 kg1 k thermal conductivity, W m1 K1 h0 ¼ Ts T0 characteristic temperature, K s final heat time, s Dimensionless variables C ¼ C=DC dimensionless concentration L ¼ L=HLCZ dimensionless length of the cavity N ¼ DCbm =bT h0 buoyancy ratio P ¼ P=ðqgbT h0 HLCZ Þ2 dimensionless pressure t ¼ tJ=HLCZ dimensionless time U ¼ V=J dimensionless horizontal velocity component ~ V dimensionless velocity vector W ¼ W=J dimensionless vertical velocity component X ¼ X=HLCZ dimensionless horizontal coordinate Z ¼ Z=HLCZ dimensionless vertical coordinate h ¼ ðT T0 Þ=h0 dimensionless temperature Dimensionless numbers Gr ¼ gbT q2 h0 H3LCZ =l2 Grashof number Nu ¼ hHLCZ =k Nusselt number Pr = lCp/k Prandtl number Ra = Gr Pr Rayleigh number Sc = l/(qD) Schmidt number
and the least dense superior layer called Upper Convective Zone (UCZ) guarantees the protection of the rest of the pond from external effects influences such as wind and rain. An efficient solar pond is one which under given ambient condition acquires the desired temperature quickly and subsequently collects maximum heat at the desired
Fig. 1. Schematic diagram of a salt gradient pond.
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the hydrodynamic and thermal structures of the transient flow developed during the storage of energy. For this purpose, the resolution of Navier–Stokes, heat and mass transfer equations governing the transfer phenomena developed in the cavity is conducted using finitevolume technique discretization in transient regime. 2. Mathematical formulation
Fig. 2. Schematic profiles of temperature and salinity in a storage pond.
temperature. This requires optimum sizing of the mentioned zones. To control the techniques of the construction of the solar ponds, a lot of research works have been developed and published: Sodah et al. (1981), Kishore and Joshi (1984), Singh et al. (1994) and Sezai and Tasdemiroglu (1995). Srinivasan and Guha (1987) considered two phases of solar pond thermal behavior. In the first phase, the pond’s liquid acquires a typical vertical temperature profile characterized by a linear temperature profile in the NCZ and constant uniform temperatures in the LCZ and the UCZ. In the second phase, the temperature of the LCZ oscillates as a function of the fluctuation of the received net solar radiation energy, which is particularly related to the season of the year. The first phase is called the maturation phase and the second phase is called the mature phase. Each of these phases depends on the meteorological parameters and the thickness of layers constituting the pond. Recently, an analytical approach has been developed by Husain et al. (2003) to determine two optimal dimensions of the NCZ giving a faster maturation phase and more performance of heat accumulation. During the maturation phase, in spite of its poor insulating performance, thin NCZ permits more sunlight penetration. However, during mature phase, big thickness of the NCZ gives a better insulation performance, but a less sunlight transmission. Therefore, these authors propose increasing with time the size of the NCZ to optimise both preparation and the mature phases. In most of the previous studies, numerical and theoretical models were developed in one-dimensional transient regime and neglected the movement of the fluid in the pond as well as mass transfer phenomena, which suppose that temperature is constant in both LCZ and UCZ. In this paper, we present a two-dimensional numerical modelling of the hydrodynamic and thermal behaviors induced in a storage cavity. The enclosure is filled with water and divided into three zones with different salinity level such as salt gradient pond (SGP). The movements are produced in the LCZ by natural convection induced by a heat source at the bottom of the cavity. Our objective is to supply by numerical simulation, a fine knowledge of
The conservation equations for a laminar incompressible buoyancy-driven flow in a two-dimensional rectangular configuration cavity are given below. The fluid properties are supposed independent of temperature and salt concentration, except the density that varies according to Boussinesq approximation (Abderrahmane, 2003). This approximation, currently used in natural convection in closed surroundings, supposes that heat or mass transfer processes arise because of the presence of temperature or concentration gradients: qðC; TÞ ¼ q0 ½1 bT ðT T0 Þ þ bm ðC C0 Þ
ð1Þ
To simplify the calculations at the same time as preserving the treated physics, we disregard the variation of density in the terms of inertia of the momentum equation, but we take it into account in the buoyancy term. Furthermore, water is heated by a heating device covering nearly the entire of the bottom of the cavity (96%) (Fig. 1). Finally, the solar radiation energy is not considered into account, so we consider in this work a storage application. 2.1. Basic equations The resulting continuity, momentum, energy and mass coupled equations can be written in dimensionless form as follows: Continuity equation ~¼0 div V U – Velocity component ! rffiffiffiffiffiffi oU Pr ! oP ~ grad U ¼ div V U ot Ra oX W – Velocity component ! rffiffiffiffiffiffi oW Pr ! oP ~ ¼ div V W þ ðh NCÞ grad W ot Ra oZ Energy equation ! oh 1 ~ p ffiffiffiffiffiffiffiffiffiffi ¼ div V h grad h ot PrRa Concentration equation ! rffiffiffiffiffiffi oC 1 Pr ! ~ ¼ div V C grad C ot Sc Ra
ð2Þ
ð3Þ
ð4Þ
ð5Þ
ð6Þ
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The above equations are subject to the following initial and boundary conditions. 2.2. Boundary and initial conditions Initially, the fluid is considered in rest condition and has ambient temperature. So the dimensionless temperature, pressure and velocities have zero initial values. The initial concentration distribution imposed in this study gives the initial conditions of mass transfer as recommended by Agha et al. (2002). Hence, the cavity is initially filled with water and subdivided into three zones in which the salinities are as follows: – The first zone (LCZ) is the deepest and is salt saturated. It occupies 50% of the total volume of the cavity. – The second zone (NCZ) is an intermediate zone in which the concentration of salt increases with the depth. This zone occupies 40% of the total volume. – The third zone (UCZ) situated above the NCZ is a zone without salt and occupies 10% of the total volume. Concerning the boundary conditions, the vertical walls are impermeable and adiabatic. At the bottom of the cavity, we impose a zero flux of mass and a fixed temperature (h = 1) on 96% of the corresponding surface, elsewhere we consider a zero heat flux to initiate convective natural movement precisely in the corner of the cavity. The free surface is impermeable and has ambient temperature. Because of the symmetry of the problem, we consider only half of the cavity. At the symmetrical vertical plane, the corresponding boundaries are: oW/oX = 0, U = 0, oh/oX = 0 and oC/oX = 0. 3. Numerical method The resolution of the equations of continuity, movement, energy and mass, is based on the finite-volume method. The cavity domain is subdivided in elementary volumes X surrounding every point of the mesh. The transport equation is then integrated in each of these volumes, expressing the balance of flux ‘JU’ of the transport parameter ‘U’ which represent the components of velocity vector, temperature or concentration: ZZZ X
oU dv ¼ ot
ZZZ
div ~ J U dv þ X
!
ZZZ
S U dv
ð7Þ
X
with ~ J U ¼ UV~ CU grad U is the flux term of U, CU is the corresponding diffusivity coefficient, SU is the corresponding source/sink term. Three different control volumes are defined for a given node point: one for each of the two vector components and one for the scalar variables. Then, each of the trans-
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port equations is integrated over its own control volume (Patankar, 1980). A computational domain consisting of 50 · 100 grid points with uniform grid spacing in O–X, and O–Z directions and a dimensionless time step 106 were found to be sufficient for producing accurate results at reasonable computed time. The derivatives in the inertia terms were discretized according to the hybrid scheme and the pressure field was dealt with via a prediction–correction procedure, all described by Patankar (1980). The temporal integration has begun using the implicit scheme of alternated directions of Douglass and Gunn (1964). 4. Results and discussion In this paper, we will present numerical results in transient regime to give a fine knowledge of the temporal evolution of the hydrodynamic and thermal characteristics induced in a storage cavity. A particular interest of this study is the controlling parameters that define the fluid flow, heat and mass transfers for double-diffusive natural convection in the storage enclosure. These parameters are the initial salinity profile, the aspect ratio and the Rayleigh number. These results are given for fixed values of Prandtl and Schmidt numbers (Pr = 6, Sc = 1000), which correspond to the averaged salt-water characteristics. The results are generated for a buoyancy ratio N equal to 103. 4.1. Stability of stratified cavity layers Without salt gradient, the convective movements prevent energy accumulation at the bottom of the cavity. In fact, the warmer fluid becoming lighter rises to the surface, where it loses some of its heat as a result of the difference between hot surface temperature and ambient, and the evaporation from the surface. As the surface fluid cools, it gets heavier and sinks to the bottom again. In this way, convection currents occur in the fluid due to the buoyancy effect. Because of this continuous mixing and, thus, the heat loss, it is impossible to store heat at the bottom of the enclosure. Imposing salinity profile suppresses this convection effect. Hence, salinity has a considerable role to maintain hot temperature at the bottom, and to reduce the thermal transfer phenomenon across the NCZ. Fig. 3 shows the concentration evolution as a function of the dimensionless vertical coordinate Z and for different dimensionless times. In this figure, we consider the average concentration in a horizontal plane for a Rayleigh number equal to 5 · 104 and a ratio L equal to 5. We can easily check that the initial concentration profile is preserved with time. This implies that solutal stratified layers are not disturbed and remain stable.
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4.2. Transient evolution of temperature and velocity distribution
Fig. 3. Temporal evolution of vertical concentration profiles for Ra = 5 · 104.
For a Rayleigh number equal to 5 · 104 and a ratio L equal to 5, Figs. 4–9 show the evolution with time of temperature distribution in the enclosure. At the beginning (Fig. 4), the liquid warms up in the lower corners of the cavity, and develop thermal gradients at the bottom of the enclosure, giving a growing up of multi-cellular thermal patterns in the LCZ (Fig. 5). Then, temperature increases with time and isotherms show many mushroom-shaped structures (Figs. 6–9) confined in the LCZ. These structures generate reversed vertical temperature gradients. In fact, in Fig. 10, showing the temporal evolution of the average temperature vertical profile, temperature gradients in the center of the LCZ are close to zero (t = 0.4 and t = 0.25) or reversed in sign at successive
Fig. 4. Temperature distribution at t = 0.025 for Ra = 5 · 104.
Fig. 5. Temperature distribution at t = 0.05 for Ra = 5 · 104.
Fig. 6. Temperature distribution at t = 0.1 for Ra = 5 · 104.
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Fig. 7. Temperature distribution at t = 0.25 for Ra = 5 · 104.
Fig. 8. Temperature distribution at t = 0.35 for Ra = 5 · 104.
Fig. 9. Temperature distribution at t = 0.4 for Ra = 5 · 104.
Fig. 10. Temporal evolution of vertical temperature profiles for Ra = 5 · 104.
times at t = 0.05 and t = 0.1. Fig. 10 also indicates that temperature is significant in the LCZ and decreases in the NCZ to reach ambient temperature in the UCZ, indicating a nearly one-dimensional conduction regime. This clearly demonstrates the role of the NCZ in the reduction of the heat transfer phenomenon, which yields an elevation of temperature only in the storage zone. In natural convective regime, thermal and hydrodynamic behaviors are coupled. Hence, to better understand the thermal behavior evolution with time, we have reproduced in Figs. 11–14, the temporal evolution of velocity distributions. Firstly, we observe two little recirculations in the lower corners of the cavity (Fig. 11). In fact, applying vertical temperature gradient across the cavity initiates flow, even for a very small temperature gradient.
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Fig. 11. Flow pattern induced at t = 0.01.
Fig. 12. Flow pattern induced at t = 0.025.
Fig. 13. Flow pattern induced at t = 0.05.
Fig. 14. Flow pattern induced at t = 0.1.
Then, a succession of small eddies (Figs. 12 and 13) is generated in the storage convective zone where the flow results from the competition between thermal and solutal
buoyancy forces. These eddies contribute to the homogenisation of the LCZ and enhance heat transfer from the bottom of the cavity.
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Finally, the flow structure reaches the same configuration shape (Fig. 14) when dimensionless temperature has nearly a value of 0.3 and only velocity amplitude will be changed. Elsewhere, in the NCZ and the UCZ, the fluid is in stagnant state. 4.3. Effect of the aspect ratio If the relative height of the LCZ is maintained constant, the dimensionless geometrical parameter characterizing the
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problem is the aspect ratio of the cavity. At t = 0.1 and for Ra = 5 · 104, Figs. 15 and 16 represent velocity and temperature fields in the cavity for two length ratios (L = 1, 2). We note that a fundamental variation in the structure of the flow and thermal patterns occurs as the length of the cavity is increasing. Therefore, for little value of dimensionless lengths (L = 1 and L = 2), Figs. 15 and 16 show two large and powerful recirculations occupying nearly the entire of the LCZ. We can particularly see in Fig. 16 corresponding to
Fig. 15. Velocity and temperature fields at t = 0.1 for L = 1 and Ra = 5 · 104: (a) flow pattern and (b) temperature distribution.
Fig. 16. Velocity and temperature fields at t = 0.1 for L = 2 and Ra = 5 · 104: (a) flow pattern and (b) temperature distribution.
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the aspect ratio L = 2, and precisely in the higher corner of the LCZ, the formation of two secondary symmetrical vortexes having a triangular shape. These secondary vortexes are associated with a strong reversed vertical temperature gradient. In fact, heat transfer in the lower fine sublayers makes temperature gradients near zero in the enclosure core, and simultaneously the upward stream of the heated fluid promote negative secondary vortex once it deviates to the center of the cavity to achieve a swirling flow. As the length of the cavity increases, multi-cellular structures are generated. For L ratio equal to 5 (Fig. 14), it appears a succession of vortexes occupying the LCZ. The generation of multi-cellular structure should improve the fluid homogenisation, and subsequently, convective transfer can be performed. In fact, conjugated cells contribute to enhance axial convection term between simultaneous cells, which support heat transfer. To check this behavior, we have reproduced in Fig. 17, the local Nusselt number as a function of the horizontal coordinate X for different dimensionless lengths
(L = 1, 2, 5 and 10) and a constant Rayleigh number (Ra = 5 · 104) at t = 0.1. The local heat transfer rate in transient regime is expressed in dimensionless form by the Nusselt number based on the height of the lower convective zone: ohðX ; tÞ NuðX ; tÞ ¼ ð8Þ oZ Z¼0 For L 6 5, the general tendency shows that heat transfer is favored when the L ratio becomes higher which is related to the appearance of several recirculations. This has been confirmed by the analysis of the axial velocity component profiles represented in the mid-height of the LCZ for different aspect ratios (Fig. 18). For L = 1, the greatest value of the Nusselt number takes place in the middle of the cavity, where the current flow stream generated between two consecutive recirculations is rising up (Fig. 15a). In this case, the maximum value of Nusselt is equal to 4.7, and this value decreases in the peripheries of the walls. This tendency is in accor-
Fig. 17. Local Nusselt number for different dimensionless lengths.
Fig. 18. Axial velocity component profiles at the middle height of the LCZ for different dimensionless lengths.
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Fig. 19. Flow pattern induced at t = 0.1 (L = 10).
dance with the dynamic profile (Fig. 18) that shows that the axial ascendant flow is the greatest in the middle of the cavity. In opposite, for L = 2, the maximum Nusselt number of 4.5 is detected symmetrically on both sides of the cavity and attain a minimum value of 2 at the center of the cavity (Fig. 17). For L = 5, Nusselt number oscillates at the frequency of the appearance of vortexes (Figs. 17 and 18). The greatest Nusselt value of 6.5 is reached at the vertical axial midplane where there is a downward stream resulting from conjunction of two consecutive opposite swirling flow (Fig. 14). The increasing of the aspect ratio contributes to the generation of typically Rayleigh–Benard multi-cellular structure. Hence, the velocity pattern reproduced in Fig. 19 for L = 10, clearly indicates the generation of multi-cellular structure along the cavity. However, it was checked that the average Nusselt number did not vary significantly beyond an aspect ratio of about 5. In fact, by considering a Rayleigh number equal to 5 · 104 and t equal to 0.1, the average Nusselt number calculated for L = 5 and L = 10 are respectively equal to 3.8 and 3.6. 4.4. Heat transfer performances Most natural convection studies are developed in steady state which does not always correspond to industrial needs. Indeed, in several industrial domains such as heat transfer in the storage cavity, transfer occurs fundamentally in transient regime. In order to characterize the evolution of the convective heat transfer coefficient through the bottom of the enclosure, obtained by integration of the local coefficient over its correspondent exchange surface, we calculate the instantaneous average Nusselt number Nu as follows: Z 1 L ohðX ; tÞ NuðtÞ ¼ dX ð9Þ L 0 oZ Z¼0 To study the Rayleigh number effect on the transient heat transfer, the average Nusselt number evolution with time is plotted in Fig. 20 for various initial Rayleigh numbers and an aspect ratio L = 5. At the beginning, the results show the expected tendency of increasing Nu with Rayleigh number. It is worth noting
Fig. 20. Temporal evolution of averaged Nusselt number for various Rayleigh numbers (L = 5).
the presence of oscillations due to the transient interaction between velocities and temperature distributions. In general, it is found that Nu(t) decreases sharply at first and then varies slowly to reach an asymptotic value of about zero which indicates the tendency to thermal equilibrium. In fact, when the heat transfer by convection makes the temperature gradients near zero in the LCZ, the heat transfer rate is reduced in all cases of the initial Rayleigh numbers. 5. Comparison with anterior results In order to check the accuracy of the modelling results obtained in the present study, we have reproduced the simulation for double-diffusive convection flow in an enclosure, with combined vertical temperature and concentration gradients in a steady state, which was reported earlier by Boussaid et al. (1999). Simulation is conducted for Pr = 0.7, Sc = 0.6 and L=H ¼ 2. Furthermore, it is assumed that the flow is incompressible, laminar and binary fluid is Newtonian. The enclosure is supposed heated by its lower base and cooled by its upper side. The constant species concentrations are adopted along the horizontal walls: high concentration at the bottom and low concentration at the top boundaries. Fig. 21, giving the local Nusselt number profiles for different buoyancy ratio N and a Rayleigh number equal to 5000, shows a satisfactory quantitative agreement between our computations and those of Boussaid et al. (1999).
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Fig. 21. Comparison of the local Nusselt number with anterior data reported by Boussaid et al. (1999).
Fig. 22. Comparison of the dimensionless vertical velocity component W with anterior data.
We also compared in Fig. 22 our results with the published data of Boussaid et al. (1999) on the dimensionless vertical velocity component W determined as a function of the dimensionless length for Rayleigh number Ra equal to 105. As well, a good agreement with the anterior results is observed. 6. Summary and conclusions This study has been developed in two-dimensional numerical modelling of the thermosolutal transfer in rectangular cavity to analyse the complex flow structure velocities and temperature distributions in transient regime. Indeed, resolution of coupled momentum, heat and mass transfer equations give interesting local information concerning evolution with time of the hydrodynamic and the thermal behaviors during the storage of energy. Firstly, we have proved the importance of the salinity gradient in the accumulation of energy and in the reduction of the thermal losses by convection. We have demonstrated that concentration stratification is stable and resistant to the flow evolution.
It is also found that for L 6 5, the convective heat transfer is improved when the cavity is lengthened as a consequence of the acceleration of the fluid in the whole volume of the LCZ. Thus, the increasing of the aspect ratio favors generation of multi-cellular structure, which improves the fluid homogenisation as a consequence of enhancement of axial current flow between simultaneous cells, which intensifies heat transfer between the heated bottom cavity and the fluid situated in the LCZ. However, it was checked that the average Nusselt number did not vary significantly beyond an aspect ratio of about 5. In the particular case of L = 2, we have noted the apparition of secondary vortex structures in the upper corners of the LCZ which are associated to a strong reversed temperature gradients. This is because the buoyancy drives the heated flow up beside the hot wall across the enclosure to the upper region of the LCZ. The average rate of heat transfer across the horizontal wall is expressed in dimensionless form by the Nusselt number based on the height of the LCZ. The dependence of Rayleigh numbers on the heat transfer were detected and identified in transient regime. It appears that heat transfer convection weakens the temperature gradients which to generate a reduction of heat transfer rate. For a long time heating, we have obtained a near zero Nusselt number independent of the Rayleigh number. Finally, numerical results have been compared with literature data and a satisfactory agreement has been found. References Abderrahmane, B., 2003. Transient natural 2D convection in a cylindrical cavity with the upper face cooled by thermoelectric Peltier effect following an exponential law. Appl. Thermal Eng. 23, 431–447. Agha, K.R., Abughres, S.M., Ramadan, A.M., 2002. Design methodology for a salt gradient solar pond coupled with an evaporation pond. Solar Energy 72 (5), 447–454. Boussaid, M., Mezenner, A., et Bouhadef, M., 1999. Convection naturelle de chaleur et de masse dans une cavite´ trape´zoı¨dale. Int. J. Thermal Sci. 38, 363–371.
M. Hammami et al. / Solar Energy 81 (2007) 476–487 Douglass, J., Gunn, J.E., 1964. A general formulation of alternatingdirection implicit methods. Numer. Math. 6, 428. Husain, M., Patil, P.S., Patil, S.R., Samdarshi, S.K., 2003. Optimum size of non-convective zone for improved thermal performance of salt gradient solar pond. Solar Energy 74 (5), 429–436. Kishore, V.V.N., Joshi, V., 1984. A practical collector efficiency equation for non-convecting solar ponds. Solar Energy 33 (5), 391–395. Patankar, S.V., 1980. Numerical heat transfer and fluid flowSeries in Computational Methods in Mechanics and Thermal Sciences. McGraw-Hill, New York.
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Sezai, I., Tasdemiroglu, E., 1995. Effect of bottom reflectivity on ground heat losses for solar ponds. Solar Energy 55 (4), 311–319. Singh, T.P., Singh, A.K., Kaushik, N.D., 1994. Investigations of thermodynamic instabilities and ground storage in a solar pond by simulation model. Heat Recov. Syst. 14 (2), 401–407. Sodah, M.S., Kaushik, N.D., Rao, S.K., 1981. Thermal analysis of three zone solar pond. Int. J. Energy Res. 5, 321–340. Srinivasan, J., Guha, A., 1987. The effect of bottom reflectivity on the performance of a solar pond. Solar Energy 39 (4), 361–367.
Transient natural convection in an enclosure with vertical solutal gradients Moez Hammami *, Mohamed Mseddi, Mounir Baccar Unite´ de Dynamique des Fluides Nume´rique et Phe´nome`nes de Transfert, De´partement de Ge´nie Me´canique, Ecole Nationale d’Inge´nieurs de Sfax, Route de Sokra, B.P. W 3038 Sfax, Tunisia Received 12 August 2005; received in revised form 4 August 2006; accepted 13 August 2006 Available online 26 September 2006 Communicated by: Associate Editor Aliakbar Akbarzardeh
Abstract Transient natural convection in an enclosure with vertical solutal gradients has been studied in this paper. Transfers in a rectangular cavity configuration translating hydrodynamic and thermal phenomena are numerically predicted by means of computational fluid dynamics (CFD) in transient regime. The objective of this numerical study is to give a fine knowledge of the hydrodynamic and thermal characteristics during energy storage in an enclosure filled with water stratified by downward salinity gradient. The enclosure is divided into three zones with different salinity level such as salt gradient pond (SGP). Water is heated by a heating device at the bottom of the cavity. The Navier–Stokes, energy and mass equations are discretized using finite-volume method, and a two-dimensional analysis of the hydrodynamic and thermal behaviors generated in transient regime in the cavity are performed. The mathematical modelling has allowed the prediction of the storage performances by developing parametrical study in view to search the convective heat transfer coefficient at the bottom of the enclosure. Velocity vector fields show the presence of recirculation zones caused only in the lower region and permit to explain the increase of the temperature in the lower convective zone (LCZ). This study shows also the importance of the salinity in the preservation of the high temperature in the bottom of the cavity, and the important reduction of the phenomenon of thermal transfer across the non-convective zone (NCZ). 2006 Elsevier Ltd. All rights reserved. Keywords: Storage pond; Salt gradient; Natural convection; Coupled heat and mass transfers; Finite-volume method
1. Introduction A solar pond is a large area solar collector using a simple technology that uses water and salt. It is an artificially constructed water pool in which a significant increase of temperature is caused in the lower region by preventing convective currents with a gradient of salinity. Hence, pond can be coupled with a thermal power station and used for *
Corresponding author. Tel.: +216 74 274 088x551; fax: +216 74 275 595. E-mail addresses: [email protected] (M. Hammami), [email protected] (M. Mseddi), [email protected] (M. Baccar). 0038-092X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2006.08.004
storage of a huge amount of energy as in nuclear reactor plant. Also, ponds present the advantage to have a long time heat storage performances throughout months, seasons or even a year. The importance of the use of this technology appears especially in the applications which need a large quantity of hot water (80–90 C), such as textile processing and food industries, production of hot air for industrial usage such as drying agricultural and chemical products, the heater of agricultural spaces, desalination of water, . . . The solar pond is made up of a cavity filled with water (Fig. 1) and subdivided into three salt water layers in which the salinity and hence the density are different. The lower layer, called lower convective zone ‘LCZ’ or storage
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Nomenclature C concentration, kg m3 DC concentration gap, kg m3 Cp specific heat, J kg1 K1 D diffusivity, m2 s1 g gravitational acceleration, m s2 H height of the cavity, m HLCZ height of the lower convective zone (LCZ), m h heatffi transfer at the bottom, W m2 K1 poverall ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ gbT h0 HLCZ characteristic velocity, m s1 L length of the cavity, m P pressure, Pa T temperature, K T0 ambient temperature, K Ts serpentine temperature, K TLCZ lower convective zone (LCZ) average temperature, K t time, s V horizontal velocity component, m s1 W vertical velocity component, m s1 X horizontal coordinate, m Z vertical coordinate, m Greek symbols q fluid density, kg m3 l dynamic fluid viscosity, Pa s bT thermal expansion coefficient, K1
convective zone is a convective layer of almost saturated salt water. The second is formed by several fine layers in which the salinity and the temperature decrease linearly from the bottom to the top (Fig. 2). In this condition, the convective phenomena are cancelled, and that is why it is called non-convective zone (NCZ). The least thick
bm mass expansion coefficient, m3 kg1 k thermal conductivity, W m1 K1 h0 ¼ Ts T0 characteristic temperature, K s final heat time, s Dimensionless variables C ¼ C=DC dimensionless concentration L ¼ L=HLCZ dimensionless length of the cavity N ¼ DCbm =bT h0 buoyancy ratio P ¼ P=ðqgbT h0 HLCZ Þ2 dimensionless pressure t ¼ tJ=HLCZ dimensionless time U ¼ V=J dimensionless horizontal velocity component ~ V dimensionless velocity vector W ¼ W=J dimensionless vertical velocity component X ¼ X=HLCZ dimensionless horizontal coordinate Z ¼ Z=HLCZ dimensionless vertical coordinate h ¼ ðT T0 Þ=h0 dimensionless temperature Dimensionless numbers Gr ¼ gbT q2 h0 H3LCZ =l2 Grashof number Nu ¼ hHLCZ =k Nusselt number Pr = lCp/k Prandtl number Ra = Gr Pr Rayleigh number Sc = l/(qD) Schmidt number
and the least dense superior layer called Upper Convective Zone (UCZ) guarantees the protection of the rest of the pond from external effects influences such as wind and rain. An efficient solar pond is one which under given ambient condition acquires the desired temperature quickly and subsequently collects maximum heat at the desired
Fig. 1. Schematic diagram of a salt gradient pond.
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the hydrodynamic and thermal structures of the transient flow developed during the storage of energy. For this purpose, the resolution of Navier–Stokes, heat and mass transfer equations governing the transfer phenomena developed in the cavity is conducted using finitevolume technique discretization in transient regime. 2. Mathematical formulation
Fig. 2. Schematic profiles of temperature and salinity in a storage pond.
temperature. This requires optimum sizing of the mentioned zones. To control the techniques of the construction of the solar ponds, a lot of research works have been developed and published: Sodah et al. (1981), Kishore and Joshi (1984), Singh et al. (1994) and Sezai and Tasdemiroglu (1995). Srinivasan and Guha (1987) considered two phases of solar pond thermal behavior. In the first phase, the pond’s liquid acquires a typical vertical temperature profile characterized by a linear temperature profile in the NCZ and constant uniform temperatures in the LCZ and the UCZ. In the second phase, the temperature of the LCZ oscillates as a function of the fluctuation of the received net solar radiation energy, which is particularly related to the season of the year. The first phase is called the maturation phase and the second phase is called the mature phase. Each of these phases depends on the meteorological parameters and the thickness of layers constituting the pond. Recently, an analytical approach has been developed by Husain et al. (2003) to determine two optimal dimensions of the NCZ giving a faster maturation phase and more performance of heat accumulation. During the maturation phase, in spite of its poor insulating performance, thin NCZ permits more sunlight penetration. However, during mature phase, big thickness of the NCZ gives a better insulation performance, but a less sunlight transmission. Therefore, these authors propose increasing with time the size of the NCZ to optimise both preparation and the mature phases. In most of the previous studies, numerical and theoretical models were developed in one-dimensional transient regime and neglected the movement of the fluid in the pond as well as mass transfer phenomena, which suppose that temperature is constant in both LCZ and UCZ. In this paper, we present a two-dimensional numerical modelling of the hydrodynamic and thermal behaviors induced in a storage cavity. The enclosure is filled with water and divided into three zones with different salinity level such as salt gradient pond (SGP). The movements are produced in the LCZ by natural convection induced by a heat source at the bottom of the cavity. Our objective is to supply by numerical simulation, a fine knowledge of
The conservation equations for a laminar incompressible buoyancy-driven flow in a two-dimensional rectangular configuration cavity are given below. The fluid properties are supposed independent of temperature and salt concentration, except the density that varies according to Boussinesq approximation (Abderrahmane, 2003). This approximation, currently used in natural convection in closed surroundings, supposes that heat or mass transfer processes arise because of the presence of temperature or concentration gradients: qðC; TÞ ¼ q0 ½1 bT ðT T0 Þ þ bm ðC C0 Þ
ð1Þ
To simplify the calculations at the same time as preserving the treated physics, we disregard the variation of density in the terms of inertia of the momentum equation, but we take it into account in the buoyancy term. Furthermore, water is heated by a heating device covering nearly the entire of the bottom of the cavity (96%) (Fig. 1). Finally, the solar radiation energy is not considered into account, so we consider in this work a storage application. 2.1. Basic equations The resulting continuity, momentum, energy and mass coupled equations can be written in dimensionless form as follows: Continuity equation ~¼0 div V U – Velocity component ! rffiffiffiffiffiffi oU Pr ! oP ~ grad U ¼ div V U ot Ra oX W – Velocity component ! rffiffiffiffiffiffi oW Pr ! oP ~ ¼ div V W þ ðh NCÞ grad W ot Ra oZ Energy equation ! oh 1 ~ p ffiffiffiffiffiffiffiffiffiffi ¼ div V h grad h ot PrRa Concentration equation ! rffiffiffiffiffiffi oC 1 Pr ! ~ ¼ div V C grad C ot Sc Ra
ð2Þ
ð3Þ
ð4Þ
ð5Þ
ð6Þ
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The above equations are subject to the following initial and boundary conditions. 2.2. Boundary and initial conditions Initially, the fluid is considered in rest condition and has ambient temperature. So the dimensionless temperature, pressure and velocities have zero initial values. The initial concentration distribution imposed in this study gives the initial conditions of mass transfer as recommended by Agha et al. (2002). Hence, the cavity is initially filled with water and subdivided into three zones in which the salinities are as follows: – The first zone (LCZ) is the deepest and is salt saturated. It occupies 50% of the total volume of the cavity. – The second zone (NCZ) is an intermediate zone in which the concentration of salt increases with the depth. This zone occupies 40% of the total volume. – The third zone (UCZ) situated above the NCZ is a zone without salt and occupies 10% of the total volume. Concerning the boundary conditions, the vertical walls are impermeable and adiabatic. At the bottom of the cavity, we impose a zero flux of mass and a fixed temperature (h = 1) on 96% of the corresponding surface, elsewhere we consider a zero heat flux to initiate convective natural movement precisely in the corner of the cavity. The free surface is impermeable and has ambient temperature. Because of the symmetry of the problem, we consider only half of the cavity. At the symmetrical vertical plane, the corresponding boundaries are: oW/oX = 0, U = 0, oh/oX = 0 and oC/oX = 0. 3. Numerical method The resolution of the equations of continuity, movement, energy and mass, is based on the finite-volume method. The cavity domain is subdivided in elementary volumes X surrounding every point of the mesh. The transport equation is then integrated in each of these volumes, expressing the balance of flux ‘JU’ of the transport parameter ‘U’ which represent the components of velocity vector, temperature or concentration: ZZZ X
oU dv ¼ ot
ZZZ
div ~ J U dv þ X
!
ZZZ
S U dv
ð7Þ
X
with ~ J U ¼ UV~ CU grad U is the flux term of U, CU is the corresponding diffusivity coefficient, SU is the corresponding source/sink term. Three different control volumes are defined for a given node point: one for each of the two vector components and one for the scalar variables. Then, each of the trans-
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port equations is integrated over its own control volume (Patankar, 1980). A computational domain consisting of 50 · 100 grid points with uniform grid spacing in O–X, and O–Z directions and a dimensionless time step 106 were found to be sufficient for producing accurate results at reasonable computed time. The derivatives in the inertia terms were discretized according to the hybrid scheme and the pressure field was dealt with via a prediction–correction procedure, all described by Patankar (1980). The temporal integration has begun using the implicit scheme of alternated directions of Douglass and Gunn (1964). 4. Results and discussion In this paper, we will present numerical results in transient regime to give a fine knowledge of the temporal evolution of the hydrodynamic and thermal characteristics induced in a storage cavity. A particular interest of this study is the controlling parameters that define the fluid flow, heat and mass transfers for double-diffusive natural convection in the storage enclosure. These parameters are the initial salinity profile, the aspect ratio and the Rayleigh number. These results are given for fixed values of Prandtl and Schmidt numbers (Pr = 6, Sc = 1000), which correspond to the averaged salt-water characteristics. The results are generated for a buoyancy ratio N equal to 103. 4.1. Stability of stratified cavity layers Without salt gradient, the convective movements prevent energy accumulation at the bottom of the cavity. In fact, the warmer fluid becoming lighter rises to the surface, where it loses some of its heat as a result of the difference between hot surface temperature and ambient, and the evaporation from the surface. As the surface fluid cools, it gets heavier and sinks to the bottom again. In this way, convection currents occur in the fluid due to the buoyancy effect. Because of this continuous mixing and, thus, the heat loss, it is impossible to store heat at the bottom of the enclosure. Imposing salinity profile suppresses this convection effect. Hence, salinity has a considerable role to maintain hot temperature at the bottom, and to reduce the thermal transfer phenomenon across the NCZ. Fig. 3 shows the concentration evolution as a function of the dimensionless vertical coordinate Z and for different dimensionless times. In this figure, we consider the average concentration in a horizontal plane for a Rayleigh number equal to 5 · 104 and a ratio L equal to 5. We can easily check that the initial concentration profile is preserved with time. This implies that solutal stratified layers are not disturbed and remain stable.
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4.2. Transient evolution of temperature and velocity distribution
Fig. 3. Temporal evolution of vertical concentration profiles for Ra = 5 · 104.
For a Rayleigh number equal to 5 · 104 and a ratio L equal to 5, Figs. 4–9 show the evolution with time of temperature distribution in the enclosure. At the beginning (Fig. 4), the liquid warms up in the lower corners of the cavity, and develop thermal gradients at the bottom of the enclosure, giving a growing up of multi-cellular thermal patterns in the LCZ (Fig. 5). Then, temperature increases with time and isotherms show many mushroom-shaped structures (Figs. 6–9) confined in the LCZ. These structures generate reversed vertical temperature gradients. In fact, in Fig. 10, showing the temporal evolution of the average temperature vertical profile, temperature gradients in the center of the LCZ are close to zero (t = 0.4 and t = 0.25) or reversed in sign at successive
Fig. 4. Temperature distribution at t = 0.025 for Ra = 5 · 104.
Fig. 5. Temperature distribution at t = 0.05 for Ra = 5 · 104.
Fig. 6. Temperature distribution at t = 0.1 for Ra = 5 · 104.
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Fig. 7. Temperature distribution at t = 0.25 for Ra = 5 · 104.
Fig. 8. Temperature distribution at t = 0.35 for Ra = 5 · 104.
Fig. 9. Temperature distribution at t = 0.4 for Ra = 5 · 104.
Fig. 10. Temporal evolution of vertical temperature profiles for Ra = 5 · 104.
times at t = 0.05 and t = 0.1. Fig. 10 also indicates that temperature is significant in the LCZ and decreases in the NCZ to reach ambient temperature in the UCZ, indicating a nearly one-dimensional conduction regime. This clearly demonstrates the role of the NCZ in the reduction of the heat transfer phenomenon, which yields an elevation of temperature only in the storage zone. In natural convective regime, thermal and hydrodynamic behaviors are coupled. Hence, to better understand the thermal behavior evolution with time, we have reproduced in Figs. 11–14, the temporal evolution of velocity distributions. Firstly, we observe two little recirculations in the lower corners of the cavity (Fig. 11). In fact, applying vertical temperature gradient across the cavity initiates flow, even for a very small temperature gradient.
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Fig. 11. Flow pattern induced at t = 0.01.
Fig. 12. Flow pattern induced at t = 0.025.
Fig. 13. Flow pattern induced at t = 0.05.
Fig. 14. Flow pattern induced at t = 0.1.
Then, a succession of small eddies (Figs. 12 and 13) is generated in the storage convective zone where the flow results from the competition between thermal and solutal
buoyancy forces. These eddies contribute to the homogenisation of the LCZ and enhance heat transfer from the bottom of the cavity.
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Finally, the flow structure reaches the same configuration shape (Fig. 14) when dimensionless temperature has nearly a value of 0.3 and only velocity amplitude will be changed. Elsewhere, in the NCZ and the UCZ, the fluid is in stagnant state. 4.3. Effect of the aspect ratio If the relative height of the LCZ is maintained constant, the dimensionless geometrical parameter characterizing the
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problem is the aspect ratio of the cavity. At t = 0.1 and for Ra = 5 · 104, Figs. 15 and 16 represent velocity and temperature fields in the cavity for two length ratios (L = 1, 2). We note that a fundamental variation in the structure of the flow and thermal patterns occurs as the length of the cavity is increasing. Therefore, for little value of dimensionless lengths (L = 1 and L = 2), Figs. 15 and 16 show two large and powerful recirculations occupying nearly the entire of the LCZ. We can particularly see in Fig. 16 corresponding to
Fig. 15. Velocity and temperature fields at t = 0.1 for L = 1 and Ra = 5 · 104: (a) flow pattern and (b) temperature distribution.
Fig. 16. Velocity and temperature fields at t = 0.1 for L = 2 and Ra = 5 · 104: (a) flow pattern and (b) temperature distribution.
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the aspect ratio L = 2, and precisely in the higher corner of the LCZ, the formation of two secondary symmetrical vortexes having a triangular shape. These secondary vortexes are associated with a strong reversed vertical temperature gradient. In fact, heat transfer in the lower fine sublayers makes temperature gradients near zero in the enclosure core, and simultaneously the upward stream of the heated fluid promote negative secondary vortex once it deviates to the center of the cavity to achieve a swirling flow. As the length of the cavity increases, multi-cellular structures are generated. For L ratio equal to 5 (Fig. 14), it appears a succession of vortexes occupying the LCZ. The generation of multi-cellular structure should improve the fluid homogenisation, and subsequently, convective transfer can be performed. In fact, conjugated cells contribute to enhance axial convection term between simultaneous cells, which support heat transfer. To check this behavior, we have reproduced in Fig. 17, the local Nusselt number as a function of the horizontal coordinate X for different dimensionless lengths
(L = 1, 2, 5 and 10) and a constant Rayleigh number (Ra = 5 · 104) at t = 0.1. The local heat transfer rate in transient regime is expressed in dimensionless form by the Nusselt number based on the height of the lower convective zone: ohðX ; tÞ NuðX ; tÞ ¼ ð8Þ oZ Z¼0 For L 6 5, the general tendency shows that heat transfer is favored when the L ratio becomes higher which is related to the appearance of several recirculations. This has been confirmed by the analysis of the axial velocity component profiles represented in the mid-height of the LCZ for different aspect ratios (Fig. 18). For L = 1, the greatest value of the Nusselt number takes place in the middle of the cavity, where the current flow stream generated between two consecutive recirculations is rising up (Fig. 15a). In this case, the maximum value of Nusselt is equal to 4.7, and this value decreases in the peripheries of the walls. This tendency is in accor-
Fig. 17. Local Nusselt number for different dimensionless lengths.
Fig. 18. Axial velocity component profiles at the middle height of the LCZ for different dimensionless lengths.
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Fig. 19. Flow pattern induced at t = 0.1 (L = 10).
dance with the dynamic profile (Fig. 18) that shows that the axial ascendant flow is the greatest in the middle of the cavity. In opposite, for L = 2, the maximum Nusselt number of 4.5 is detected symmetrically on both sides of the cavity and attain a minimum value of 2 at the center of the cavity (Fig. 17). For L = 5, Nusselt number oscillates at the frequency of the appearance of vortexes (Figs. 17 and 18). The greatest Nusselt value of 6.5 is reached at the vertical axial midplane where there is a downward stream resulting from conjunction of two consecutive opposite swirling flow (Fig. 14). The increasing of the aspect ratio contributes to the generation of typically Rayleigh–Benard multi-cellular structure. Hence, the velocity pattern reproduced in Fig. 19 for L = 10, clearly indicates the generation of multi-cellular structure along the cavity. However, it was checked that the average Nusselt number did not vary significantly beyond an aspect ratio of about 5. In fact, by considering a Rayleigh number equal to 5 · 104 and t equal to 0.1, the average Nusselt number calculated for L = 5 and L = 10 are respectively equal to 3.8 and 3.6. 4.4. Heat transfer performances Most natural convection studies are developed in steady state which does not always correspond to industrial needs. Indeed, in several industrial domains such as heat transfer in the storage cavity, transfer occurs fundamentally in transient regime. In order to characterize the evolution of the convective heat transfer coefficient through the bottom of the enclosure, obtained by integration of the local coefficient over its correspondent exchange surface, we calculate the instantaneous average Nusselt number Nu as follows: Z 1 L ohðX ; tÞ NuðtÞ ¼ dX ð9Þ L 0 oZ Z¼0 To study the Rayleigh number effect on the transient heat transfer, the average Nusselt number evolution with time is plotted in Fig. 20 for various initial Rayleigh numbers and an aspect ratio L = 5. At the beginning, the results show the expected tendency of increasing Nu with Rayleigh number. It is worth noting
Fig. 20. Temporal evolution of averaged Nusselt number for various Rayleigh numbers (L = 5).
the presence of oscillations due to the transient interaction between velocities and temperature distributions. In general, it is found that Nu(t) decreases sharply at first and then varies slowly to reach an asymptotic value of about zero which indicates the tendency to thermal equilibrium. In fact, when the heat transfer by convection makes the temperature gradients near zero in the LCZ, the heat transfer rate is reduced in all cases of the initial Rayleigh numbers. 5. Comparison with anterior results In order to check the accuracy of the modelling results obtained in the present study, we have reproduced the simulation for double-diffusive convection flow in an enclosure, with combined vertical temperature and concentration gradients in a steady state, which was reported earlier by Boussaid et al. (1999). Simulation is conducted for Pr = 0.7, Sc = 0.6 and L=H ¼ 2. Furthermore, it is assumed that the flow is incompressible, laminar and binary fluid is Newtonian. The enclosure is supposed heated by its lower base and cooled by its upper side. The constant species concentrations are adopted along the horizontal walls: high concentration at the bottom and low concentration at the top boundaries. Fig. 21, giving the local Nusselt number profiles for different buoyancy ratio N and a Rayleigh number equal to 5000, shows a satisfactory quantitative agreement between our computations and those of Boussaid et al. (1999).
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Fig. 21. Comparison of the local Nusselt number with anterior data reported by Boussaid et al. (1999).
Fig. 22. Comparison of the dimensionless vertical velocity component W with anterior data.
We also compared in Fig. 22 our results with the published data of Boussaid et al. (1999) on the dimensionless vertical velocity component W determined as a function of the dimensionless length for Rayleigh number Ra equal to 105. As well, a good agreement with the anterior results is observed. 6. Summary and conclusions This study has been developed in two-dimensional numerical modelling of the thermosolutal transfer in rectangular cavity to analyse the complex flow structure velocities and temperature distributions in transient regime. Indeed, resolution of coupled momentum, heat and mass transfer equations give interesting local information concerning evolution with time of the hydrodynamic and the thermal behaviors during the storage of energy. Firstly, we have proved the importance of the salinity gradient in the accumulation of energy and in the reduction of the thermal losses by convection. We have demonstrated that concentration stratification is stable and resistant to the flow evolution.
It is also found that for L 6 5, the convective heat transfer is improved when the cavity is lengthened as a consequence of the acceleration of the fluid in the whole volume of the LCZ. Thus, the increasing of the aspect ratio favors generation of multi-cellular structure, which improves the fluid homogenisation as a consequence of enhancement of axial current flow between simultaneous cells, which intensifies heat transfer between the heated bottom cavity and the fluid situated in the LCZ. However, it was checked that the average Nusselt number did not vary significantly beyond an aspect ratio of about 5. In the particular case of L = 2, we have noted the apparition of secondary vortex structures in the upper corners of the LCZ which are associated to a strong reversed temperature gradients. This is because the buoyancy drives the heated flow up beside the hot wall across the enclosure to the upper region of the LCZ. The average rate of heat transfer across the horizontal wall is expressed in dimensionless form by the Nusselt number based on the height of the LCZ. The dependence of Rayleigh numbers on the heat transfer were detected and identified in transient regime. It appears that heat transfer convection weakens the temperature gradients which to generate a reduction of heat transfer rate. For a long time heating, we have obtained a near zero Nusselt number independent of the Rayleigh number. Finally, numerical results have been compared with literature data and a satisfactory agreement has been found. References Abderrahmane, B., 2003. Transient natural 2D convection in a cylindrical cavity with the upper face cooled by thermoelectric Peltier effect following an exponential law. Appl. Thermal Eng. 23, 431–447. Agha, K.R., Abughres, S.M., Ramadan, A.M., 2002. Design methodology for a salt gradient solar pond coupled with an evaporation pond. Solar Energy 72 (5), 447–454. Boussaid, M., Mezenner, A., et Bouhadef, M., 1999. Convection naturelle de chaleur et de masse dans une cavite´ trape´zoı¨dale. Int. J. Thermal Sci. 38, 363–371.
M. Hammami et al. / Solar Energy 81 (2007) 476–487 Douglass, J., Gunn, J.E., 1964. A general formulation of alternatingdirection implicit methods. Numer. Math. 6, 428. Husain, M., Patil, P.S., Patil, S.R., Samdarshi, S.K., 2003. Optimum size of non-convective zone for improved thermal performance of salt gradient solar pond. Solar Energy 74 (5), 429–436. Kishore, V.V.N., Joshi, V., 1984. A practical collector efficiency equation for non-convecting solar ponds. Solar Energy 33 (5), 391–395. Patankar, S.V., 1980. Numerical heat transfer and fluid flowSeries in Computational Methods in Mechanics and Thermal Sciences. McGraw-Hill, New York.
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Sezai, I., Tasdemiroglu, E., 1995. Effect of bottom reflectivity on ground heat losses for solar ponds. Solar Energy 55 (4), 311–319. Singh, T.P., Singh, A.K., Kaushik, N.D., 1994. Investigations of thermodynamic instabilities and ground storage in a solar pond by simulation model. Heat Recov. Syst. 14 (2), 401–407. Sodah, M.S., Kaushik, N.D., Rao, S.K., 1981. Thermal analysis of three zone solar pond. Int. J. Energy Res. 5, 321–340. Srinivasan, J., Guha, A., 1987. The effect of bottom reflectivity on the performance of a solar pond. Solar Energy 39 (4), 361–367.