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Saturated solar ponds: 1. Simulation procedure
Solar EnergyVol. 50, No. 3, pp. 275-282, 1993
0038-092X/93 $6.00 + .00 Copyright © 1993 Pergamon Press Ltd.
Printed in the U.S,A.
SATURATED
S O L A R P O N D S : 1. S I M U L A T I O N P R O C E D U R E
D. SUBHAKARand S. SRINIVASAMURTHY Refrigeration and Air-Conditioning Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Madras, 600 036 India Abstract--The mass and energy balances on the upper convective zone, nonconvective zone, and lower convectivezone of a saturated solar pond are written to yield a set of nonlinear partial differentialequations. These are solved numerically to predict the thermal performance of the pond over a long period of time for various initial and boundary conditions. This model considers external parameters such as hourly variation of incident solar radiation, ambient temperature, air velocity, and relative humidity. Temperature and concentration dependence of density, thermal conductivity, specific heat, and mass diffusivityare taken into account. Heat transfer modes considered between the upper convectivezone and the ambient are convection, evaporation, and radiation, Ground heat losses from the lower convective zone are also considered. This model is used to study the development of temperature and concentration profiles inside a saturated solar pond. This model can also be used to predict the long-term performance of a saturated solar pond for various heat extraction temperatures and rates.
1. I N T R O D U C T I O N
A solar pond collects and stores solar energy in the form of heat over a long period. In a salinity gradient solar pond, the thermal convection is suppressed by imposing a density gradient by dissolving some salts. The density gradient allows the formation of a temperature gradient, which eventually results in storage of heat at the bottom of the pond. However, these ponds have problems of maintenance due to salt diffusion from the bottom and gradient instability due to atmospheric and other disturbances. These problems, which exist in a conventional solar pond, can, at least in theory, be overcome by making the pond saturated at all levels with a salt whose solubility increases with temperature. Such saturated ponds have no apparent diffusion problems and the gradients are self-sustaining, depending on local temperature. Thus the main advantage of such a pond is its inherent stability. The performance of a saturated solar pond is influenced by many parameters that depend on the pond as well as the ambient. Experimental investigations to study the performance of the pond are time consuming and expensive. Any change in pond parameters necessitates a complete change in the experimental setup. Due to long time requirements for the pond to heat up and be ready for heat removal, and also due to the ability of the pond to store heat over seasons, computer simulation provides an alternative approach to study the effect of parameters on pond performance. Ochs and Bradley[1 ] have defined the stability criteria for saturated solar ponds and have identified many salts. They have demonstrated the stability criteria of a borax pond. Jain and Metha[2]have successfully demonstrated the self-creation, self-maintenance, and self-correction criteria of gradients in a saturated disodium phosphate pond. Rothmeyer [ 3 ]has made a detailed study over the stability and Soret effect on a potassium nitrate pond. Recently, Vitner et al. [4 ] analyzed the stability conditions, using potassium 275
aluminum sulphate, and evaluated the effect of operational parameters on stability. The main criteria for the selection of a suitable salt is that it should dissolve less at lower temperatures, and its solubility should increase steeply with temperature. Also it should be stable, colorless in solution, nontoxic, cheap, easily available, and ecologically acceptable. A pond should satisfy the stability criteria given by dp dx
-->o.
(1)
Because density of a salt solution depends on its concentration and temperature,
dp
OpdC+OpdT>o"
dx
OC d x
(2)
OT d x
In case of saturated solutions, because the saturation concentration is a strong function of temperature, the stability condition is approximated as[l] dp
dx
-
Op d T
aTdx
>0.
(3)
This shows that when the temperature gradient is positive, for a pond to be stable the saturation density of the solution should increase with temperature. Various salts like magnesium chloride, potassium nitrate, and ammonium nitrate satisfy the stability criteria and hence are considered suitable for a saturated polar pond. Various models are available to predict the performance of unsaturated solar ponds. Most models consider the pond to be nonconvective and the only mode of heat transfer to be conduction [ 5 ]. A few models consider the solar pond to be analogous to a flat plate collector. Its efficiency is derived in the form of a Hot-
D. SUBHAKAR and S. SRINIVASAMURTHY
276
tel-Whillier-Bliss equation[6]. Analytical[5,6], finite elements[7], and finite differences [ 8-10 ]techniques have been applied to simulate the pond performance. Many models are based on assumptions such as steadystate performance; absence of salt diffusion; constancy of thermophysical properties; neglect of the influence of the upper convective zone (UCZ); consideration of UCZ temperature to be the same as the wet or dry bulb of the ambient temperature; absence of or approximations on heat losses; constancy or sinusoidal variations of incident radiation, ambient temperature, humidity, and air velocity; neglect of bottom losses; and approximation that transmittance of salt solution is the same as that of water. In this paper, a transient model of a saturated solar pond is solved by making energy balances, including salt diffusion. This model takes a variety of initial and boundary conditions and heat removal methods. All ambient parameters are varied hourly. All thermophysical properties are considered to be dependent on both temperature and concentration.
o,
I
x -- 0 UCZ X=X 1
NCZ X
dx I x+~x
XN
x--x 2
LCZ
• OR
XL
2. ANALYSIS
2.1 Physical model and coordinate system For the purpose of modeling, the pond is considered to have three zones, as in Fig. 1. The UCZ has a thickness o f x u , and the lower convective zone (LCZ) has a thickness of XL. The total depth of the pond is L. The nonconvective zone (NCZ) is divided into Ngrids of equal distances of t~x. There is an isothermal layer xg distance below the pond bottom, maintaining a temperature of 7",. Depth is measured from the top of the pond. x = 0 indicates the surface of the pond; x = x~ indicates the layer separating the UCZ from the NCZ, and x = x2 indicates the layer separating the NCZ from the LCZ.
x=L
Xg
Te
2.2 Assumptions It is assumed that the saturated solar pond is so large that a one-dimensional model is valid to represent its performance. All the sides of the pond are assumed to be well insulated. There is no shade at the bottom of the pond due to side walls. A part of the radiation that is not absorbed at the bottom is assumed not available and lost to ambient. The refractive index of the salt solution is assumed to be constant. The UCZ and LCZ boundary movements are not considered. The ground temperature is assumed to be a function of mean annual ambient temperature. Energy and mass balances are carried out per unit area of the pond. As the heat of dissolution for the salts considered here is small, it is neglected in the energy balance calculations. 2.3 Governing equations Energy balance over the NCZ yields
(kaTI Ox \ Ox ] 0
OI Ox
with boundary conditions as
O(pCoT ) Ot '
Isothermal layer
Fig. 1. Schematic of a saturated solar pond.
T] .... = Tu
(5)
Tlx=x2 = TL.
(6)
Here p, Cp, and k are functions of temperature and concentration. However, temperature and concentration depend on each other and also on depth and time. Insolation depends on depth and time. Because the UCZ and LCZ temperatures are not known, separate energy balances are carried out on each, using lumped parameters. Thus, energy balance over the UCZ yields
k OT ~xxlx= .t + I[ x=0 - I1 .... _ Q,
O(pCpT) x o . (7) Ot
(4) Energy balance over the LCZ yields
277
Saturated solar ponds: Part 1
- k ~¢,O~x=x2+ kgO~x~=x~ + l l . = x 2 - Ilx=L
where sky temperature is given by[l 1] T'sks = Ta[0.8 + ( T'~/250)] °'25
O(pCpT)XL. Ot
q- O~bl[x=L -- QR -
(8)
The mass-balance equation needs to be solved only during the period of destabilization. For this duration, the mass balance over the NCZ yields
(22)
and dew point temperature [ 13 ] is Th = 273.15 + ( - 3 5 . 9 5 7 - 1.8726A2 + 1.1689A~).
(23)
Heat transfer between the ground and LCZ is calculated Ox
~xx = - ~ - '
(9)
Qg=
with boundary conditions CI,,,., = C d T l , . . d
(I0)
Cl~=., = c d TIx~.,).
(II)
Here D depends on both concentration and temperature. The initial conditions are TIx-o =
T.,~t
(12)
Clx.0 = Cs(Tlx,0). 2.4
as
(13)
Heat loss calculations
Total heat loss that occurs from the surface of the pond is given by
Qt = Q~ + Q~ + Q,.
(14)
ks(TL- Tg)/xg,
(24)
where x s is the depth of isothermal layer below the pond bottom maintained a t Tg and Tg = 0.83Ta + 3.7.
(25)
2.5. Radiation absorption inside the pond The a m o u n t of radiation reaching any point inside the pond is a function of many factors like position of sun, time of day, location, distance traveled by light inside the pond, wavelength of solar spectrum, type of salt solution, its temperature, and concentration. The incident angle is found from the properties of solar radiation as cos0~ = c o s b c o s 4 ~ c o s w + s i n 6 s i n 4 ~ ,
(26)
where the declination is given by [ l 1]
The convective heat loss is[11] 6 = 23.45 sin[360(284 + na)/365] Q¢ = hc(Tu - Ta). where hc = 5.7 + 3.8va.
(27)
15)
16)
and ~0 and q~ are the hour angle and the longitude, respectively. The a m o u n t of radiation reflected from the surface of the pond is calculated from the angle of reflectance using Snell's law,
The evaporative heat loss is [ 12 ] sin 02/sin 01 =
nl/n2,
(28)
Q¢ = 0.0144(A, - A2){Tv - Ta and Fresnel's equation [ 11 ], + T ~ ( A I - A2)/(268,900 - Al)} °'33
17) r = 0.5 {sin~(02 - 01)/sin2(02 + 05)
A~ = p d T u )
18)
A2 = PdT,)Rh.
19)
Here, saturation pressure is given as a function of temperature by [ 13 ]
0.048602T + 0.417648E - 4 T 2 - 0.144521E - 7 T 3 + 6.5459673 In T).
(20)
The radiative heat loss is Qr = a~( T~j - Tsky), ,4
(21 )
(29)
Thus, the actual radiation that crosses the pond surface is ( 1 - r)lo. The a m o u n t of radiation absorbed by a solution is given by Beer's law[14]as I(X, x) = I(X, 0)e -"t~)x.
p~ = e x p ( - 5 , 8 0 0 . 2 2 0 6 / T + 1.3914993 -
+ tan2(02 - 0~)/tan2(02 + 01)}.
(30)
The actual path length is obtained by dividing the vertical depth by cos 02. Many models, based on this equation, are available to predict the radiation absorption in pure water. Rabl and Nielsen [15 ]divide the solar spectrum into four zones and assign one spectral absorption coeflScient for each spectrum as
D. SUBHAKARand S. SRINIVASAMURTHY
278 4
I ( x ) = Io ~ #ie -"i~ .
(31)
i=0
Bansal and Kaushik[16]have used a five-term model, Hull [ 17 ] has used a 40-term transmission function, and recently Cengel and Ozisik[18]have used a 20-term model, all to various accuracies. Bryant and Colbeck[19 ] have suggested a much simpler expression for water:
I ( x ) = 0.36 - 0.08 In(x).
(32)
Some of the above correlations are based on data from transmission of light through distilled water. However, for a solar pond, the transmission coefficient will be different. Ideally one should use data from actual ponds, but only very few actual measured values are available[20,21]. The following relations are used to calculate the transmissivity of a saturated solar pond in this model: For a MgCl2 pond,
r(x) = exp(-0.4837523 + 0.004513878/x - 1.176989x - 2.476999x 2 + 3.31534x 3 -- 1.155307x4);
(33)
for a KNO3 pond, r(x) = exp(-O.1405344 - 0.002854687/x - 3.044335x + 2.046293x 2 - 0.550894x3). (34)
2.6 Estimation of thermophysical properties Although the variations of density as a function of temperature and concentration are available around ambient temperatures, for many salts the variation of density, specific heat, thermal conductivity, and diffusion coefficient are not available for the full range of temperatures (ambient to boiling point) and concentrations (very dilute to saturation conditions). Available data in the literature are collected, and graphic extrapolation confirmed by regression analysis is used to express the thermophysicai properties as a function of temperature and concentration. In the absence of actual experimental data, the relations given in the Appendix are used in the simulation of saturated solar pond performance.
3. S O L U T I O N
PROCEDURE
To solve the four coupled, nonlinear partial differential eqns (4), (7), (8), and (9), the weighted-average finite differences technique is used [ 22 ]. First, the four equations are written in a suitable finite differences form, considering appropriate boundary conditions. For this purpose, the UCZ and LCZ are considered as single grid points. The NCZ is divided into grids of N
equal sizes. A space step of 6x and time step of 6t are used. The numerical stability criterion [ 22 ] is
abt/ax 2 <_ 0.5.
(35)
Because the weighted average method is used, the equations are of explicit form when the weightage is zero, implicit form when the weightage is one, and Crank-Nicolson form when the weightage is 0.5. Depending on the boundary conditions and heat removal methods, an optimum weightage based on the time taken per time step is used to solve all the equations. Equations (4), (7), and (8), when written in the finite differences form, yield a set of simultaneous equations that are of tridiagonal matrix form. This matrix is inverted using the Thomas algorithm[23]to obtain the temperature profile inside the pond. Because this profile has to be corrected for the nonlinear thermophysical properties, all these equations are iterated over a time step until convergence is obtained. The number of iterations taken for convergence is around eight for the explicit, 60 for the Crank-Nicolson, and 120 for the implicit method, when no heat is removed from the LCZ. These values vary with x, N, t, and boundary conditions. During heat removal, the number of iterations taken for convergence increases significantly. With this temperature profile, eqn (9) is solved using the same procedure, and a corresponding concentration profile is obtained. Again the temperature profile is corrected with the new concentration profile until both profiles satisfy eqns (4), (7), (8), and (9). These new profiles should also satisfy the stability criteria given in eqn (3). This procedure is repeated for each time step. An energy and mass balance check for each grid is done for each time step using the new temperature and concentration profiles. A time step of 1 h is used, because hourly values of meteorological data are normally available. Numerical oscillations are noticed at the beginning of heat removal. In those cases, heat removal is not considered and postponed to the next time step, so that the pond could have gained some more heat and be ready for heat removal. To simulate the pond performance with a given set of parameters over a period of 18 months it takes nearly 3,400 CPU seconds with the Siemens 7-580E computer when the explicit method is used.
4. R E S U L T S A N D D I S C U S S I O N
Using the model described above, the long-term performance of saturated magnesium chloride ponds having 1.05, 1.50, and 2.00 m depth are computed. The meteorological data considered here apply to conditions in Madras, India (18°N, 84°E)[24]. Starting time of the pond is taken as 1 March. Figure 2 shows the variation in LCZ temperature with time for various pond depths. The pond storage temperature increases rapidly and reaches a maximum of 85°C in summer but maintains a temperature of 65°C until the beginning of the next summer. During the next summer,
Saturated solar ponds: Part 1
279
120 _
MgClz
_ _
_
K N O a
100 o
=
Depth (2.00 m//)/
80-
/ / /
//
60r//
N
rl
UCZ = 0 . 0 5 m NCZ = 0 . 5 0 m I
I
I
I
I
3
6
9
12
15
Time
( Months
18
)
Fig. 2. Effect of depth on pond performance.
the pond attains a maximum of 95°C without heat removal. Such information helps to decide heat extraction temperature and duration for useful applications. The development of temperature profile inside the 1.5-m-deep, near-saturated solar pond is shown in Fig.
20 0.0
3. At the beginning, the maximum temperature occurs just below the UCZ, and the portion of the NCZ is slowly heating up. Later the peak temperature shifts toward the LCZ, as the only major heat loss from the pond is through the UCZ. Once the maximum temperature reaches the LCZ, the storage zone temperature
Temperature, 40
°C 60
I
80
MgCI~ Madras
0.4
0.8 i
1.2 i
-247"1~~ f l a t~~erI tirax(e- _ t . . . . . . . 86~
Fig. 3. Temperature profile development inside a saturated solar pond. UCZ = 0.05 m, NCZ = 0.50 m, LCZ = 0.95 m, starting time l March. Profiles are drawn at l, 24, 96, 456, and 863, h of elapsed time.
280
D. SUBHAKAR and S. SRINIVASA M U R T H Y
35
37 I
0.0
Concentration, 39 I
wt 41 I
43 I
MgClz adras
0.4
0.8-
Elapsed time (hr) 1.2
/
- -
,/
5136
I I '
I" I I
13896
I 2208
10224
Fig. 4. Concentration profile development inside a saturated solar pond. UCZ = 0.05 m, NCZ = 0.50 m, LCZ = 0.95 m, starting time 1 March. Profiles are drawn at 2,208, 5,136, 10,224, and 13,896 h of elapsed time.
starts increasing with time. Until then the p o n d will be left unsaturated to keep it stable. Saturation o f the p o n d is achieved only after the m a x i m u m o f the temperature profile reaches the LCZ. Figure 4 shows the m o v e m e n t s o f the c o n c e n t r a t i o n profile over one a n d o n e - h a l f years. As the t e m p e r a t u r e of the LCZ increases, the c o n c e n t r a t i o n increases to supersaturation a n d the profile becomes more stable. W h e n the temperature o f the p o n d drops in the subsequent winter, correspondingly the c o n c e n t r a t i o n profile adjusts itself to a new saturation level t h a t is lower t h a n the s u m m e r c o n c e n t r a t i o n profile b u t still m a i n t a i n s the stability. Thus, this model is capable o f predicting the temperature a n d c o n c e n t r a t i o n profile d e v e l o p m e n t a n d m o v e m e n t inside a saturated solar pond, given its size, location, a n d a m b i e n t conditions, 5. C O N C L U S I O N S
A m a t h e m a t i c a l model of a saturated solar p o n d is presented by m a k i n g the mass a n d energy balances. This model is capable of e n c o m p a s s i n g a variety of b o u n d a r i e s as well as initial conditions. It considers the variation in pond parameters, a m b i e n t parameters, a n d salt solution parameters a n d can be used to study the effect of various parameters o n the useful heat stored in the LCZ. This model can predict the temperature a n d c o n c e n t r a t i o n profiles that exist inside the p o n d at any time a n d helps to study its development with time. W i t h this model, the capacity o f a saturated solar p o n d to store solar energy in the form o f heat over seasons is explained.
NOMENCLATURE A~ A2 C Cp D Dc D% Dr2 Fa hc I K k L M, m nd //l /12
p Q R Rh r T T' 7~ t V v= V X x, x2 Z+, Z_
constant as defined in eqn (18) constant as defined in eqn (19) solution concentration, weight percentage specific heat (J kg -~ °C -Z) diffusion coefficient (m 2 s -~) dielectric constant diffusion coefficient at infinite dilution (m 2 s -l) diffusion coefficient at higher concentrations ( m 2 s-~ ) Faraday convective heat transfer coefficient (W m -2 °C-l) solar radiation flux at any point inside the pond (W m -2 ) constant as in eqn (AI6) thermal conductivity (W m -j °C -1) depth ( m ) moles solvent in V cm 3 solution molality number of the day of the year refractive index of air refractive index of water partial pressure, Pa heat flux (W m-:) gas constant (8.316 J K -~ g mo1-1 ) relative humidity reflectivity temperature (°C) temperature (K) Mean temperature (°C) time (s) volume of solution (cm 3) air velocity ( m s -~ ) partial molal volume of solvent (cm 3 g mo1-1) depth (m) boundary between UCZ and NCZ boundary between NCZ and LCZ valency
281
Saturated solar ponds: Part 1
Greek a thermal diffusivity (m 2 s -~ ) or ab pond bottom absorptivity constant used in eqn (A7) declination angle (°) e emissivity tr Stefan-Boltzmann constant (5.669E-8 W m -2 K -4) 0~ solar radiation incident angle over pond surface (o) 02 angle of refraction at pond surface (°) 4~ lontitude ( ° ) hour angle ( ° ) ~, wavelength (m -~) ~ o ~,o limiting ionic conductance (cm A V -j g-equivalent -~ ) ~, spectral absorption coefficient for the ith band of solar radiation u viscosity, poise ~, fraction of solar radiation with absorption coefficient r/i r transmissivity p density (kg m 3) y+ activity coefficient
Subscripts a c d e g k N R r s sky sol T u U w wet 0
ambient convective dew point evaporative ground lower convective zone Nonconvective zone heat removed radiative saturation sky solution top useful upper convective zone water wet bulb pond surface
REFERENCES
1. T. L. Ochs and J. O. Bradley, Stability criteria for saturated solar ponds, Proceedings t f l 4 t h 1ECEC, Boston, 53-55 (1979). 2. S. C. Jain and G. D. Metha, Laboratory demonstration of self-creation, self-maintenance and self-correction of saturated solar ponds, Pr~'eedings ofl 5th IECEC. Seattle, 1448-1451 (1980). 3. M.K. Rothmeyer, Saturated solar ponds: Modified equations and results of laboratory experiments. M. S. Thesis, University of New Mexico, Albuquerque, NM (1979). 4. A. Vitner, S. Sarig, and R. Reisfeld, The self-generation mechanism of a laboratory scale saturated solar pond, Solar Energy41, 133-140 (1988). 5. R. Weinberger, The physics of the solar pond, Solar Energy8, 45-56 (1964).
6. C. F. Kooi. The steady state salt gradient solar pond, Solar Energy 23, 37-45 (1979). 7. Z. Panahi, J. C. Batty, and J. P. Riles, Numerical simulation of the performance of a salt-gradient solar pond, ASME J. Solar Energy Eng. 105, 369-374 (1983). 8. J. R. Hull, Computer simulation of solar pond thermal behavior, Solar Energy 25, 33-40 (1980). 9. Y. F. Wang and A. Akbarzadeh, A study on the transient behavior of solar ponds, Energy 7, 1005-1017 (1982). 10. H. Rubin, B. A. Benedict, and S. Bachu, Modeling the performance of a solar pond as a source of thermal energy, Solar Energy 32, 771-778 (1984). 11. J. A. Duffle and W. A. Beckman, Solar engineering of thermalprocesses, John Wiley & Sons, New York (1980). 12. P. i. Cooper, The maximum efficiency of single-effect solar stills, Solar Energy 15, 205-217 (1973). 13. American Society of Heating, Refrigerating, and Air Conditioning Engineers, 1981 fundamentals handbook, ASHRACE, Atlanta, GA ( 1981 ). 14. P. T. Tsilingiris, An accurate upper estimate for the transmission of solar radiation in salt gradient ponds, Solar Energy 40, 41-48 (1988). 15. A. Rabl and C. E. Nielsen, Solar ponds for space heating, Solar Energy 17, 1-12 (1975). 16. P. K. Bansal and N. D. Kaushik, Salt gradient stabilized solar pond collector, Energy Cony. Mgmt. 21, 81-95 (1981). 17. J. R. Hull, Calculation of solar pond thermal energy efficiency with a diffusely reflecting bottom, Solar Energy 29, 385-389 (1982). 18. Y. A. Cengel and M. N. Ozisik, Solar radiation absorption in solar ponds, Solar Energy 33, 581-591 (1984). 19. H. C. Bryant and I. Colbeck, A solar pond for London? Solar Energy 19, 321-322 (1977). 20. A. Almanza and H. C. Bryant, Observations of the transmittance in two solar ponds, ASME Z Solar Energy Eng. 105, 378-379 (1983). 21. Yu. Usmanov, V. Eliseev, and G. Umarov, Optical characteristics of a solar reservoir, Geliotechnika 7, 28-32 (1971). 22. G. D. Smith, Numerical solution oJpartial differential equations, Clarendon Press, London (1978). 23. D. U. von Rosenberg, Methods for the numerical solution q/partial differential equations, Appendix A, p. 113, Elsevier, New York (1969). 24. A. Mani, Handbook of solar radiation--Data for India, Allied, New Delhi ( 1981 ). 25. H. Stephen and T. Stephen, Solubilities oJinorganic and organic compounds, Pergamon Press, New York ( 1963 ). 26. T. lsono, Density, viscosity and electrolytic conductivity of concentrated aqueous electrolyte solutions at several temperatures, J. Chem. Eng. Data 29, 45-52 (1984). 27. International Critical Tables, Vol. 2, pp. 327-328, McGraw-Hill, New York (1928). 28. R. C. Reid and T. K. Sherwood, The properties of gases and liquids, 2nd ed., pp. 561-566, McGraw-Hill, New York (1958). 29. R. A. Robinson and R. H. Stokes, Electrolytic solutions, p. 465, Butterworth, Stoneham, MA (1959). 30. Handbook of Chemistry and Physics, 60th ed., CRC Press, Boca Raton, FL (1979).
APPENDIX
1. Saturation concentration as a fimction of temperature
Cs = 6.097778 + 0.9312499T - 0.00296313T 2
The expression for saturation concentration [25]for MgCI2 + 0.1325556E - 5T 3.
is
(A2)
Cs = 33.56694 + 0.091710133T - 0.7433568E - 3T 2 + 0.7092859E - 5T 3 and for KNO3 is
(AI)
2. Estimation t?[o, Cp, and k Density, expressed as a function of temperature and concentration [26], for MgC12 is of the form
D. SUBHAKAR and S. SRINIVASA MURTHY
282
,~{~g = 15.73337 + 1.492666T
P = 987.8076 + 9 . 6 3 1 3 6 3 C - 0 . 0 3 1 6 1 9 8 7 C T
~c~ = 38.93973 + 1.41499T + 0.3165063E - 2 T z
+ 0.1734767E - 3 C T 2 - 0.4419498E - 2CT L5
,~, = 38.23607 + 1.355312T + 0.2124153E - 2 T 2 + 0.8534297E - 2 C t S T
(A3) ~ o 3 = 39.70862 + 1.175739T + 0.37718E - 2 T 2. ( A I 0 )
and for KNO3 For higher concentrations, the l i m i t i n g diffusion coelficient is correlated using G o r d o n ' s s e m i e m p i r i c a l equation [ 28 ] as
P = 987.0859 + 7 . 7 7 4 6 9 2 C - 0 . 0 6 0 6 5 7 2 9 C T - 0.8392187E - 4 C T 2 + 0 . 0 0 8 5 7 4 9 4 8 C T LS.
(A4) o
V
#w
The variation o f Cp with c o n c e n t r a t i o n [ 2 7 ] f o r MgCI2 is expressed as Cp = (4.185 - 0 . 0 6 5 7 C + 421.2E - 6 C 2 ) , 1,000
(A5)
W a t e r viscosity is corrected for a given t e m p e r a t u r e using the following relation [ 301: I o g ( u w / l . 0 0 2 ) = 11.3272(20 - T )
a n d for KNO3 as
- 0.001053(T-
Cp = 1 , 0 0 0 . ( 4 . 2 6 0 9 1 4 - 0 . 0 5 4 3 3 8 C + 0.770462E - 3 C 2 - 0.3764739E - 5C3).
(A6)
The d e p e n d e n c e of t h e r m a l c o n d u c t i v i t y on c o n c e n t r a t i o n and t e m p e r a t u r e is expressed a s [ 2 7 ]
20)21/(T+
105).
The activity coelficient of the solution is estimated [ 29 ] as - I n 3'_* = 4,202,796.774 × { p w / ( T ~ D ~ ) } ° s Z + Z _ K °'5,
k = kw(l - 1.0E - 53C)
(A12)
(AI3)
(A7) where density o f water is expressed[30]as
and kw = 0.5608 + 0.1986E - 2 T - 0.7765E - 5 T 2,
(A8)
ow = (999.8395 + 1 6 . 9 4 5 2 T -
7.987E - 3 T 2
- 46.17E - 6 T 3 + 105.563E - 9 T 4 - 280.542E
where 3 for MgCI2 is 488 and for KNO3 is 347. -
12TS)/(I.0+0.0168798T),
(A14)
3. Estimation q f d~ffhsion co~[licient In case of a saturated solar pond, the mass diffusion coefficient is to be e s t i m a t e d at or near saturated conditions. Because actual data are not available, the following procedure is used to estimate the diffusion coetficient at a desired temperature and concentration. The l i m i t i n g diffusion coelficient D°2 is e s t i m a t e d using the Nernst e q u a t i o n [ 2 8 ] a s
D°2 = O . O 0 0 2 R T F a 2 / { ( I / X ° ) + ( I / X ° _ ) I .
(A9)
The l i m i t i n g ionic c o n d u c t a n c e ( ~ ) values for various ions are corrected for a given t e m p e r a t u r e [ 2 9 ] a s
T h e dielectric constant for water is e s t i m a t e d [ 3 0 ] a s Dc = 78,5411.0 - 4.579E - 3 ( T - 5(T-
25) + 1,19E
25)2- 2.8E- 8(T-
25)31
(AI5)
and K = 0.5 ~ MiZ-2,.
(AI6)
0038-092X/93 $6.00 + .00 Copyright © 1993 Pergamon Press Ltd.
Printed in the U.S,A.
SATURATED
S O L A R P O N D S : 1. S I M U L A T I O N P R O C E D U R E
D. SUBHAKARand S. SRINIVASAMURTHY Refrigeration and Air-Conditioning Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Madras, 600 036 India Abstract--The mass and energy balances on the upper convective zone, nonconvective zone, and lower convectivezone of a saturated solar pond are written to yield a set of nonlinear partial differentialequations. These are solved numerically to predict the thermal performance of the pond over a long period of time for various initial and boundary conditions. This model considers external parameters such as hourly variation of incident solar radiation, ambient temperature, air velocity, and relative humidity. Temperature and concentration dependence of density, thermal conductivity, specific heat, and mass diffusivityare taken into account. Heat transfer modes considered between the upper convectivezone and the ambient are convection, evaporation, and radiation, Ground heat losses from the lower convective zone are also considered. This model is used to study the development of temperature and concentration profiles inside a saturated solar pond. This model can also be used to predict the long-term performance of a saturated solar pond for various heat extraction temperatures and rates.
1. I N T R O D U C T I O N
A solar pond collects and stores solar energy in the form of heat over a long period. In a salinity gradient solar pond, the thermal convection is suppressed by imposing a density gradient by dissolving some salts. The density gradient allows the formation of a temperature gradient, which eventually results in storage of heat at the bottom of the pond. However, these ponds have problems of maintenance due to salt diffusion from the bottom and gradient instability due to atmospheric and other disturbances. These problems, which exist in a conventional solar pond, can, at least in theory, be overcome by making the pond saturated at all levels with a salt whose solubility increases with temperature. Such saturated ponds have no apparent diffusion problems and the gradients are self-sustaining, depending on local temperature. Thus the main advantage of such a pond is its inherent stability. The performance of a saturated solar pond is influenced by many parameters that depend on the pond as well as the ambient. Experimental investigations to study the performance of the pond are time consuming and expensive. Any change in pond parameters necessitates a complete change in the experimental setup. Due to long time requirements for the pond to heat up and be ready for heat removal, and also due to the ability of the pond to store heat over seasons, computer simulation provides an alternative approach to study the effect of parameters on pond performance. Ochs and Bradley[1 ] have defined the stability criteria for saturated solar ponds and have identified many salts. They have demonstrated the stability criteria of a borax pond. Jain and Metha[2]have successfully demonstrated the self-creation, self-maintenance, and self-correction criteria of gradients in a saturated disodium phosphate pond. Rothmeyer [ 3 ]has made a detailed study over the stability and Soret effect on a potassium nitrate pond. Recently, Vitner et al. [4 ] analyzed the stability conditions, using potassium 275
aluminum sulphate, and evaluated the effect of operational parameters on stability. The main criteria for the selection of a suitable salt is that it should dissolve less at lower temperatures, and its solubility should increase steeply with temperature. Also it should be stable, colorless in solution, nontoxic, cheap, easily available, and ecologically acceptable. A pond should satisfy the stability criteria given by dp dx
-->o.
(1)
Because density of a salt solution depends on its concentration and temperature,
dp
OpdC+OpdT>o"
dx
OC d x
(2)
OT d x
In case of saturated solutions, because the saturation concentration is a strong function of temperature, the stability condition is approximated as[l] dp
dx
-
Op d T
aTdx
>0.
(3)
This shows that when the temperature gradient is positive, for a pond to be stable the saturation density of the solution should increase with temperature. Various salts like magnesium chloride, potassium nitrate, and ammonium nitrate satisfy the stability criteria and hence are considered suitable for a saturated polar pond. Various models are available to predict the performance of unsaturated solar ponds. Most models consider the pond to be nonconvective and the only mode of heat transfer to be conduction [ 5 ]. A few models consider the solar pond to be analogous to a flat plate collector. Its efficiency is derived in the form of a Hot-
D. SUBHAKAR and S. SRINIVASAMURTHY
276
tel-Whillier-Bliss equation[6]. Analytical[5,6], finite elements[7], and finite differences [ 8-10 ]techniques have been applied to simulate the pond performance. Many models are based on assumptions such as steadystate performance; absence of salt diffusion; constancy of thermophysical properties; neglect of the influence of the upper convective zone (UCZ); consideration of UCZ temperature to be the same as the wet or dry bulb of the ambient temperature; absence of or approximations on heat losses; constancy or sinusoidal variations of incident radiation, ambient temperature, humidity, and air velocity; neglect of bottom losses; and approximation that transmittance of salt solution is the same as that of water. In this paper, a transient model of a saturated solar pond is solved by making energy balances, including salt diffusion. This model takes a variety of initial and boundary conditions and heat removal methods. All ambient parameters are varied hourly. All thermophysical properties are considered to be dependent on both temperature and concentration.
o,
I
x -- 0 UCZ X=X 1
NCZ X
dx I x+~x
XN
x--x 2
LCZ
• OR
XL
2. ANALYSIS
2.1 Physical model and coordinate system For the purpose of modeling, the pond is considered to have three zones, as in Fig. 1. The UCZ has a thickness o f x u , and the lower convective zone (LCZ) has a thickness of XL. The total depth of the pond is L. The nonconvective zone (NCZ) is divided into Ngrids of equal distances of t~x. There is an isothermal layer xg distance below the pond bottom, maintaining a temperature of 7",. Depth is measured from the top of the pond. x = 0 indicates the surface of the pond; x = x~ indicates the layer separating the UCZ from the NCZ, and x = x2 indicates the layer separating the NCZ from the LCZ.
x=L
Xg
Te
2.2 Assumptions It is assumed that the saturated solar pond is so large that a one-dimensional model is valid to represent its performance. All the sides of the pond are assumed to be well insulated. There is no shade at the bottom of the pond due to side walls. A part of the radiation that is not absorbed at the bottom is assumed not available and lost to ambient. The refractive index of the salt solution is assumed to be constant. The UCZ and LCZ boundary movements are not considered. The ground temperature is assumed to be a function of mean annual ambient temperature. Energy and mass balances are carried out per unit area of the pond. As the heat of dissolution for the salts considered here is small, it is neglected in the energy balance calculations. 2.3 Governing equations Energy balance over the NCZ yields
(kaTI Ox \ Ox ] 0
OI Ox
with boundary conditions as
O(pCoT ) Ot '
Isothermal layer
Fig. 1. Schematic of a saturated solar pond.
T] .... = Tu
(5)
Tlx=x2 = TL.
(6)
Here p, Cp, and k are functions of temperature and concentration. However, temperature and concentration depend on each other and also on depth and time. Insolation depends on depth and time. Because the UCZ and LCZ temperatures are not known, separate energy balances are carried out on each, using lumped parameters. Thus, energy balance over the UCZ yields
k OT ~xxlx= .t + I[ x=0 - I1 .... _ Q,
O(pCpT) x o . (7) Ot
(4) Energy balance over the LCZ yields
277
Saturated solar ponds: Part 1
- k ~¢,O~x=x2+ kgO~x~=x~ + l l . = x 2 - Ilx=L
where sky temperature is given by[l 1] T'sks = Ta[0.8 + ( T'~/250)] °'25
O(pCpT)XL. Ot
q- O~bl[x=L -- QR -
(8)
The mass-balance equation needs to be solved only during the period of destabilization. For this duration, the mass balance over the NCZ yields
(22)
and dew point temperature [ 13 ] is Th = 273.15 + ( - 3 5 . 9 5 7 - 1.8726A2 + 1.1689A~).
(23)
Heat transfer between the ground and LCZ is calculated Ox
~xx = - ~ - '
(9)
Qg=
with boundary conditions CI,,,., = C d T l , . . d
(I0)
Cl~=., = c d TIx~.,).
(II)
Here D depends on both concentration and temperature. The initial conditions are TIx-o =
T.,~t
(12)
Clx.0 = Cs(Tlx,0). 2.4
as
(13)
Heat loss calculations
Total heat loss that occurs from the surface of the pond is given by
Qt = Q~ + Q~ + Q,.
(14)
ks(TL- Tg)/xg,
(24)
where x s is the depth of isothermal layer below the pond bottom maintained a t Tg and Tg = 0.83Ta + 3.7.
(25)
2.5. Radiation absorption inside the pond The a m o u n t of radiation reaching any point inside the pond is a function of many factors like position of sun, time of day, location, distance traveled by light inside the pond, wavelength of solar spectrum, type of salt solution, its temperature, and concentration. The incident angle is found from the properties of solar radiation as cos0~ = c o s b c o s 4 ~ c o s w + s i n 6 s i n 4 ~ ,
(26)
where the declination is given by [ l 1]
The convective heat loss is[11] 6 = 23.45 sin[360(284 + na)/365] Q¢ = hc(Tu - Ta). where hc = 5.7 + 3.8va.
(27)
15)
16)
and ~0 and q~ are the hour angle and the longitude, respectively. The a m o u n t of radiation reflected from the surface of the pond is calculated from the angle of reflectance using Snell's law,
The evaporative heat loss is [ 12 ] sin 02/sin 01 =
nl/n2,
(28)
Q¢ = 0.0144(A, - A2){Tv - Ta and Fresnel's equation [ 11 ], + T ~ ( A I - A2)/(268,900 - Al)} °'33
17) r = 0.5 {sin~(02 - 01)/sin2(02 + 05)
A~ = p d T u )
18)
A2 = PdT,)Rh.
19)
Here, saturation pressure is given as a function of temperature by [ 13 ]
0.048602T + 0.417648E - 4 T 2 - 0.144521E - 7 T 3 + 6.5459673 In T).
(20)
The radiative heat loss is Qr = a~( T~j - Tsky), ,4
(21 )
(29)
Thus, the actual radiation that crosses the pond surface is ( 1 - r)lo. The a m o u n t of radiation absorbed by a solution is given by Beer's law[14]as I(X, x) = I(X, 0)e -"t~)x.
p~ = e x p ( - 5 , 8 0 0 . 2 2 0 6 / T + 1.3914993 -
+ tan2(02 - 0~)/tan2(02 + 01)}.
(30)
The actual path length is obtained by dividing the vertical depth by cos 02. Many models, based on this equation, are available to predict the radiation absorption in pure water. Rabl and Nielsen [15 ]divide the solar spectrum into four zones and assign one spectral absorption coeflScient for each spectrum as
D. SUBHAKARand S. SRINIVASAMURTHY
278 4
I ( x ) = Io ~ #ie -"i~ .
(31)
i=0
Bansal and Kaushik[16]have used a five-term model, Hull [ 17 ] has used a 40-term transmission function, and recently Cengel and Ozisik[18]have used a 20-term model, all to various accuracies. Bryant and Colbeck[19 ] have suggested a much simpler expression for water:
I ( x ) = 0.36 - 0.08 In(x).
(32)
Some of the above correlations are based on data from transmission of light through distilled water. However, for a solar pond, the transmission coefficient will be different. Ideally one should use data from actual ponds, but only very few actual measured values are available[20,21]. The following relations are used to calculate the transmissivity of a saturated solar pond in this model: For a MgCl2 pond,
r(x) = exp(-0.4837523 + 0.004513878/x - 1.176989x - 2.476999x 2 + 3.31534x 3 -- 1.155307x4);
(33)
for a KNO3 pond, r(x) = exp(-O.1405344 - 0.002854687/x - 3.044335x + 2.046293x 2 - 0.550894x3). (34)
2.6 Estimation of thermophysical properties Although the variations of density as a function of temperature and concentration are available around ambient temperatures, for many salts the variation of density, specific heat, thermal conductivity, and diffusion coefficient are not available for the full range of temperatures (ambient to boiling point) and concentrations (very dilute to saturation conditions). Available data in the literature are collected, and graphic extrapolation confirmed by regression analysis is used to express the thermophysicai properties as a function of temperature and concentration. In the absence of actual experimental data, the relations given in the Appendix are used in the simulation of saturated solar pond performance.
3. S O L U T I O N
PROCEDURE
To solve the four coupled, nonlinear partial differential eqns (4), (7), (8), and (9), the weighted-average finite differences technique is used [ 22 ]. First, the four equations are written in a suitable finite differences form, considering appropriate boundary conditions. For this purpose, the UCZ and LCZ are considered as single grid points. The NCZ is divided into grids of N
equal sizes. A space step of 6x and time step of 6t are used. The numerical stability criterion [ 22 ] is
abt/ax 2 <_ 0.5.
(35)
Because the weighted average method is used, the equations are of explicit form when the weightage is zero, implicit form when the weightage is one, and Crank-Nicolson form when the weightage is 0.5. Depending on the boundary conditions and heat removal methods, an optimum weightage based on the time taken per time step is used to solve all the equations. Equations (4), (7), and (8), when written in the finite differences form, yield a set of simultaneous equations that are of tridiagonal matrix form. This matrix is inverted using the Thomas algorithm[23]to obtain the temperature profile inside the pond. Because this profile has to be corrected for the nonlinear thermophysical properties, all these equations are iterated over a time step until convergence is obtained. The number of iterations taken for convergence is around eight for the explicit, 60 for the Crank-Nicolson, and 120 for the implicit method, when no heat is removed from the LCZ. These values vary with x, N, t, and boundary conditions. During heat removal, the number of iterations taken for convergence increases significantly. With this temperature profile, eqn (9) is solved using the same procedure, and a corresponding concentration profile is obtained. Again the temperature profile is corrected with the new concentration profile until both profiles satisfy eqns (4), (7), (8), and (9). These new profiles should also satisfy the stability criteria given in eqn (3). This procedure is repeated for each time step. An energy and mass balance check for each grid is done for each time step using the new temperature and concentration profiles. A time step of 1 h is used, because hourly values of meteorological data are normally available. Numerical oscillations are noticed at the beginning of heat removal. In those cases, heat removal is not considered and postponed to the next time step, so that the pond could have gained some more heat and be ready for heat removal. To simulate the pond performance with a given set of parameters over a period of 18 months it takes nearly 3,400 CPU seconds with the Siemens 7-580E computer when the explicit method is used.
4. R E S U L T S A N D D I S C U S S I O N
Using the model described above, the long-term performance of saturated magnesium chloride ponds having 1.05, 1.50, and 2.00 m depth are computed. The meteorological data considered here apply to conditions in Madras, India (18°N, 84°E)[24]. Starting time of the pond is taken as 1 March. Figure 2 shows the variation in LCZ temperature with time for various pond depths. The pond storage temperature increases rapidly and reaches a maximum of 85°C in summer but maintains a temperature of 65°C until the beginning of the next summer. During the next summer,
Saturated solar ponds: Part 1
279
120 _
MgClz
_ _
_
K N O a
100 o
=
Depth (2.00 m//)/
80-
/ / /
//
60r//
N
rl
UCZ = 0 . 0 5 m NCZ = 0 . 5 0 m I
I
I
I
I
3
6
9
12
15
Time
( Months
18
)
Fig. 2. Effect of depth on pond performance.
the pond attains a maximum of 95°C without heat removal. Such information helps to decide heat extraction temperature and duration for useful applications. The development of temperature profile inside the 1.5-m-deep, near-saturated solar pond is shown in Fig.
20 0.0
3. At the beginning, the maximum temperature occurs just below the UCZ, and the portion of the NCZ is slowly heating up. Later the peak temperature shifts toward the LCZ, as the only major heat loss from the pond is through the UCZ. Once the maximum temperature reaches the LCZ, the storage zone temperature
Temperature, 40
°C 60
I
80
MgCI~ Madras
0.4
0.8 i
1.2 i
-247"1~~ f l a t~~erI tirax(e- _ t . . . . . . . 86~
Fig. 3. Temperature profile development inside a saturated solar pond. UCZ = 0.05 m, NCZ = 0.50 m, LCZ = 0.95 m, starting time l March. Profiles are drawn at l, 24, 96, 456, and 863, h of elapsed time.
280
D. SUBHAKAR and S. SRINIVASA M U R T H Y
35
37 I
0.0
Concentration, 39 I
wt 41 I
43 I
MgClz adras
0.4
0.8-
Elapsed time (hr) 1.2
/
- -
,/
5136
I I '
I" I I
13896
I 2208
10224
Fig. 4. Concentration profile development inside a saturated solar pond. UCZ = 0.05 m, NCZ = 0.50 m, LCZ = 0.95 m, starting time 1 March. Profiles are drawn at 2,208, 5,136, 10,224, and 13,896 h of elapsed time.
starts increasing with time. Until then the p o n d will be left unsaturated to keep it stable. Saturation o f the p o n d is achieved only after the m a x i m u m o f the temperature profile reaches the LCZ. Figure 4 shows the m o v e m e n t s o f the c o n c e n t r a t i o n profile over one a n d o n e - h a l f years. As the t e m p e r a t u r e of the LCZ increases, the c o n c e n t r a t i o n increases to supersaturation a n d the profile becomes more stable. W h e n the temperature o f the p o n d drops in the subsequent winter, correspondingly the c o n c e n t r a t i o n profile adjusts itself to a new saturation level t h a t is lower t h a n the s u m m e r c o n c e n t r a t i o n profile b u t still m a i n t a i n s the stability. Thus, this model is capable o f predicting the temperature a n d c o n c e n t r a t i o n profile d e v e l o p m e n t a n d m o v e m e n t inside a saturated solar pond, given its size, location, a n d a m b i e n t conditions, 5. C O N C L U S I O N S
A m a t h e m a t i c a l model of a saturated solar p o n d is presented by m a k i n g the mass a n d energy balances. This model is capable of e n c o m p a s s i n g a variety of b o u n d a r i e s as well as initial conditions. It considers the variation in pond parameters, a m b i e n t parameters, a n d salt solution parameters a n d can be used to study the effect of various parameters o n the useful heat stored in the LCZ. This model can predict the temperature a n d c o n c e n t r a t i o n profiles that exist inside the p o n d at any time a n d helps to study its development with time. W i t h this model, the capacity o f a saturated solar p o n d to store solar energy in the form o f heat over seasons is explained.
NOMENCLATURE A~ A2 C Cp D Dc D% Dr2 Fa hc I K k L M, m nd //l /12
p Q R Rh r T T' 7~ t V v= V X x, x2 Z+, Z_
constant as defined in eqn (18) constant as defined in eqn (19) solution concentration, weight percentage specific heat (J kg -~ °C -Z) diffusion coefficient (m 2 s -~) dielectric constant diffusion coefficient at infinite dilution (m 2 s -l) diffusion coefficient at higher concentrations ( m 2 s-~ ) Faraday convective heat transfer coefficient (W m -2 °C-l) solar radiation flux at any point inside the pond (W m -2 ) constant as in eqn (AI6) thermal conductivity (W m -j °C -1) depth ( m ) moles solvent in V cm 3 solution molality number of the day of the year refractive index of air refractive index of water partial pressure, Pa heat flux (W m-:) gas constant (8.316 J K -~ g mo1-1 ) relative humidity reflectivity temperature (°C) temperature (K) Mean temperature (°C) time (s) volume of solution (cm 3) air velocity ( m s -~ ) partial molal volume of solvent (cm 3 g mo1-1) depth (m) boundary between UCZ and NCZ boundary between NCZ and LCZ valency
281
Saturated solar ponds: Part 1
Greek a thermal diffusivity (m 2 s -~ ) or ab pond bottom absorptivity constant used in eqn (A7) declination angle (°) e emissivity tr Stefan-Boltzmann constant (5.669E-8 W m -2 K -4) 0~ solar radiation incident angle over pond surface (o) 02 angle of refraction at pond surface (°) 4~ lontitude ( ° ) hour angle ( ° ) ~, wavelength (m -~) ~ o ~,o limiting ionic conductance (cm A V -j g-equivalent -~ ) ~, spectral absorption coefficient for the ith band of solar radiation u viscosity, poise ~, fraction of solar radiation with absorption coefficient r/i r transmissivity p density (kg m 3) y+ activity coefficient
Subscripts a c d e g k N R r s sky sol T u U w wet 0
ambient convective dew point evaporative ground lower convective zone Nonconvective zone heat removed radiative saturation sky solution top useful upper convective zone water wet bulb pond surface
REFERENCES
1. T. L. Ochs and J. O. Bradley, Stability criteria for saturated solar ponds, Proceedings t f l 4 t h 1ECEC, Boston, 53-55 (1979). 2. S. C. Jain and G. D. Metha, Laboratory demonstration of self-creation, self-maintenance and self-correction of saturated solar ponds, Pr~'eedings ofl 5th IECEC. Seattle, 1448-1451 (1980). 3. M.K. Rothmeyer, Saturated solar ponds: Modified equations and results of laboratory experiments. M. S. Thesis, University of New Mexico, Albuquerque, NM (1979). 4. A. Vitner, S. Sarig, and R. Reisfeld, The self-generation mechanism of a laboratory scale saturated solar pond, Solar Energy41, 133-140 (1988). 5. R. Weinberger, The physics of the solar pond, Solar Energy8, 45-56 (1964).
6. C. F. Kooi. The steady state salt gradient solar pond, Solar Energy 23, 37-45 (1979). 7. Z. Panahi, J. C. Batty, and J. P. Riles, Numerical simulation of the performance of a salt-gradient solar pond, ASME J. Solar Energy Eng. 105, 369-374 (1983). 8. J. R. Hull, Computer simulation of solar pond thermal behavior, Solar Energy 25, 33-40 (1980). 9. Y. F. Wang and A. Akbarzadeh, A study on the transient behavior of solar ponds, Energy 7, 1005-1017 (1982). 10. H. Rubin, B. A. Benedict, and S. Bachu, Modeling the performance of a solar pond as a source of thermal energy, Solar Energy 32, 771-778 (1984). 11. J. A. Duffle and W. A. Beckman, Solar engineering of thermalprocesses, John Wiley & Sons, New York (1980). 12. P. i. Cooper, The maximum efficiency of single-effect solar stills, Solar Energy 15, 205-217 (1973). 13. American Society of Heating, Refrigerating, and Air Conditioning Engineers, 1981 fundamentals handbook, ASHRACE, Atlanta, GA ( 1981 ). 14. P. T. Tsilingiris, An accurate upper estimate for the transmission of solar radiation in salt gradient ponds, Solar Energy 40, 41-48 (1988). 15. A. Rabl and C. E. Nielsen, Solar ponds for space heating, Solar Energy 17, 1-12 (1975). 16. P. K. Bansal and N. D. Kaushik, Salt gradient stabilized solar pond collector, Energy Cony. Mgmt. 21, 81-95 (1981). 17. J. R. Hull, Calculation of solar pond thermal energy efficiency with a diffusely reflecting bottom, Solar Energy 29, 385-389 (1982). 18. Y. A. Cengel and M. N. Ozisik, Solar radiation absorption in solar ponds, Solar Energy 33, 581-591 (1984). 19. H. C. Bryant and I. Colbeck, A solar pond for London? Solar Energy 19, 321-322 (1977). 20. A. Almanza and H. C. Bryant, Observations of the transmittance in two solar ponds, ASME Z Solar Energy Eng. 105, 378-379 (1983). 21. Yu. Usmanov, V. Eliseev, and G. Umarov, Optical characteristics of a solar reservoir, Geliotechnika 7, 28-32 (1971). 22. G. D. Smith, Numerical solution oJpartial differential equations, Clarendon Press, London (1978). 23. D. U. von Rosenberg, Methods for the numerical solution q/partial differential equations, Appendix A, p. 113, Elsevier, New York (1969). 24. A. Mani, Handbook of solar radiation--Data for India, Allied, New Delhi ( 1981 ). 25. H. Stephen and T. Stephen, Solubilities oJinorganic and organic compounds, Pergamon Press, New York ( 1963 ). 26. T. lsono, Density, viscosity and electrolytic conductivity of concentrated aqueous electrolyte solutions at several temperatures, J. Chem. Eng. Data 29, 45-52 (1984). 27. International Critical Tables, Vol. 2, pp. 327-328, McGraw-Hill, New York (1928). 28. R. C. Reid and T. K. Sherwood, The properties of gases and liquids, 2nd ed., pp. 561-566, McGraw-Hill, New York (1958). 29. R. A. Robinson and R. H. Stokes, Electrolytic solutions, p. 465, Butterworth, Stoneham, MA (1959). 30. Handbook of Chemistry and Physics, 60th ed., CRC Press, Boca Raton, FL (1979).
APPENDIX
1. Saturation concentration as a fimction of temperature
Cs = 6.097778 + 0.9312499T - 0.00296313T 2
The expression for saturation concentration [25]for MgCI2 + 0.1325556E - 5T 3.
is
(A2)
Cs = 33.56694 + 0.091710133T - 0.7433568E - 3T 2 + 0.7092859E - 5T 3 and for KNO3 is
(AI)
2. Estimation t?[o, Cp, and k Density, expressed as a function of temperature and concentration [26], for MgC12 is of the form
D. SUBHAKAR and S. SRINIVASA MURTHY
282
,~{~g = 15.73337 + 1.492666T
P = 987.8076 + 9 . 6 3 1 3 6 3 C - 0 . 0 3 1 6 1 9 8 7 C T
~c~ = 38.93973 + 1.41499T + 0.3165063E - 2 T z
+ 0.1734767E - 3 C T 2 - 0.4419498E - 2CT L5
,~, = 38.23607 + 1.355312T + 0.2124153E - 2 T 2 + 0.8534297E - 2 C t S T
(A3) ~ o 3 = 39.70862 + 1.175739T + 0.37718E - 2 T 2. ( A I 0 )
and for KNO3 For higher concentrations, the l i m i t i n g diffusion coelficient is correlated using G o r d o n ' s s e m i e m p i r i c a l equation [ 28 ] as
P = 987.0859 + 7 . 7 7 4 6 9 2 C - 0 . 0 6 0 6 5 7 2 9 C T - 0.8392187E - 4 C T 2 + 0 . 0 0 8 5 7 4 9 4 8 C T LS.
(A4) o
V
#w
The variation o f Cp with c o n c e n t r a t i o n [ 2 7 ] f o r MgCI2 is expressed as Cp = (4.185 - 0 . 0 6 5 7 C + 421.2E - 6 C 2 ) , 1,000
(A5)
W a t e r viscosity is corrected for a given t e m p e r a t u r e using the following relation [ 301: I o g ( u w / l . 0 0 2 ) = 11.3272(20 - T )
a n d for KNO3 as
- 0.001053(T-
Cp = 1 , 0 0 0 . ( 4 . 2 6 0 9 1 4 - 0 . 0 5 4 3 3 8 C + 0.770462E - 3 C 2 - 0.3764739E - 5C3).
(A6)
The d e p e n d e n c e of t h e r m a l c o n d u c t i v i t y on c o n c e n t r a t i o n and t e m p e r a t u r e is expressed a s [ 2 7 ]
20)21/(T+
105).
The activity coelficient of the solution is estimated [ 29 ] as - I n 3'_* = 4,202,796.774 × { p w / ( T ~ D ~ ) } ° s Z + Z _ K °'5,
k = kw(l - 1.0E - 53C)
(A12)
(AI3)
(A7) where density o f water is expressed[30]as
and kw = 0.5608 + 0.1986E - 2 T - 0.7765E - 5 T 2,
(A8)
ow = (999.8395 + 1 6 . 9 4 5 2 T -
7.987E - 3 T 2
- 46.17E - 6 T 3 + 105.563E - 9 T 4 - 280.542E
where 3 for MgCI2 is 488 and for KNO3 is 347. -
12TS)/(I.0+0.0168798T),
(A14)
3. Estimation q f d~ffhsion co~[licient In case of a saturated solar pond, the mass diffusion coefficient is to be e s t i m a t e d at or near saturated conditions. Because actual data are not available, the following procedure is used to estimate the diffusion coetficient at a desired temperature and concentration. The l i m i t i n g diffusion coelficient D°2 is e s t i m a t e d using the Nernst e q u a t i o n [ 2 8 ] a s
D°2 = O . O 0 0 2 R T F a 2 / { ( I / X ° ) + ( I / X ° _ ) I .
(A9)
The l i m i t i n g ionic c o n d u c t a n c e ( ~ ) values for various ions are corrected for a given t e m p e r a t u r e [ 2 9 ] a s
T h e dielectric constant for water is e s t i m a t e d [ 3 0 ] a s Dc = 78,5411.0 - 4.579E - 3 ( T - 5(T-
25) + 1,19E
25)2- 2.8E- 8(T-
25)31
(AI5)
and K = 0.5 ~ MiZ-2,.
(AI6)