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Solar EnergyVol. 50, No. 3, pp. 275-282, 1993 0038-092X/93 $6.00 + .00Printed in the U.S,A. Copyright 1993 Pergamon Press Ltd.

S A T U R A T E D 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 ED. SUBHAKARand S. SRINIVASA MURTHYRefrigeration 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 lowerconvective zone 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 forvarious initial and boundary conditions. This model considers external parameters such as hourly variationof incident solar radiation, ambient temperature, air velocity, and relative humidity. Temperature andconcentration dependence of density, thermal conductivity, specific heat, and mass diffusivityare taken intoaccount. 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. Thismodel is used to study the development of temperature and concentration profiles inside a saturated solarpond. This model can also be used to predict the long-term performance of a saturated solar pond forvarious heat extraction temperatures and rates.

1. INTRODUCTIONA solar pond collects and stores solar energy in theform o f heat over a long period. In a salinity gradientsolar pond, the thermal convection is suppressed byimposing a density grad ient by dissolving some salts.The density gradient allows the formation of a tem-perature gradient, which eventually results in storageof heat at the bottom of the pond. However, theseponds have problems of mai ntenan ce due to salt dif-fusion from the bottom and gradient instability due toatmospheric and other disturbances. These problems,which exist in a con vent iona l solar pond, can, at leastin theory, be overcome by making the pond saturatedat all levels with a salt whose solubility increases withtemperature. Such saturated ponds have no apparentdiffusion problems an d the gradients are self-sustaining,depending on local temperature. Thus the main ad-vantage of such a pon d is its inherent stability.

The per formance of a saturated solar pond is influ-enced by many parameters that depend on the pondas well as the ambien t. Expe rimental investigat ions tostudy the performance of the pond are time consum ingand expensive. Any change in pond parameters ne-cessitates a complete change in the experimental setup.Due to long time requirements for the pond to heatup and be ready for heat removal, and also due to theability of the pond to store heat over seasons, computersimulation provides an alternative approach to studythe effect of parameters on pond performance.

Ochs and Bradley[1 ] have defined the stability cri-teria for saturated solar ponds and have identified manysalts. They have dem onstr ated the stability criteria ofa borax pond. Jain and Metha[2]have successfullydemonstrat ed the self-creation, self-maintenance , andself-correction criteria of gradients in a saturated di-sodium phosphate pond. Rothmeyer [ 3 ]has made adetailed study over the stability and Soret effecton a potassium nitrate pond. Recently, Vitner etal.[4 ] analyzed the stability conditions, using potassium

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aluminum sulphate, and evaluated the effect of oper-ational parameters on stability.

The m ain criteria for the selection of a suitable saltis that it should dissolve less at lower temperatures,and its solubility should increase steeply with temper-ature. Also it should be stable, colorless in solution,nontoxic, cheap, easily available, and ecologically ac-ceptable.

A po nd should satisfy the stability criteria given bydp- - > o . ( 1 )dx

Because density of a salt solution depends on its con-centration and temperature,

dp OpdC+OpdT>o" ( 2 )dx OC dx OT dx

In case of saturated solutions, because the saturationconcentration is a strong function of temperature, thestability condition is approximated as[l]

dp Op dT- > 0 . ( 3 )dx aTdxThis shows that when the temperatu re gradient is pos-itive, for a pond to be stable the saturation density ofthe solution should increase with temperature. Varioussalts like magnesium chloride, potassium nitrate, andammonium nitrate satisfy the stability criteria andhence are considered suitable for a saturated polarpond.

Various models are available to predict the perfor-mance of unsatu rated solar ponds. Most models con-sider the pond to be nonco nvective and the only modeof heat transfer to be c ondu ction [ 5 ]. A few modelsconsider the solar pond to be analogous to a flat platecollector. Its efficiency is derived in the form o f a Hot-

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Saturated solar ponds: Part 1- k ~,O~x=x2+ kgO~x~=x~ + l l . = x 2 - I lx = L w h e r e s k y t e m p e r a t u r e i s g i v e n b y [ l 1]

q- O ~b l [ x=L - - QR - O ( p C p T ) X L . ( 8 )OtT h e m a s s - b a l a n c e e q u a t i o n n e e d s t o b e s o lv e d o n l yd u r i n g t h e p e r i o d o f d e s ta b i l i z a ti o n . F o r t h i s d u r a t i o n ,t h e m a s s b a l a n c e o v e r th e N C Z y i e l d s

O x ~xx = - ~ - ' ( 9 )w i t h b o u n d a r y c o n d i t i o n s

C I, ,, ., = C d T l , . . d ( I 0 )C l ~ = ., = c d T I x ~ . , ) . ( I I )

H e r e D d e p e n d s o n b o t h c o n c e n t r a t i o n a n d t e m p e r -a t u r e . T h e i n i t ia l c o n d i t i o n s a r eT I x - o = T . , ~ t ( 1 2 )C l x .0 = C s ( T l x , 0 ) . ( 1 3 )

2 .4 Heat los s ca lcu la t ionsT o t a l h e a t l o s s t h a t o c c u r s f r o m t h e s u r f a c e o f t h e

p o n d i s g i v e n b yQt = Q~ + Q~ + Q , . ( 1 4 )

T h e c o n v e c t i v e h e a t l o s s i s [ 1 1 ]Q = h c ( T u - T a ) .

w h e r e1 5 )

h c = 5 .7 + 3 .8 v a . 1 6 )T h e e v a p o r a t i v e h e a t l o s s i s [ 1 2 ]Q = 0 . 0 1 4 4 ( A , - A 2 ) { T v - T a

+ T ~ ( A I - A 2 ) / ( 2 6 8 , 9 0 0 - A l ) } '3 3 1 7 )A ~ = p d T u ) 1 8 )A 2 = P d T , ) R h . 1 9 )

H e r e , s a t u r a t i o n p r e s s u r e i s g i v e n a s a f u n c t i o n o f t e m -p e r a t u r e b y [ 1 3 ]p~ = e x p ( - 5 , 8 0 0 . 2 2 0 6 / T + 1 .3 9 14 9 93

- 0 . 0 4 8 6 0 2 T + 0 . 4 1 7 6 4 8 E - 4 T 2 - 0 . 1 4 4 5 2 1 E- 7 T 3 + 6 . 5 4 5 9 6 7 3 I n T ) .

T h e r a d i a t i v e h e a t l o s s i sQr a~( T~j - ,4= T s k y ) ,

27 7

T 'sk s = Ta [0 .8 + ( T ' ~ /2 50 ) ] '25a n d d e w p o i n t t e m p e r a t u r e [ 1 3 ] i s

( 2 2 )

T h = 2 7 3 . 1 5 + ( - 3 5 . 9 5 7- 1 . 87 2 6 A 2 + 1 . 1 6 8 9 A ~ ) . ( 2 3 )

H e a t t r a n s f e r b e t w e e n t h e g r o u n d a n d L C Z i s c a l c u la t e da s

Q g = k s ( T L - T g ) / x g , ( 2 4 )w h e r e x s is t h e d e p t h o f i s o t h e r m a l l a y e r b e l o w t h ep o n d b o t to m m a i n t a i n e d a t T g a n d

T g = 0 . 8 3 T a + 3 . 7. ( 2 5 )2 .5 . R a d i a t i o n a b s o r p ti o n i n s i d e t h e p o n d

T h e a m o u n t o f ra d i a t io n r e a c h i n g a n y p o i n t i n si d et h e p o n d i s a f u n c t i o n o f m a n y f a c to r s l ik e p o s i t io n o fs u n , t i m e o f d a y , l o c a t i o n , d i s t a n c e t r a v e l e d b y l i g h ti n s i de t h e p o n d , w a v e l e n g t h o f s o la r s p e c t r u m , t y p e o fs a lt s o l u ti o n , i ts te m p e r a t u r e , a n d c o n c e n t r a t i o n . T h ei n c i d e n t a n g l e i s f o u n d f r o m t h e p r o p e r t i e s o f s ol a rr a d i a t i o n a s

co s0 ~ = c o s b c o s 4 ~ c o s w + s i n 6 s i n 4 ~ , ( 2 6 )w h e r e t h e d e c l i n a t i o n i s g i v e n b y [ l 1

6 = 2 3 .4 5 s i n [ 3 6 0 ( 2 8 4 + n a ) / 3 6 5 ] ( 2 7 )a n d ~0 a n d q~ a r e t h e h o u r a n g l e a n d t h e l o n g i t u d e ,r e s p e c t i v e l y .

T h e a m o u n t o f r a d i a t i o n r e f le c t ed f r o m t h e s u rf a c eo f t h e p o n d i s c a l c u l a t e d f r o m t h e a n g l e o f r e f le c t a n ceu s i n g S n e l l ' s la w ,

s in 0 2 / s i n 0 1 = n l / n 2 , ( 2 8 )a n d F r e s n e l ' s e q u a t i o n [ 11 ] ,r = 0 .5 {s in ~ (0 2 - 0 1 ) / s i n 2 (0 2 + 0 5 )

+ t a n 2 ( 0 2 - 0 ~ ) / t a n 2 ( 0 2 + 0 1 ) } . ( 2 9 )T h u s , t h e a c t u a l r a d i a t i o n t h a t c r o s s e s t h e p o n d

su r f ace i s ( 1 - r ) l o . T h e a m o u n t o f ra d i a t io n a b s o r be db y a s o l u t i o n i s g iv e n b y B e e r ' s l a w [ 1 4 ] a s

I ( X , x ) = I ( X , 0 ) e - " t ~ )x . ( 3 0 )T h e a c t u a l p a t h l e n g t h is o b t a i n e d b y d i v i d i n g t h e v e r -t i c a l d e p t h b y c o s 0 2 .

( 2 0 ) M a n y m o d e l s , b a s e d o n t h i s e q u a t i o n , a r e a v a i l a b l et o p r e d i c t th e r a d i a t i o n a b s o r p t i o n i n p u r e w a t e r . R a b la n d N i e l s e n [ 15 ] d i v i d e t h e s o l a r s p e c t r u m i n t o f o u rz o n e s a n d a s s i g n o n e s p e c t r a l a b s o r p t i o n c o e f l S c i e n t

( 2 1 ) f o r e a c h s p e c t r u m a s

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278 D. SUBHAKARand S. SRINIVASAMURTHY4I ( x ) = I o ~ #ie -"i~ . (31)

i = 0

Bansal and Kaushik[16]have used a five-term model,Hull [ 17 ] has used a 40-term transmission function, andrecently Cengel and Ozisik[18]have used a 20-termmodel, all to various accuracies. Bryant and Col-beck[19 ] have suggested a muc h simpler expression forwater:

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 coefficientwill be different. Ideally one should use data from actualponds, but only very few actual measured values areavailable[20,21]. The following relations are used tocalculate the transmissivity of a saturated solar pondin this model:For a MgCl2 pond,

r ( x ) = exp(-0.4837523 + 0.004513878/x- 1.176989x - 2.476999x 2 + 3.31534x 3

for a KNO3 pond,- - 1 . 1 5 5 3 0 7 x 4 ) ; (33)

r(x) = exp(-O.1405344 - 0.002854687/x- 3.044335x + 2.046293x 2 - 0.550894x3). (34)

2 . 6 E s t i m a t i o n o f t h e r m o p h y s i c a l p r o p e r ti e sAlthough the variations of density as a function of

temperature and concentration are available aroundambie nt temperatures, for many salts the variation ofdensity, specific heat, thermal conductivity, and dif-fusion coefficient are not available for the full range oftemperatures (ambient to boiling point) and concen-trations (very dilute to saturation condi tions). Avail-able data in the literature are collected, and graphicextrapolat ion confi rmed by regression analysis is usedto express the thermophysicai properties as a funct ionof temperature and concentration. In the absence ofactual experimental data, the relations given in theAppendix are used in the simula tion of saturated solarpond performance.

3 . S O L U T I O N P R O C E D U R ETo solve the four coupled, non linear partial differ-

ential eqns (4), (7), (8), and (9), the weighted-averagefinite differences technique is used [ 22 ]. First, the fourequations are written in a suitable finite differencesform, considering appropriate boundary conditions.For this purpose, the UC Z a nd LCZ are considered assingle grid points. The NCZ is divided into grids of N

equal sizes. A space step of 6x and time step of 6t areused. The numer ical stability criterion [ 22 ] is

a b t / a x 2

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Saturated solar ponds : Part 1 2791 2 0

_ _ M g C l z_ _ K N O a

1 0 0o

= 80-60-

N r / /r l

//

Depth2 . 0 0 m / / ) // //

U C Z = 0 . 0 5 mN C Z = 0 . 5 0 mI3 I I I6 9 12T i m e ( M o n t h s )

Fig. 2. Effect of depth on pond performance.

I1 5 1 8

t h e p o n d a t ta i n s a m a x i m u m o f 9 5 C w i th o u t h e a tr e m o v a l . S u c h i n f o r m a t i o n h e l p s t o d e c i d e h e a t e x -t r a c t io n t e m p e r a t u r e a n d d u r a t i o n f o r us e fu l a p p li c a -t ions.

T h e d e v e l o p m e n t o f t e m p e r a t u r e p r o f il e in s i d e th e1 .5 -m -deep , nea r - sa tu ra t ed so l a r pon d i s show n in F ig .

3 . A t th e b e g i n n in g , t h e m a x i m u m t e m p e r a t u r e o c c u r sj u s t b e l o w th e U C Z , a n d t h e p o r t i o n o f th e N C Z i ss lowly hea t i ng up . L a t e r t he peak t em pe ra tu re sh i f t st o w a r d t h e L C Z , a s t h e o n l y m a j o r h e a t l o ss f r o m t h ep o n d is t h r o u g h th e U C Z . O n c e t h e m a x i m u m t e m -pe ra tu re r eaches t he L C Z , t he s t o rage zone t em pe ra tu re

0 .0

0 .4

0.8

1.2

T e m p e r a t u r e , C2 0 4 0 6 0 8 0I MgCI~Madras

i

i -247"1~ ~ f l a t~~er t i rax (e - _ t . . . . . . .8 6 ~

Fig. 3. Temperature profile development inside a saturated solar pond. UC Z = 0.05 m, NCZ = 0 .50 m,LCZ = 0.95 m, start ing t ime l March. Profiles are drawn at l , 24, 96, 456, and 863, h o f elapsed t ime.

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2 8 0 D . S U B H A K A R a n d S. S R IN I VA S A M U R T H Y

0.0

0.4

0 . 8 -

1.2 - -

3 5C o n c e n t r a t i o n , w t3 7 3 9 4 1 4 3

I I I IMgClza d r a s

/,/

5 1 3 6

E l a p s e dt i m e ( h r )I I "I I' I

1 3 8 9 6 I2208

1 0 2 2 4

F ig . 4 . C o n c e n t r a t i o n p r o f il e d e v e l o p m e n t in s i d e a s a t u r a te d s o l a r p o n d . U C Z = 0 .0 5 m , N C Z = 0 .5 0 m ,LC Z = 0 .95 m , s ta r t ing t ime 1 March . P ro f i l e s a re d rawn a t 2 ,208 , 5 ,136 , 10 ,224 , and 13 ,896 h o f e lapsedt i m e .

s t a rt s i n c r e a s in g w i t h t im e . U n t i l t h e n t h e p o n d w i llb e l e ft u n s a t u r a t e d t o k e e p i t s t a b l e. S a t u r a t i o n o f t h ep o n d i s a c h i e ve d o n l y a f te r t h e m a x i m u m o f t h e t e m - A~A2p e r a t u r e p r o fi l e r e a c h e s t h e L C Z . F i g u r e 4 s h o w s t h e Cm o v e m e n t s o f t h e c o n c e n t r a t i o n p r of il e o v e r o n e a n d C po n e - h a l f y e a rs . A s t h e t e m p e r a t u r e o f t h e L C Z i n - DDcc r ea s e s, t h e c o n c e n t r a t i o n i n c r e as e s t o s u p e r s a t u r a t i o n D %a n d t h e p r o f il e b e c o m e s m o r e s t a b le . W h e n t h e t e m - Dr2p e r a t u r e o f t h e p o n d d r o p s i n t h e s u b s e q u e n t w i n t e r , F ac o r r e s p o n d i n g l y t h e c o n c e n t r a t i o n p r o fi l e a d j u s t s i t se l f hct o a n e w s a t u r a t i o n l ev e l t h a t i s l o w e r t h a n t h e s u m m e r Ic o n c e n t r a t i o n p r o f i l e b u t s t i ll m a i n t a i n s t h e s t a b il i t y .T h u s , t h i s m o d e l i s c a p a b l e o f p r e d i c t in g t h e t e m p e r - Kka t u r e a n d c o n c e n t r a t i o n p r of il e d e v e l o p m e n t a n d Lm o v e m e n t i n s i d e a s a t u r a t e d s o l a r p o n d , g i v e n i ts s iz e, M ,l o c a t io n , a n d a m b i e n t c o n d i t i o n s , m

n d/ / l

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p r e s e n t e d b y m a k i n g t h e m a s s a n d e n e r g y b a l a n c e s. RT h i s m o d e l i s c a p a b l e o f e n c o m p a s s i n g a v a r i e t y o f R hb o u n d a r i e s a s w e l l a s i n i t i a l c o n d i t i o n s . I t c o n s i d e r s rt h e v a r i a t io n i n p o n d p a r a m e t e r s, a m b i e n t p a r a m e t e rs , TT 'a n d s a lt s o l u t i o n p a r a m e t e r s a n d c a n b e u s e d t o s t u d y 7t h e e f fe c t o f v a r i o u s p a r a m e t e r s o n t h e u s e fu l h e a t ts t o r e d i n t h e L C Z . T h i s m o d e l c a n p r e d i c t t h e t e m - Vp e r a t u r e a n d c o n c e n t r a t i o n p r o fi le s t h a t e x is t i n s id e v=t h e p o n d a t a n y t i m e a n d h e l p s t o s t u d y i ts d e v e l o p m e n t V

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NOMENCLATUREc o n s t a n t a s d e f i n e d in e q n ( 1 8 )c o n s t a n t a s d e fi n e d i n e q n ( 1 9 )s o l u t i o n c o n c e n t r a t i o n , w e i g h t p e r c e n t a g especif ic h eat (J kg -~ C -Z)d i f fus ion coef f ic ien t (m 2 s -~)d i e l ec t r ic c o n s t a n td i f fus ion coef f ic ien t a t in f in i t e d i lu t ion ( m 2 s - l )diffusion coefficient at high er conc entr at io ns ( m 2 s-~ )F a r a d a yc o n v e c t i v e h e a t t r a n s f e r c o e f fi c ie n t ( W m - 2 C - l )s o l a r r a d i a ti o n f l ux a t a n y p o i n t i n s i d e th e p o n d ( Wm -2 )c o n s t a n t a s i n e q n ( A I 6 )t h e r m a l c o n d u c t i v i ty ( W m - j C - 1 )d e p t h ( m )m o l e s s o l v e n t i n V c m 3 s o l u t i o nm o l a l i t yn u m b e r o f t h e d a y o f th e y e a rr e f r a ct i v e i n d e x o f a i rr e f r a ct i v e i n d e x o f w a t e rpa r t i a l p ressure , Pah e a t flu x ( W m - : )gas cons tan t (8 .316 J K -~ g mo1-1 )r e l a ti v e h u m i d i t yreflect ivi tyt e m p e r a t u r e ( C )t e m p e r a t u r e ( K )M e a n t e m p e r a t u r e ( C )t i m e ( s )v o l u m e o f s o l u ti o n ( c m 3)air veloci ty ( m s -~ )p a r t ia l m o l a l v o l u m e o f s o l v e n t ( c m 3 g m o 1 - 1 )d e p t h ( m )b o u n d a r y b e t w e e n U C Z a n d N C Zb o u n d a r y b e t w e e n N C Z a n d L C Zv a l e n c y

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S a t u r a t e d s o l a r p o n d s : P a r t 1 28 1G r e e ka t h e r m a l d i f f u s i v it y ( m 2 s - ~ ) or

a b p o n d b o t t o m a b s o rp t i v i tyc o n s t a n t u s e d i n e q n ( A 7 )d e c l i n a t i o n a n g l e ( )e e m i s s i v i t ytr S t e f a n - B o l t z m a n n c o n s t a n t ( 5 .6 6 9 E - 8 W m - 2 K - 4 )0~ s o l a r r a d i a t i o n i n c i d e n t a n g l e o v e r p o n d s u r f a c e ( o )02 a n g l e o f r e f r a c t i o n a t p o n d s u r f a c e ( )4~ lon t i tu de ( )h o u r a n g l e ( )~, w a v e l e n g t h ( m - ~ )~ o ~,o l i m i t i n g i o n ic c o n d u c t a n c e ( c m A V - j g - e q u i v a -

len t -~ )~ , s p e c t r a l a b s o r p t i o n c o e f f i c i e n t f o r t h e i t h b a n d o fs o l a r r a d i a t i o nu v i scos i ty , po i se~ , f r a c t i o n o f s o l a r r a d i a t i o n w i t h a b s o r p t i o n c o e f f i c ie n tr/ir t r a n s m i s s i v i t yp d e n s i t y ( k g m 3 )y+ ac t iv i ty coe f f ic ien t

S u b s c r i p t sa a m b i e n tc c o n v e c t i v ed d e w p o i n te e v a p o r a t i v eg g r o u n dk l o w e r c o n v e c t i v e z o n eN N o n c o n v e c t i v e z o n eR h e a t r e m o v e dr rad ia t ives s a t u r a t i o nsky skys o l s o l u t i o nT t o pu u se fu lU u p p e r c o n v e c t i v e z o n ew w a t e rw e t w e t b u l b0 p o n d s u r f a c e

REF ER ENCE S

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A P P END I X

1. Saturation concentration as a f imction of temperatureT h e e x p r e s s i o n f o r s a t u r a t i o n c o n c e n t r a t i o n [ 2 5 ] f o r M g C I2is

C s = 6 .0 9 7 7 7 8 + 0 . 9 3 1 2 4 9 9 T - 0 . 0 0 2 9 6 3 1 3 T 2+ 0 . 1 3 2 5 5 5 6 E - 5 T 3. ( A 2 )

C s = 3 3 .5 6 6 9 4 + 0 . 0 9 1 7 1 0 1 3 3 T - 0 . 7 4 3 3 5 6 8 E

a n d f o r K N O 3 i s- 3 T 2 + 0 . 7 0 9 2 85 9 E - 5 T 3 ( A I ) 2. Estim ation t?[o, Cp, and kD e n s i t y , e x p re s s e d a s a f u n c t i o n o f t e m p e r a t u r e a n d c o n -c e n t r a t i o n [ 2 6 ] , f o r M g C 1 2 i s o f t h e f o r m

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8/8

2 8 2 D . S U B H A K AR a n d S . S R I N IV A S A M U R T H YP = 9 8 7 . 8 0 7 6 + 9 .6 3 1 3 6 3 C - 0 . 0 3 1 6 1 9 8 7 C T

+ 0 . 1 7 3 4 7 6 7 E - 3 C T 2 - 0 . 4 4 1 9 4 9 8 E - 2 C T L5+ 0 . 8 5 3 4 2 9 7 E - 2 C t S T ( A 3 )

a n d f o r K N O 3P = 9 8 7 .0 8 5 9 + 7 . 7 7 4 6 9 2 C - 0 . 0 6 0 6 5 7 2 9 C T

- 0 . 8 3 9 2 1 8 7 E - 4 C T 2 + 0 . 0 0 8 5 7 4 9 4 8 C T LS. ( A 4 )T h e v a r i a ti o n o f C p w i t h c o n c e n t r a t i o n [ 2 7 ] f o r M g C I2 i s e x -p r e s s e d a s

C p = ( 4 . 1 8 5 - 0 . 0 6 5 7 C + 4 2 1 . 2 E - 6 C 2 ) , 1 , 0 0 0 ( A 5 )

,~{ ~g = 1 5 . 7 3 3 3 7 + 1 . 4 9 2 6 6 6 T~c~ = 3 8 . 9 3 9 7 3 + 1 . 4 1 4 9 9 T + 0 . 3 1 6 5 0 6 3 E - 2 T z, ~, = 3 8 . 2 3 6 0 7 + 1 . 3 5 5 3 1 2 T + 0 . 2 1 2 4 1 5 3 E - 2 T 2

~ o 3 = 3 9 . 7 0 8 6 2 + 1 . 1 7 5 7 3 9 T + 0 . 3 7 7 1 8 E - 2 T 2. ( A I 0 )F o r h i g h e r c o n c e n t r a t i o n s , t h e l i m i t i n g d i f fu s i o n c o e l f ic i e n t i sc o r r e l a t e d u s i n g G o r d o n ' s s e m i e m p i r i c a l e q u a t i o n [ 2 8 ] a s

o V #w

W a t e r v i s c o s i t y is co r r e c t e d f o r a g i v e n t e m p e r a t u r e u s i n g t h ef o l l o w i n g r e l a t i o n [ 301:

a n d f o r K N O 3 a sC p = 1 , 0 0 0 . ( 4 . 2 6 0 9 1 4 - 0 . 0 5 4 3 3 8 C

+ 0 . 7 7 0 4 6 2 E - 3 C 2 - 0 . 3 7 6 4 7 3 9 E - 5 C 3 ) . ( A 6 )

I o g ( u w / l . 0 0 2 ) = 1 1 . 3 2 7 2 (2 0 - T )- 0 . 0 0 1 0 5 3 ( T - 2 0 ) 2 1 / ( T + 1 0 5) . ( A 1 2 )

T h e a c t i v i t y c o e l f ic i e n t o f t h e s o l u t i o n i s e s t i m a t e d [ 2 9 ] a s

T h e d e p e n d e n c e o f t h e r m a l c o n d u c t i v i ty o n c o n c e n t r a t i o na n d t e m p e r a t u r e i s e x p r es s e d a s [ 2 7 ]

k = k w ( l - 1 . 0 E - 5 3 C ) ( A 7 )a n d

k w = 0 . 5 6 0 8 + 0 . 1 9 8 6 E - 2 T - 0 . 7 7 6 5 E - 5 T 2, ( A 8 )w h e r e 3 f o r M g C I2 is 4 8 8 a n d f o r K N O 3 i s 3 4 7 .

3. Es tim at ion q f d~ffhsion co~[lic ientI n c a s e o f a s a t u r a t e d s o l a r p o n d , t h e m a s s d i f fu s i o n c o e f-

f i c ie n t is t o b e e s t i m a t e d a t o r n e a r s a t u r a t e d c o n d i t i o n s . B e -c a u s e a c t u a l d a t a a r e n o t a v a i l a b l e , th e f o l l o w i n g p r o c e d u r ei s u s e d t o e s t i m a t e t h e d i f f u s i o n c o e t f i ci e n t a t a d e s i re d t e m -p e r a t u r e a n d c o n c e n t r a t i o n . T h e l i m i t i n g d i f f u s i o n c o e l f ic i e n tD 2 i s e s t i m a t e d u s i n g th e N e r n s t e q u a t i o n [ 2 8 ] a s

- I n 3'_* = 4 , 2 0 2 , 7 9 6 . 7 7 4 { p w / ( T ~ D ~ ) } s Z + Z _ K '5 , ( A I 3 )

w h e r e d e n s i t y o f w a t e r i s e x p r e s s e d [ 3 0 ] a s

o w = ( 9 9 9 . 8 3 9 5 + 1 6 . 9 4 5 2 T - 7 . 9 8 7E - 3T 2- 4 6 . 1 7 E - 6 T 3 + 1 0 5 . 5 6 3 E - 9T 4 - 2 8 0 . 5 4 2 E

- 1 2 T S ) / ( I . 0 + 0 . 0 1 6 8 7 9 8 T ) , ( A 1 4 )

T h e d i e le c t r ic c o n s t a n t f o r w a t e r i s e s t i m a t e d [ 3 0 ] a s

D c = 7 8 , 5 4 1 1 . 0 - 4 . 5 7 9 E - 3 ( T - 2 5 ) + 1 ,1 9 E- 5(T - 25) 2- 2.8 E- 8(T - 25)31 ( A I 5 )

D 2 = O . O 0 0 2 R T F a 2 / { ( I / X ) + ( I / X _ ) I . ( A 9 ) a n dT h e l i m i t i n g i o n i c c o n d u c t a n c e ( ~ ) v a l u e s f o r v a r i o u s i o n s K = 0 . 5 ~ MiZ-2 , . ( A I 6 )a r e c o r r e c t e d f o r a g i v e n t e m p e r a t u r e [ 2 9 ] a s