THE ADVANCED SOLAR POND (ASP): BASIC THEORETICAL ASPECTS *

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..'
SolarEMrgyVol.43.No.I,pp.35-44,1989
PrintedintheU.S.A.
~
0038-092X/89 S3.00+.00
CopyrightC>1989MaxwellPergamonMacmillanpic
THE ADVANCED SOLAR POND (ASP):BASIC THEORETICAL
ASPECTS*
HILLEL RUBINand GIORGIOA.BEMPORAD
CoastalandMarineEngineeringResearchInstitute(CAMERI) Departmentof Civil Engineering,
Technion-IsraelInstituteof Technology,Haifa32000,Israel
Abstract-This manuscriptconcernsthepossibleimprovementof theconventionalsolarpond(CSP)
performanceby applyinga multiselectiveinjectionandwithdrawalprocedure.We applythetermad-
vancedsolarpond(ASP),forasolarpond(SP) inwhichsuchaprocedureisapplied.Themultiselective
injectionandwithdrawalprocedurecreatesin theSP a stratifiedthermallayer,namelyaflowinglayer
whichis subjecttosalinityandtemperaturestratification.This phenomenonis associatedwithreduction
of heatlossesintotheatmosphereandanincreaseof thetemperatureof thefluid layeradjacentto the
SP bottom.
In theframeworkof thisstudytransportphenomenain theASP areanalyzedandsimulatedby ap-
plyinga simplifiedmathematicalmodel.The analysisandsimulationsindicatethatthemultiselective
andwithdrawalproceduremaysignificantlyimprovetheperformanceof theSP.
1.INTRODUCTION
The solarpond(SP) is a shallowwaterbodybeing
virtuallyatrapfor solarradiation.Thetrappedsolar
radiationis convertedinto thermalenergywhichis
accumulatedinthedeepwaterlayersof theSP.The
thermalenergycanbe accumulateddueto thesta-
bilizingsalinitygradientsexistingin theSP,which
~ preventthermalconvectionin thewaterbody.
Properoperationof theSP dependsontheability
towithdrawhotwaterbyaselectivewithdrawalwhile
preservingthedensityprofileof thepond.
In theconventionalsolarpond(CSP) weidentify
threemajorfluidlayersasshowninFig.lea):surface
layer,barringlayer,andthermallayer.
The surfacelayeris completelymixeddueto at-
mosphericeffects.Thebarringlayeris comprisedof
a stagnantfluid;it separatesthethermallayerfrom
thesurfacelayer.Heatis accumulatedinthethermal
layer;thislayeris subjecttohorizontalflow needed
in'orderto utilizethethermalenergy.The thermal
layeris almostcompletelymixedduetotheselective
withdrawal,injection,andthermalconvection.
In theadvancedsolarpond(ASP)[I] thereis an
additionalstratifiedthermallayeras shownin Fig.
l(b).This layeris comprisedof severalsublayers.
Eachsyblayeris equippedwithinjectionandwith-
drawalports.Thereforeamultiselectiveinjectionand
withdrawalcharacterizestheASP.By makingsome
basiccalculationsit wasclaimedthattheASP overall
efficiencycanbemuchhigherthanthatof theCSP[l].
HowevertheanticipatedconfigurationshowninFig.
1(b) requiresadequatefacilitiesthatshouldbe de-
veloped.Thephysicalrationalefor theASP,asstated
by oneof thereviewersof thispaper,is to create
flowinglayersin theupperportionsof thethermal
/'"""'0..
*Partsof thisstudywerepresentedattheAnnualMeet-
ing of theAmericanSolar EnergySociety,June 20-24,
1988.
layerandlowerportionsof thebarringlayerof aCSP,
soastoremoveheatfromthelayers,lowertheirtem-
perature,reducediffusivelossesthroughthebarring
layer,andleavemorethermalenergyfor storageand
extraction.
This studyconcernsthebasictheoreticalaspects
of theASP performance.Wedevelopamathematical
approachleadingtoanumericalmodelbywhichthe
engineeringfeasibilityof theASP canbeevaluated.
2.ABSORPTION OF THE SOLAR RADIATION
The solarradiationis absorbedin theSP andcon-
vertedintothermalenergy.Theheatingprocesscan
berepresentedasaneffectgeneratedbyalinesource
whosestrengthis distributedexponentiallyalongthe
SP depth[2-5].The strengthof thethermalenergy
source,qT,eventuallyrepresentstherateof absorp-
tionof thesolarenergyin thewaterbody.
The solarradiationarrivingat thebottomof the
SP is completelyabsorbedbythepondbottom,pro-
videdthatit is completelyblack.If thepondbottom
is insulatedtheenergyabsorbedin thebottomleads
to heatflux whichentersthethermallayer.
We applyasteadystatesimulationof theSP per-
formanceby referringto theaverageannualvalues
of thephysicalparametersgoverningtheSP opera-
tion.It wasshownby variousstudies[6]thatsteady
statesimulationsbasedonaveragevaluesof param-
etersarequiteaccuratefor calculationsreferringto
longtimesof operationof theSP.
Ourcalculationsof solarradiationrefertotheDead
SeaareainIsrael,wheresomeCSPsareoperational.
We considerthatthesolarradiationenergypenetrat-
ing theSP surfaceis 200Wm-2.This valueis ob-
tainedby assumingthathalfof thedailyenergyar-
rivesattheSP surfaceduringthemiddlethirdof the
day,andthattheradiationat 2 p.m.of October21
canrepresentthestrengthof thesolarradiationof the
35
36
H.RUBINand Go A.BEMPORAD
Plasticnet
Velocity
Salinity
temperature
°t
Salinity
temperature
it
---L-
-r
Surface
layer
Barring layer
Thermal
layer
( a)
Plasticnet
~
-..2..-
-r
Surface
layer
Barring
layer
Stratifiedthermallayer
Homogeneousthermallayer
(b)
Fig.1.Distributionof velocity,salinity,temperatUreanddensityin solarponds.(a) theconventional
solarpond.(b) theadvancedsolarpond.
middlethirdof theday.The refractionangleis as-
sumedtobe32.5degrees.
3.THE FLOW FIELD
Thesurfacelayer.This layeris subjectto atmo-
sphericeffectsandthewashflow.Somewaterquan-
titiesevaporatefromthislayerintotheatmosphere.
Thereforetheflow'rateof thesurfacelayerdecreases
alongthepondasfollows
Q(T)
=
Q~)
-
qx
whereQ(T)is thesurfaceflow-rateper unit width;
Q~) is theentrancevalueof Q(T);q is therateof
evaporationfromtheASP surface;x is thehorizontal
coordinate.It shouldbenotedthatwithregardtosome
aspectsthenonuniformvelocityprofileof thesurface
layershouldbetakenintoaccount.However,in the
presentstudy,wemainlyconcernthedifferencesbe-
tweentheCSP andASP.For suchconsiderationsthe
assumptionof uniformsurfaceflow is acceptable.
Somestudies[7,8]referto mixingeffectsgener-
atedby theatmosphere.Sucheffectscausethesur-
faceflowtobeassumedforpracticalpurposesasbeing
(1)
almostuniformlydistributed.Longexperiencein the
laboratoryandfieldoperationsshowedthatthesur-
faceof theSP shouldbeprotectedagainstwindef-
fects.Sucheffectsincludewavesandvariouskinds
of currents.Plasticnets,asshownin Figs.l(a) and
l(b) werefoundtobeexcellentmeansfortheSP sur-
faceprotection.Eventuallyin everyoperationalSP
suchmeansareutilized.
Thebarringlayer.This layeris stagnant.Large
salinitygradientsexistingin this layerinsulatethe
thermallayersfromthemixingeffectsexistinginthe
surfacelayer.
Thestratifiedthermallayer.This layeris subject
tohorizontalflow,anditssalinitygradientavoidsthe
formationof circulatingcurrentsof thermohaline
convection.Thereforeineachsublayerthefollowing
conditionshouldbesatisfied[9,1O];
-
ac
~
(
v -t K
)
apaT
j
ap
ay
v+D aTay ac
(2)
whereC is salinity;T is temperature;p is density;v
is kinematicviscosity;Kis heatdiffusivity;D is salt
diffusivity;y is theverticalcoordinate.
...
The flowin thestratifiedthermallayeris carried
..
.<
'f/"
4
.
t
'.It
..
'"
?
~
~
~
'*
~
'.
.
~
II
!
~:
11
~.
~
t
i
"1
,
I
.
f
r
,-
.,
I
I
I
I
II
f~
t
It
J
-'if;;
.;:~
r
Ii'
A
;1'
~~
~
'"
.~
,~
Theoreticalaspectsof theASP
outbyinjectionandwithdrawalportscreatingseveral
sublayers.The totalthicknessof thestratifiedther-
mallayeris givenasfollows:
M
deS)
=2:
dj
i-I
wheredj is thethicknessof eachflowingsublayer;M
is thenumberof sublayers.At theinjectionsports
eachsublayerhasits particulartemperatureandsa-
linity.However,dueto thesmall thicknessof the
sublayers,it is expectedthat,aftera shortdistance,
continuoussalinityandtemperatureprofilesarees-
tablishedin thestratifiedthermallayer.
Theshearstressdistributioninthei-thsublayeris
givenasfollows:
T
=
TiB
-
Y(TiB
-
T/T)/dj
whereTiBandT,Tareshearstressesatthebottomand
topof thesublayer,respectively;Y is alocalvertical
coordinate,namelyY = 0 at thebottomof thesub-
layer,andY = dj atthetopof thesublayer.
Thefollowingconditionsshouldbesatisfiedatthe
interfacesexistingbetweenthevarioussublayers:
T,T= T(i+I)B UiT= U(i+I)B
Accordingtovariousstudies[11-13]wemayassume
thatif thethermallayersaresubjecttoa continuous
laminarvelocityprofilethenthefollowingcondition
is satisfied:
TMT
= -ttTw tt ==0.62
whereTwis theshearstressatthebottomof theSP;
- TMTistheshearstressatthetopof thestratifiedther-
mal layer;tt is a coefficient.
The thicknessof eachflowingsublayerdepends
on its flow-rateQ(i)andits densitygradientasfol-
lows[I4]:
~
=
K
(
~
)
O.27
Lj gO.sLl's;Lj
=
p(i)
(iJp/iJY)j
wherepmis theaveragedensityof thesublayerfluid;
Lj is thebuoyancycharacteristiclength;K is a con-
stant;g is thegravitationalacceleration.Someother
expressionsfor dj weresuggestedfor laminaraswell
asinertialbuoyantlayers[I5,I6].However,all these
expressionsprovidesimilarresultsin therangeof
Reynoldsnumbersabout1000whichis of ourinter-
estwithregardto theSP.
Accordingto variousexperimentalstudies[17]
performedinReynoldsnumbersof approximately1000
it isobtainedthatK
==
0.2.
Reynoldsnumberof thethermallayersflow can
bedefinedbyvariousexpressions.Hereweadoptthe
followingdefinition:
Re=R
v""
(3)
,whereQ is thedischargeper unit width;v""is the
averagekinematicviscosity.Consideringlaminarflow
of thethermallayersanintegrationof eqn(4) yields
thefollowingvelocityproflle:
I
[
y2
]
U
=-
TiBY
- -
(TiB
-
TiT)
+
UiB
J.Lj 2dj
whereIJ.jis theviscosityof thefluid comprisingthe
i-thsublayer;UiBisthevelocityexistingatthebottom
of thesublayer.
By integrationof (9) we obtainthe following
expressionfor thesublayerflow-rate:
(4)
[
I
tf
Q(i)
=
udY
=
~ (2TiB
+
T,T)
+u;sdj
° 6IJ.j
(5)
Thehomogeneousthermallayer.This layeris sub-
ject tohorizontalflowandthermalconvectionstem-
mingfromheatabsorptionbytheblackbottomof the
pond.In our calculationswe considerthattheflow
is basicallylaminar.Howevertheeffectof thermal
convectionwhichenhancesmomentumtransferinthe
fluid layeris representedby an increasedeffective
viscosity.
The shearstressandvelocitydistributionin the
homogeneousthermallayeraregivenrespectivelyas
follows:
T
= Tw
-
Y(Tw
-
TIB)/do
(6)
1
[
y2
]
U
=
J.Ltff TwY- 2do(Tw- TIB)
wheredois thehomogeneouslayerthickness;IJ.tffis
theeffectiveviscosity.By integrating(12) overthe
thicknessof thehomogeneousthermallayerwe ob-.
tainthefollowingexpressionfor itsflow-rate:
(7)
Q
(O)
=
..!
(
d~ d~
)
Tw
+
TIB-
fl-tff
3 6
Thethicknessofthehomogeneoustherma1layershould
becontrolledartificallyby variousmeans,like tem-
poraryincreaseof theflow-rateof thestratifiedther-
mal layer.In ourcalculationsweassumethatdohas
a givenvalue.By applyingeqns(4)-(13) wedeter-
minethedistributionof shearstressesandvelocities
in thethermallayersasshownin theAppendix.
4.THE TEMPERATURE FIELD
We assumethat,in eachelementaryfluidvolumeof
the waterbody,heatconvectionis the dominant
transportmechanismin thehorizontaldirection,and
molecularheatdiffusionis thedominanttransport
mechanismin theverticaldirection.In ordertocon-
37
(8)
(9)
(10)
(11)
(12)
(13)
38
H.RUBINand G.A.BEMPORAD
sidereffectsof thennalconvectionon heattransfer
in thehomogeneousthennallayer,it is possibleto
assumethatheatdiffusivityis increasedbythether-
mal convection[18].Calculationsindicatethatheat
transferfromthepondbottomintothehomogeneous
layerincreasesheatdiffusivityin severalordersof
magnitude.Thereforewe mayassumethatwithre-
gardto heattransferthis layeris eventuallyfully
mixed.The barringlayeris stagnant.Thus in this
layeronly moleculardiffusionof heattakesplace.
Duetocomparativelylargetemperaturegradientsin
theverticaldirectionandsmalltemperaturegradients
in thehorizontaldirectionwe ignoreheatdiffusion
in thelatterdirection.Mixing effectsin thesurface
layerandintimatecontactwiththeatmospherecause
itstemperaturetobeunifonnlydistributedinthever-
ticalandhorizontaldirections.
Fonnulatingthe assumptionsrepresentedin the
precedingparagraphweobtainthefollowingexpres-
sionsfor thesurface,barring,stratifiedthennaland
homogeneousthennallayers,respectively:
T(T) = canst
tiT qT
-
0
-+--
ayZ KpCp
"aT a2T qT
--=-+-
K ax ayZ KpCp
do
(0)aTo
=
1
qTdy
pCpQ ax 0 aT
)
+J<f)
-
(
KpCpay Y-do
whereT(T)is thesurfacelayertemperature;Cpis the
specificheat;Tois thehomogeneouslayertempera-
ture;K is heatdiffusivity;J}B)is heatflux fromthe
SP bottom.Expressions(14)-(17) are employed
hereafterin orderto developa numericalmodelof
heattransferin theASP.
5.TIlE SALINITY FIELD
With regardto convectionanddiffusiondominance
we utilizethesameassumptionappliedin thepre-
cedingsectionwithregardtoheattransfer.We also
assumethatthehomogeneousthennallayeris fully
mixedwith regardto salinitydistribution.Vertical
moleculardiffusionis theonlysalinitytransfermech-
anismconsideredin thestagnantbarringlayer.Due
totheverticalsalinitydiffusionprocess,thesalinity
of thehomogeneousthennallayerdecreasesalong
theSP,andthesalinityof thesurfacelayerincreases
alongtheSP.
Fonnulatingtheassumptionsrepresentedin the
precedingparagraphweobtainthefollowingexpres-
sionsfor thesurface,barring,stratifiedthennaland
homogeneousthennallayersrespectively:
(
aC(T»
)
=
-
Q(T)acm+qeT)
ay y-h D ax.D
a2c
-=0
ayZ
(18)
(19)
"ac a2c
vax=ayZ
(20)
(
aco
)
=-
Q(O)aco
ay y-do D ax
whereeT) is thesurfacelayersalinity;Cois theho-
mogeneousthennallayersalinity;D is saltdiffusiv-
ity;h is thedistancebetweentheSP bottomandthe
interfaceexistingbetweenthesurfaceandbarring
layers.
Theexpressionsrepresentedbyeqns(18)-(21)are
employedhereafterin thedevelopmentof thenu-
mericalmodelwhichisabletosimulatesalinitytransfer
in theASP.
(21)
(14)
6.THE NUMERICAL MODEL
Expressions(14)-(21) aresubjectto initial con-
ditions,whicharephysicallytheconditionsexisting
attheSP entrance.Expressions(15)-(16)and(19)-
(20)areparabolicdifferentialequations.In orderto
solvetheseequationsweapplyanimplicitfinitedif-
ferencenumericalmodelwhichutilizesvariablemesh
sizein theverticaldirection.Applyingsuchanap-
proachfor (15)and(16)weobtainthefollowingset
of linearequations:
.
(15)
(16)
(17)
[
2
]
[
"
i
_..,.(m+1)
+
T
(m+1)
-
lJ-1 J
(ilYj +ilYj-l)ilYj_,Kjilx
2 2
]
+ +
(ilYj
+
ilYj-l)ilYj (ilYj
+
ilYj-l)ilYj-1
-..,.(m+I)
[
2
]
1'+1
J
(ilYj
+
ilYj-l)ilYj
=T;m).-!i..+ qTj
~
KjPCp
(22)
wheremis asuperscriptreferringtothelongitudinal
positionof thenodalpoint;j is a subscriptreferring
to theverticalpositionof thenodalpoint.
Expressions(14) and (17) representboundary
conditionsof thenumericalgridbeingexpressedre-
spectivelyasfollows
~m)=T(T) atj =N
(23)
[Q
<O).1
]
1
T~m+l)
- - -
+
Tjm+I)-
Kilx ilYI ilYI
= Tdm)Q(O)+ (<p)y=do
Kilx PKCpQ(O)
.where <Pis the intensityof the solar radiation.
(24)
~
1!~.
f.".
~i~:
~'
~~
Theoreticalaspectsof theASP
Expressions(22)-(24)in conjunctionwithgiven
initial conditions,representingthetemperaturepro-
file attheSP entrance,yieldthedevelopmentof the
temperatureprofilealongtheSP.
Applyingthevariablemeshsizefor thefinitedif-
ferenceapproximationof (19)and(20)weobtainthe
followingsetof linearequations:
[
2
] [
U.
-
c
~m+I)
+
C
(m+I)
-L
)-1
j
(~Yj
+
~Yj-I)~Yj-1 DJu
2 2
]
+ +
(~Yj+I1Yj-I)~Yj (I1Yj+I1Yj-I)~Yj-1
[
2
]
u.
-
C
(m+1)
-
C
(mJ-L
j+1 - j
(~Yj
+
I1Yj-I)~Yj Dlu
Expressions(18) and(21) representboundarycon-
ditionsof thenumericalgridbeingexpressedrespec-
tivelyasfollows:
1
[
1
Q
(7)(mJ
]
C
(m+l)-
+
C
.(m+IJ
+
q
N-I N - - --
~YN-I I1YN-1 DNtu DN
Q(7)(mJ
='C(mJ-
N DNtu
[
Q(OJ
1
]
1
c~m+1)
- - -
+c\m+IJ-
D,ju ~YI I1YI
Q(OJ
-c(mJ-
- 0D,ju
whereN isthenumberof nodalpointsinthevertical
direction.As asubscriptN referstotheinterfacebe-
tweenthesurfaceandbarringlayers.
Expressions(25)-(27)inconjunctionwithagiven
salinityprofileat theSP entranceyield thedevel-
opmentof thesalinityprofilealongtheASP.
7.SIMULATION OF THE ADVANCED SOLAR POND
PERFORMANCE
Theoreticallythenumericalmodeldevelopedinthis
studycan be appliedfor any velocitydistribution
stemmingfromdifferentinjectionprocedures.How-
everonlyaprocedureformingthesmoothandcon-
tinuouslaminarprofileavoidstheKelvin-Helmholtz
typeof instability[18]generatingcirculatingcurrents
andmixingphenomenabetweenthemovingsublay-
ers.Thereforewe referhereto smoothlaminarve-
locityprofiles.
Practicallythevalueof theReynoldsnumberde-
pendson theavailabilityof thermalenergyandthe
temperatureof its exploitationstemmingfromthe
strengthof solarradiationandthesizeof theSP.For
longSP theReynoldsnumbershouldbehighenough
toallowappropriateuseof theaccumulatedthermal
energy.We applyhereseveralexamplesreferringto
Re
=
500.However thereferencetohigher Reynolds
numberdoesnotchangethebasicimplicationsof this
study.
39
~~
Figure2showsthevelocityprofileforthermallayer
dischargeof 0.5 litlseclm in whichReynoldsnum-
beris 500.ThisprofIleof velocitiesis associatedwith
thedevelopmentof temperatureandsalinityprofiles
asshownrespectivelyin Figs.3 and4.In thesefig-
urestheprofilesreferringto entrancerepresentthe
initial conditionsof thenumericalmodel.
Figures3providesomeinformationaboutthepos-
sibleadvantagesof theASP withregardtotheCSP.
It is assumedthatthesurfacelayertemperatureis al-
mostidenticaltotheatmospherictemperature.There-
fore,heatlossesintotheatmospheredependon the
temperaturegradientsexistingin thebarringlayer.
We havealsotoconsiderthatsomesolarradiationis
obsorbedin thebarringlayer.Most of it is lostinto
theatmosphere.As a resultof thephenomenadis-
cussedintheprecedingsentences,by increasingthe
averagetemperatureof thethermallayerexistingin
a CSP we causeanincreasedheatlossinto theat-
mosphere.We applyFig.3,andcomparetheper-
formanceof a CSP andan ASP,whoseReynolds
numberandtotalthicknessof thethermallayersare
identical.If thefluid adjacentto theSP bottomis
subjecttothesametemperaturein bothponds,then
thetemperatureexistingin theinterfacebetweenthe
thermallayersandthebarringlayeris higherin the
CSP thanin theASP.Thereforeidenticaltempera-
tureexistingatthebottomof bothpondsleadstolarger
heatlossesin theCSP thanin theASP.
Therearevariousmannersto calculatetheeffi-
ciencyof theSP performance[19].Herewerepresent
thisparameterastheratiobetweenthethermalen-
ergygainedin thethermallayersandtheenergyof
thesolarradiationwhichpenetratestheSP surface.
The thermalenergygainedin thethermallayers
is equaltothedifferencebetweentheheatflux con-
vectedatthepondexitandthisfluxconvectedatthe
pondentrance.Intheparticularexamplesrepresented
by Fig.3(a)andFig.3(b)wereferredtobottomen-
trancetemperatureof 80°Cin a CSP andanASP,
andsurfacetemperatureof 35°C.Thefiguresindicate
thattheexit,bottomtemperaturesfor theCSP and
ASP are85°C~and94°C,respectively.Thenetenergy
outputof the CSP and ASP are 15.7kW1mand
21.7kW1m,respectively.Theseenergyoutputsare
obtainedwithefficienciesof 8 and11percent,re-
spectively.Followingthe-suggestionof oneof the
reviewersof thismanuscript,weperformedsimula-
tionwithCSP subjecttothesameinitialandbound-
aryconditionsasthoseof Fig.3,whosethermallayer
thicknessis 25cm.Theoutputtemperaturewas88°C
andtheefficiencywas14percent.This phenomenon
is typicaltotheCSP operation,wheresignificantin-
creaseinefficiencycanbeobtainedprovidedthatlow
outputtemperatureandsmall heatstorageare ac-
ceptable.Howeverthegeneraloutcomeof all sim-
ulationswasthatwiththeASP it ispossibletoobtain
significantincreasein thecombinationof themain
basicparametersof theSP utilization:outputtem-
perature,efficiencyandheatstorage.
Withregardtosalinitytransfertherearesomedif-
(26)
~n
1.2
H.RUBINand G.A.BEMPORAD
0.0
o
2
40
:3 4
Velocity,(mm/sec)
5
Fig.2.Velocitydistributioninthesolarpond,Re=500.
ferencesbetweenthedevelopmentof thesalinitypro-
file alongtheCSP andASP asindicatedby Fig.4.
Howeverthesedifferenceshaveaminoreffectonthe
SP performance.It shouldbenotedthatthesignifi-
cantdifferencebetweentheincreaseof salinityof the
surfacelayeranddecreaseof salinityof thethermal
layerstemsfromwaterevaporation.
8.DISCUSSION
The simulationsrepresentedin theprecedingsec-
tiondemonstratesomepossibleadvantagesof theASP.
Suchadvantagescanbe summarizedasanincrease
of thepondbottomtemperature,anincreaseof the
SP efficiencyandan increasein theheatstorage.
Howeverintheprecedingsectionweonlyconsidered
a singledesignprocedureof theASP,in whichthe
stratifiedthermallayeris createdon accountof the
upperportionof thehomogeneousthermallayerof
theCSP.Howevervariousotherdesignandutiliza-
tionproceduresarealsoattractive.It is possibleto
expandthestratifiedthermallayeronaccountof the
barringlayer.Insuchamannerweincreasetheamount
of solarradiationwhichcanbeutilized.We alsoin-
creasetheheatstorageof theSP in sucha manner.
A veryattractiveproceduresuggestsa very thick
stratifiedthermallayerbeingcomprisedof several
sublayers.The flow-ratesof all sublayersareiden-
tical.Theflow-ratewithdrawnfromthefirstsublayer
isinjectedintothesecondsublayerandsoonasshown
schematicallyin Fig.5.Theflowof thelowestsub-
layercanbeeithertransferredintotheheatexchanger
of theheatutilizingsystem,or injectedintotheho-
mogeneousthermallayer(Fig.5).Howeverin such
acasetheinterfaceexistingbetweenthestratifiedand
homogeneousthermallayersmayrepresentalocation
of discontinuityin thevelocityprofileas shownin
Fig.5.Thereforethis interfacemaybe subjectto
Kelvin-Helmholtzinstability.Then it is eventually
representedby a thinmixinglayer.The schematics
of thefiguresuggestthattheheatexchangerssystem
withdrawshotwaterfromthehomogeneousthermal
layer,anddivertswaterof comparativelylow tem-
peratureintothetopof thestratifiedthermallayer.
HoweverFig.5 alsoshowssomeof thepracticaldif-
ficultiesassociatedwiththeASP.TheASP requires
a lot of pipingandpumping,inletsandoutletsar-
rangedlaterallytoinducelaterallyuniformflowand
minimizemixing.All thesetopicshavebeenbeyond
thescopeof thisstudy.Howeversomeotherpositive
issuesof theASP shouldalsobeconsidered.
Thehightemperature'ofthehomogeneousthermal
layerenablesits salinityto be veryhigh,provided'
that salts like magnesiumchlorideare utilized.
Thereforethesurfacelayersalinitycanalsobehigher
thanthesalinityof thatlayerin theCSP.The in-
creasedsalinityof thesurfacelayerdecreasestherate
of evaporationfromtheSP surface.This phenome-
f
..
't!
j
~
*
j
,
I
I
,.'
',.<,
'.W
.,
..
t
e
,
t
v,
;!
t
1
!
1
\,
t:'
t
i
1
!
6
,
-,,\1
.
",
-
1.01
Om
<Entrance)
E
-
..
E
.
----
200m
,g
Ol
-.-.-
400m
-..-.-
600m
...
0.6.
E -...-
800m'
0
It
I
----
JOOOm(Exit)
Q)
0.4
u
c
c
-
CI)
0
0.2
1.2
E
1.0
-
e-
o
~
0.8
CD
CI)
t=
E 0.6
o
at
g 0.4
1
1
2
en
Q
0.2
0.0
20
(a)
1.2
-
E
-
..
E 1.0
o
-
-
o
CD
0.8
CI)
.c:
.-
E 0.6
e
IJ...
~
0.4
c:
o
-
en
Q
0.2
0.0
20
(b)
Theoreticalaspectsof theASP
----
Om (Entrance)
200m
-.-.-
400 m
600 m
-..-..-
-...-
800 m
1000 m(Exit)
----
ave.~radient 80CYm
100
o m(Entrance)
--- -
200m
400m
-.-.-
-..-..-
600m
800m
80
90 100
Fig.3.Temperatureprof1ledevelopmentalongthesolarpond,Re =500.(a) intheconventionalsolar
pond.(b) in theadvancedsolarpond.
41
~
~
42
H.RUBINand G.A.BEMPORAD
1.2
Om
(Entrance)
-
1.0
200m
400m
-.-.-
E
-
..
E
o 0.8
-
-
o
m
Q)
.c.0.6
l-
E
e
lL.0.4
-..---
600m
800m
-...-
----
1000m(Exit)
Fig.4.Salinityprofiledevelopmentalongthesolarpond,Re =500.(a)in theconventionalsolarpond.
(b) in theadvancedsolarpond.
Q)
0
c
0
0.2
-
en
a
0.0
0
10 20 30 40
50
(a)
Concentration
.,(0/0)
1.2
.
-
Om(Entrance)
----
200m
-
1.0.
E
I
-.-.-
400m
-
..
E
I
-..-..-
600 m
0.81
-..--
800m
0
m
r-
----
1000 m(Exit)
Q)
-{:.0.6
E
e
LL
0.4
Q)
0
c
0
-
.0.2
a
0.0
30 40 50
0 10 20
(b)
Concentration.,(0/0)
-
,
"
)
..
.
I,
if
<,
k
~
,
,
~
It.
"
~
I'"
,
.
Theoreticalaspectsof theASP
'V
--:r
Fromheat
exchangers
--c-
Additionalsalt
Fig.5.
A schematicof anadvancedsolarpondinwhichthewithdrawnflow-rateof aparticularsublayer
is injectedintotheadjacentlowersublayer.
nonis associatedwithanincreaseinthesurfacelayer
temperature;namelyit reducesthetemperaturegra-
dientexistingin thebarringlayer,andtherebyit re-
ducesheatlossesintotheatmosphere.
The presentstudycoveredonly somebasicas-
pectsof theASP performance.Somemorecareful
studiesaccompaniedwith experimentalinvestiga-
tionsshouldbeperformedbeforeanypilot plantof
suchaSP isdesigned.Howeververyimportantprac-
ticalissuesshouldbetakenintoaccountlikepiping,
mixingsystem,settlementandsolutionof salts,etc.
All suchsubjectsshouldbedealtbeforea costef-
fectiveASP canbeenvisioned.
9.SUMMARY AND CONCLUSIONS
Thereis a possibilitytoimprovetheperformance
of theCSPbyapplyingamulti-injection-withdrawal
procedure.SuchaprocedurecreatesintheSP anad-
ditionalstratifiedthermallayer.TheSP withsucha
layeristermedanASP.Thebasicaspectsof theASP
operationareanalyzedin thispaperby applyinga
simplifiedmathematicalmodel consideringmajor
transportphenomenain thesolarpond.This model
leadsto analyticalcalculationsof the momentum
transferandnumericalsimulationsof heatandsalin-
itytransferintheSP.Suchsimulationsindicatethat
theASP advantagesareimpliedby higherbottom
temperaturesandhigherefficiency.Furthermorethe
ASP suggestsa varietyof proceduresfor its utiliza-
tion,someof themarediscussedin thispaper.The
positivetheoreticalresultsof thisstudywithregard
totheASP operationsuggesttheperformanceof some
laboratorystudiesrelevanttothissubject.
NOMENCLATURE
C salinity,dimensionless
Co salinityof the homogeneousthennallayer,dimen-
sionless
c:n salinityof thesurfacelayer,dimensionless
Cp specificheat,Jkg"'°e"'
d,thicknessof thei-ththennalsublayer,m
do thicknessof thehomogeneousthennallayer,m
D massdiffusivity,m2s"'
g gravitationalacceleration,ms-2
h distancebetweenthepondbottomandthesurfacelayer,
m
JT diffusiveheatflux,Wm-2
J'f) diffusiveheatflux at thesolarpondbottom,Wm"2
K coefficient,dimen$ionless
L,characteristicbuoyancylengthof thei-thsublayer,m
q rateof evaporation,ms"1
qT strengthof theheatsource,Wm-3
Q volumetricflow-rateper unitwidth,m2s-'
(to flow-rateof thei-thsublayer,m2s"'
.
Q(T) flow-rateof thesurfacelayer,m2s"1
Q?;>entranceflow-rateof thesurfacelayer,m2s"1
Q'OI flow-rateof thehomogeneousthermallayer,m2s"'
Re Reynoldsnumberof thethennallayersfow,dimen-
sionless
T temperature,°e
T(T) temperatureof thesurfacelayer,°e
To temperatureof thehomogeneousthermallayer,°e
U flow velocity,ms"1
U/B flowvelocityatthebottomof thei-thsublayer,ms"'
U,T flow velocityat thetopof thei-thsublayer,ms-I
x horizontalcoordinate,m
y verticalcoordinate,m
Y local verticalcoordinate,m
a ratiobetweenshearstresses,dimensionless
K heatdiffusivity,m2s.1
1.1.viscosity,Pas
1.1..//effectiveviscosity,Pas
v kinematicviscosity,m2s-'
v...averagekinematicviscosity,m2s-1
p density,kg/m-3
43
44
H.RUBINand G.A.BEMPORAD
p(Qaveragedensityof thei-thsublayer,kg/m-J
T shearstress,Pa
Tw shearstressat thesolarpondbottom,Pa
TiS shearstressat thebottomof thei-thsublayer,Pa
TiT shearstressat thetopof thei-thsublayer,Pa
c/I energyof thesolarradiation,Wm-2
Acknowledgment-Thisresearchwas supportedby the
Ministryof EnergyandInfrastructure,Israel.
REFERENCES
1.A.Ozdor,Methodof trappingandutilizingsolarheat,
U.S.patentNo.4,462,389(1984).
2.H.Weinberger,Thephysicsof thesolarpond,Solar
Energy8,45(1964).
3.R.A.Tybout,A recursivealternatetoWeinberger's
model of thesolar pond,Solar Energy 11,109(1966).
4.A.Rabl andC.F.Nielsen,Solarpondfor spaceheat-
ing,Solar Energy17,1(1975).
5.H.Rubin,B.A.BenedictandS.Bachu,Modelingthe
performanceof a solarpondas a sourceof thermal
energy,Solar Energy32,771(1984).
6.V.Joshi andV.V.N.Kishore,Applicabilityof steady
stateequationsforsolarpondthermalperformancepre-.
dictions,Solar Energy11,821(1986).
7.J.R.Hull,Computersimulationof solarpondthermal
behavior,Solar Energy25,33(1980).
8.J.F.AtkinsonandD.R.F.Harleman,A windmixed
layer modelfor solar ponds,Solar Energy31,243
(1983).
9.G.Veronis,On finite amplitudeinstabilityin ther-
mohalineconvection,J.MarineRes.23,1(1965).
10.D.A.Nield,The thermohalineRayleigh-Jeffreys
.problem,J.Fluid Mech.29,545(1967).
11.A.T.IppenandD.R.F.Harleman,Steadystatechar-
acteristicsof subsurfaceflow,U.S.Nat.Bur.of Stan-
dards,Cire.521,Symp.onGravityWaves,79(1951).
12.J.P.Raymond,Etudedescourantsd'eauboueusedans
lesretenues,4thCongressonLargeDams,NewDelhi,
Trans.vol.4,R48(1951).
13.K.Lofquist,Flow andstressnearinterfacebetween
stratifiedliquids,ThePhysicsofFluids3,158(1960).
14.P.Gariel,Recherchesexperimentalessuel'ecoulement
decouchessuperposeesdefluidesdedensitesdifferentes,
La HouilleBlanche4,56(1949).
15.G.A.Lawrence,Selectivewithdrawalsthrougha point
sink,2ndInternationalSymposiumonStratifiedFlows,
Trondheirn,Norway,411,(1980).
16.J.Imberger,Selectivewithdrawals:areview,2ndIn-
ternationalSymposiumon StratifiedFlows,Tron-
dheirn,Norway,411(1980).
17.O.LevinandC.Elata,Selectiveflowof densitystrat-
ifiedfluid,Tech.ReportNo.5/133/62,Dept.of Civil
Eng.,Technion,Haifa,Israel(1962).
18.S.Chandrasekhar,Hydrodynamicandhydromagnetic
stability,OxfordattheClarendonPress,London(1961).
19.C.F.Kooi,Thesteadystatesaltgradientsolarpond,
Solar Energy23,37(1979).
APPENDIX:
CALCULATION OF THE THERMAL LAYERS PHYSICAL PARAMETERS
Accordingtoeqns(5)-(13) weobtainfor eachsublayer
of thestratifiedthermallayerandthehomogeneousthermal
layerthefollowingexpressions:
d,d,Q(o)
-
T;a
+
-
T(I+I}8
+
U(l-UT
=
-
31L1 61L,d,
d,d,
UiT
=
-
T;a
+
-
T(I+I}8
+
U(I-UT
21-4 21L,
(A.1)
(A.2)
1'08
='Tw;TMB= -aT...
(A.3)
where the subscript i
=
0 refers to the homogeneousther-
mal layer.
The expressionsrepresentedin (A.I)-(A.3) are em-
ployedin ordertoobtaina setof M + 1linearequations.
Fromthemoneequationreferstothehomogeneousthermal
layer(i
=
0) andM equationsrefertothevarioussublayers
of thestratifiedthermallayer.TheM + 1equationsare
representedasfollows:
~
(
~
dJ-,
) (
d,d,_,
)
L.J -
+
-
TJS
+
-
+
-
T;a
+
J-O 2ILJ 21LJ-'3ILl 21L,_,
d,Q(Q.
+-T{/+'}8
=
-
O:s I:SM
-
1
61L1 d,
At-I
(
)
(
)
~ dJ dJ-,d",d"'-I
L.J -+- TJB+-+- T",s+
J-O 2ILJ 21LJ-'3ILAt 21L"'-1
d",Q(/o()
-a--rOB=-
61L",d",
Thesetof equationsrepresentedby (A.4) and(A.5) is as-
sociatedwitheqn(7) toproviderelationshipsbetweenthe
shearstressdistribution,thepartialflow-rates,thicknesses
of thevarioussublayersof thestratifiedthermallayerand
thedensitygradientsexistingin thesesublayers.
If wealsorefertoacontinuouslaminarvelocityprofile
thenvaluesof thesublayersthicknessesandflow-ratesare
directlyconnected.
(A.4)
(A.5)