updraft towers Modeling, simulation and optimization of general solar Applied Mathematical Modelling

ContentslistsavailableatScienceDirect

Applied Mathematical Modelling

journalhomepage:www.elsevier.com/locate/apm

Modeling, simulation and optimization of general solar updraft towers

Hannes von Allwörden

^a

, Ingenuin Gasser

^a,∗

, Muhammad Junaid Kamboh

^b

aDepartment Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg 20146, Germany

bInstitute of Micro System Technology, Hamburg University of Technology, Eißendorfer Straße 42 (M), Hamburg 21073, Germany

a r t i c l e i n f o

Article history:

Received 4 October 2017 Revised 2 July 2018 Accepted 11 July 2018 Available online 20 July 2018 Keywords:

Solar updraft tower Sloped collector field Humidity

Small Mach number

a b s t r a c t

Amodeltodescribeasolarchimneypower plantwithagenerallyslopedcollectorfield andforthegeneralsituationofhumidairispresented.Thisisasignificantdevelopment ofexistingsimplemodelsforsolarupdrafttowerswithplanarcollectorfieldsforthesit- uationofpurelydryair. Themodeldescribing thegasdynamicsinthecollectorand in thechimneyincludes aturbinemodel,friction andheattransferlosses,evaporationand condensationmodelsetc.However,therelevantphysicscanbemodeledinonespacedi- mension.It is theresult ofafully compressible gasdynamic modelinthe smallMach numberlimit.Anumericalalgorithmisdefinedwhichadmitsveryfastsimulations.There- foreoptimizationprocedurescaneasilybeapplied.Numericalresultsonoptimizationwith respecttogeometricandphysicalparameterswhichmaybeconsideredbothintheplan- ningandtheoperationalphasearepresented.Theresultsarecomparedqualitativelyand – ifavailable– quantitativelytoprototypedataandtosimulationsfromtheliterature.

© 2018TheAuthors.PublishedbyElsevierInc.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The globaldemand forelectrical energycontinues to grow due to increasing globalpopulation and industrialization.

Ontheother hand,themineralresources e.g. oil,naturalgas,andcoal, onwhichwe havetraditionallyreliedto produce electricity,aredepleting.Thisisonlyoneofthereasonsthatthepricesoftheseresourcesarerising.Inaddition,nowadays, environmentalconcernslikeemissionsandgreenhouseeffect,safetyandhealthhazardspertainingtoelectricityproduction arebecomingmoreimportant,particularlyinthehighlyindustrializedcountries.Numerousexamplesofdisastersrelatedto nuclearenergyhavebeenwitnessed,themostrecentofthembeingatFukushima,Japan2011.Therefore,sustainableand environmentfriendly methodsto produceelectricityare gainingmoreandmoreimportance. Developedcountriesaround theworldaredevotingmoreandmorespecialeffortstothischallenge.

Thefocusofthisstudyistoanalyzesocalled“solarchimneypowerplantswithsloped collectors” whicharebasedon thesolarthermalprinciple.Theideahasbeenpioneeredby BilgenandRheaultin[1]andoriginatedasan offspringofso called“solarupdrafttowers” (SUT).ForanoverviewonSUTssee[2–6].

∗ Corresponding author.

E-mail address: ingenuin.gasser@uni-hamburg.de (I. Gasser).

https://doi.org/10.1016/j.apm.2018.07.023

0307-904X/© 2018 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Fig. 1. General design of solar updraft towers.

TheSUTconsistsofthreecomponents,namelycollector,chimney,andaturbine.Thecollectorsectionislikearoofmade up ofglassor some transparentmaterial whichstandsata fewmeters height fromtheground. Ahighvertical chimney standsinthecenterofthecollectorsection,or,incaseofaslopedcollectorfield,atitsuphillend. Aturbineisinstalledat thebottomofthechimneywhereitmeetsthecollector.Thepowerplantoperatesontwowell-knownphysicalphenomena, i.e.thegreenhouseeffectandthechimneyeffect.Theairinsidethecollectorisheatedupduetogreenhouseeffectasthe solarradiationfallsontheglassroof.Thelessdenseheatedairtendstoriseupduetobuoyancyforcesandthereforeaflow ofairisgeneratedtowardsthechimney.Thisresultingairflowdrivestheturbine.Clearly,theoutputincreasesheightening chimneyandenlargingthecollectorarea.SeeFig.1forcomponentsandworkingprincipleofasolarupdrafttower.

Inthispaperwediscusstwo importantissuesregardingextendedmodeling andapplicationsofSUTs. Oneisavariant proposedin[1]toconsidermountainousregions.Thesecondissueistogeneralizethemodeltothesituationofhumidair inordertounderstandthepossibleinfluencesofhumidityontheperformance.

Let us start with the possibility of mountainous regions. This implies building a collector field along the slope of a mountainorahillwhichthenalsofunctionsaschimneysuchthatontoponlyasmallerverticalchimneyisneeded.Asan examplewecanimagine atriangularcollectorareaonahilldirectedversus south.SeeFig.1(b)fortheworkingprinciple forsuch asolarchimneypowerplant.Thisvarianthasbigadvantages.Ononesidetheconstructioncostsforthe(smaller) chimneyaremuchlowerandontheotherside– supposethedirectionofthecollectorfieldisappropriatelychosenversus south– theangleofincidenceoftheradiationonthecollectorfield(dependingonthelatitude)ishigherandthereforea higheroutputofpowerisobtained.Inadditioninmanyregionsitiseasiertofindhills ormountainsinsteadoflargeflat areas(asitisneededfortheclassicalvariantofthesolarupdrafttower).Thereforethisideawasproposedinparticularfor mountainousareasathigherlatitude[1].

Now letuscometotheissueofhumid air.Consideringthetemperaturerangetheairexperiences onitswaythrough theplantanditsinfluenceonthematerialproperties,itisnaturaltoaskhowwatervaporaffectstheoperationofaSUTby condensationandevaporation.ThisisdonebyKrögerandBlainein[7],whereitisfoundthat“moistairgenerallyimproves thedraftandthatcondensationmayoccurinthechimneyundercertainconditions”.Thepaperdoes,however,onlyconsider thechimney,i.ethedifferentamountsofenergyneededforheatingandevaporationinthecollectorarenotaccountedfor.

Thereforetheinfluenceofhumidityforthewholepowerplantisnotyetstudiedandnotobvious.Mostmodelsneglect watervapour completely forthe sakeof simplicity.The usual argumentforthe restriction todrymodels is theassump- tion thattypical areasofoperation,e.g. deserts,wouldbehighlyarid.Ontheother hand,especially herethe recovery of condensatedwatercouldbeadesirableside-effect,makingplantsmorecosteffective.Tothisend,severalvariationstouse SUTsforseawaterdesalinationhavebeenproposed,eitheraddingclosedstillstothecollectorground[8,9] orattempting tore-gainwatervapouratthetopofthechimneybyahigh-efficiencycondenser[10].

Givenitsenormousareademand,agriculturaluseofpartsofthecollectorisdesirable.Inhisdissertation,Pretoriusinves- tigatestheroleofevapotranspirationinanelaboratemodelandfindsthepotentialpowerproductionloweredsignificantly comparedtoclassicconditions[11].

Theideaofthispaperistomodel,simulateandtooptimizeaSolarUpdraftTowerwithslopedcollectorinthegeneral situationofhumidair.Themainfocusisonusingamodelwhichallowsvery fastsimulations suchthat optimization(i.e.

with respect to certain parameters) can easily be done. In Section 3 the model is presented, in Section 3.5 the crucial small Mach numberasymptotics is explained,in Section 4 differentnumerical schemes are proposed and tested, andin Section5therelatedoptimizationswithrespecttocertainparametersareperformed.

Wesummarizethemainfeaturesofthemodel:

• Themodelisderivedfromthefullycompressiblefluiddynamicequations.

• ThesmallMachnumberisusedtosimplifythemodel.

• Therearenoassumptionsonthedensityorthetemperatureprofilesinthecollectororthechimney.

• Herethecoreisthefluiddynamicpart,in[1]theattentionismainlygiventothethermaltransfermodel.

• Itisacontinuoustimedependentmodel.Withthisitallowstransientsimulationsandweobtainautomaticallythestable stationarysolutions(aslongtimebehavior).

• Itiseasytoincludetimedependenciesinthedata,e.g.adailyradiationprofile.Withthisitiseasytosimulatea(longer) timeperiodandtoaverageoverit.

• Forevenfastersimulations,thesteadystatesystemcanbeconsidered.Formanyquestionsthestationarysolutioncon- tainssufficientinformation.

• Weapplythemodelto(easily)optimizethepoweroutput.

2. Historyandmodelevolution

Inthecontextofrenewableandsustainableenergies,overtheyearsamultitudeofproposalshavebeenmade.Aswith anytechnology,thereisneverendingroomforimprovementandinnovationinthisfield.However,veryspecialconsidera- tionisbeinggiventoelectricalenergyproductionfromsolarradiation,thereasonbeingsimplythatthesolarradiationis virtuallyinfiniteovertime.Moreover,thereareseveralregions wheresunlightisreadilyavailableinquantitativelyaccept- ablelevelsofpowerdensity,particularlydesertareas.

Currentlytherearetwo mainapproachestoproduceelectricityfromsolarradiation:Thephotovoltaic technology,con- verting radiationdirectly intoelectrical current, andthe solarthermaltechnology, converting solarenergy intoelectrical energyindirectly.Thesolarupdrafttowertechnologyconsideredinthispaperisasolarthermalpowerplant.

Historically,theconceptofsolarupdrafttowershasbeenaroundforoveracenturynow.Oneoftheearliestdescriptions ofthesolarchimneyconceptwaspresentedbySpanishColonelIsidoroCabanyesin1903[12],formoredetails,alsosee[13]. In1926,FrenchEngineerBernardDubosproposedasolarchimneyplantwithitschimneyrestingonamountainslope[3]. In1931,theGermanHannsGüntherillustratedsuchasolarpowerplantmoreconcretely[14].Somepicturesofeachofthese proposalscanbefoundin[3].Finally,thefirstexperimentalprototypeofasolarupdrafttoweronan“industrial” scalewas constructedatManzanares,Ciudad Real,Spain in1982.TheprojectwasfundedbyGermangovernmentandsupervisedby JörgSchlaich[5,15,16].However,therehavebeenbuiltmanyothersmallerprototypesinthelastdecades[17–19].

Solarupdrafttowershavecertainmeritswhichmakethemworthconsideringandcomparabletootherrenewableenergy proposalsatpresent.No sophisticated technologyisinvolved,itisalmost freeofdeterioration, theiroperation andmain- tenancecostsare lowandthetechnologypermitstowork evenoverdecades.However, fora fullscalecommercialpower planttohavedesirablepoweroutput,largelandareaisrequiredforthecollectorfieldandconstructionofahighchimney canlead to civilengineeringproblems andthereforehighcosts [20,21].The associatedproblems can besolved to a rea- sonableextent.Nevertheless,thetechnologyisindirect competitionwiththe hightechphotovoltaictechnologyandother solarthermalapproaches.Apartfromthealreadymentionedadvantages,thesolarupdrafttechnologyhasthepossibilityto store(againinalowtechmanner) apartoftheabsorbedenergyovertherangeofuptoaday.Thisisdonebyinstallinga waterstoragesysteminthecollectorgroundwhichabsorbs energy(heat)duringthedayandreleasesitduringthenight tomaintaincontinuous(non-constant)supplyofelectricalenergy[2].

Therearemanyvariantsandideasaroundthistechnology.Foranoverviewsee[22,23].

Asomehowrelatedidea,basedonaninverteddownwardflow,isthesocalledenergytower.Theretheenergyproduction originatesintheevaporationenergyoftheemployedwater(see[24,25]fordetails).Thisisimportantbecausefortheenergy toweradetaileddescriptionofthehumidityandtheevaporationisneeded.

Apartfromtheonesmentionedbeforetherearepapersconcerningthemodelingapproachestostudysolarupdrafttow- ersfromarangeofdifferentperspectives.Somepapersconcernspecific(modeling)questionsrelatedtothecollector[26,27], thechimney [7,28–30], theturbine[31] orthe poweroutput [32]. Inthe literature whichconsiders the full powerplant therearesomeapproacheswhichdosignificantsimplifications,i.e.nolossesintheenergybalance[17,18,33,34],linearden- sityprofiles [1],stationaryconditionsonly[35,36]etc.Otherpapersfocusonfull3dimensionalfluiddynamicsimulations oftheairflowina SolarUpdraftTower[37,38].Theseare clearlyveryexpensivesimulationsandnot appropriateforfast optimizationissues.Generallywe canconcludethatthereisstillaneedforreasonablycomplexmodels,whichadequately describethephysicalfeaturesofthesystem,permit fastsimulations,andallowforoptimizationprocedures.Inthecontext ofsolarchimneypowerplantswithslopedcollectors,thereisalmostnoliteratureexceptthefundamentalideain[1]. 3. Modeling

In thissection, we explain how the basic model introduced in [39] fora SUTwith flat collectorand dry aircan be extendedtoaccountforthegeneralsettingofslopedcollectorfieldsandcondensationorevaporationonsurfaces.Inaddi- tionweexplore furtherpossiblerefinementsandsimplificationsintheapproach.Oneistoconsiderthecollector-turbine- chimneyunitasasimplenetwork.Theotheroneistheuseofthesteadystateequationsfordescribingthequasi-stationary situationwherechangesintheexternaldata(radiation,outsidetemperatureetc.)occuronaslowtimescale.

InSection 3.1wesetup the2dEulerequationsforwetairina SUTwithavery generalgeometry.Thecorresponding initialandboundaryconditionsarediscussedinSections3.2and3.3,respectively.Byintegrationperpendiculartothemain airmovement,we obtaina1d modelinSection3.4,wheresourceandsinktermsforevaporationandcondensationarise fromthe surfaceboundary conditions.This modelis furthersimplified in Section 3.5andthe steadystate equationsare consideredinSection3.6.

TheexperimentaldatafromtheManzanaresprototype[15,16]showthatthetemperatureriseinthecollectorisapproxi- matelyaround20Kelvinsandtheflowvelocitiesareoforder10m/s[2].Comparablevaluesareobtainedin[1]forthecase ofpowerplantswithslopedcollector,andin[35]and[7]forSUTswithhumidity.

Inlightofthesmallvelocities(comparedtothespeedofsound),we havetodealwitha smallMachnumberflowand acompressiblemodelseemsnottobesuitable,anincompressiblemodelmightseemappealing.Butontheotherhand,the chimney effectisbasedondensitychanges andthereforetheflowcannot be treatedasincompressible. Thefact thatthe flowisdrivenbybuoyancyforcesasaresultofsmalldensitychangesmaypointtothepossibilityofusingtheBoussinesq approximation [40]. Thereare examples wheretheBoussinesq approximation isproved not towork well, i.e.inthe case ofa simplechimney withfire asa heatsource [41].We donot generallyexcludethe possibilityofusing theBoussinesq approximation forsolarupdrafttowers, butherewe usea moregeneralapproach.The smallMachnumberapproach we choose inthispaperallows usto identifywhetherthe typical features oftheBoussinesq approximation – linear profiles in pressure, densityand temperature– are confirmed by simulations ornot. In this sense we reduce the uncertainness introduced by using an approximativemodel like theBoussinesq approximation. Similar considerations concerning small Machnumberapproximationsarediscussedin[30]incaseofhighchimneys.

Theaimofthisstudyistopresentasimpleone-dimensionalmodeltodescribegas/airdynamicsinsidethepowerplant withslopedcollectorfield andhumidity.Atypicalairparticleentersthecollectorsection,travelsthroughtheturbineand leavesthesystematthetopofchimney.Theparticlepathcanbeconsideredasonespatialdimensioninmodelinganalysis.

Thecrosssectionalareaperpendiculartotheparticlepathisrectangularinthecollectoropeningwhichbecomes,assuming constant roof height,linearlysmallandeventually becomesconstant andcircularin thechimney duetoconstant radius.

Thisone dimensionalapproach wasalreadyusedsuccessfullyin[39] to modelthesimplestcasewithcompletelydry air andaflatcollector.Thereisnoindicationthatmultidimensionaleffectsarecrucial(orcannotbemodeledintheonedimen- sionalapproach).Moreover,a onedimensionalmodelhasmanysignificant advantageswithrespecttoamultidimensional approach.Themodelpresentedin[39] considersthefull powerplantandisaimed topermitvery fastnumericalsimula- tions.Thisisnecessarytobeappliedforoptimisingapowerplantwithrespecttoparametersorintheoperationalphase.

3.1. 2dEulerequations

Weassume theSUTtobe symmetrictoits centeraxisandtheflowto haveno angularcomponent.Therefore wecan denotethepositionofaparticlewithintheplantincylindercoordinatesby (^r˜,h˜)^{.Whilethechimneyisassumedtohave} a cylindricalshape for simplification,we have more freedom of choice in the collector, as long as all “typical” particle trajectoriesremaincomparable.Thisallowsforthedeviationfromstrictlycylindricalcollectorse.g.to(notnecessarilylinear)

“funnel” approachessuitable for sloped collectorsor for a modificationof the collector roofprofile in order to address the changeofdensityresultingfromthecross-sectionalchange.Inthismanner,wecandescribeavarietyofpossiblecollector geometries.Forinstance,thecollectorlengthL_collistobeunderstoodasthearclengthofthetypicalparticlepath,i.e.the collectorradiusinthecaseofaplanarcircularcollector,orthetriangleheightinthecaseofanisoscelestriangle.

Collectorandchimney arecoupledbyaninner“box” inwhichtheairisguidedfromhorizontaltoverticalmotionand energyisextractedbyoneormoreturbines(seeFig.2(a)forgeneralmodelgeometrynomenclature).

Theairincollectorandchimneycanbedescribedbythewell-known2dEulerequationsonacylinderforamixtureof dryairandwatervapourwithmolaramountsn˜a,n˜_h,respectively,onthedomain=coll∪chim(seeFig.2(a)).Asusual, unscaledquantitiesaredenotedbytildas.TheEulerequationsareasetoffourbalanceequationsandanadditionalclosing relationforthefiveunknownsn˜a,n˜_h,velocityu,pressure p˜,andtemperatureT˜:

D

Dt˜

(

^r˜n˜

)

^{=0 (1a)}

D

Dt˜

(

^r˜n˜h

)

−

∇

˜

Dn˜r˜

∇

˜

n˜h

n˜

=0 (1b)

D

Dt˜

(

^r˜

ρ

^˜u˜

)

^+r˜

∇

^˜p˜−gr˜

ρ

^˜−

μ

∇

˜

˜ r

∇

˜^u˜

+1

3

∇

˜²

(

^r˜u˜

)

=0 (1c)

D Dt˜

r˜

ρ

˜

cT˜+u˜²/2+h˜g

+

∇

˜

(

^r˜p˜u˜

)

⁻

∇

^˜

k

∇

˜^T˜

=0 (1d)

RT˜

(

^n˜a+n˜h

)

^{= p˜. (1e)}

Fig. 2. Nomenclature.

Table 1 Derived variables.

Symbol Definition Meaning

˜

n n ˜ a+ ˜ n h Molar amount of wet air ρ˜ M an ˜ a+ M hn ˜ h Mass density

c (c an ˜ a+ c hn ˜ h)/ n ˜ Spec. heat capacity of wet air

Table 2 Operators.

Symbol Definition

∇˜

∂/ ∂r ˜

∂/ ∂^˜ h

D

D˜t ∂/ ∂t ^˜ + ˜ u ∇˜

Table 3 Fixed parameters.

Symbol Dimension Value Meaning

M a M ¹N ⁻¹ 0.0289 kg/mol Molar mass of dry air M h M ¹N ⁻¹ 0.018 kg/mol Molar mass of water

D L ²T ⁻¹ 2 . 3 ·10 ⁻⁵m ²/ s Diffusion coefficient water vapour - air g L ¹T ⁻² 9.81 m/s ² Gravitational acceleration

μ M ¹L ⁻¹T ⁻¹ 17 . 1 ·10 ⁻⁶Pa s Dynamic viscosity of dry air c a L ²T ⁻²θ⁻² 718 J/(kg K) Specific heat capacity of dry air c h L ²T ⁻²θ⁻¹ 1556 J/(kg K) Specific heat capacity of water k M ¹L ¹T ⁻³θ⁻¹ 0.0262 W/(m K) Thermal conductivity of dry (!) air R M ¹L ²T ⁻²N ⁻¹θ⁻¹ 8.314 J/(mol K) Gas constant

Thenewlyintroducedvariables,operatorsandparametersareexplainedinTables1,2,and3,respectively.

Afterscaling,weobtain D

Dt

(

^rn

)

^{=0 (2a)}

D

Dt

(

^rnh

)

−

∇

rn

∇

_n

h

n

=0 (2b)

D

Dt

(

^r

ρ

^u

)

⁺¹

^r

∇

^{p− e2}

Fr²r

ρ

⁻_Re¹

∇ (

^r

∇

^u

)

⁺¹₃

∇

²

(

^ru

)

=0 (2c)

D Dt

r

(

^c1n_h+na

)

^T+

( γ

⁻¹

) ρ

^u₂²⁺

( γ

⁻¹

)

Fr²

ρ

^h

+

( γ

⁻¹

) ∇ (

^rup

)

⁻

∇

rRe Pr

∇

^T

= 0 (2d)

Table 4

Reference values for Manzanares test plant.

Symbol Dimension Value Interpretation

l r L ¹ 320 m Collector length + chimney height u r L ¹T ⁻¹ 10 m/s Typical gas velocity at turbine p r M ¹L ⁻¹T ⁻² 101328 Pa Typical pressure at ground level T r θ¹ 300 K Typical temperature at ground level

Table 5

Dimensionless numbers.

Symbol Definition Value Name

_lr^D_ur 7 . 54 ·10 ⁻⁹ Diffusion coefficient 1

= _γMa¹2

pr

u²rρr 861 related to Mach number

1 Fr²

glr

u²r 29.9 Froude number

1

Re μ

lrurρr 2.09 ·10 ⁸ Reynolds number (γ−1) c^Ra 0.29 Adiabatic exponent (dry)

Re Pr

k

urcaρrlr 7 . 27 ·10 ⁻⁹ Reynolds over Prandtl number

c 1 chMh

caMa 1.28 quotient of specific heat capacities

c 2 Mh

Ma 0.62 quotient of molar masses

nT = p (2e)

wherethereferencevaluesforscalingandthedefinitionsandvaluesofthedimensionlessnumberscanbefoundinTables4 and5.NotethatthegivenvaluesrefertotheManzanarestestplantandwillvaryforsomeoftheexamplesconsideredlater.

Ofcourse,(2a)or(2b)couldbereplacedbyananalogousequationforna.

Let us make a few comments on the modeling of condensation phenomena. We assume that surface condensa- tion/evaporationis ofparticularimportance becauseof thecooled chimney wallsandthehighavailable surfacearea, es- pecially for comparablysmall chimney radii. Therefore generalcondensation mechanisms such asdroplets inside theair streamare assumedtobe oflowerrelevance andtobecome importantonlyforhighervaluesofhumiditywhichare not ofparticularinterest forSUTs.Thus potential condensationenergycarriedby n_h can be neglectedinenergyconservation (2d)andourmodelbydesigncannotshowprecipitationeffectsinsidethechimneyaspredictedbyKrögerandBlaine[7].

DropletcondensationinSUTshasbeenconsideredin[42].Inourmodelitcouldbeincorporatedbyappropriatesource andsinktermsandadditionalconservationlaws.Werefrainfromdoingsoduetothementioned argumentsandinfavor ofmodelsimplicity,althoughwecannotcompletelyruleouttheformingofcloudsinthechimney.Acomparablemodelfor down-draft“energytowers” includingevaporationfromdropletsinsteadoffilms hasbeenproposedin[25].Furtherideas fortheimplementationofdropletscanbefoundin[43,44].

3.2. Initialconditions

For transientsimulations, we have to prescribe appropriate initial conditions in t=0 for the unknowns na, n_h, T, u, andp.Note that,because ofrelation(2e), onlyfourconditions havetobe specified. Anaturalchoice are theprofiles for temperature, pressureanddensitygiveninthesurroundingatmosphere. Theseprofilesvary verymuchdependingonthe circumstances. Forconsistency, we choose theadiabatic atmosphereformulasobtained bysolving a stationaryversion of (2a)–(2d)asshownin[39].

3.3. Boundaryconditions

We nowimposeboundaryconditions forthe solidsurfaces,i.e.chimney walls

∂

chim.wall,collectorroof

∂

coll.roof and -ground

∂

coll.ground(seeFig.2(a)):

u_⊥:=n

(

^n·u

)

^{=u (3a)}

n_h=n^sat_h := 1 Texp

11.96− 3984 T_{1 K}^Tr −38.15

1bar pr

(3b)

Ja := nau_⊥−

ⁿ

∂

∂

ⁿ

_n

a

n

=0 (3c)

Table 6 Parameters.

Symbol Value Meaning

h 43900 J/mol Enthalpy of vaporization α˜ ^{10 W/(m 2K) Heat transfer coefficient}

˜

q 0.3 ·10 ³W/m ² Solar radiation energy

Table 7

Dimensionless numbers.

Symbol Definition Value

1

Ja hnr

caρrTr 5.03 α _ca^α˜_ρ^t_r^r_lr 1 . 14 ·10 ⁻³ Q _caρ^q˜_r^tr_lrTr 1 . 14 ·10 ⁻⁴

∂

^T

∂

^{n =Ja1RePrJh−}

α (

^Tex−T

)

^on

∂

chim.wall∪

∂

coll.roof

Qcos

( θ

−

β )

^on

∂

coll.ground, (3d)

wherethenewparametersanddimensionlessnumbersaregiveninTables6and7.

Condition(3a)representstheusualno-slipcondition.Ifcondensationorevaporationoccurs,therewillbeatleastathin layerofliquidwateronthesurfaces,saturatingtheairinthevicinitywithwatervapour.Weassumethatduringoperation, thesurfacesare drainedwherenecessary sothat theplantisnot floodedandthe waterlayerremains thin,not affecting thermalconductivityetc.TheparametrizationofsaturationvapourpressurebytheAntoineequationin(3b)wastakenfrom [45].In (3c) weexpresstheassumption that thedryportionofthe aircannot passthrough walls,whereas we willhave negativeJ_h forcondensation andpositive J_h forevaporation. Due tothis unambiguousness,we will write Jinstead ofJ_h. TheRobin-typecondition(3d)representsNewton’scoolinglaw,complementedbytheterm _Ja¹_Re^PrJrepresentingreleasedor absorbedcondensation energy.Forthe collectorroof andchimney wall, theheat transfer coefficient

α

˜ ^{models bothheat} conductionthroughthematerialandthe(wind-dependent)transfertotheambientair.

ThesolarradiationisgivenbyQ.Itenters throughthecollectorground,becausemostofthesolarenergyisabsorbed bythe groundandtransferred tothe airconvectively(note that theimposed initialconditiondoesnot account forthis).

Thisallocationoftheheatsourceturnsouttobecrucialforthecalculationofevaporationrates.Ofcourse,Qisafunction oftime andcould beenhanced– if knownandavailable – withfactorsforspatialdependentshading,reflectivity,etc.For simplicityand dueto lack of available data we assume Q only to be time dependent. Since Q changes only slowlyand significantlyonlyonatimescaleofhours,formanyofourshorttimesimulationsQdoesnotevenchangeintime.Butthere isnodifficultytoincludeitstimedependencewhenrunningsimulationsovermanyhours.

Weresignfromestablishingamoreelaboratetime-dependentheattransitionmodel,i.e.incorporatingtime-andspatial dependentsubmodels forQand

α

^{,or more}sophisticated setupsincludinge.g. waterbags ormultipleglazing. These are describedindetailin[11]and[35]andcouldeasilybeintegrated,butdonotlieinthefocusofthiswork.

Next,wemodelthetransitionfromthecollectoroutlettothechimney entrance.TheSUTcanbe regardedasasystem ofcoupledpipes,suggestingatreatmentsimilarto[46].Theturbinesection istreatedasa“blackbox” insteadoftryingto preciselymodelthe streamlinesinside bymeans ofbalanceequations.Theseconnect the valuesintegratedoverchimney entrance

∂

chim.inandcollectorexit

∂

coll.outwithareaA_coll.out.Theresultingtimedelayisneglected.

∂^coll.out

r f

(

^x

)

^dx=

∂^chim.in

r f

(

^x

)

^dx

∀

^{f ∈}

{

^T,n,

ρ

^,na,nh

}

^(4a)

∂coll.out

r

ρ

^u

(

^x

)

^dx=S

π

2 +

θ

· ∂chim.in

r

ρ

^u

(

^x

)

^dx. (4b)

∂coll.out

r p

(

^x

)

^dx−A_coll.out

^p=

∂chim.in

r p

(

^x

)

^dx. (4c)

(4a)statesthat whilethepropertiesshould bepreservedoverall,wedonotattempttoconstructapoint-to-pointmap betweencollectorexitandchimney entrance.In(4b),whereSdenotesarotationmatrix,wedescribetheairbeingguided fromitsmotionparalleltothegroundtoverticalascend.ThepressureEq.(4c)aswellasthelackofasinktermin(4b)are dueto thefact that the turbines usedin SUTs are pressure-staged[47].Note that instead ofa pressure lossfactor fthe absolutepressuredrop^{pisusedinthismodel.}

TheSUT iscoupled toits surroundings at

∂

coll.in and

∂

chim.out by theusual inflowconditions for

ρ

^{,n,and T,and}

Dirichletboundaryconditionforp[25,39]:

p=pex+

^pwind (5a)

ρ

=

ρ

^ex,n=nex,T=Tex,if u·n>0. (5b) Theterm^pwind=^pwind(^x,t)^{isintroducedtoaccountfora}potentiallyhigherorlowereffectivepressureduetoam- bientwind.Due tothe assumedradialsymmetry, thisisonly directlyapplicable tocollectorgeometries inwhichthe air canonlyenterinonedirection,suchasthedescribedtriangularslopedcollectorfield.Ifthisisnotthecase,ourmodeling approachallowstosplitthecollectorintosectorswhichcouldthenbetreatedseparately.

3.4. 1dmodel

In order to simplifyour model,we now integrate over collectorheight andchimney radius. By doing this, boundary conditions (3a)–(3d) will be transformed to source terms. In the conservationof momentum (2c), the no-slip condition (3a)willbereplacedby thewell-knownfrictionterm

ξ

^{u|u|.TodeterminethemassflowJ,follow[48]:Firstsolve (3c)for}

u_⊥andplugitintotheanalogousdefinitionofJ: Jh=−

1

1−ⁿ_n^h

n

∂

∂

ⁿ

_n

h

n

.

AssumingJtobeconstantoverathinboundarylayerofthickness

δ

^yields

J_h=1

δ

δ 0

J_hdx_⊥=−

ⁿ

δ

δ 0

1

1−ⁿ_n^h

∂

∂

ⁿ

_n

h

n

dx_⊥=

ⁿ

δ

^ln

1−^nh_n^(δ) 1−^nh_n⁽⁰⁾

. (6)

Anexpressionfortheenergysourceterm

:=c1TJ_h−Re Pr

∂

^T

∂

^{n (7)}

inconservationlaw(2d)canbeachievedinasimilarmanner:Theansatz()n=0leadsto T = T

(

⁰

)

+

(

^T

( δ )

−T

(

⁰

) )

^exp

c_1Re^PrJ_hx_⊥

−1 exp

c1Pr ReJh

δ

−1 .

Evaluating^atx⊥=0andx_⊥=

δ

^{yieldsthesymmetricexpression}

=c1J_hT0+T_δ

2 −c1J_hT_δ−T0

2 exp

c1Pr ReJ_h

δ

+1 exp

c1Pr ReJ_h

δ

−1. (8)

Thefirsttermin(8)canbeinterpretedasenergycarriedbythemassfluxJ,whereasthesecondrepresentstheheattransfer by convectionwhichisaugmented orobstructedbyJvia afactoroftheshape f(^x)=^x(_exp^exp₍⁽_x^x₎⁾₋⁺₁¹⁾.Straightforward analysis showsthatfhasaremovablesingularityandniceasymptoticbehavior:

f

(

^x

)

^{= f}

(

^−x

)

^{; lim}_x→0^f

(

^x

)

^{=2; lim}_x→0^f

(

^x

)

^=0; _x→±∞^{lim f}

(

^x

)

⁻

|

^x

|

^{=0. (9)}

Plug(3d)into(7): 0 =

+J_h

₁

Ja−c1T0

−Re Pr

α (

^Tex−T0

)

^on

∂

chim.wall∪

∂

coll.roof

Qcos

( θ

⁻

β )

^on

∂

coll.ground . (10)

For the collectorroof and the chimney wall, (10) is equivalentto the corresponding equations presentedin [45,48]; for thecollectorgrounditcanbeviewedasamodifiedPenmanequation [49].Thepropertiesof^{derivedin(9)facilitatean} asymptoticanalysisof(10):While|J_h|→0implies

^=Re_Pr

α (

^Tex−T₀

)

^on

∂

chim.wall∪

∂

coll.roof

Qcos

( θ

⁻

β )

^on

∂

coll.ground

(11)

and

T₀−T_δ

δ

⁼

α (

^Tex−T₀

)

^on

∂

chim.wall∪

∂

coll.roof

Qcos

( θ

⁻

β )

^on

∂

coll.ground, (12)

i.e.regularheatflowasinthedryscenario,for|J_h|→∞weget 1

Ja Pr ReJ_h =

α (

^Tex−T0

)

^on

∂

chim.wall∪

∂

coll.roof

Qcos

( θ

⁻

β )

^on

∂

coll.ground, (13)

whereallenergyexchangethroughsurfacesisusedforcondensationheat.

The derivationof Jand ^{wasbased on theassumption of wetsurfaces in(3b), implyingthat} evaporation isalways possible.Toavoidunrealisticallyhighrates,introduce ascalarfunction Jmax onthe surfacesasan upperbound.Jmax will usually be0 onthe wallsandtheroof, butcanbe positive forthecollectorgroundin casewaterisprovided constantly byappropriateirrigation¹.Thisirrigationcanbethoughte.g.ascomingfromadditionaluseofthecollectorfordesalination or agriculture. In the latter case, water consumption is a cost factor so restriction of water accessappears natural. If J exceedsJmax,setJ:=Jmax,findthecorrespondingT₀bysolving(6),andsolve(10)or(12)for^{.Thisprocedureallowsusto} determinethemassandenergysourcetermsJ_w,w fromthechimney wallandthecollectorroofaswellasJ_g,gfrom thegroundforanygivenstateofna,n_h,TandpintheSUT.TheeffectofvaryingJmaxwillbestudiedinSection5.6.

After integration, the Euler equations for collector and chimney are simplified respectively to become the non- conservativesystem

(

^na

)

t+

(

^nau

)

x =−Ax

Anau

(

^nh

)

t+

(

^nhu

)

x =−Ax

Anhu+

surf

L AJ

ρ (

^ut+uux

)

⁺¹

^{px =−u}

^surf

L A

M_h Ma

J−Ax

A

ρ

^u2+

ρ

Fr²sin

θ

⁺

surf

L A

ξρ

^u

|

^u

|

pt+

γ (

^pu

)

x =

( γ

⁻¹

)

^upx−

( γ

⁻¹

)

u

surf

ρ

_A^L

ξ

^u

|

^u

|

^−u₂²

surf

L A

M_h Ma

J

−

( γ

⁻¹

)

¹ Fr²

surf

L A

M_h Ma

J−Ax

Ac₄up+

surf

L A

p=T

(

ⁿa+nh

)

. (14)

Notethatonthetransitionfrom2dto1d,handrhavebeenreplacedbythenewvariablex∈(x_coll.in,x_coll.out)∪(x_chim.in, x_chim.out).A(x)andL(x)denotethecross-sectionalarea andthesurfacelengthatpositionx,whilesurfisa reminderthat thesourceterms(i.e.friction

ξρ

^u|u|andcondensation/evaporationJ)havetobeevaluatedandsummatedoverthetypesof surfacesrelevantatx.

Thesystemsontheintervalsarecoupledbyintegratedversionsof(4a)–(4c),whereequalityofthecross-sectionalareas A_coll.out=A_chim.in,whichcanbeachievedbyslightlyshiftingthetransitionbetweencollectorandturbinesection,allowsfor simplification:

f

(

^xcoll.out

)

= f

(

^xchim.in

) ∀

^f∈

{

^T,nh,na,u

}

^(15a)

p

(

^xcoll.out

)

−

^p=p

(

^xchim.in

)

^(15b)

(10)isusedinordertodetermineTgandTwandtheouterboundaryconditions(5a)and(5b)arestillvalid.

3.5.LowMachnumberasymptotics

AfterperforminglowMachnumberasymptoticsasin[25,39,50],thesystemscaninfirstorderbeapproximatedby:

u

(

^x

)

^{= A}_A^coll

(

^x.out

)

^ucoll.out− 1 A

(

^x

)

xcoll.out x

L

(

^y

)

c4

( (

^Tw

)

⁺

(

^Tg

) )

^{dy (16a)}

(

^na

)

t+u

(

^na

)

x=−L

(

^x

)

A

(

^x

)

nL

c4

( (

^Tw

)

+

(

^Tg

) )

^(16b)

(

^nh

)

t+u

(

^nh

)

x= L

(

^x

)

A

(

^x

)

J

(

^Tg

)

+J

(

^Tw

)

−nh

c4

( (

^Tw

)

+

(

^Tg

) )

(16c)

(

^ut

)

t=− 1

coll

ρ

^dx+chim

ρ

^dx

_chim

coll

λρ

^u

|

^u

|

^−sin

⁽ θ )

Fr²

ρ

^ex

ρ

⁻¹

+

ρ

^u

(

^u

)

xdx

+

coll

uc₂L

A

(

^J

(

^Tg

)

^+Tw

) )

⁻

ρ

1

A

xcoll.out x

L

c₄

( (

^Tw

)

⁺

(

^Tg

) )

^dy

t

dx

+

chim

uc₂L

AJ

(

^Tw

)

⁻

ρ

1

A

x xchim.in

L

c₄

(

^Tw

)

^dy

t

dx+

^p

, (16d)

1For brevity, we will also write J max= y, with y ∈ R ⁺ denoting the constant rate of irrigation on the collector ground, or J max= f(x ) for sector-wise irrigation.

where

c₃:=c₁n_h+na

n ; c₄:=

γ

+c₃−1 c₃ .

Forachimneywithconstantradius²,suchasintheManzanaresprototypeortheplantsproposedbyBilgenandRheault [1],(16a)–(16c)become

u

(

^x

)

=ut+ ^x

xchim.in

2 r_chim

(

^Tw

)

c4

dy. (17a)

(

^na

)

t+u

(

^na

)

x=− 2 r_chim

na

c₄

(

^Tw

)

^(17b)

(

ⁿh

)

t+u

(

ⁿh

)

x= 2 r_chim

J

(

^Tw

)

−n_h c4

(

^Tw

)

. (17c)

In(16d),the termchim coll

sin(θ ) Fr²

_ρex

ρ −1

dxdenotes theupdraft,whereintegrationover thecollectorcanbe omittedfor theflatsetup.Theturbinepressurelossfactorf,whichisoftenemployedasameansforplantoperationcontrol,describes thefractionoftheupdraftthatisextractedforelectricalenergyproductionintheturbineandthereforecannotbeusedfor accelerationoftheairflow,i.e.wedefine

f:=

^p

_chim

coll sin(θ )

Fr²

_ρex

ρ −1

dx. (18)

Fromthecouplingconditions(15a),only

na

(

^xcoll.out

)

=na

(

^xchim.in

)

^andnh

(

^xcoll.out

)

=nh

(

^xchim.in

)

⁽¹⁹⁾

arestillrelevant,(10)stillapplies.Theinitialconditionsaretakenfirstorderin

ε

^{,too,whichyields}

ρ (

^0,x

)

⁼

ρ

^{ex≡1 and T}

(

^0,x

)

^{=Tex≡1 (20)}

inourscaling;notethattheinitialandboundaryconditionsonpstillappearaspartof(16d).

It is readilyshown that for thedry case, i.e.Jmax=0 and(n_h)ex sufficiently low so that nocondensation occurs, the modelsimplifiestobeverysimilarto[39].

3.6. Steadystate

Insteadystate,wherethetimederivativesvanish,thesystem(16a)–(16d)becomes ux= 1

c4

surf

L

A

−uAx

A (21a)

(

^na

)

x=−1 u

na

c₄

surf

L

A

^(21b)

(

ⁿh

)

x=1 u

surf

L AJ−n_h

c4

surf

L A

(21c)

^p= ¹

0 −

surf

L

A

ξρ

^u

|

^u

|

−u

surf

L A

M_h Ma

J−

ρ

^uux+sin

( θ )

Fr²

(

¹−

ρ )

^dx. (21d)

4. Numericalsimulations 4.1. Algorithms

Inthissection,wediscussthenumericalsimulationandoptimizationofthetransientmodel(16a)–(16d)and the steady- statemodel(21a)–(21d).Eqs.(16b),(16c)and(16d)fromthelast sectionareevolutionequations.Fortheunknown

ρ

^(x,t)

we haveatransport partialdifferentialequation (PDE)andforv(t) wehavean ordinary differentialequation (ODE).Note thattheboundaryconditionsforpressureandtheturbinepressuredrop(5a)and(15b)arealreadyincorporatedin(16d)as

2For non-constant radii r chim= r chim(x ), u tin (17a) only gets an additional Bernoulli-like factor corresponding to the one in (16a)

Table 8

Comparison of simulation times for a run in dry air conditions (with fixed parameters).

Transient Simulation Steady-state simulation u 0given p given

0.64 s 0.60 s 12.7 s

Table 9

Comparison of simulation times for a steady state simulation for dry and wet conditions (with fixed parameters).

Dry Wet

u 0given 0.60 s 0.64 s p given 12.60 s 12.66 s

parameter.Wehavetheboundaryconditions(5b)for

ρ

^{andtheinitialconditions(20)for}

ρ

^{andv.Thefollowingisasketch}

ofournumerical(explicitforwardintime)strategytosolvethesystem(16a)–(16d): 1.Updateufromvin(16a).

2.MakeanexplicitupwindschemeinthePDEs(16b),(16c)forna,n_h. 3.MakeanexplicittimestepintheODE(16d)forv.

4.Repeat1.–3.untilsteadystateisreached.

Thisisaverysimplebutreasonablealgorithm.ThestabilityisensuredbytheCourant–Friedrichs–Lewy(CFL)condition.

Notethat the CFLconditiondoesnot implyvery restrictive time steps since duetoEqs. (16b)and(16c)it dependsonly ontheflowvelocity.Thisisincontrasttoafullynonlinearfluidmodel(like(14))wherethetimestepsinthesmallMach numberregime becomeveryrestrictive sincetheretheCFLconditionisgovernedby thespeedofsound.Amoredetailed analysisrelatedtoefficiencyfromanumericalpointofviewinasimilarmodelcanbefoundin[46].Moreover,itispossible todefinemoresophisticatedschemesaswellbut– aswewillsee– thereseemstobenostrongneed.

Inmanysituationsonly thesteadystate solutionisofdeeper interest.There ismorethan onereasonforthat. Oneis thatthetransientsolutionsinmostoftherealisticsituationsconvergerapidly(orderofmagnitudeis10min)toastationary state. Asecond reasonis that, underthe assumptions made, the changes in time ofthe data (boundary condition, solar radiationrateetc.)areveryslowoveratypical24hcycle.Thereforeitisareasonablealternativetosolveovera24hcycle afew stationaryproblems,insteadof runninga transientproblemover such along time period.Ontopof that,later on wewillinvestigatequalitativelyhowvariationinoneparameteraffectsperformanceataparticularpointintimewhilethe other parameters arekept constant. Thenaturalchoice forthisis toconsider equilibriumsolutions. Thuswe have,asan alternative,asimpleschemetosolvethesteady-statesystem(21a)–(21d):

1.Makeaninitialguessforthevelocityatthecollectorentranceu₀.

2.Integrate (21a)–(21c)with boundarycondition (^nexa,n^ex_h,u₀)^{over collectorand chimney employinga standard Runge–}

Kutta-method.

3.Calculatethecorrespondingturbinepressureloss^p(u0)from(21d).

4.Tomatchagiventurbinepressureloss^{p,useaniterativesolveronsteps2.-3.tofindu}0satisfying^p(^u0)−^p =0. Inthefollowingwe willuseinafew examplesboththelongtimetransientandthestationaryapproachtoobtainthe stationarysolution.Wewillseethatthetwoapproacheslead– asexpected– tothesameresult.Representativecomputation timesonaPC(Intel(R)Core(TM)i5-2400CPU@3.10GHz,8GBmemory)aregiveninTables8and 9.

Inthe application optimizationis an important issue. Thereforein a second step we wouldlike to optimize.For the optimizationofthepoweroutputweuse

1.–4. Runtransientsimulationfromformeralgorithmtoobtaincorresponding turbinevelocityu_turb(^{p)forgiventurbine} pressureloss^p

5. UsestandardMatlaboptimizationtool

fminsearch

on1.–4.tofind minp>0(−u_turb(^p)·^p)^.

Again,asanalternative(andasacrosscheck)forfindingtheoptimalturbinepressurelosswecanuseaslightlychanged approach:

1. Makeaninitialguessforvelocityatcollectorentranceu₀.

2.-3. Findtheturbinepressureloss^p(u0)tothegiveninitialvelocityu₀asbefore.

4. UsestandardMatlaboptimizationtool

fminsearch

on2.-3.tofind min

u₀>0(−u_turb·^p(^u0))^.