JOURNALOFTHERMOPHYSICSANDHEATTRANSFER Vol.25,No.1,January–March2011 Analysis of Solar-Heated Thermal Wadis to Support Extended-Duration Lunar Exploration R.Balasubramaniam,∗S.Gokoglu,†andK.Sacksteder‡ NASAJohnH.GlennResearchCenteratLewisField,Cleveland,Ohio44135 R.Wegeng§ PacificNorthwestNationalLaboratory,Richland,Washington99352 and N.Suzuki¶ NASAHeadquarters,Washington,DistrictofColumbia20546 DOI:10.2514/1.49843 Therealizationoftherenewedexplorationofthemoonpresentsmanytechnicalchallenges;amongthemisthe survivaloflunar-surfaceassetsduringperiodsofdarknesswhenthelunarenvironmentisverycold.Thermalwadis areengineeredsourcesofstoredsolarenergyusingmodifiedlunarregolithasathermalstoragemassthatcansupply energytoprotectlightweightroboticroversorotherassetsduringthelunarnight.Thispaperdescribesananalysisof theperformanceofthermalwadisbasedontheknownsolarilluminationofthemoonandestimatesofproducible thermalpropertiesofmodifiedlunarregolith.Analysishasbeenperformedforthelunarequatorialregionandfora potentialoutpostlocationnearthelunarsouthpole.Thecalculationsindicatethatthermalwadiscanprovidethe desiredthermalenergyandtemperaturecontrolforthesurvivalofroversorotherequipmentduringperiodsof darkness. Nomenclature t = onehalfofthesynodicperiodonthemoon 0 C ,C0 = specificheatofthewadiandregolith,respectively, (354h),s p p x = coordinateintheverticaldirectionintothewadiand J=(cid:1)kg(cid:2)K(cid:3) theunderlyingregolith,m d = depthofthewadi,m (cid:1),(cid:1)0 = thermaldiffusivitiesofthewadiandregolith, d0(cid:4)d = depthoftheregolithlayerunderlyingthewadi,m f,f = periodicsurfaceheatfluxinthemediumasafunction respectively,m2=s j oftimeandinatimeintervalj,W=m2 (cid:1)abs = radiativeabsorptivityofthewpad(cid:1)(cid:1)i(cid:1)surfacep(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:4) = dimensionlessparameter,(cid:1)1= 2(cid:3)(cid:1)k0=k(cid:3) (cid:1)=(cid:1)0 k,k0 = thermalconductivityofthewadiandregolith, (cid:2) = radiativeemissivityofthewadisurface respectively,W=(cid:1)m(cid:2)K(cid:3) N,N0 = scaleddepthofthewadi(d=p(cid:1)(cid:1)(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)(cid:1))andtheregolith (cid:2)(cid:1) = (cid:2)=(cid:2)0,scaledemissivityofthewadisurface layer[(cid:1)d0(cid:4)d(cid:3)=p(cid:1)(cid:1)(cid:1)(cid:1)0(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)],respectiv0ely (cid:2)0,(cid:2)1 = daytimeandnighttimeemissivity,respectively,ofthe q = solarfluximpingingo0nthewadisurface,W=m2 wadisurface q(cid:1) = q=q ,scaledsolarflux (cid:5),(cid:5)0 = scaledtemperaturedistributioninthewadi(T=Tref) q = peakmsaxolarflux,W=m2 andtheunderlyingregolithlayer(T0=Tref) qmax = heatfluxsuppliedtotherover,W=m2 (cid:6) = dimpensionlessparameter, q(cid:1)rov = q =(cid:1)(cid:1) q (cid:3),scaledroverheatflux k=(cid:5) (cid:1)(cid:1)(cid:1)(cid:1)t(cid:1)(cid:1)0(cid:1)(cid:1)(cid:1)(cid:2)0(cid:3)(cid:3)1=4(cid:1)(cid:1)absqmax(cid:3)3=4(cid:6) rov rov abs max (cid:7) = scaledcoordinateintheverticaldirection, T,T0 = temperaturedistributionasafunctionofxandtinthe (cid:1)d(cid:4)x(cid:3)=dfor0(cid:7)x(cid:7)dand(cid:1)x(cid:4)d(cid:3)=(cid:1)d0(cid:4)d(cid:3)for wadiandtheunderlyingregolithlayer,K d(cid:7)x(cid:7)d0 T = environmenttemperature,K T(cid:1)a = T =T ,scaledenvironmenttemperature (cid:8),(cid:8)0 = densitiesofthewadiandregolith,respectively, a a ref kg=m3 T = initialtemperatureofthewadi,K i (cid:3) = Stefan–Boltzmannconstant, T = referencetemperature,(cid:1)(cid:1) q =(cid:2) (cid:3)(cid:3)1=4,K ref abs max 0 5:67(cid:8)10(cid:4)8 W=(cid:1)m2(cid:2)K4(cid:3) T = surfacetemperatureofthewadi,K s (cid:9) = scaledtime,t=t t = time,s 0 (cid:10) = periodicscaledsurfaceheatfluxinthemediuminthe j timeintervaljinJaeger’smethod Presented as Paper 2009-1339 at the 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, I. Introduction Orlando,FL,5–8January2009;received12March2010;revisionreceived 26July2010;acceptedforpublication29July2010.Thismaterialisdeclared NASA plans to establish a permanent, manned outpost on the aworkoftheU.S.Governmentandisnotsubjecttocopyrightprotectionin moon, and many systems are under development as part of theUnitedStates.Copiesofthispapermaybemadeforpersonalorinternal NASA’splanstodeployinsituresourceutilizationcapabilitiesatthe use,onconditionthatthecopierpaythe$10.00per-copyfeetotheCopyright lunaroutpost.Thereisaneedtoprotecttheexplorationsystemsfrom ClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923;include theextremecoldofthelunarsurface.Largetemperatureswingsare thecode0887-8722/11and$10.00incorrespondencewiththeCCC. experiencedatmostlocationsonthemoon,inpartbecauseofitsslow ∗Research Associate Professor, National Center for Space Exploration rotationrate(thediurnalcycleofthemoonisabout29Earth-days Research;CaseWesternReserveUniversity,Cleveland,Ohio. †SeniorAerospaceEngineer. long) and in part because native lunar regolith is very resistant to ‡SeniorCombustionScientist.AssociateFellowAIAA. thermal conduction. Heat does not penetrate very deeply: the §ChiefEngineer.SeniorMemberAIAA. characteristic penetration depth is approximately 10 cm during ¶Program Manager, Exploration Systems Mission Directorate. Member periodsofsunlightand,asaresult,temperaturestypicallyrangefrom AIAA. ahighofabout400Ktoalowofabout100K.Atthepresenttime,for 130 BALASUBRAMANIAMETAL. 131 Fig.1 Thelunarthermalwadiconcept.Ontheleft,asun-trackingreflectordirectssunlightontoathermalmassduringperiodsofsolarillumination whileroversconductlunar-surfaceoperations.Ontheright,roversarethermallycoupledtothethermalmasstostaywarmduringperiodsofdarkness andarefurtherprotectedbyaheat-lossshieldtolimitradiativelossestospace. planningpurposes,theoutpostisassumedtobelocatedinveryclose ametal-enrichedproduct.Severaltechnicalissuesmustberesolved proximitytothesouthpoleofthemoon,ontherimofShackleton inordertoshowthatthermalwadiscanbeavaluableadjunctelement crater.Thislocationisvaluedlargelybecauseitexperiencesnearly of the lunar-surface architecture, including whether thermal-mass constantsunlight.∗∗Althoughthesunmustalwaysbeatornearthe materialcanbereadilyproducedatareasonablecostonthelunar horizoninthislocation,thesmalltiltofthemoon’saxiswithrespect surface. totheeclipticresultsinshortperiodsofdarkness,primarilywhen Our objective in this paper is to analyze the extent to which a nearbytopographicfeaturescastshadowsonthislocation.Presently, thermal wadi can provide temperature control for a robotic rover thelongestperiodofdarknessisestimatedtobe52h. basedonseveralsimplifiedconfigurationsofoperationalhardware In this paper, we present the model formulation, details of the andmodificationstothenativeregolithforcreatingthewadithermal thermalanalysis,andresultsofanefforttoexamineoneaspectofthe mass. We present the model formulation and results of a thermal technicalfeasibilityofusingthermalwadis(engineeredsourcesof analysisofvariouswadiconceptstodeterminethetemperatureofthe heatandpower[1,2])thatcanprotectexplorationsystemsfromthe thermalmass.Asthewadisurfaceisthesimplest interfacetoany extremecoldofthelunarsurface.Theideaisthatmoderatetemper- hardwarethewadiwouldsupportduringperiodsoflunardarkness, ature cycles would be obtained with materials that have a higher the surface temperature is especially ofinterest. The analysis was thermal conductivity than the native regolith. Accordingly, it has performedforsimulatedconditionsattwolocations:1)nearthelunar been hypothesized that materials with acceptable properties for equator,wherethesolarilluminationofthesurfacecanbeconsidered thermal energy storage can be produced by using solar energy to periodicandsimilaroverwideareas,and2)asingleselectedsitenear sinterand/ormeltlunarregolith,allowingittocoalesceintoacontin- the lunar south pole, where the illumination is quite irregular but uousmass.Indeed,theuseofmoltenregolithasameansofthermal consideredapromisingsitefortheplannedlunaroutpost. energy storage on the moon was considered for a solar-dynamic power system [3]. Along with additional hardware to regulate the absorption and loss of thermal energy, the resulting thermal mass II. Model wouldexperienceareducedtemperatureswingandcouldserveasa Theobjectiveofthethermalanalysisistodeterminethetemper- warming pad for robotic rovers or other explorationassets during atureofthewadi,especiallyhowthewadi-surfacetemperaturevaries periodsofdarknessonthelunarsurface. with time, how the maximum and minimum surface temperatures Thebasicconceptofathermalwadiis illustrated inFig.1and dependontheincidentsolarflux,thedepthofthewadi,itsthermal consists of a thermal mass plus one or more energy reflectors for properties(suchasitsthermalconductivity,diffusivity,andradiative a)reflectingsolarenergyontothethermalmassduringperiodsof properties),andcoverageofthewadisurfacebyathinlayerofdust. sunlightandb)reflectingradiantenergybacktothethermalmass Also ofinterest are the contributions of operational hardware that during periods of darkness. During periods of sunlight, thermal might accompany the thermal wadi, including a reflector of solar energy is absorbed and stored within the thermal mass. During illuminationthattracksthemovementofthesunwithrespecttothe periodsofdarkness,thestoredenergyisusedtoprovidetemperature wadi surface and a reflector that limits the radiativeloss from the controlforroversandotherexplorationassets.Thethermalproperty surfaceofthethermalmasstospace,aswellasatypicalroverthat values of the thermal mass are critical to the effectiveness of the mayneedtobeheatedovernight.Throughoutouranalysis,weuse thermalwadi.Initsnativestate,lunarregolithisapoormaterialfor thethermalproperties,summarizedinTable1,ofnativeregolithand thermalenergystorage.Becauseofitsverylowthermaldiffusivity, basaltrock.Thelatteristheparentmaterialfromwhichregolithis about6:6(cid:8)10(cid:4)9 m2=spermeasurementsmadeduringtheApollo formed. As mentioned before, various processing methods can be program[4],heatdoesnotpenetratethelunarsurfaceverydeeplyand usedtoimprovethethermalpropertiesoftheregolithclosetothoseof islostrapidlyduetoradiationduringperiodsofdarkness.Weshow solidbasalt. laterthatthelargesurface-temperatureswingdurinpg(cid:1)(cid:1)t(cid:1)h(cid:1)(cid:1)emoon’s27- Weassumethatthesolarfluxincidentonthesurfaceisspatially daydiurnalcycleiscontrolledbythevalueofk0= (cid:1)0(mayalsobe p uniformandvarieswithtime(e.g.,Fig.2fortheequatorialregion). written as p(cid:1)k(cid:1)(cid:1)0(cid:1)(cid:8)(cid:1)(cid:1)(cid:1)0(cid:1)C(cid:1)(cid:1)(cid:1)p(cid:1)0(cid:1)(cid:1)), which is approximately 123W (cid:1)s(cid:1)=(cid:1)m2(cid:2)K(cid:3) Thedepthofthewadiislikelytobeshallowcomparedwithitslateral for native regolith. The regolith, however, contains the elemental extent, so that the wadi can be easily constructed. The thermal materialsfromwhichareasonablethermalenergystoragemedium conductivityofthewadimaterialisexpectedtobemuchlargerthan canbefabricated,andexperimentsonEarthhavedemonstratedthat that of native regolith. Within the bulk of the wadi, we therefore solarand/ormicrowaveenergycanenablethenecessaryconversion expectnegligiblelateral heat loss tothesurrounding regolith, and processes. Examples of regolith processing methods that can heatwillbetransferredprimarilyinthedownwarddirection.Aone- producethermalmasseswithimprovedthermalpropertiesinclude dimensional analysis is therefore sufficient to discern the thermal compactingandsintering[5],meltingandsolidificationofprocessed behavior of the wadi. A one-dimensional analysis has an added or unprocessed regolith, inclusion of materials with high-thermal benefitofbeingrelativelysimplecomparedwithamultidimensional conductivityand/orhigh-thermalcapacity,andchemicallyreducing analysis,anditissufficienttodeterminethesensitivityofvarious regolithbythermochemicalorelectrochemicalmeans[6]toproduce design options and system parameters on the maximum and minimumsurfacetemperaturesofthewadi. ∗∗Data available at http://www.nasa.gov/pdf/163896main_LAT_GES_ OurthermalmodelofthewadiissimilartotheWesselinkmodelof 1204.pdf[retrieved23September2010]. thermalenergytransferinplanetaryregolith[7].Inthismodel,the 132 BALASUBRAMANIAMETAL. Table1 Physicalpropertiesofwadimaterials nighttimesurfaceheatflux(atypicalvalueis25W=m2,asshown later).Inreality,thetemperatureoftheroverisexpectedtobetime Properties Nativeregolith Basaltrock dependentasitlosesheattothesurroundings;therefore,therateof Thermaldiffusivity 6:6(cid:8)10(cid:4)9 m2=s 8:7(cid:8)10(cid:4)7 m2=s energy transfer between the wadi and the rover would be time Density 1800kg=m3 3000kg=m3 dependent as well. Noattempt is madein this studyto model the Specificheat 840J=(cid:1)kg(cid:2)K(cid:3) 800J=(cid:1)kg(cid:2)K(cid:3) thermalresponseoftherover. Thermalconductivity 0:01W=(cid:1)m(cid:2)K(cid:3) 2:1W=(cid:1)m(cid:2)K(cid:3) Within the bulk of thewadi, there is a balance between energy storage(duetosensibleheat)andenergytransferbyconduction.The bottomboundaryofthewadiisassumedtobeincontactwithnative solarfluxthatimpingesonthematerialiseitherabsorbedorreflected regolith.Atthisinterface,thetemperatureandheatfluxareassumed byitssurface.Thermalenergytransportbeneaththesurfaceissolely to be continuous. To model the unbounded extentof regolith that by conduction, and there is no radiative flux within the material. existsbeneaththewadi,afiniteregolithlayerofsuitablethickness Radiativeheatlossisallowedonlyfromthesurfaceintospace.Jaeger must be chosen in the numerical calculations. In most of the [8] has used such a model to predict the periodic steady-state calculations that we report, the native regolith-layer thickness temperature of the surface of the moon. In Jaeger’s model, the beneaththewadiwasassumedtobe20cm,whichisshowntobe incidentsolarfluxhasasinusoidaldependencewithtimeduringthe adequatelater.Thecharacteristicthermalpenetrationdepthinnative p lunardayandavanishingfluxduringthelunarnight.Thephysical regolithis (cid:1)(cid:1)(cid:1)(cid:1)0(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1),wheret isonehalfofthesynodicperiodonthe 0 0 propertiesoftheregolithlayer(treatedtobeinfinitelydeep)andthe moon(t (cid:11)354h)andisapproximately9cm.Thebottomboundary 0 radiativepropertiesofitssurfaceareassumedconstantwithrespect ofthelayerofnativeregolithisconsideredtobeadiabatic. tobothspaceandtime.BothWesselink[7]andJaeger[8]assumed Basedonthepreviousmodel,theequationsgoverningthetransfer thattheemissivityofthelunarsurfacewasunityintheirmodels.The ofenergyinthewadiandtheunderlyingregolith,respectively,are reflectanceoflunarmarebasaltsamplesandlunarregolithsamples fromvariouslocationshasbeenmeasuredinthevisibleandnear- @T @2T (cid:9)(cid:1) 0(cid:7)x(cid:7)d (1) infraredspectrum[4]andisgenerallyintherangeof0.05to0.3.In @t @x2 ourmodel,wewillassumethattheabsorptivityofthewadisurfaceis constant and equal to 0.9 over the wavelength spectrum of the @T0 @2T0 (cid:9)(cid:1)0 d(cid:7)x(cid:7)d0 (2) incidentsolarradiation.Thenativeemissivityofthewadisurfaceis @t @x2 also assumed to be a constant and equal to 0.9 in thewavelength spectrumoftheemittedradiation(infraredregion).Ourvaluesforthe Theboundaryconditionsare absorptivityandemissivityareconsistentwiththoseusedinthermal @T models in the literature for the surface of Mercury (Mitchell and (cid:4)k (cid:9)(cid:1) q(cid:1)t(cid:3)(cid:4)(cid:3)(cid:2)(cid:1)t(cid:3)(cid:1)T4(cid:4)T4(cid:3)(cid:4)q (cid:1)t(cid:3) at x(cid:9)0 (3) de Pater [9], who used 0.88 and 0.9 for the absorptivity and @x abs a rov emissivity,respectively)andthesurfaceofMercuryandthemoon (Vasavadaetal.[10],whoused0.9and0.95fortheabsorptivityand @T @T0 emissivity,respectively).MitchellanddePater[9]pointoutthatthe T(cid:9)T0; k (cid:9)k0 at x(cid:9)d (4) @x @x precisevaluesoftheabsorptivityandemissivityarenotcritical,since the surface temperature dependsonthe fourthroot ofthesequan- @T0 tities.Inmostofthecasesthatwemodel,weassumethataradiation (cid:9)0 atx(cid:9)d0 (5) shieldisusedduringthenighttoreducetheradiativelossfromthe @x wadisurface.Insuchcases,weassumethattheeffectoftheradiation Theemissivityofthewadisurfaceis(cid:2) (typically0.9)duringtheday shieldcanberepresentedbyreducingthesurfaceemissivityofthe 0 whentheincidentsolarfluxisnonzero,anditis(cid:2) (typically0.25) waditoaneffectivevalue.Ifthewadisurfaceandtheradiationshield 1 are flat with dimensions that are large relative to their separation during the nighttime when a heat-loss-limiting radiation shield is used.Similarly,q iszeroduringthedayandisaspecifiedconstant distanceandareisothermal(butatunequaltemperatures),andthe rov (typically 25W=m2) during the night. As an initial condition, we environmenttowhichtheshieldisexposedisregardedasablack- assumethatT(cid:9)T0(cid:9)T,whereT isaconstant,typicallytakentobe body,itcanbeshownfromenergyconsiderationsthattheeffectof i i 100K.T istheskyorenvironmenttemperatureforradiativeheat theradiationshieldisaccommodatedbyaneffectiveemissivityofthe a loss, and it is set to zero in the calculations. The equations and wadisurfacethatisgivenby(cid:2) (cid:9)1=(cid:1)1=(cid:2) (cid:10)2=(cid:2) (cid:4)1(cid:3).Ifthe eff wadi shield boundary conditions were solved numerically, as an initial value wadisurfacebehavesclosetoablackbody(weassume(cid:2) (cid:9)0:9in wadi problem,usingthesoftwareMathematica,whichusesthemethodof this study)andthe shield has an emissivity ofabout 0.5,then the linesforthesolutionofpartialdifferentialequations. effectivewadi-surfaceemissivityisabout0.25.Smalleremissivity values of thewadi surface or the shield result in smaller effective wadi-surfaceemissivities.Wechoosetousearelativelyconservative A. DimensionlessFormulationoftheGoverningEquations effectivewadi-surfaceemissivityof0.25inthisstudy.Inmostofthe To determine the dimensionless parameters that control the calculations,weassumethatenergyissuppliedfromthesurfaceof behaviorofthesystemofequationsandboundaryconditionsgiven the wadi to a rover to keep it warm during the lunar night. We previously, we cast the system in terms of the following dimen- representtheenergytransferfromthewaditotheroverbyaconstant sionlessvariables: T T0 t (cid:5)(cid:9) ; (cid:5)0(cid:9) ; (cid:9)(cid:9) 1. T T t ref ref 0 d(cid:4)x x(cid:4)d 0.8 (cid:7)(cid:9) for 0(cid:7)x(cid:7)d; (cid:7)(cid:9) for d(cid:7)x(cid:7)d0 d d0(cid:4)d max 0.6 q(cid:1)(cid:9) q ; (cid:2)(cid:1)(cid:9) (cid:2) ; q(cid:1) (cid:9) qrov / q q 0.4 qmax (cid:2)0 rov (cid:1)absqmax (cid:2)(cid:1) q (cid:3)1=4 0.2 Tref(cid:9) ab(cid:2)s(cid:3)max 0 500 1000 1500 2000 2500 t (h) @(cid:5) 1 @2(cid:5) (cid:9) (6) Fig.2 Equatorialsolarflux.Thesolarfluxprofileisasemisinusoid. @(cid:9) N2@(cid:7)2 BALASUBRAMANIAMETAL. 133 @(cid:5)0 1 @2(cid:5)0 thermally thick solid fuels [11], where the relationship between @(cid:9) (cid:9)N02 @(cid:7)2 (7) ignitiondelaytimeapnd(cid:1)(cid:1)(cid:1)i(cid:1)n(cid:1)(cid:1)(cid:1)c(cid:1)(cid:1)i(cid:1)d(cid:1)(cid:1)entradiativeheatfluxcanbeusedto inferfuelproperties( k(cid:8)C ).Betweennativeregolithandbasalt,(cid:8) p (cid:6) @(cid:5) @(cid:5)0 variesbyafactoroflessthantwo,andCpisaboutthesame.Thus,the (cid:9)q(cid:1)(cid:1)(cid:9)(cid:3)(cid:4)(cid:2)(cid:1)(cid:1)(cid:9)(cid:3)(cid:1)1(cid:4)T(cid:1)4(cid:3)(cid:4)q(cid:1) (cid:1)(cid:9)(cid:3); (cid:9)0 at(cid:7)(cid:9)1 thermalconductivitykisthekeythermalpropertythatcontrolsthe N @(cid:7) a rov @(cid:7) differenceintheirthermalresponses. (8) @(cid:5) @(cid:5)0 (cid:5)(cid:9)(cid:5)0; (cid:9)(cid:4)(cid:4)N at(cid:7)(cid:9)0 (9) B. IncidentSolarFluxProfiles @(cid:7) @(cid:7) The equatorial solar flux model is shown in Fig. 2. The flux is assumedtobesinusoidalduringthedaytimeandvanishesduringthe Thedimensionlessparametersareasfollows: 1)Theparameter(cid:6)(cid:9)k=p(cid:1)(cid:1)(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:3)(cid:3)1=4(cid:1)(cid:1) q (cid:3)3=4 isameasure lunarnight.Theanglebetweenthesurfacenormalandthesunatthe ofthetransferofenergyfromthe0in0cidentsuabrsfamceaxheatfluxintothe peakisfurtherassumedtobethesamefromlunarmonthtolunar wadibyconduction.With(cid:2) (cid:9)(cid:1) (cid:9)0:9andq (cid:9)1300W=m2, month.Thetimeperiodofthesolarfluxisthesynodicperiod,which 0 abs max is approximately 708 h. The peak heat flux is taken to be the value of (cid:6) is 0.04 and 0.66 for native regolith and basalt, q (cid:9)1300 W=m2. Figure 3 shows how the solar heating of the respectively. max 2) N(cid:9)d=p(cid:1)(cid:1)(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)(cid:1) is the ratio of the wadi depth to the thermal thermalmasscanbeenhancedbytheuseofareflectorthattracksthe 0 suntodirectthefullsolarfluxtothesurfacethroughoutthelunarday. penetrationdepthinit.Forawadidepthof50cmofbasalt,N(cid:9)0:48. 3)N0(cid:9)(cid:1)d0(cid:4)d(cid:3)=p(cid:1)(cid:1)(cid:1)(cid:1)0(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)isaratioofthedepthoftheunderlying Thisconstitutesthemaximumintensitysourceofheatingwithoutthe 0 additional complexity of concentrating or focusing the solar layerofnativeregolithtoitsthermalpenetrationdepth.Asmentioned before, while the underlying regolith is of infinite extent, a finite illuminationonthesurface.Thesolarfluxisasquarewaveinthis case.Fornumericalpurposes,thescaledheatfluxisrepresentedas regolith-layerthicknessmustbechosenforcomputationalpurposes. 0:5(cid:10)0:5tanh(cid:5)25sin(cid:1)(cid:11)t=t (cid:3)(cid:6).Inreality,thereflectormayhavesome Foralayerthatis20cmthick,N0(cid:9)2:2. 0 p(cid:1)(cid:1)(cid:1) p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) losses,andthefluxprofileonthewadisurfacemightbeinbetween 4)The parameter(cid:4)(cid:9)(cid:1)1= 2(cid:3)(cid:1)k0=k(cid:3) (cid:1)=(cid:1)0 is ameasure ofthe whatisshowninFigs.2and3. importanceofheatlossfromthewaditotheunderlyingregolithlayer. Thesolarilluminationatthelunarpoleisquitedifferentfromthat 5)T(cid:1) (cid:9)T =T isthedimensionlessenvironmenttemperature. a a ref attheequator.Figure4showsanimageofthelunarsouthpole{2006 Inadditiontothepreviousparameters,thethermalresponseofthe Goldstone Solar System Radar (GSSR) background radar back- wadidependsonthevariationwithtimeoftheincidentheatflux,the scatterimage[12]}.ThelargemountainatthetopisMalapert,the effective surface emissivity, and the heat transferred to the rover. bluearrowpointstothesouthpole,andtheyellowarrowpointstoa Notethatin1)and4),thethermalconductivityanddiffusivitycanbe sitelabeledsiteAontherimofShackletoncrater.Inthispaper,we p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) combined and written as k(cid:8)C . This quantity appears in many p explorethepossibilityofconstructingathermalwadiatthispartic- otherthermalapplications,suchasfireresearchfortheignitionof ularsite.Thesolarilluminationdataarebasedonthe2006GSSRJet Propulsion Laboratory digital elevation model (DEM) [12]. It is pixel(14,828,5711)intheDEM.Itisthebest-illuminatedsitenear 1. thesouthpolebasedonalowestcontinuouseclipsetimemetric. Figure5showsthesolarilluminationatthissiteoveracompressed 0.8 timescale.Thedatastarton1January2008andarein1hincrements. ax 0.6 The illumination is considered to be 100% when the sun angle is m largerthanthelocalterrainelevationangle.Whenthesunangleis q / q 0.4 smallerthantheterrainelevationangle,thesunisblocked,andthe illuminationis0%.Partialsunblockagesarealsopossible.Theleft 0.2 panel in this figure shows the illumination for about 2.8 years (25,000 h) and includes three polar winters (the dark regions of 500 1000 1500 2000 2500 illumination).Therightpanelshowstheilluminationinaportionof t (h) thefirstwinter(from2000to3000h).Itisevidentthat,evenduring Fig.3 Equatorial solarfluxobtainedwiththe useofasun-tracking winter,substantialsolarilluminationisavailableatthissite.Inthese reflector.Thesolarfluxprofileisasquarewave. data,inFig.5,thetimeperiodwiththelongesteclipseperiod(less Fig.4 SiteA(yellowarrow)nearthelunarsouthpole(bluearrow)ontherimofShackletoncrater. 134 BALASUBRAMANIAMETAL. 1. 1. 0.8 0.8 ax 0.6 ax 0.6 m m q q / q 0.4 / q 0.4 0.2 0.2 5000 10 000 15 000 20 000 25 000 2000 2200 2400 2600 2800 3000 t (h) t (h) Fig.5 Solarfluxatthepolarsitewiththeuseofareflector. than100%solarenergy)occursfromhour3256to3327(approxi- III. ResultsandDiscussion mately71h).Thelongest0%solarflux(approximately52h)occurs The results for the temperature in the wadi, chiefly its surface inthesameperiod.Weusedtheilluminationprofileshownbetween temperature,arepresentednext,inbothlunarequatorialandpolar t(cid:9)0and6500h,inFig.5,inourthermalmodeltodeterminethe regions. We first show the influence of the various dimensionless surfacetemperatureofthewadi.Weexpectthetemperatureresultsin parametersonthescaledwadi-surfacetemperature,inSec.III.A,for thistimeperiodtorepresenttheworst-casescenarioatthissite. periodicincidentsolarflux.Theseresults,tabulatedinTable2,arefor thescaled(dimensionless)surfacetemperature.Theintentistoshow theinfluenceofthevariousparameters,whereachosenparameteris C. RoverHeat-LossConsiderations variedoneatatimewhiletherestarefixed.SectionsIII.BandIII.C Astheprincipaluseofathermalwadiistoprovideasourceof discussselectedresults forthewadi-surface temperatures nearthe sensibleheattoaroverduringthelunarnight,thedesignofathermal equatorandinthevicinityofthepole,respectively.Theseresultsare wadi entails the knowledge of the rate at which energy must be forthephysicalsurfacetemperature.Thisisbecause,fromthepoints suppliedtoarover.Shortlyafternightfall,theroverisanticipatedto ofviewofthermalwadidesignandoperation,itisofinterestwhether arriveatthelocationofathermalwaditoretireforthenight.Atthis requirementsonminimumandmaximumtemperaturesaremetand time,theroverwouldbequitewarm.Aradiationshieldisexpectedto whateffectsthesizeandthermalpropertiesofthewadihaveonthe bedeployedoverthewadiduringthenighttominimizetheheatloss. temperature and amount of energy stored. Since changes in wadi Therefore,forpracticallytheentirelunarnight,theenergyexchange thermal properties affect many of the dimensionless parameters betweenthewadi,therover,theradiationshield,andfreespaceareall simultaneously,andnotjustasingleparameter,wehavechosento intimatelycoupled.Thisisaformidableheat-transferproblem,with present thewadi-surface-temperature results andtheir dependence time-dependenttemperaturefieldsineachoftheseobjects.Inthis onvariousdesignparametersintermsofphysicalquantities. study, we simplify the analysis and represent the rate of energy The wadi-surface temperatures in both equatorial and polar exchangebetweenthewadiandtheroverbyaconstantnighttime regionsaretabulatedinTable3.Itisseenthatthemaximumwadi heatflux.Toestimatethisheatflux,weassumethatitisdesiredto temperatureisquiteinsensitivetovariouswadidesignparameters. maintaintheroveratagiventemperature,andtheroverisprotected The theoretical maximum surface temperature is T , which was ref bythermalinsulationmaterial(suchassilicaaerogel)ofasuitable definedearlier,andrepresentsabalancebetweentheabsorptionof thickness.Thereisheatlossbyradiationfromtheexposedsurfaceof thepeaksolarfluxandthesimultaneousradiativeemissionwithout theinsulationtofreespace.Atsteadystate,theconductionheatflux heat conduction into the wadi. For (cid:1) (cid:9)(cid:2) (cid:9)0:9, T (cid:9)389 K abs 0 ref throughtheinsulationlayermustequaltheradiativeheatfluxfromits whenthepeakheatfluxis1300 W=m2.T (cid:9)321K,closetothe ref surface. We choose the rover temperature to be 220 K, which skin-touchtemperatureof318K,whenthepeakheatfluxisreduced corresponds to the low-temperature tolerance of many electronics to600W=m2.Forasunangleintherangeof1.5to6(cid:12)nearthepole, components.Assumingathermalconductivityof0:01W=(cid:1)m(cid:2)K(cid:3) the peak heat flux is reduced to 1300sin(cid:1)1:5(cid:11)=180(cid:3)(cid:9)34 W=m2 fortheinsulationmaterial,andvaryingitsthicknessbetween1and and1300sin(cid:1)6(cid:11)=180(cid:3)(cid:9)136W=m2,respectively,andT isinthe ref 3in.anditssurfaceemissivitybetween0.2and0.8,theroverheat range of 157 to 221 K. With regard to the minimum surface lossisestimatedbyaone-dimensionalheattransferanalysistobein temperature, a limit of 220 K corresponds to the low-temperature the range 10 to 25 W=m2. In subsequent calculations, we have tolerancelimitofmanyelectronicscomponents.Wewillrefertothis chosenaconstantnighttimesurfaceheatfluxofq (cid:9)25W=m2to temperaturelimittojudgetheadequacyofvariouswadidesignop- rov representtherateofenergyexchangebetweenthewadiandtherover. tions.Theaveragesurfacetemperatureforequatorialenvironmentsis Table2 Scaledmaximum,minimum,andaveragewadi-surfacetemperaturesforequatorialsolarflux (cid:5) (cid:5) (cid:5) max min av N N0 (cid:4) (cid:6) Fourthcycle Eighthcycle Fourthcycle Eighthcycle Fourthcycle Eighthcycle Nominal 0.50 2 0.04 0.65 0.997 0.997 0.668 0.670 0.868 0.869 Varying(cid:6) 0.50 2 0.04 0.50 0.998 0.998 0.632 0.634 0.855 0.855 0.50 2 0.04 0.30 0.999 0.999 0.563 0.564 0.828 0.828 0.50 2 0.04 0.10 1.000 1.000 0.427 0.428 0.767 0.768 0.50 2 0.04 0.05 1.000 1.000 0.355 0.356 0.732 0.732 Varying(cid:4) 0.50 2 0 0.65 0.999 0.999 0.656 0.656 0.866 0.866 0.50 2 0.02 0.65 0.998 0.998 0.662 0.663 0.867 0.867 0.50 2 0.06 0.65 0.996 0.996 0.673 0.676 0.869 0.870 0.50 2 0.08 0.65 0.995 0.996 0.677 0.682 0.869 0.870 VaryingN 0.25 2 0.04 0.65 0.998 0.998 0.584 0.586 0.837 0.838 1.00 2 0.04 0.65 0.998 0.999 0.730 0.735 0.881 0.882 1.50 2 0.04 0.65 0.993 0.995 0.725 0.741 0.875 0.880 VaryingN0 0.50 1 0.04 0.65 0.998 0.998 0.664 0.664 0.868 0.868 0.50 3 0.04 0.65 0.995 0.996 0.665 0.675 0.866 0.869 BALASUBRAMANIAMETAL. 135 obtainedbyintegrationofthetime-dependentsurfacetemperature 400 duringthediurnalcycle.Itisnotthearithmeticmeanofthemaximum 350 andminimumsurfacetemperatures.Forawadiofagivensizeand Tmax = 377 K thermalproperties,theaveragesurfacetemperatureisindicativeof 300 theamountofenergythatisstoredinit.Indeed,itcanbeshownunder ()K 250 a periodic steady state (where the net surface heat flux in a cycle Ts 200 vanishes) that the time-averaged temperature at any location is independentofdepthwithinthewadi. 150 T = 182 K min A. InfluenceofDimensionlessParametersonthe 500 1000 1500 2000 2500 t (h) SurfaceTemperature Fig.7 Influenceofnighttimeheat-loss-limitingradiationshieldonthe Thedependenceofthemaximum,minimum,andaveragescaled thermalwadi-surfacetemperaturevstimefora50-cm-deepwadiwith wadi-surfacetemperaturesonthedimensionlessparametersN,N0,(cid:4), thethermalpropertiesofbasaltrock.(cid:1) (cid:1)0:9duringdaytimeforall and (cid:6) is summarized in Table 2. We have assumed that the solar curves (blue curve: (cid:1) (cid:1)0:9; green 0curve: (cid:1) (cid:1)0:5; red curve: illuminationprofileistheequatorialprofilewithareflector,shownin (cid:1) (cid:1)0:25). 1 1 1 Fig.3,(cid:2) =(cid:2) (cid:9)0:3andq(cid:1) (cid:9)0.Foraninitialcondition,wehave 0 1 rov used(cid:5)(cid:9)(cid:5)0(cid:9)0:3.InTable2,therangeof(cid:6)and(cid:4)coversthethermal propertyrangefromnativeregolithtobasalt.Nisthedimensionless temperatureof100K,conditionsclosetosteady-stateoscillationsare thermalpenetrationdepthinthewadi,andonewouldexpectitsvalue achievedinfourdiurnalcycles.Foralltheequatorialcases,wehave to be less than one in order to minimize thewadi mass. N0 is the used four diurnal cycles in the calculations unless otherwise dimensionlesspenetrationdepthintheunderlyingregolithlayer,and specified. thecalculationsareexpectedtobeinsensitivetoitwhenN0islarger thanone.Insuchcases,however,thecalculationsmustbeperformed 2. BasaltRockwithSinusoidalHeatFlux formorediurnalcyclestoachievenear-periodicsteady-stateresults. ThebluecurveinFig.7showsthesurface-temperaturevariation Table 2 includes results for four and eight diurnal cycles that with the time of what we describe as a nominal thermal wadi demonstrate that periodic steady-state surface-temperature con- configuration. The nominal equatorial thermal wadi assumes the ditions,asmostcasesarereachedwithinafewcycles.Themaximum thermalpropertiesofbasaltrock,thethermalconductivityofwhichis waditemperatureisclosetooneinallcases,anditisquiteinsensitive about100timesthatofnativeregolith,andawadidepthof50cm. toanyoftheparametersintherangesshowninTable2.Thevaluesof The characteristic thermal penetration depth p(cid:1)(cid:1)(cid:1)(cid:1)t(cid:1)(cid:1)(cid:1)(cid:1) in basalt is (cid:6)andNhavethemosteffectontheminimumsurfacetemperature, 0 approximately 100 cm. The surface thermal emissivity is set to a and (cid:6) has the most effect on the average surface temperature. constant, (cid:2)(cid:9)0:90. Compared with that of native regolith, the Betweenthefour-cycleandeight-cycleresults,theminimumsurface maximumtemperatureisreducedslightlyfrom387to375K,andthe temperature shows the most change, especially for N and N0>1. minimum temperature increases from 117 to 182 K, but it is not Hence,ittakesmorediurnalcyclestoachievetheperiodicsteady- sufficientlywarmduringmuchofthelunarnighttopassivelyprovide stateminimumtemperatureforrelativelydeeplayers. heattoaroverorotherequipmentsusceptibletocold.Figure7also showshowthesurface-temperaturevariationisalteredbytheuseof B. ThermalWadisinLunarEquatorialEnvironments theradiationshieldthatreducestheheatradiatedtospacefromthe 1. NativeRegolithwithSinusoidalHeatFlux surfaceofthethermalmassofthewadiduringthelunarnight.To Asareferenceforthevariousthermalwadiconfigurationstobe simulatetheheat-loss-limitingshield,themodeladoptsaneffective surfaceemissivityreducedfrom0.9duringtheunshieldeddaytime explored, the variation with time of the surface temperature of unalteredlunarregolithexposedtothesolarflux,describedinFig.2, configuration to a lower value during the night. The maximum surfacetemperatureisbarelyaffected,whiletheminimumsurface wascalculatedandisshowninFig.6.TheblackcurveinFig.6isthe temperatureissignificantlyelevatedbytheuseofaradiationshield result of the present numerical model. The regolith-layer depth is (T (cid:9)233K for (cid:2) (cid:9)0:25). The abrupt change in the surface takentobe20cm,basedonthethermalpenetrationdepthargument min 1 givenearlier.Weassumedthatthepeakheatfluxis1300W=m2,and emissivitybetween daytime andnighttime leadstokinksofsmall magnitudeinthesurface-temperaturevariation.Thesefeaturesare the surface absorptivity, daytime emissivity, and nighttime alsoevidentintheresultscalculatedbyJaeger’smethod[8]foran emissivityareequalto0.9.TheoverlaidbluecurveinFig.6isthe infinitelydeepwadi(seetheAppendix). periodic steady-state surface temperature that is calculated using Jaeger’smethod[8],describedintheAppendix.Itisevidentfrom Fig.6thatthenumericalcalculationwitharegolith-layerdepthof 3. BasaltRockwithReflector,RadiationShield,andRoverHeating 20cmisinexcellentagreementwiththeperiodicsteady-stateresult Figure 8 shows how the surface temperature of the nominal for an infinite-depth layer. The maximum and minimum surface thermalwadiconfigurationischangedwiththeuseofasun-tracking temperaturesare387and117K,respectively.Withaninitialuniform solarilluminationreflector(withthesolarfluxdescribedinFig.3) fromthatshowninFig.7forasinusoidalflux.Fromsunrise,ittakes 400 400 350 300 Tmax = 387 K 350 Tmax = 388 K )K 300 (Ts 220500 T = 117 K ()TKs250 min 200 150 150 T = 181 K 500 1000 1500 2000 2500 min t (h) 500 1000 1500 2000 2500 Fig.6 Surfacetemperaturevstimefornativeregolith.Theequatorial t (h) solarfluxisasshowninFig.2(blackcurve:presentcalculationsfora20- Fig.8 Surfacetemperatureofanequatorialthermalwadiwithasolar cm-deeplayer;bluecurve:periodicsteady-stateresultcalculatedusing reflector,aheat-loss-limitingradiationshield,androverheatingfora50- Jaeger’smethod[8]foraninfinitelydeeplayer). cm-deepwadi[(cid:1) (cid:1)0:9(blue),(cid:1) (cid:1)0:5(green),and(cid:1) (cid:1)0:25(red)]. 1 1 1 136 BALASUBRAMANIAMETAL. approximately270hofabsorbingheatforthethermal-masssurface surface temperature. Note that the small kinks in the temperature to reach the maximum temperature. The surface remains at this variationinFig.7justafterdaylightandnightfallareabsentinFig.8. temperaturefor80additionalhours,untilsunset.Figure8includes Thehighheatfluxprovidedbythesun-trackingreflectorimmedi- heatsupplytoaroveratnightandcomparesthesurfacetemperature atelyaftersunriseandjustbeforetheadventofdarknesssignificantly of the wadi for various values of the effective nighttime surface elevatesthesurfacetemperatureandtendstosuppresstheinfluence emissivity.Whilethemaximumtemperatureattainedispractically ofabruptchangeinthesurfaceemissivity. thesame,variationsintheabilityoftheradiationshieldtoreduce radiativeheatlossgreatlyaffecttheminimumtemperaturesattained duringthenight,whichisintherange181K,whenthesurfaceis 4. InfluenceofWadiDepthVariation unshielded,to247K,when(cid:2) (cid:9)0:25.Withtheuseoftheshield,the Akeycharacteristic ofthethermalwadi conceptisitsdepth:it 1 reflector,andthethermalpropertiesofbasaltrock,thethermalmass affectsthemassofmaterialthatmustbemodifiedtobuildthewadi maintainstemperatureshighenoughtosupplyheattoaroverorother andtheamountofenergythatthewadicanstore.Figure9showsthe equipmentthroughouttheequatoriallunarnight.FromTable3,the resultsforthewadi-surfacetemperature,inwhichthedepthofthe averagesurfacetemperaturefollowsthesametrendastheminimum wadiisvariedaboutthenominaldepthof50cm,from25to100cm. Table3 Maximum,minimum,andaveragewadi-surfacetemperaturesinequatorialandnear-polarenvironments Wadidepth, Regolith-layer Wadi Peak Solar Radiation (cid:2) Rover T ,K T ,K T ,K night max min av m depth,m thermal heatflux, reflector shield heatflux, diffusivity W=m2 W=m2 Equatorial —— 0.20 Regolithvalue 1300 No No 0.90 0 387 117 232 —— 1a Regolithvalue 1300 No No 0.90 0 387 117 232 1a —— Basaltvalue 1300 No No 0.90 0 365 215 277 1a —— Basaltvalue 1300 No Yes 0.50 0 368 237 290 1a —— Basaltvalue 1300 No Yes 0.25 0 371 257 305 1a —— Basaltvalue 1300 Yes No 0.90 0 380 230 311 1a —— Basaltvalue 1300 Yes Yes 0.50 0 382 258 327 1a —— Basaltvalue 1300 Yes Yes 0.25 0 384 290 344 0.50 0.20 Basaltvalue 1300 No No 0.90 0 375 182 268 0.50 0.20 Basaltvalue 1300 No Yes 0.50 0 376 207 283 0.50 0.20 Basaltvalue 1300 No Yes 0.25 0 377 233 298 0.50b 0.30 Basaltvalue 1300 No Yes 0.25 0 378 234 298 0.50 0.20 Basaltvalue 1300 Yes No 0.90 0 388 192 303 0.50 0.20 Basaltvalue 1300 Yes Yes 0.25 0 388 261 339 0.50 0.20 Basaltvalue 1300 Yes Yes 0.25 25 388 247 334 0.50b 0.30 Basaltvalue 1300 Yes Yes 0.25 25 388 247 334 0.25 0.20 Basaltvalue 1300 Yes Yes 0.25 25 389 202 318 1.00c 0.20 Basaltvalue 1300 Yes Yes 0.25 25 385 279 341 0.50 0.20 Basalt/5 1300 Yes Yes 0.25 25 386 236 324 0.50c 0.20 Basalt/10 1300 Yes Yes 0.25 25 386 219 315 0.50 0.20 Basaltvalue 1300 Yes No 0.90 25 388 181 299 0.50 0.20 Basaltvalue 1300 Yes Yes 0.50 25 388 211 316 0.50 0.20 Basaltvalue 1300 Yes Yes 0.10 25 389 292 354 0:50(cid:10)1mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 386 255 332 dustd 0:50(cid:10)2mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 381 260 330 dustd 0:50(cid:10)3mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 376 264 327 dustd 0:50(cid:10)4mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 371 267 324 dustd 0:50(cid:10)5mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 365 268 321 dustd 0:50(cid:10)7mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 355 270 315 dustd 0:50(cid:10)10mm 0.20 Basaltvalue 1300 Yes Yes 0.25 25 341 269 307 dustd Polare —— 0.20 Regolithvalue 1300 Yes Yes 0.25 25 389 171 —— 0.50 0.20 Basaltvalue 1300 Yes No 0.90 25 389 261 —— 0.50 0.20 Basaltvalue 1300 Yes Yes 0.25 25 389 320 —— 0.50 0.50 Basaltvalue 1300 Yes Yes 0.25 25 389 318 —— 0.50 0.75 Basaltvalue 1300 Yes Yes 0.25 25 389 319 —— 0.50 0.20 Basaltvalue 600 Yes Yes 0.25 25 320 273 —— 0.25 0.20 Basaltvalue 600 Yes Yes 0.25 25 321 266 —— 0.10 0.20 Basaltvalue 600 Yes Yes 0.25 25 321 241 —— 0.50 0.20 Basalt/5 600 Yes Yes 0.25 25 321 255 —— 0.50 0.20 Basalt/10 600 Yes Yes 0.25 25 321 242 —— 0.50 0.20 Basaltvalue 600 Yes Yes 0.10 25 321 293 —— 0.50 0.20 Basaltvalue 600 Yes Yes 0.50 25 321 251 —— 0.50 0.20 Basaltvalue 600 Yes Yes 0.90 25 321 229 —— aJaeger’smethod. b12thcycleresults. cSixthcycleresults. dFifthcycleresults. eSolarilluminationprofileisasshowninFig.5;0(cid:7)t(cid:7)6500h. BALASUBRAMANIAMETAL. 137 400 layer.Attheinterfacebetweenthedustlayerandthewadi,weassume 350 T = 389 K that temperature and heat flux are continuous. For dust-layer max thicknessof5mmorless,weseethatthewadisurfacehasasmaller 300 temperatureswingduringthelunardiurnalcyclecomparedwitha ()K 250 situationwithoutanydust.Thus,thedustlayeractstoshieldthewadi Ts surface from the environment at both temperature extremes. It 200 appearsthatadustlayerwithathicknessofafewmillimeterdoesnot 150 Tmin = 202 K degradetheperformanceofthewadiandmightactuallyhelpmitigate thetemperatureswingduringthediurnalcycle. 500 1000 1500 2000 2500 t (h) 7. TemperatureVariationwithDistanceBeneathWadiSurface Fig.9 The effect of wadi depth on the surface temperature of an equatorial thermal wadi with a solar reflector, a heat-loss-limiting Thetemperaturevariationwithdistancebeneaththesurfaceina 50-cm-deep wadi with a 30-cm-deep underlying regolith layer is radiationshield,androverheating[25cm(blue),50cm(green),and 100cm(red),eachwiththethermalpropertiesofbasalt]. shown in Fig. 11 for constant daytime illumination of the wadi surface.Similarprofilesareobtainedforsinusoidaldaytimeillumi- nationaswell.Thetemperaturedistributionisshownattimescloseto Themaximumtemperatureisnotverysensitivetothedepthofthe theextremaofthesurfacetemperature,aswell asanintermediate wadi.Theminimumsurfacetemperatureisquitesensitiveandvaries time(seelegendinFig.11fortheexacttimes;seeFig.8foranalogous from202K,foradepthof25cm,to279K,foradepthof100cm.The surfacetemperaturesvstime).Inthesecalculations,thethicknessof correspondingaveragesurfacetemperatures(fromTable3)are318 theunderlyingregolithlayerwasincreasedfrom20to30cmtobetter and341K.Theshallowerthewadi,thecolderthesurfacegets,and approximatetheunboundedextentofregolithunderneaththewadi. the average surface temperature is lower. Noting that the average Also,12diurnalcycleswereused,becauseittakeslongertoachievea temperature is independent of location within the wadi at steady periodicsteadystatefordeeperlayers,asmentionedearlier.From state,fromanenergystoragepointofview,deeperwadisareprefer- Table3,itcanbeseenthatthesurface-temperatureresultsfora50- able,sincethemassofthewadi(representedbyitsdepth)andthe cm-deepwadiarepracticallythesamewitha20-cm-deepora30-cm- energystoredperunitmass(representedbytheaveragetemperature) deepunderlyingregolithlayer.Figure11showsthattheinfluenceof arebothincreased. thediurnalcycleisnotfeltsignificantlybeyondapproximately20cm intheunderlyingregolithlayer.Also,a50-cm-deepwadihasvery 5. InfluenceofVariationofThermalDiffusivity little temperature variation with depth, and it can be considered practicallyisothermalatalltimes. Thethermaldiffusivityofthewadimaterialalsohasasignificant effect on the minimum surface temperature. A thermal diffusivity thatisafactorof10smallerthanthatofbasalt(whichisaboutafactor C. ThermalWadisinLunarNear-PolarEnvironments of10greaterthanthatofnativeregolith)resultsinaminimumsurface Inthevicinityofthelunarpoles,themaximumnaturallyoccurring temperature of about 219 K, as compared with the minimum temperatureofthelunarsoilisfarlowerthanthatatlowerlatitudes temperatureof247Kforbasalt.Figure10showsthewadi-surface becausethesunangleisalwaysveryoblique(asmentionedbefore, temperaturewhenthethermaldiffusivitywasreducedtoaslowas thetheoreticalmaximumtemperatureisintherangeof157to221K onetenththatofbasalt. forasunanglebetween1.5and6(cid:12)).Thus,whilethetrackingsolar reflector could be considered optional in the equatorial sites, it is 6. InfluenceofDustCoveringWadiSurface essentialnearthepolestoheatthewadiadequately.Figure12shows theperformanceofthethermalwadiconfigurationsimilarinanear- A potential issue of concern is the effect of dust covering the polar environment similar to the nominal wadi shown by the red surfaceofthewadi.Itiswellknownthatdustisubiquitousonthe curveinFig.8.Inthiscalculation,thescaledsolarfluxvariationwith moon.Assumingthatthethermalpropertiesofdust(thatweregardas timeq(cid:1)(cid:1)t(cid:3),showninFig.5,hasbeenused.Theemissivityofthewadi fine particles of lunar regolith) are the same as those of native surfaceistakentobe(cid:2)(cid:1)t(cid:3)(cid:9)(cid:1)(cid:2) (cid:4)(cid:2) (cid:3)q(cid:1)(cid:1)t(cid:3)(cid:10)(cid:2) ,andtheheatlossto regolith,onecaninferthatasurfacedustlayerinexcessof20cm(of the rover is defined as q (cid:1)t(cid:3)0(cid:9)C(cid:5)11(cid:4)q(cid:1)(cid:1)t(cid:3)(cid:6),1with C(cid:9)25 W=m2. theorderofthethermalpenetrationdistanceinnativeregolith)would rov Themaximumsurfacetemperatureis389K,whichisthetheoretical renderthewaditobecompletelydysfunctional.Resultsarereported maximumsurfacetemperature.Theminimumsurfacetemperatureis inTable3forthetemperatureofthewadisurfacethatisincontact about320K,anditisobtainedattwoseparatetimes.Thisismuch withdust(andnotthetemperatureatthetopofthedustlayerthatis higher than the equatorial counterparts because of the extended exposedtofreespace)fordust-layerthicknessupto10mm.Forthese illuminationofthewadi.Figure13isasimilarplotwheretheheat- calculations, theformalismdescribedearlierhasbeenextendedto loss-limitingshieldhasbeeneliminated.Withthesurfaceofthewadi accommodate regolithlayersabove andbelow thewadi. Theheat allowedtoradiatetospaceduringperiodsofdarkness,theminimum fluxbalance,giveninEq.(3),isappliedatthesurfaceofthedust temperature falls to approximately 261 K. The maximum and 400 400 350 T = 388 K max 350 300 )K 300 ( 250 Ts )K 250 125000 Tmin = 219 K (Ts 200 150 500 1000 1500 2000 2500 100 t (h) 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Fig.10 Theeffectofthermaldiffusivityonthesurfacetemperatureof x(m) anequatorialthermalwadiwithasolarreflector,aheat-loss-limiting Fig.11 Temperaturedistributionvsdistancebelowthesurfacefora radiationshield,androverheatingfora50-cm-deepwadi[basaltvalue 50-cm-deepwadianda30-cm-deepunderlyingregolithlayer.Thesolar (red), one fifth of basalt value (green), and one tenth of basalt value flux is as shown in Fig. 3, with 12 diurnal cycles, qrov(cid:1)25W=m2 (blue)]. [t(cid:1)8127h(red),t(cid:1)8307h(green),andt(cid:1)8486h(blue)]. 138 BALASUBRAMANIAMETAL. 400 Tmax = 389 K IV. Conclusions The objective of this study was to determine the answer to the 350 question:canengineeredthermalwadisfunctionassafehavensfor 300 explorationassets,protectingthemfromtheextremecoldoflunar ()K 250 Tmin = 320 K darkness?Thepresentanalysisprovidesanaffirmativeanswertothis Ts question.Thatis,theanalysisshowsthatthermalwadisofthepresent 200 concept,ifconstructedonthelunarsurfaceusingmodifiedregolithas thethermalmass,canprotectexplorationassetsfromtheextremely 150 coldconditionsthatareotherwiseencounteredonthelunarsurface. Asignificantadvantageoftheconceptistheuseofmodifiedlunar 1000 2000 3000 4000 5000 6000 t (h) regolithasthethermal-massmaterial,therebysubstantiallyreducing themassthatmustbebroughtfromEarthtomanufactureit. Fig.12 Surfacetemperatureofthethermalwadiofdepth50cm,with Otherthaninapermanentlyshadowedcraterateitherofthelunar thermalpropertiesofbasaltrockatthelunarpolarsiteusingthesun- trackingreflectorandtheheat-loss-limitingshield.Heatissuppliedtoa poles,thetwocasesconsideredrepresentreasonableboundsofthe roverduringtheperiodsofdarkness. bestcase(Shackletoncraterrim,withamaximumperiodofdarkness ofabout52h)andtheworstcase(equatoriallocationswithmaximum periodsofdarknessofabouttwoEarthweeks).Ineithercase,ifthe minimumwadi-surfacetemperaturesinthenear-polarenvironment thermalmasshasareasonabledepth,ifenergyisstoredinthethermal arealsosummarizedinTable3forvariouscases. massinsufficientquantitiesthroughtheuseofareflectingsurface,if Sincethesummer–winterilluminationcycleshowninFig.5hasa heatlossbyradiationiscontrolledthroughtheuseofaradiantenergy timeperiodthatisanorderofmagnitudegreaterthanthetimeperiod reflector, and if suitable thermophysical properties are achieved ofthediurnalcycleattheequator,thethermalpenetrationdepth(that whenproducingthethermalmass,theanalysisshowsthatthermal- isproportionaltothesquarerootofthetimeperiod)isapproximately masstemperaturescanbemaintainedtypicallygreaterthan230K. three times as large at the pole than at the equator. We therefore Engineeringtradeoffscanbemadewithrespecttothedesignofthe studiedtheinfluenceofthethicknessoftheunderlyingregolithlayer thermalwadiandthedesignofthesystemthatmustbeplacedon (0.2,0.5,and0.75m;seeTable3)onthesurfacetemperatureofthe themooninordertoproducethermal-massmaterialsandassemble wadi. The maximum deviation in the wadi-surface temperature thethermalwadisoastomeetspecifictemperaturerequirementsand betweenthe0.2and0.75mregolith-layerthicknessesis1.6K,and optimizethemassofwhatmustbelaunchedfromEarthwhilealso themaximumdeviationbetweenthe0.5and0.75mregolith-layer taking into account other requirements related to reliability and thicknesses is 0.3 K. We have used results for the wadi-surface performance. temperaturewith0.2mastheregolith-layerthicknessinFigs.12and We can also contemplate improvements to the design of the 13.Wealsocheckedthesensitivityoftheminimumtemperaturein thermalwadithatprovideagreateramountofenergystorage.For Fig. 13 by using the initial condition T(cid:9)100K at t(cid:9)1000h example,solarconcentratorscanbeprovidedthatfocusmoreenergy (insteadoft(cid:9)0h)andtheinitialconditionT(cid:9)300K(insteadof fluxandchargethethermal mass with moreenergythan wehave T(cid:9)100K)att(cid:9)0 hr.Inbothcases,thechangeintheminimum consideredinouranalyses,andtheradiantenergyreflectorcanbe temperatureislessthan0.5K. designedtobemoreefficient,yieldinglowereffectiveemissivities Itmightbesaferforsuitedastronautsandequipmenttoreducethe and further reducing heat loss. Likewise, if a small amount of maximumtemperaturetobeclosetotheskin-touchtemperaturelimit encapsulatedphasechangematerial(e.g.,paraffinwax)isbrought of318K.Table3showsthatwhenthepeakheatfluxisreducedto from Earth and incorporated in the thermal mass, greater energy 600W=m2whilekeepingtheloss-limitingshieldduringperiodsof storagecapacitywillbeenabledcoincidentwithreducedtemperature darkness, the maximum and minimum surface temperatures ranges.Whilethesefeatureshavenotbeenconsideredinourthermal achievedare320and273K,respectively.Wealsonotethatforthe analysis,it is neverthelessclearthattheywould provide improve- equatorialcase,iftheradiationshieldisusedapproximately100h mentstothefunctionalityofthethermalwadi. before sunset, then thewadi-surface temperature at sunset will be lowerthanthetouchtemperatureof318K.Thisstrategyisdifficultto simulateinthepolarcase,sincethedurationoflightedperiodsand Appendix:PeriodicSteadyState theonsetofdarknessareirregular. Whenthesolarfluxisperiodic,weexpectthesurfacetemperature Theeffectofbuildingthewaditoashallowerdepthinthelunar of the wadi to eventually achieve a periodic steady state. As polarenvironmentandrelaxingthethermaldiffusivitytofractional mentionedearlier,Jaeger[8]hasdevelopedamodeltocalculatethe valuesofthediffusivityofbasaltrockarealsopresentedinTable3. periodicsteady-statetemperatureofthesurfaceofthemoon.While Reducingthewadidepthreducestheminimumsurfacetemperature. Jaegerassumedthatthebulkthermalpropertiesofregolithandthe Forawadidepthof10cm,theminimumsurfacetemperatureisabout surface radiative properties are constant in space and time in the 241K.Whenthewadithermaldiffusivityisreducedbyafactorof10 calculationsthathereported,hismodelisactuallymoregeneraland from the basalt value, the minimum surface temperature is about canaccountforvariationsinthesurfaceradiativepropertiesduring 242K.Thesetemperaturesareabovethetolerancelimitof220Kfor thediurnalcycle.Therefore,weareabletocapturetheinfluenceofa electronicequipment. radiationshieldusedduringthelunarnightbyemployinghismodel. Anextensionofthemodel,toaccountforthermalpropertiesthatare varyinginspace(suchasinawadiofafinitedepthwithaninfinite T = 389 K amountofregolithbelowit),doesnotappeartobeavailableinthe 400 max literature. We therefore assume that the medium is infinitely deep 350 withconstantproperties.Theperiodicsteadystate,withtimeperiod 300 2t0,isdividedintoJ uniformintervalsj(cid:9)1,J.Thefundamental )K result derived by Jaeger is that if the surface temperature of the (Ts220500 Tmin = 261 K mtheedrieummaiinsimngaiinnttaeirnveadlsajt(cid:9)un2it,yJinintheveefiryrspteinritoedrv,athlejn(cid:9)at1staenaddyzesrtoatien, the average heat flux into the medium in the jth interval is 150 p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)k(cid:8)C =2(cid:11)t (cid:3)(cid:10),where p 0 j 1000 2000 30t0 0(h)4000 5000 6000 (cid:10) (cid:9)2p(cid:1)J(cid:1)(cid:1)(cid:4)p2J Z 1(cid:1)1(cid:4)e(cid:4)z2=J(cid:3)(cid:5)e(cid:4)(cid:1)J(cid:4)1(cid:3)z2=J(cid:4)e(cid:4)z2(cid:6)dz (A1) Fig.13 SameasFig.12butwithouttheheat-loss-limitingshield. 1 (cid:1)(cid:11)(cid:1)(cid:1)(cid:1) 0 z2(cid:1)1(cid:4)e(cid:4)z2(cid:3) BALASUBRAMANIAMETAL. 139 400 400 350 T = 371 K 350 T = 384 K 300 max max 300 )K )K (Ts250 (Ts250 200 200 150 Tmin = 215 K 150 Tmin = 230 K 500 1000 1500 2000 2500 500 1000 1500 2000 2500 t (h) t (h) a) b) Fig.A1 Periodicsteady-statesurfacetemperaturecalculatedemployingJaeger’smethod[8]foraninfinitelydeepbasaltlayerfora)sinusoidalheatflux andb)asquarewaveheatflux.Peakflux(cid:1)1300W=m2.(cid:1) (cid:1)0:9duringdaytimeforallcurves(bluecurve:(cid:1) (cid:1)0:9;greencurve:(cid:1) (cid:1)0:5;redcurve: 0 1 1 (cid:1) (cid:1)0:25). 1 andforj(cid:9)2,J, GlennResearchCenteratLewisFieldbyprovidingdatatousfrom p thenear-polarlunarilluminationmodelhedevelopedbasedonthe (cid:1)(cid:1)(cid:1) p(cid:1)(cid:1)(cid:1) p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) p(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:10)j(cid:9)2 J(cid:1) j(cid:10) j(cid:4)2(cid:4)2 j(cid:4)1(cid:3) digitalelevationmodeldiscussedpreviously.Wehopethatbyusing 2J Z 1e(cid:4)(cid:1)j(cid:4)1(cid:3)z2=J(cid:1)1(cid:4)e(cid:4)z2=J(cid:3)(cid:5)e(cid:4)(cid:1)J(cid:4)1(cid:3)z2=J(cid:4)e(cid:4)z2(cid:6) thisstandardizedmodel,theutilityofourresultscanbecompared (cid:4)p dz (A2) withotherfeaturesofthelunaroutpost. (cid:1)(cid:11)(cid:1)(cid:1)(cid:1) 0 z2(cid:1)1(cid:4)e(cid:4)z2(cid:3) IfthesurfacetemperaturehasvaluesT ;T ;... ;T forj(cid:9)1,J, thentheheatfluxintothemediuminthse1jths2intervalsiJs References s(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) [1] Wegeng,R.S.,Mankins,J.C.,Balasubramaniam,R.,Sacksteder,K., f (cid:9) k(cid:8)CpXJ T (cid:10) (A3) Gokoglu,S.A.,Sanders,G.B.,andTaylor,L.A.,“ThermalWadisin j 2(cid:11)t si j(cid:4)i(cid:10)1 SupportofLunarScienceandExploration,”6thInternationalEnergy 0 i(cid:9)1 Conversion Engineering Conference, AIAA Paper 2008-5632, (Notethat,inthisexpression,(cid:10) ;(cid:10) ;(cid:2)(cid:2)(cid:2)areequalto(cid:10) ;(cid:10) ;(cid:2)(cid:2)(cid:2)). July2008. Assumingthattheincidents0olar(cid:4)1fluxisasshowninJFigJ(cid:4).12,the [2] Balasubramaniam,R.,Wegeng,R.S.,Gokoglu,S.,Suzuki,N.,and surface heat flux can bewritten during the day and night, respec- SEaxcteknstdeedde-rD,uKr.a,ti“oAnnaLlyusnisarofESxoplalor-rHateioatnesd,”Th4e7rtmhalAWIAadAistAoeSrouspppaocret tively,as SciencesMeetingandExhibit,AIAAPaper2009-1339,Jan.2009. [3] Colozza,A.J.,“AnalysisofLunarRegolithThermalEnergyStorage,” f(cid:1)t(cid:3)(cid:9)(cid:1) q sin(cid:1)(cid:11)t=t (cid:3)(cid:4)(cid:2)(cid:3)(cid:1)T4(cid:4)T4(cid:3) for 0(cid:7)t(cid:7)t (A4) abs max 0 s a 0 NASACR189073,1991. [4] Heiken,G.H.,Vaniman,D.T.,andFrench,B.M.(eds.),TheLunar f(cid:1)t(cid:3)(cid:9)(cid:4)(cid:2)(cid:3)(cid:1)T4(cid:4)T4(cid:3) for t (cid:7)t(cid:7)2t (A5) Sourcebook: A User’s Guide to the Moon, Cambridge Univ. Press, s a 0 0 NewYork,1993. [5] Taylor,L.A.,andMeeks,T.T.,“MicrowaveSinteringofLunarSoil: Wehaveused(cid:2)(cid:9)(cid:2) inEq.(A4)and(cid:2)(cid:9)(cid:2) inEq.(A5)todetermine theinfluenceofara0diationshielduseddur1ingthenight.Once(cid:10)jis VProol.p1er8t,ieNs,oT.3h,eo2r0y0,5a,npdp.P1ra8c8t–ic1e9,”6.JournalofAerospaceEngineering, determined for each j, Eqs. (A3–A5) yield a set of nonlinear doi:10.1061/(ASCE)0893-1321(2005)18:3(188) afolgrejb(cid:9)rai1c,eJq.uWaetiounsesdfoNretwhetounn’skmnoewthnodtetmopsoerlvateutrheesnTosn1;liTnse2a;r.s.y.s;tTemsJ [6] TAanylOovr,eLrv.iAew.,aannddCEavrarileura,tWio.nD,”.,RIeIIs,o“uOrcxeysgoefnNPeroadr-uEcatirotnhoSnpathcee,MUonoinv:. ofequations,withJ(cid:9)1501(JaegerusedJ(cid:9)21[8]). OfArizona,Tucson,AZ,1993,pp.69–108. The periodic steady-state results for native regolith with a [7] Wesselink, A. J., “Heat Conductivity and the Nature of the Lunar sinusoidal heat flux are shown in Fig. 6. Similar results for an Surface Material,” Bulletin of the Astronomical Institutes of The infinitelydeepbasaltlayerinthepresenceofanighttimeradiation NetherlandsVol.10,1948,pp.351–363. [8] Jaeger,J.C.,“ConductionofHeatinaSolidwithPeriodicBoundary shieldareshowninFigs.A1aandA1bforasinusoidalandasquare Conditions, with an Application to the Surface Temperature of the waveheatflux,respectively.Resultsforafinitebasaltlayerafterfour Moon,”ProceedingsoftheCambridgePhilosophicalSociety(Mathe- diurnal cyclesare shownearlier inFig. 7. Comparing Figs. 7and maticalandPhysicalSciences),Vol.49,No.2,1953,pp.355–359. A1a,weseethatthemaximumsurfacetemperatureiselevated,and doi:10.1017/S0305004100028450 theminimumsurfacetemperatureisreducedforawadiofafinite [9] Mitchell,D.L.,anddePater,I.,“MicrowaveImagingofMercury’s depthcomparedwiththoseforaninfinitelydeepwadi.Notethatthe Thermal Emission at Wavelengths from 0.3 to 20.5 cm,” Icarus, abruptchangeinemissivitybetweendayandnightleadstokinksof Vol.110,No.1,1994,pp.2–32. smallmagnitudeinthesurfacetemperaturethatareevidentinboth doi:10.1006/icar.1994.1105 Figs.7andA1a.Thesearesuppressedforthesquarewaveheatflux [10] Vasavada, A. R., Paige, D. A., and Wood, S. E., “Near-Surface resultsinFig.A1b,aswellasforafinite-depthbasaltwadiinFig.8. TemperaturesonMercuryandtheMoonandtheStabilityofPolarIce Deposits,”Icarus,Vol.141,No.2,1999,pp.179–193. doi:10.1006/icar.1999.6175 Acknowledgments [11] Tewarson,A.,“GenerationofHeatandChemicalCompoundsinFires,” TheSFPEHandbookofFireProtectionEngineering,3rded.,National ThisworkwassupportedbytheDirectorateIntegrationOfficeof FireProtectionAssoc.,Quincy,MA,2002,pp.3–161. theNASAExplorationSystemsMissionDirectorate,forwhichwe [12] Fincannon,H.J.,“LunarPolarIlluminationforPowerAnalysis,”6th are very grateful. The authors also gratefully acknowledge the International Energy Conversion Engineering Conference, AIAA valuablecontributionsmadebyH.J.FincannonofNASAJohnH. Paper2008-5631,July2008.