Table Of ContentAppliedThermalEngineering119(2017)541–552
ContentslistsavailableatScienceDirect
Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Optimum design of ablative thermal protection systems for atmospheric
entry vehicles
Aniello Riccioa, Francesco Raimondoa, Andrea Sellittoa, Valerio Carandenteb,⇑, Roberto Sciglianob,
Domenico Tescioneb
aDepartmentofIndustrialandInformationEngineering,SecondUniversityofNaples,Italy
bStructuresandMaterialsDepartment,ItalianAerospaceResearchCentre(CIRA),Italy
h i g h l i g h t s
(cid:1)ImplementaFEMbasedmodelforthethermalanalysisofablativeTPSs.
(cid:1)Presentanoptimizationprocedureforthedesignofablativeheatshields.
(cid:1)ApplythenumericalmethodtotheablativeTPSofthereentrycapsuleStardust.
a r t i c l e i n f o a b s t r a c t
Articlehistory: The Thermal Protection System (TPS) provides spacecrafts entering the atmosphere with the thermal
Received8January2016 insulationfromtheaerothermodynamicheating.Thedesignofsuchasubsystemisverycritical,consid-
Revised3March2017 eringthatitsdamagecanleadtoacatastrophicfailureofthewholeentrysystem,inparticularifablative
Accepted11March2017
materialsareconsidered.InordertodesignanablativeTPS,infact,areliablenumericalprocedure,ableto
Availableonline20March2017
computesurfacerecessionrate,pyrolysisandinternaltemperaturehistoriesundersevereheatingcondi-
tions,isnecessary.Indeed,theTPSneedstobesizedtoeffectivelyshieldthespacecraftfromthehighheat
Keywords:
fluxesactingduringtheatmosphericentryphase.Atthesametime,itsweighthastobetheminimum
TPS
valueabletoguaranteeasuitableprotection.
Ablativematerials
Thisarticleaimstodescribeanoptimizationprocedureforthedesignofablativeheatshields.Inpar-
Thermalanalysis
FEM ticular,inthepresentwork,thenumericalmethodisappliedtotheablativeTPSofthehypersonicreentry
Optimization capsuleStardust.
(cid:1)2017ElsevierLtd.Allrightsreserved.
1.Introduction describethisphenomenonarestillindevelopment[1,2].Thecom-
plexityofthephenomenaunderconsiderationsisremarkedbythe
Duringtheatmosphereentry,hypersonicvehiclesaresubjected unsatisfying analytical methods proposed [3,4]. Moreover, in the
to strong shocks, equilibrium or non-equilibrium gas chemistry, experimental field, tests are not easy to implement considering
large heat fluxes, and, as consequence, very high temperatures thecomplexenvironmentalconditionsencountered.Thusfar,the
are reached on the structure. Those conditions require a proper mostcompletemethodologyfordealingwiththeproblemsrelated
designed Thermal Protection System (TPS). For very high entry to ablation is the numerical resolution of differential equations
speeds, in particular, the use of ablative material is mandatory. governingthephenomenaofinterest.Forthispurpose,onepossi-
TodesignanablativeTPS,areliablenumericalprocedureisneeded bleapproachassumesthatthesoliddecomposesifacriticaltem-
to compute surface recession rate, pyrolysis effects, and internal perature is reached. This critical temperature is a material
temperaturehistoriesundersevere heatingconditions. Ascanbe propertyanditisindependentfromtheincomingheatflow.Afur-
seenfromahistoricalliteratureoverview,consideringthecomplex ther approach considers the chemical reactions in the char layer
phenomenologyoftheablativeprocess,numericalmodelsthatcan accordingtothe‘‘frozenchemistry”modelorthe‘‘chemicalequi-
librium” model [5]. Despite the wide amount of research carried
outsofar,thedesignmethodologiesofablativeheatshieldsneed
⇑ Correspondingauthorat:StructuresandMaterialsDepartment,ItalianAero- not-negligible improvements. A fundamental aspect that these
spaceResearchCentre(CIRA),ViaMaiorisesnc,81043Capua,Italy. methodshavetoconsiderisrelatedtothefactthattheTPSusually
E-mailaddress:[email protected](V.Carandente).
http://dx.doi.org/10.1016/j.applthermaleng.2017.03.053
1359-4311/(cid:1)2017ElsevierLtd.Allrightsreserved.
542 A.Riccioetal./AppliedThermalEngineering119(2017)541–552
representsalargefractionofthetotalspacecraftvolume[6].The areencountered,butreactionsareexclusivelylocalizedonthesur-
objectiveofthisworkistoestimatetheheatshieldminimumvol- faceofthematerial[13].Inconclusion,ablationperformsitsfunc-
ume able to keep the mission requirements for a blunt-nosed tion of thermal barrier through the dissipation of the incoming
sphere-coniccapsule.ThecapsuleinexamisinparticulartheStar- thermalenergythankstothe‘‘sacrifice”ofapartoftheprotecting
dustprobe.StardustmissionwasthefirstU.S.missionsolelyded- material,whichabsorbsaconsiderableamountofheatand,atthe
icatedtoacometstudyandwasthefirsttoreturntoEarthin2006 sametime,createsabarriereffectduetothegeneratedgasflow.A
cosmicdustandsamplesfromawell-preservedcometcalledWild- furtherclassificationcanbemadeaccordingtotheoperatingper-
2 [7]. The heat shield of this probe was composed by a material formanceof the surface layer: ‘‘melting” and ‘‘non-melting” [14].
calledPICA(PhenolicImpregnatedCarbonAblator),whichispart Thefirsttypeconsistsofthermoplasticmaterials:aliquidsurface
of the Lightweight Ceramic Ablators (LCAs) family. This material layerproducedbyfusionisremovedimmediatelybytheaerody-
consists of a commercially available middle density carbon fiber namic shear stresses. The non-melting materials can be further
matrixsubstrateimpregnatedwithaphenolicresin[8]. divided into two sub-categories: ‘‘Low Temperatures Ablators”
The work is organized as follows. In Section 2 the theoretical (LTA) and ‘‘High Temperatures Ablators” (HTA). The carbon-
background for the ablative phenomena, ablative classification carbon and carbon-silicon ablatives are examples of HTA, whose
and optimization process are presented. Section 3 reports the main characteristic is to preserve mechanical properties even at
numericalmodelimplemented.InSection4,theresultsareillus- hightemperatures.FortheLTAs,thechemicalablationispreceded
trated and properly commented.Section 5 reports the main con- bythedeteriorationofthemechanicalpropertieswithincreasing
clusionsofthework. temperature, while the thermal ablation (sublimation) becomes
appreciable only from around 2500K. This type of material is
mainly used in return missions where very high heat fluxes are
2.Theoreticalbackground reached.AnexampleofLTAistheablativefamilyconsistingofcar-
bonfibersandphenolicresins,suchasPICA.Theyarerecognizable
TheTPSisasubsystemprovidingprotectionandinsulationtoa for their excellent ability to form char and to ignite pyrolysisfor
spacecraft from the aerodynamic heating due to high enthalpy highvaluesoftheheatflux.
flows encountered in the atmospheric entry phase at hypersonic Thechoiceofanablativematerialrequiresacarefulevaluation
speeds. Thermal protection systems for spacecrafts make use of oftheentryconditions,lookingforthebestcompromisebetween
different physicochemical mechanisms for their thermal energy ablation effectiveness and heat insulation performance. In this
management:dissipation,cooling,insulationandablation.Leaving wayitispossibletoidentifythemostfittingmaterialforapartic-
asidetheisolationandcoolingmechanisms,theheatshieldscanbe ularmission.
dividedintotwobigcategories:reusableandablative. ThePICAisinparticulararelativelynewmaterial,developedat
ThereusableTPSsdonotchangethemassandcompositionof NASA Ames Research Center, and is characterized by a lower-
materialsduringtheexposuretotheaerothermodynamicenviron- density value than other ablative materials, in particular with
ment.Theireffectivenessisbasedpartiallyontheabsorptionand respecttothecarbonphenolic,whilemaintainingahighablative
partiallyonthere-radiationoftheincomingheatflux.Thisfeature capacity for high heat fluxes [8]. Moreover, it presents a lower
guarantees the possibility to reuse these materials for different thermal conductivity of other ablative materials for same levels
missions. Such systems, however, are utilized for relatively low of incoming heat fluxes [8]. PICA is made with a thermosetting
thermal loads. The main characteristic of such heat shields is to resin, a commercial phenol-formaldehyde (SC 1008) [15]. This
re-radiate a large part of the energy coming from the convective resin acts as a matrix for carbon fibers chopped as shown in
andtheradiativecomponentsoftheaerothermodynamicheatflux, Fig.1[16].
andtoaccommodateonlyarelativelysmallamountofenergyby Itpresentsafinaldensityof0.22–0.32g/cm3andischaracter-
conduction within the material. Another key factor is linked to izedbythefollowingmasscomposition:92%carbon,4.9%oxygen,
the use of insulating materials. They are often inorganic because 2.2% hydrogen and 0.9% nitrogen [17]. Its ablation rate increases
of their low thermal conductivity and the need to minimize the withtheincomingheatflowandthechemicalspeciespredominant
overallmassoftheheatshield[9]. inthecharareCandCO.Thehighporositypossessedbysuchcom-
Theterm‘‘ablative”referstoanumberofphysicochemicalpro- positeisthemainreasonforthelowdensityandconductivityval-
cesses, including vaporization, chemical reactions and erosion, ues.Completepropertiesofthismaterialarereportedin[8].
leadingtothesurfacematerialremoval.Forthisclassofmaterials,
the incoming thermal loads are restrained thanks to the phase 2.1.PhysicalphenomenainanablativeTPS
changeofthematerial[10].Aclassificationfromaphenomenolog-
ical point of view divides ablative materials in two classes: non- The ablation process involves multiple physical and chemical
charringandcharring[11,12].Thecharringtypeablativesarebest phenomena, in particular for charring type materials considered
usedforatmosphericentrymissions.Inthisclass,oncetheresinis in this study. During hypersonic flight conditions, the formation
heated,adecompositionphenomenoncalledpyrolysistakesplace ofthebowshock,interactingwiththeboundarylayer,leadstoa
inthebulkofthematerial.Thepyrolysisgeneratesgaseousprod- temperature increase in the proximity of the vehicle surface. In
ucts,usuallyhydrocarbons,whichblowthroughtheheatedsurface additiontoconvectiveheating,andradiationproducesnonnegligi-
into the boundary layer, with the consequent reduction of the bleeffects,inparticularatsuper-orbitalentryspeeds.Thisresult-
effectiveconvectivefluxaffectingthespacecraft.Thepyrolysisof ing heat flux is partially transferred by conduction within the
the resin also produces a carbonaceous residue (char), which is materialoftheshield.Forcharringtypematerials,decomposition
depositedonthecompositefibrousreinforcement. Moreover,the reactionsoriginategaseousproducts,leavingacarbonizedporous
chemical reactions between the material at the surface and the residue. The pyrolysis gases originated by the underlying virgin
boundary layer species can lead to a recession of the aforesaid material,generateegaseousflowthat,passingthroughtheporous
material surface. These reactions can be endothermic (vaporiza- structure of the char, blows in the boundary layer. One of the
tion, sublimation) or exothermic (oxidation). The interaction of effectsofthesegasesistoreducetheeffectiveconvectiveheatflow
theablativematerialsand theirproductswiththeambientgases actingontheTPS.Ontheotherhand,thesegasescantriggernew
is much more complex than here described. On the other hand, corrosiveprocessesduetotheirinteractionwithcarbonresidues.
inanon-charringtypeablative,nointernalpyrolysisphenomena The carbonaceous residues may endure oxidation processes due
A.Riccioetal./AppliedThermalEngineering119(2017)541–552 543
Fig.1. MicrographsofPICAatdifferentmagnifications[16].
tothesurroundingfluidand,finally,willalsobepresentmechan-
icalerosionphenomena.Theconsequenceofallthesephenomena
isathicknessreductionoftheablativeandshapevariationofthe
heatshield.Inconclusion,theablationphenomenonconsistsinto
a number of physicochemical interaction between the TPS and
theexternalenvironment.
2.2.Numericalmodelforthermalanalyses
Inthepresentworkthecomplexphysicochemicalphenomena
occurring in ablative TPSs are numerically modelled assuming a
numberofsimplifyinghypotheses,widelyimplementedinseveral
works reported in literature [17–20]. As already discussed, PICA
material is considered. Figs. 2–4 report thermal experimentally
measuredproperties[8].
The material decomposition, or the density variation, is com-
putedexplicitlyasamaterialproperty.Thematerialdensityvaria-
tion has been implemented applying the Arrhenius relation for
each ithcomponentof thePICAas in [18,19]. In Eq. (1), Band w
q q
areconstants,Eistheactivationenergy, and arethedensi-
0 ch
tiesofthevirginandcharmaterial.
@@qti¼(cid:3)(cid:1)Bi(cid:4)e(cid:3)RETi(cid:3)(cid:4)q0i(cid:4)(cid:4)qiq(cid:3)qci(cid:5)wii¼A;B;C ð1Þ Fig.3. PICAspecificheatasafunctionofthematerialtemperature[8].
0i
ThevaluesassumedforcalculationsarereportedinTable1.
Fig.4. PICAeffectiveheatofablationasafunctionoftheconvectiveheatfluxacting
Fig.2. PICAthermalconductivityasafunctionofthematerialtemperature[8]. onthematerial[8].
544 A.Riccioetal./AppliedThermalEngineering119(2017)541–552
Table1
ConstantparametersimplementedinEq.(1).
Component B(1/s) W(–) E/R(K) q0(kg/m3) qc(kg/m3)
A 1.40E4 3 8555.6 229 0
B 4.48E9 3 20444.4 972 792
C 0 0 0 160 160
q r
The material density has been therefore obtained as the cient,calculatedonthebasisofthemodelreportedin[13], the
weightedmeanofitscomponents(inEq.(2),Cistheresinvolume Stefan-Boltzmann constant, e the material emissivity, T1 the
fraction,AandBrepresenttheresincomponentsandCthecarbon asymptoticflowtemperature.
Fiberform).Theseparametershavebeensetaccordingto[18,20]. Asfarasthematerialrecessionisconcerned,theassumptionof
steady-stateablationhasbeenconsidered.Ateachinstantoftime
q¼CðqA(cid:3)qBÞþð1(cid:3)CÞqC ð2Þ theablationratehasbeenestimatedbyEq.(9),inwhichQ⁄isthe
socalledheatofablation,qisthematerialdensityandq_ isthe
The internal decomposition transforms part of the solid in hw
hotwallconvectiveheatflux,evaluatedfromEq.(8)as:
pyrolysis gas. Moreover, thanks to the hypothesis of mono-
(cid:4) (cid:5)
dwiimthenthseiovniarlgiflnomw,aqteuraiasli-asrteaatisc,tahnedmimaspseflrmoweaobfiltihtyeopfytrhoelyisnistegrafascies q_hw¼q_cw 1(cid:3)cHpTw ð8Þ
tot
connectedtothedecompositionbythesimplerelationreportedas
Eq.(3). q_
@m_ @q s_¼qQhw(cid:5) ð9Þ
g ¼ ð3Þ
@x @t
According to the structure scheme reported in Fig. 5, the in- 2.3.Verificationandvalidationofthenumericalmodel
depthtemperatureresponse ofthematerial has beencarried out
setting as boundary conditions the expressions reported in Eqs. Ashighlightedin[22]themainobjectiveofthewell-established
(4–7).Inparticular,Eq. (4)refers totheexternalsurfaceexposed Verification and Validation (V&V) methodology is to progress in
to the aerothermodynamic environment (y=hi). This approach the development of predictive numerical models. Oberkampf
hasbeenwidelydiscussedin[21]. et al. [23] published a very detailed review paper about V&V. In
(cid:4) (cid:5) (cid:4) (cid:5) (cid:1) (cid:3) thisreferencepaper,theauthorsstated:‘‘Howshouldconfidence
q_ ¼(cid:3)k @T ¼q_ 1(cid:3)cpTw / (cid:3)re T4 (cid:3)T4 ð4Þ in modelling and simulation be critically assessed? Verification
in @y y¼hi cw Htot blow w 1 andValidation(V&V)ofcomputationalsimulationsaretheprimary
methodsforbuildingandquantifyingthisconfidence’’.Inapaper
(cid:4) (cid:5)
q_ ¼(cid:3)k @@Ty ¼0 ð5Þ dmeedcihcaatneidcstmootdheels,reTlhaaticoknerb[e2t4w]epenresuenncteedrtaailnstoietshiasnpdroVc&esVs ffoorr
y¼0 the‘‘developmentofmodelsthatcanbeusedtomakeengineering
(cid:4) (cid:5) predictionswithhighconfidence’’.
q_ ¼(cid:3)k @T ¼0 ð6Þ Theverificationphasedealswithonlynumericalaspects.Dur-
@x x¼0 ingtheverificationstage,onedistinguishesbetweencodeverifica-
tion and solution verification. Code verification leads to the
(cid:4) (cid:5)
q_ ¼(cid:3)k @@Tx ¼0 ð7Þ ifidleens,tiifinctahtieonnuomfperriocgarlaamlgmorinitghmersr,oirnsitnhethceominppiuletrasnadnoduotppuertadtiantga
x¼b systems. The present study has been performed using Matlab
InEq.(4),q_ istheincomingnetheatflux,q_ thecoldwallcon- [25] and APDL Ansys languages [26]. Therefore, code verification
in cw
vectiveheatflux,c isthespecificheatatconstantpressure,T the is assumed to be completed. As far as solution verification is
p w
walltemperature,H thetotalenthalpy,/ theblowingcoeffi- concerned, the main source of errors is due to the spatial
tot blow
Fig.5. Structureschemeandheatfluxesforcharringablative.
A.Riccioetal./AppliedThermalEngineering119(2017)541–552 545
discretization.Inordertoensurethetimeandmesh-independency Table3
afullconvergenceanalysishasbeenperformedonthemodel[21]. Numberofelementsinradialandaxialdirectionsforthedifferentanalyzedcases.
In addition, the model has been also validated crosschecking Case# Numberofelements
numerical results with analogous outcomes of experimental and
Axialdirection(n) Radialdirection(m)
numericaltestsreportedinliterature[21,27].
1 10 20
For reader convenience, we briefly report, what is extensively
2 25 45
presented in [21], i.e. a full convergence analysis realized repro- 3 50 95
ducing the same experimental conditions adopted by Covington 4 75 135
[27]. A cylindrical material sample geometry with a radius of 5 100 185
5.08cm and an initial height of 2.74cm has been taken into
account with a material emissivity of 0.9 for the evaluation of
the re-radiated heat flux. Reference test conditions are reported
inTable2,wheret isthetransientanalysisduration.
max
The mesh has been realized using a growing number of ele-
mentsinradialandaxialdirections,accordingtovaluesreported
inTable3.
Twocomparisonshavebeentakenintoaccounttoevaluatethe
modelrobustness.Thefirstisacomparisonintermsoferrorofpre-
dicted final thickness and the second is in terms of final surface
temperature.
Inconclusion,resultsintermsofbulkandsurfacetemperature
can be considered sufficiently accurate even for a relatively low
number of elements (Fig. 6), while a larger number of elements
(Fig.7)isrequiredtoincreaseaccuracyonthediscreteevaluation
of the material thickness and to reduce the surface temperature
Fig.6. Erroronthefinalsurfacetemperaturefortheanalyzedcases.
dropconnectedwiththeelementkilling.
Finally, the results obtained with the n=100 have been com-
pared with both numerical and experimental results presented
by Covington in [27] showing a very satisfying results cross-
checking.
2.4.Optimizationprocess
Ageneticoptimizationprocedurehasbeenappliedtothether-
malprotectionshieldinordertoevaluatetheminimumweightfor
thecapsuleTPSabletoguaranteeaneffectivethermalprotection.
GeneticAlgorithms(GAs)fallwithinnaturalmethodsandcanbe
used to solve problems of research and optimization. GAs are
applicable to the resolution of a wide variety of problems not
appropriate for classic optimization algorithms, including those
in which the objective function is discontinuous, non-
differentiable,stochastic,orstronglynonlinear[28–30].
GAsworkwithapopulationofindividuals,eachofwhichrepre- Fig.7. Erroronthefinalthicknessfortheanalyzedcases.
sentsapossiblesolutionoftheanalyzedproblem.Eachindividual
is assigned a score, called ‘‘fitness”, representing the goodness of
3.Numericalmodelling
thesolutiontotheproblem.Thespecimenwiththebestfeatures
(i.e.thosewithhigherfitnessvalues),havegreateropportunityto
CalculatingthethermalresponseofanentryvehicleTPSisvery
‘‘mate”withotherindividualsofthepopulationthatwillgenerate
important for its correct sizing. On one hand the instantaneous
otherspecimensofthepopulation.Iftheresultingindividualshave
materialrecessionratehastobeestimatedand,ontheotherhand,
asufficientlylargefitnessvalue,theywillreplacetheworstindi-
thein-depthtemperatureresponseoftheTPShastobecomputed
viduals of the previous generation. In such a way, in the further
inordertoevaluatetheamountofmaterialrequiredasinsulation
generation there will be a higher number of characteristics pos-
to keep the bond-line temperature (i.e., the temperature of the
sessedbythebestindividualsofthepreviousgeneration.Thiswill
materialsurfacenotdirectlyexposedtotheflow)belowaspecified
ensurethatthebestfeatureswillbepropagatedtomoreindividu-
limit.
alsinthepopulation.Moreover,featuresnotincludedamongthose
Inordertoappreciatethematerialablationoftheexternalsur-
of the original genetic species can sporadically be created by the
face of the heat shield, a Finite Element Analysis has been per-
introductionofrandommutationsinthegenerationofnewindi-
formed within this work. An advanced finite element model has
viduals. If the optimization is well designed, the population will
beenintroducedtopredictthefinalthicknessandshapeoftheheat
convergetoanoptimalsolutionoftheproblem.
shield,attheendoftheatmosphericentryphase.
Table2
Testconditionformodelverificationandvalidation. 3.1.Analyzedgeometry
Q_cw(MW/m2) p(atm) Htot(MJ/kg) tmax(s) In this work the Stardust probe has been analyzed. It has six
5.80 0.450 29.5 15 majorcomponents:a heatshield,abackshell,a samplecanister,
546 A.Riccioetal./AppliedThermalEngineering119(2017)541–552
Fig.8. SchematicdiagramoftheStardustSampleReturnCapsule(SRC)[31].
Fig.10. ConvectiveandradiativeheatfluxesprofilesestimatedalongtheSRCentry
trajectory.
Fig.9. GeometryoftheanalyzedcapsuleinFEmodel.
a parachute system and avionics. The Stardust Return Capsule
(SRC) is a short truncated cone, having a 0.811m diameter,
0.499mhighandatotalmassof45.8kg,includingtheparachute
system[7].InFig.8a2DschematicdiagramoftheSRCisreported.
Fig.11. Designvariables.
The geometry described above was discretized in the axisym-
metricmodeldepictedinFig.9.
TheAnsys(cid:3)MechanicalFEcode[26]hasbeenusedtoapplythe
itwasnecessarytonumericallysolvetheequationcharacterizing
numerical procedure hereinafter discussed to the computational
the dynamics of the entry capsule [32], taking advantage of the
mesh shown in Fig. 9. In particular, 2D elements PLANE55 have
Runge-Kuttamethod[33].Oncethedataofthere-entrytrajectory
been selected to perform thermal calculations. Each element has
have been calculated, the convective heating at the stagnation-
four nodes with a single degree of freedom, the temperature, at
point of axisymmetric blunt bodies has been estimated by
eachnode.
Sutton-Graves formulation [34]. This model is characterized by
q
Eq. (10), according to [34], where V is the velocity in m/s, is
3.2.Ablationandoptimizationmodellingboundaryconditions theatmosphericdensityinkg/m3,R isthenoseradiusinmand
n
1.73(cid:4)10(cid:3)4isaconstantderivedfortheEarthatmosphere.
rffiffiffiffiffi
The cold wall convective heat flux, the total enthalpy and the q
radiation flux profiles along the entry trajectory of the SRC have q_conv ¼1:73(cid:4)10(cid:3)4(cid:4)V3 R ð10Þ
beenimportedasexternalarrays.Inordertocalculatethesearrays, n
A.Riccioetal./AppliedThermalEngineering119(2017)541–552 547
Fig.12. ObjectfunctionVsIDnumber.
Fig.13. ObjectfunctionVsIDnumber–DetailofthebestID.
However,convectiveheatfluxisnottheonlythermalloadact- A preliminary estimation, valid in hypersonic regime for high
ingonare-enteringcapsule:radiativeheatfluxesarealsosignifi- Machnumbers,ofthenon-dimensionalheatfluxdistributionalong
cant for the thermal protection system. Among the different the capsule surface has been obtained implementing the Lees’
engineering models for radiative heating available in literature, theory [36]. The corresponding dimensional values have been
theoneprovidedbyTauberandSutton[35]hasbeenconsidered obtainedmultiplyingthenon-dimensionalheatfluxspatialdistri-
in the present work. Tauber and Sutton’s engineering correlation butionbythetemporalvariationofthestagnation-pointheatflux
is reported in Eq. (11), being C a constant that depends on the alongthetrajectory.
atmosphere and f(V) tabulated values, functions of both flight Once these arrays containing the external loads have been
velocitiesVandatmosphericcomposition.Theexponentsaandb imported,thenetheatfluxisappliedateachinstantoftime,taking
can be either constants or functions of density and free stream intoaccountthesurfacetemperaturecalculatedassolutionofthe
velocity,asspecifiedin[35]. thermaltransientanalysisattheprevioustimeinstant.Forsurface
q_ ¼CRaqbfðVÞ ð11Þ temperatureshigherthanathresholdvalue,theablationrates_has
rad n been estimated by Eq. (9), according to [8,18,21,34]. Then, the
InFig.10,convectiveandradiativeheatfluxesatthestagnation thicknessoftheheatshieldisreducedofthequantitys_(cid:4)dt.When
pointestimatedalongtheSRCentrytrajectoryareshown. thethicknessreductionissuchthatmorethanahalfofanelement
548 A.Riccioetal./AppliedThermalEngineering119(2017)541–552
Fig.14. Peakbond-linetemperatureVsID.
Fig.15. ScatterMatrix.
thickness is not included in the computational domain, this ele- Thisnumericalmodelhasbeenimplementedinanoptimization
ment is disabled by the ‘‘ekill” function of the Ansys(cid:3) code [26]. tool,allowingtoobtaintheconfigurationwiththelowestablative
Oncetheanalysisiscompleted,thevalueofthenodestemperature volume,whileobservingthetemperatureconditionsofthebond-
onthebondlinearecomparedwiththedesignlimits[37,38]. line[37,38].Inparticular,thisoptimizationprocessconsidersthe
A.Riccioetal./AppliedThermalEngineering119(2017)541–552 549
Fig.16a. InitialcomputationalmeshforID0. Fig.17a. FinalcomputationalmeshforID0.
Fig.16b. InitialcomputationalmeshforID1336.
Fig.17b. FinalcomputationalmeshforID1336.
volumeoftheablativeshieldastheobjectfunction,whilethecon-
straintfunctionisthemaximumtemperatureonthebond-line.An
initialpopulationof50individualshasbeenconsideredforthefirst Each individual of the population is described with six design
generation,andatotalof50generationshavebeenanalyzed.The variables. These variables represent the value of thickness of the
choiceofthefirst50individualsisbasedona randomsequence, PICAheatshield,measuredstartingfromthebondlinetotheouter
fillingrandomly,withauniformdistribution,thedesignspace[39]. surface,insevencontrolpoints,asshowninFig.11.
550 A.Riccioetal./AppliedThermalEngineering119(2017)541–552
Fig.19a. Finaltemperatureonthebond-linenodesforID0.
Fig.18a. FinalnodaltemperaturedistributionforID0.
Fig.19b. Finaltemperatureonthebond-linenodesforID1336.
Astheindividualskeepgrowing,theytendtoconcentratealong
acertainpathresultinginastabilizationoftheobjectivefunction
(seeFig.12).Fig.13isadetailofFig.12clearlyshowingthebest
estimatedindividual.
InFig.14thevaluesofmaximumtemperatureonthebond-line
(constraintfunction)foreachIDareshown.Inthiscasethevalues
ofthepeaktemperaturetendstoapproachthelimitsetto394K
Fig.18b. FinalnodaltemperaturedistributionforID1336.
[37,38],indicatedintheFig.14bythedottedline.
4.Numericalresults Fig.15showsthescattermatrix,representingtheinfluenceof
optimizationparametersontheanalysis.
In this section, the results of the FE model optimization are Scatter matrix shows how the variable sp9 (see Fig. 12) has
shown.Inparticular,inFig.12,theobjectfunctionvariationversus moreinfluencewithrespecttotheothervariablesontheobjective
theDesignIDisshown.Theminimumoftheobjectfunctionoccurs function. The ID ‘‘0” refers to the initial configuration while ID
fortheindividual‘‘1336”. ‘‘1336” is the configuration with the lowest volume. In Figs. 16a
