Ci a ion: Ga mendia, I.; Anglada, E.
In luence o he Measu emen s
Unce ain ies in he Co ela ion o
Spacec a The mal Models agains
The mal Resul s. Ae ospace 2022,9,
821. h ps://doi.o g/10.3390/
ae ospace9120821
Academic Edi o : Paolo To o a
Recei ed: 26 Oc obe 2022
Accep ed: 12 Decembe 2022
Published: 14 Decembe 2022
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ae ospace
A icle
In luence o he Measu emen s Unce ain ies in he Co ela ion
o Spacec a The mal Models agains The mal Resul s
Iñaki Ga mendia 1,* and E a Anglada 2
1Mechanical Enginee ing Depa men , Enginee ing School o Gipuzkoa, Uni e si y o he Basque Coun y
UPV/EHU, Plaza de Eu opa, 1, E-20018 Donos ia-San Sebas ián, Spain
2TECNALIA, Basque Resea ch and Technology Alliance (BRTA), Mikele egi Pasealekua, 2,
E-20009 Donos ia-San Sebas ián, Spain
*Co espondence: [email p o ec ed]
Abs ac :
G ound he mal es s a e always manda o y be o e any space mission is lown in o space.
The collec ed esul s o hese es s a e mainly empe a u es o he di e en pa s o he spacec a
(nodes) o di e en mission scena ios. The measu ed empe a u es always show di e ences wi h
he expec ed alues coming om he compu e he mal ma hema ical models. The o igin o hese
di e ences is pa ially ela ed o he inhe en e o coming om physical measu emen s. The he mal
pa ame e s ha compose he compu e he mal ma hema ical models mus always be co ela ed
wi h he esul s coming om es s. This pape s udies, h ough h ee he mal models, he di icul ies
ha a e ound in he co ela ion p ocess when he measu ed empe a u es each a ce ain le el o
e o . The mal pa ame e s become mo e di icul o be iden i ied when he measu emen e o le el
inc eases. Howe e , he empe a u e ields ob ained wi h hese poo he mal pa ame e s a e good
enough o he mission he mal analysis. Se e al e o le els, di e en load cases and co ela ion o
s eady-s a e and ansien cases a e s udied o p obe hese indings.
Keywo ds:
model co ela ion; he mal ma hema ical model; measu emen s e o ; he mal con ol;
g adien based algo i hm
1. In oduc ion
The The mal Con ol Subsys em is a undamen al componen o he enginee ing
design wo k o any spacec a . I s pu pose is clea : o main ain all he componen s o he
spacec a and payloads inside he sa e ange o empe a u es de ised o he mission. The
design ask equi es o know he empe a u es dis ibu ions expec ed o he spacec a
o any mission scena io (cold case, ho case, anspo , expe imen a ion, e c.) [
1
–
5
]. This
ask equi es de ining he hea inpu s ha he spacec a unde goes om ex e nal sou ces
(sola , in a ed, albedo, e c.) as well as he hea p oduced inside he spacec a (elec onics,
hea e s, e c.).
To be able o p edic hese empe a u es dis ibu ions, The mal Ma hema ical Models
(TMM) a e buil , conside ing he geome y, he ma e ials he mal p ope ies, he p esence o
he mal insula ion (Mul i-Laye Insula ion, MLI), he mos a s, hea e s, e c. The empe a u e
o each node, in which he TMM is disc e ized, is gi en by Equa ion (1), whe e nis he
numbe o nodes o he TMM, GL(i, j) is he conduc i e conduc ance (W/m) be ween nodes
iand j,
σ
is he S e an–Bol zmann cons an (5.67
·×
10
−8
W/(m
2·
K
4
)), GR(i, j) is he adia i e
conduc ance (m
2
) be ween nodes iand j,T
i
and T
j
a e he empe a u es (K) o nodes iand
j,
MiCi
is he p oduc o he inode mass (kg) imes he hea capaci y (J/(kg
·
K)) and q
i
is
he powe (W) ha en e s in o node i. The subsc ip s iand jgo om 1 o n. I is usual
o call he mal ine ia o he p oduc
MiCi
as i desc ibes he “opposi ion” o change he
empe a u e o inode when subjec ed o a powe inpu . The e o e, he solu ion o he se
o nonlinea equa ions co esponding o he nodes sol es he TMM. This p ocess is called
he The mal Lumped Pa ame e me hod (TLP) and p o ides he empe a u e dis ibu ion
Ae ospace 2022,9, 821. h ps://doi.o g/10.3390/ae ospace9120821 h ps://www.mdpi.com/jou nal/ae ospace
Ae ospace 2022,9, 821 2 o 13
o he di e en pa s o he spacec a (nodes), as well as he hea lows (W) be ween he
nodes [
6
,
7
]. The TMM is sol ed o he di e en load cases and scena ios, in o de o p edic
he he mal beha io in e e y si ua ion expec ed du ing he mission.
j=n
∑
j=1
GL(i,j)Ti−Tj+
j=n
∑
j=1
σGR(i,j)T4
i−T4
j+MiCi
dTi
d =qi(1)
As any ma hema ical model, he TMM ep esen s eali y in an app oxima e way. The
empe a u es p edic ed by he TLP me hod will be co ec i he simpli ying assump ions
made o build he TMM we e app op ia e and easonable. O he wise, p edic ed esul s
will be poo . Fo his eason, he TMMs mus be alida ed.
Conside ing he ideas men ioned in he p e ious pa ag aphs, he need o he mal
es s on g ound is clea . Thei pu pose is o ep oduce on g ound he he mal scena ios
ha he spacec a will ind in o bi . These es s a he labo a o y scale will p oduce a se o
measu ed empe a u es ha could be compa ed wi h he p edic ed empe a u es calcula ed
wi h he TMMs. I he empe a u es measu ed and he empe a u es p edic ed a e close
enough, he TMM ep esen s well he eali y, and he he mal enginee s ha e a ool o
p edic wi h eliabili y o he he mal scena ios ha could no be es ed in he labo a o y.
Howe e , he e is always di e ences be ween he measu ed empe a u es in he
labo a o y and he calcula ed ones wi h he TMMs. These di e ences could be a ibu ed
o wo di e en sou ces. On he one hand, he TMMs cons uc ion is an app oxima e
p ocess and simpli ying assump ions o e en e o s could be done. On he o he hand,
measu emen de ices and, in gene al, measu emen echniques a e, by hemsel es, an
impe ec p ocess, which implies unce ain ies and e o s.
The e o e, he TMMs mus be co ela ed. Tha is, he he mal pa ame e s ha compose
he TMM (GLs, GRs and MCs) mus be modi ied in o de o p edic empe a u es as close
as possible o he measu ed ones. Much wo k has al eady been de o ed o his ask by
esea che s and di e en me hods and app oaches ha e been used, bu a ully ope a ional
semiau oma ic solu ion o he p oblem has no been achie ed ye . Fo ins ance, Klemen [
8
]
used quasi-New on algo i hms o he class de ined by C. G. B oyden and s a ed ha
his app oach educes he numbe o i e a ions by se e al o de s o magni ude. Fu he
wo k by Klemen , Anglada and Ga mendia [
9
] compa ed he quasi-New on app oach
wi h he gene ic algo i hm solu ions, showing a be e pe o mance o quasi-New on
algo i hms. In e e ence [
10
], Ga mendia and Anglada p esen ed hei ini ial wo k on
co ela ion, based on gene ic algo i hms. In he Ph.D. wo k o I. To albo [
11
], a comple e
ma hema ical de elopmen o he co ela ion p oblem was ackled based on a Jacobian
ma ix o mula ion and a Moo e–Pen ose pseudo-in e sion on he non squa e ma ix.
Finally, Anglada, Ma ínez-Jiménez and Ga mendia [
12
] analyzed he wo k o M.J.D.
Powell on op imiza ion and compa ed esul s based on g adien -based algo i hms wi h
esul s ob ained wi h gene ic algo i hms.
The me hod ha is p esen ed in his pape deals wi h he co ela ion p oblem in
a semiau oma ed, ma hema ical way, ins ead o doing i in a manual way, based on he
expe ience o he mal enginee s. O he app oaches, like s a is ical o e en machine lea ning,
a e ou o he scope o his s udy.
The TMM co ela ion p ocedu e is basically an in e se he mal p oblem whe e he
alues o he model pa ame e s a e es ima ed based on he empe a u e da a. The main
d awback o his app oxima ion is ha he p oblem is ill-posed due o he absence o a
unique solu ion. Tha is, di e en combina ions o he TMM pa ame e s could p o ide
he co ec empe a u es, being he main isk, he loss o he physical sense due o he
alues ma hema ically assigned o hese pa ame e s. One op ion o a enua e his p oblem
is o include se e al load cases in he co ela ion as was s a ed in p e ious wo ks o he
au ho s [13,14].
An addi ional di icul y o in e se he mal p oblems is ha he e ec o changes in
bounda y condi ions a e usually dumped, causing a change in sys em empe a u e o
Ae ospace 2022,9, 821 3 o 13
lowe magni ude han hose changes in bounda y condi ions. The e o e, du ing he in e se
p oblem esolu ion, small changes in he sys em empe a u e caused, o example, by
measu emen unce ain ies, can o igina e om big changes in he adjus ed pa ame e s.
Fo his eason, he objec i e o his pape is o s udy he in luence o he measu emen
unce ain ies and e o s, in he p ocess o co ela ing he TMM agains he measu ed
empe a u es.
2. Co ela ion Me hodology and Handling o Measu emen s Unce ain ies
P e ious wo ks by he au ho s ha e shown he possibili y o doing an adequa e
co ela ion o he mal pa ame e s, bo h in he s eady s a e and ansien si ua ions, o
small size TMMs using minimiza ion algo i hms [
13
,
14
]. These wo ks we e de eloped
using he same h ee small TMMs used in he wo k p esen ed he ea e , composed by 4
nodes, 7 nodes and 16 nodes, espec i ely. The las wo models we e de i ed om he TMM
o he T ibolab ins umen , a space ibome e ha was lown on boa d he In e na ional
Space S a ion [15].
The co ela ion me hodology ollowed could be summa ized saying ha he objec i e
is o minimize he di e ences be ween he empe a u es measu ed in he es s and he
empe a u es p edic ed by he TMMs. Fo doing his, he TMM he mal pa ame e s
(GLs, GRs, MCs) a e changed h ough a g adien -based se o minimiza ion sub ou ines
(TOLMIN), de eloped by P o esso M. J. D. Powell [16,17].
In an ideal case, empe a u es p edic ed o each node would be compa ed wi h
he co esponding measu ed alue. Howe e , he numbe o measu emen poin s in he
he mal es a e usually lowe han he numbe o nodes o he TMMs.
In his case, ins ead o using eal es empe a u es as e e ence alues, he nex
p ocedu e has been ollowed. Fo each model, a e e ence TMM was se up and sol ed.
The GLs, GRs and MCs used in ha model a e conside ed he e e ence pa ame e s ( he
co ec pa ame e s) and he empe a u es p o ided he e e ence empe a u es. Then, his
model has been modi ied (GLs, GRs and MCs ha e been andomly al e ed) and has been
called he base model. The empe a u es ob ained wi h his base model a e he p edic ed
empe a u es (o base empe a u es), which should ma ch he e e ence (co ec ) ones. Tha
is, he base model ep esen s he model ha he mal enginee s would p oduce wi h CAD
ools, ma e ial p ope ies, e c., and ha mus be co ela ed agains e e ence empe a u es.
The main ad an age o using his p ocedu e in his s udy ins ead o he empe a u es
measu ed in he mal es s is ha in his way, we ha e a ailable he co ec empe a u e
alues in e e y node o he model, and we ha e also a ailable he co ec alues o he
model pa ame e s (GLs, GRs and MCs). The e o e, we can e alua e no only he e o
le el in he p edic ed empe a u es bu also he e o in he alues assigned o he TMM
pa ame e s (GLs, GRs and MCs) du ing he co ela ion.
Wha has been explained as a as now does no ake in o accoun possible e o
measu emen s, ins ead i is assumed ha empe a u es would be “pe ec ly” measu ed.
In o de o s udy he e ec o ha ing some deg ee o e o in he measu emen o he
empe a u es, he e e ence empe a u es ha e been modi ied ollowing Equa ion (2) (see
e e ence [18]).
T∗
e e ence =T e e ence +ω·σ(2)
whe e
ω
is a andom a iable wi h no mal (Gaussian) dis ibu ion, ze o mean, and uni a y
s anda d de ia ion. σis he s anda d de ia ion o he measu emen e o s.
Real p ecision o empe a u e measu emen s is di icul o es ima e, as he in o ma ion
gi en by he measu emen de ices supplie s is no comple e. To add mo e unce ain y,
di e en ypes o he mocouples o he mis o s a e used o he empe a u e measu e-
men s. The scena io complica es u he i we ake in o accoun he ac ha he mal es s
equipmen is always buil on an indi idual clien basis ( he e a e no s anda d he mal
acuum chambe s). Consequen ly, we will y o esume hese ac s in a unique pa ame e ,
he s anda d de ia ion
σ
, and we will use di e en alues o i anging om
σ=
0.1 o
Ae ospace 2022,9, 821 4 o 13
σ=
0.0001 o see he in luence i has on he esul s’ accu acy. Assuming 99% con idence
o he measu ed empe a u e, ωlies in he ange shown in Equa ion (3).
−2.57583 <ω<2.57583 (3)
To gene a e alues o ωin he men ioned ange, he andom_seed and andom_numbe
sub ou ines o he Fo an language ha e been used. The pseudo andom numbe e u ned,
x
, is a eal alue be ween 0 and 1, so a linea ans o ma ion is used o ob ain he alues o
ωin he adequa e ange ollowing Equa ion (4).
ω=5.15166x−2.57583 (4)
A g aphical desc ip ion o he me hodology can be seen in Figu e 1.
Ae ospace 2022, 9, x FOR PEER REVIEW 4 o 14
acuum chambe s). Consequen ly, we will y o esume hese ac s in a unique pa ame-
e , he s anda d de ia ion 𝜎, and we will use di e en alues o i anging om 𝜎=0.1
o 𝜎=0.0001 o see he in luence i has on he esul s’ accu acy. Assuming 99% con i-
dence o he measu ed empe a u e, 𝜔 lies in he ange shown in Equa ion (3).
− 2.57583 < ω < 2.57583
.
(3)
To gene a e alues o 𝜔 in he men ioned ange, he andom_seed and andom_numbe
sub ou ines o he Fo an language ha e been used. The pseudo andom numbe e-
u ned, 𝑥, is a eal alue be ween 0 and 1, so a linea ans o ma ion is used o ob ain he
alues o 𝜔 in he adequa e ange ollowing Equa ion (4).
𝜔=5.15166𝑥−2.57583
(4)
A g aphical desc ip ion o he me hodology can be seen in Figu e 1.
Figu e 1. G aphical desc ip ion o he me hodology.
3. Resul s Ob ained o Di e en Case S udies and Discussion
Now, he esul s ob ained o he h ee di e en case s udies (4, 7 and 16 nodes
TMMs) will be p esen ed. The numbe o calcula ions done is e y high, he e o e, he
ables ha ollow will y o esume he mos impo an poin s needed o e alua e he
co ela ion p ocess when measu emen e o s a e p esen . The employed cases will be
hose p esen ed p e iously elsewhe e (see e e ences [13,14]), o allow he in e es ed
eade o ha e a comple e idea o he p ocess o co ela ion, i s possibili ies and he di i-
cul ies p esen .
Fo he sake o cla i y, be o e displaying he ables wi h esul s, i is in e es ing o
no e ha he e a e wo able ypes: hose showing esul s o he mal pa ame e s and
hose showing esul s o empe a u es.
Resul s co esponding o he mal pa ame e s ep esen he a i hme ic mean o he
di e ences be ween he e e ence he mal pa ame e s and he alues assigned o hem by
he minimiza ion algo i hm. I we call NPAR, he numbe o unknown pa ame e s ha
mus be co ela ed (GLs, GRs and MCs), and 𝑃𝑖, each one o hese pa ame e s, he e o
would be calcula ed ollowing Equa ion (5).
𝐸𝑟𝑟𝑜𝑟= 1
𝑁𝑃𝐴𝑅 ∑ [|𝑃𝑖𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒−𝑃𝑖𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑
𝑃𝑖𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 |·100]
𝑁𝑃𝐴𝑅
𝑖=1
(5)
In he case o empe a u e esul s, he alues co espond o he a i hme ic mean be-
ween he e e ence empe a u es and he empe a u es ob ained wi h he TMM once co -
ela ed, ha is, using he he mal pa ame e s ob ained om he co ela ion. I we call
NNOD, he numbe o nodes o he model, NSTEP, he numbe o ime s eps used in he
Figu e 1. G aphical desc ip ion o he me hodology.
3. Resul s Ob ained o Di e en Case S udies and Discussion
Now, he esul s ob ained o he h ee di e en case s udies (4, 7 and 16 nodes TMMs)
will be p esen ed. The numbe o calcula ions done is e y high, he e o e, he ables ha
ollow will y o esume he mos impo an poin s needed o e alua e he co ela ion
p ocess when measu emen e o s a e p esen . The employed cases will be hose p esen ed
p e iously elsewhe e (see e e ences [
13
,
14
]), o allow he in e es ed eade o ha e a
comple e idea o he p ocess o co ela ion, i s possibili ies and he di icul ies p esen .
Fo he sake o cla i y, be o e displaying he ables wi h esul s, i is in e es ing o no e
ha he e a e wo able ypes: hose showing esul s o he mal pa ame e s and hose
showing esul s o empe a u es.
Resul s co esponding o he mal pa ame e s ep esen he a i hme ic mean o he
di e ences be ween he e e ence he mal pa ame e s and he alues assigned o hem by
he minimiza ion algo i hm. I we call NPAR, he numbe o unknown pa ame e s ha
mus be co ela ed (GLs, GRs and MCs), and
Pi
, each one o hese pa ame e s, he e o
would be calcula ed ollowing Equa ion (5).
E o =1
NPAR
NPAR
∑
i=1"
P e e ence
i−Pco ela ed
i
P e e ence
i
·100#(5)
In he case o empe a u e esul s, he alues co espond o he a i hme ic mean
be ween he e e ence empe a u es and he empe a u es ob ained wi h he TMM once
co ela ed, ha is, using he he mal pa ame e s ob ained om he co ela ion. I we call
NNOD, he numbe o nodes o he model, NSTEP, he numbe o ime s eps used in he
solu ion o he he mal case and NCASE, he numbe o load cases aken in o accoun in he
calcula ion, we can see ha NTEMP, he numbe o empe a u es p esen in he co ela ion,
Ae ospace 2022,9, 821 5 o 13
is gi en by Equa ion (6). Fu he , he e o would be calcula ed ollowing Equa ion (7)
(calling Tieach one o hese empe a u es).
NTEMP =NNOD·NSTEP·NCASE (6)
E o =1
NTEMP
NTEMP
∑
i=1T e e ence
i−Tco ela ed
i(7)
3.1. 4 Nodes Model
As a i s simple model, a heo e ical 4 nodes model (nodes 1 o 4) has been used.
The he mal model, which can be seen in Figu e 2, has h ee linea conduc ances (GLs),
h ee adia i e conduc ances (GRs) and h ee he mal ine ias (MCs). Powe is applied in
node numbe 1 and a cons an empe a u e o 20
◦
C is main ained in sink node 4, o all
he load cases. The ansien case ex ends 7200 s (2 h), and he ini ial empe a u e o all
he nodes is 20
◦
C. The ime s ep used is 600 s (10 min). The beha io and co ela ion o
his model was s udied in dep h in e e ences [
13
,
14
], which can be consul ed o mo e
de ailed in o ma ion. Now, di e en e o le els ha e been in oduced o he e e ence
empe a u es, as can be seen in he di e en alues assigned o he s anda d de ia ion (SD)
in he ables included in nex sec ions. Thei in luence has been s udied o s eady s a e
cases and o ansien analysis.
Ae ospace 2022, 9, x FOR PEER REVIEW 5 o 14
solu ion o he he mal case and NCASE, he numbe o load cases aken in o accoun in
he calcula ion, we can see ha NTEMP, he numbe o empe a u es p esen in he co e-
la ion, is gi en by Equa ion (6). Fu he , he e o would be calcula ed ollowing Equa ion
(7) (calling 𝑇𝑖 each one o hese empe a u es).
𝑁𝑇𝐸𝑀𝑃=𝑁𝑁𝑂𝐷·𝑁𝑆𝑇𝐸𝑃·𝑁𝐶𝐴𝑆𝐸
(6)
𝐸𝑟𝑟𝑜𝑟= 1
𝑁𝑇𝐸𝑀𝑃 ∑ |𝑇𝑖𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒−𝑇𝑖𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑|
𝑁𝑇𝐸𝑀𝑃
𝑖=1
(7)
3.1. 4 Nodes Model
As a i s simple model, a heo e ical 4 nodes model (nodes 1 o 4) has been used. The
he mal model, which can be seen in Figu e 2, has h ee linea conduc ances (GLs), h ee
adia i e conduc ances (GRs) and h ee he mal ine ias (MCs). Powe is applied in node
numbe 1 and a cons an empe a u e o 20 °C is main ained in sink node 4, o all he load
cases. The ansien case ex ends 7200 s (2 h), and he ini ial empe a u e o all he nodes
is 20 °C. The ime s ep used is 600 s (10 min). The beha io and co ela ion o his model
was s udied in dep h in e e ences [13,14], which can be consul ed o mo e de ailed in-
o ma ion. Now, di e en e o le els ha e been in oduced o he e e ence empe a-
u es, as can be seen in he di e en alues assigned o he s anda d de ia ion (SD) in he
ables included in nex sec ions. Thei in luence has been s udied o s eady s a e cases
and o ansien analysis.
Figu e 2. The 4 nodes model.
3.1.1. 4 Nodes Model. Co ela ion o S eady S a e Cases
Table 1 shows he e o s in he TMM pa ame e s bo h a he ini ial s a e and a e he
co ela ion, o di e en le els o measu emen e o s (di e en s anda d de ia ion al-
ues). The analysis has been done including di e en load cases in he co ela ion. Th ee
load cases esul s (ho , s ay ali e and cold) we e a ailable o he co ela ions. Co ela ion
2(a) was done using ho and s ay ali e cases, co ela ion 2(b) was done using ho and cold
cases and co ela ion 2(c) was done using s ay ali e and cold cases.
Table 1. E o s ob ained o he TMM he mal pa ame e s. S eady s a e 4 nodes model.
SD = 0.01
SD = 0.001
SD = 0.0001
SD = 0.00001
SD = 0.0
No. o
Load Cases
Ini ial E o
E o
E o
E o
E o
E o
3
67.92%
61.53%
120.20%
6.60%
0.04%
0.02%
2 (a)
67.92%
888.09%
62.28%
48.31%
2.16%
0.00%
2 (b)
67.92%
85.39%
57.73%
18.30%
0.67%
0.00%
2 (c)
67.92%
64.80%
45.79%
41.46%
0.65%
0.00%
Figu e 2. The 4 nodes model.
3.1.1. 4 Nodes Model: Co ela ion o S eady S a e Cases
Table 1shows he e o s in he TMM pa ame e s bo h a he ini ial s a e and a e he
co ela ion, o di e en le els o measu emen e o s (di e en s anda d de ia ion alues).
The analysis has been done including di e en load cases in he co ela ion. Th ee load
cases esul s (ho , s ay ali e and cold) we e a ailable o he co ela ions. Co ela ion 2(a)
was done using ho and s ay ali e cases, co ela ion 2(b) was done using ho and cold cases
and co ela ion 2(c) was done using s ay ali e and cold cases.
Table 1. E o s ob ained o he TMM he mal pa ame e s. S eady s a e 4 nodes model.
SD = 0.01 SD = 0.001 SD =
0.0001
SD =
0.00001 SD = 0.0
No. o
Load Cases Ini ial E o E o E o E o E o E o
3 67.92% 61.53% 120.20% 6.60% 0.04% 0.02%
2 (a) 67.92% 888.09% 62.28% 48.31% 2.16% 0.00%
2 (b) 67.92% 85.39% 57.73% 18.30% 0.67% 0.00%
2 (c) 67.92% 64.80% 45.79% 41.46% 0.65% 0.00%
The mean ini ial e o in he pa ame e s (67.92%) can be educed i he s anda d
de ia ion e o in he measu emen s is 0.0001 o lowe .
A e ob aining he co ela ed he mal pa ame e s wi h he di e en SD e o le -
els, TMMs we e e-buil using he co ela ed pa ame e s. Then, hese co ela ed models
Ae ospace 2022,9, 821 6 o 13
we e un and new p edic ed empe a u es we e ob ained. Table 2collec s he absolu e
empe a u e e o le el o hese models.
Table 2. E o s ob ained o p edic ed empe a u es o s eady s a e 4 nodes model.
SD = 0.01 SD = 0.001 SD =
0.0001
SD =
0.00001 SD = 0.0
No. o
Load Cases Ini ial E o E o E o E o E o E o
3 1.6511 0.0174 0.0051 0.0001 0.0000 0.0000
2 (a) 1.6511 0.0151 0.0041 0.0017 0.0000 0.0000
2 (b) 1.6511 3.5935 0.0038 0.0001 0.0000 0.0000
2 (c) 1.6511 0.0185 0.0070 0.0003 0.0000 0.0000
The mean ini ial e o o p edic ed empe a u es goes clea ly down i he s anda d
de ia ion SD in he measu ed empe a u es is equal o lowe han 0.001.
As i can be seen, he smalle he s anda d de ia ion o he e o in he empe a u es
measu emen s, he be e he esul s bo h o he he mal pa ame e s and o empe a u es.
3.1.2. 4 Nodes Model: Co ela ion o T ansien Cases
The simple 4 nodes model has been also s udied o ansien cases. The numbe
o empe a u es is now much highe , as alues in each ime s ep a e a ailable o each
node. Again, di e en le els o s anda d de ia ion SD we e conside ed in he co ela ion
o he he mal pa ame e s. Table 3collec s he esul s o ansien cases o he he mal
pa ame e s, when wo load cases o one load case a e conside ed. As expec ed, he lowe
he s anda d e o in he measu emen o empe a u es, he be e he he mal pa ame e s
co ela ion. Addi ionally, when wo load cases a e used ins ead o one, imp o ed esul s
a e ob ained.
Table 3. E o s ob ained o he TMM he mal pa ame e s. Co ela ion o ansien 4 nodes model.
SD = 0.01 SD = 0.001 SD =
0.0001
SD =
0.00001 SD = 0.0
No. o
Load Cases Ini ial E o E o E o E o E o E o
2 56.94% 27.60% 13.69% 1.49% 0.11% 0.03%
1 56.94% 58.92% 47.58% 2.46% 0.12% 0.06%
Once he co ela ed he mal pa ame e s we e ob ained, a calcula ion o p edic ed
empe a u es was done o he di e en load cases conside ed. Resul s a e collec ed in
Table 4, which shows he mean e o alue o he empe a u es p edic ed e sus he co ec
empe a u es.
Table 4. E o s ob ained o p edic ed empe a u es o ansien 4 nodes model.
SD = 0.01 SD = 0.001 SD = 0.0001 SD = 0.00001 SD = 0.0
No. o
Load Cases Load Case Ini ial
E o E o E o E o E o E o
2 cold 0.9414 0.0031 0.0011 0.0001 0.0000 0.0000
2 s ay ali e 1.5025 0.0053 0.0010 0.0001 0.0000 0.0000
2 ho 2.2464 0.0087 0.0006 0.0001 0.0000 0.0000
1 cold 0.9414 0.0061 0.0007 0.0001 0.0000 0.0000
1 s ay ali e 1.5025 0.0107 0.0035 0.0002 0.0000 0.0000
1 ho 2.2464 0.0183 0.0108 0.0005 0.0000 0.0000
Ae ospace 2022,9, 821 7 o 13
I is wo h o no e ha , e en o he highes le els o s anda d de ia ion e o , he
empe a u e alues ma ch e y well wi h he co ec empe a u e alues. This happens
e en o he TMMs whose he mal pa ame e s show bigge e o s.
Finally, he e ec o he measu emen e o s has been also s udied in he si ua ion
when he empe a u e o one node is unknown (node 2), which implies a highe di icul y
o achie e a good co ela ion. The mal pa ame e s esul s ob ained in his case a e collec ed
in Table 5. As i could be expec ed, esul s a e wo se han hose shown in Table 3. Only he
h ee load cases wi h a minimum s anda d de ia ion (SD = 0.00001) show a good ma ch
be ween he co ela ed pa ame e s and he eal ones.
Table 5.
E o s ob ained o he TMM he mal pa ame e s. Co ela ion o ansien 4 nodes model
wi h 1 unknown empe a u e.
SD = 0.01 SD = 0.001 SD =
0.0001
SD =
0.00001 SD = 0.0
No. o
Load Cases Ini ial E o E o E o E o E o E o
2 56.94% 76.20% 66.41% 65.46% 63.95% 3.37%
1 56.94% 266.90% 75.47% 63.76% 66.44% 33.31%
3 56.94% 65.11% 65.41% 65.35% 5.96% 4.58%
Once again, he new se s o p edic ed empe a u es we e ob ained wi h he new
co ela ed he mal pa ame e s. The esul s a e shown in Table 6, whe e he mean absolu e
empe a u e e o s a e collec ed.
Table 6.
E o s ob ained o p edic ed empe a u es o ansien 4 nodes model, 1 unknown empe a u e.
SD = 0.01 SD = 0.001 SD = 0.0001 SD = 0.00001 SD = 0.0
No. o
Load Cases Load Case Ini ial E o E o E o E o E o E o
2 cold 0.9414 0.1089 0.0326 0.0161 0.0150 0.0001
2 s ay ali e 1.5025 0.1761 0.0530 0.0254 0.0243 0.0023
2 ho 2.2464 0.2678 0.0815 0.0379 0.0373 0.0035
1 cold 0.9414 0.3811 0.1248 0.0800 0.0133 0.0124
1 s ay ali e 1.5025 0.6265 0.2008 0.1355 0.0212 0.0206
1 ho 2.2464 0.9785 0.3049 0.2182 0.0345 0.0322
3 cold 0.9414 0.1718 0.0538 0.0158 0.0024 0.0019
3 s ay ali e 1.5025 0.2784 0.0879 0.0248 0.0039 0.0031
3 ho 2.2464 0.4287 0.1357 0.0378 0.0060 0.0047
Close examina ion o he new p edic ed empe a u es in Table 6shows a d ama ic
imp o emen o he p edic ed alues e sus he ini ial e o . E en o he poo es he mal
pa ame e s co ela ion ( hose wi h he highes s anda d de ia ion), he new se o empe a-
u es ma ch well wi h he co ec empe a u es. Howe e , esul s when conside ing h ee
load cases ins ead o wo load cases o he co ela ion a e wo se, which is an unexpec ed
esul . The au ho s ha e no clea explana ion o his ac .
3.2. 7 Nodes Model
The p e ious 4 nodes model showed he ends and limi a ions o he co ela ion
me hod when applied o a heo e ical small he mal model. I is in e es ing o make an
equi alen s udy o a bigge model, which co esponds o a eal de ice. In his sec ion,
we s udy a educed 7 nodes model o he T ibolab ins umen , a space ibome e ha was
lown on boa d he In e na ional Space S a ion [
15
]. Th ee o he nodes a e sink nodes:
wo adia ion sink nodes (nodes 99,241 and 99,271) and one conduc ion sink node (node
10,000). The model consis s o ou linea conduc ances, wo adia ion conduc ances and
Ae ospace 2022,9, 821 8 o 13
ou he mal ine ias, o he ansien cases, which can be seen in Figu e 3. In he ansien
cases, he calcula ions un o 86.400 s ( ha is, one day) and ime s ep
∆ =
600
s
. This
makes a o al o 144 ime s eps. The ini ial empe a u e conside ed o = 0 is T= 20 ◦C.
Ae ospace 2022, 9, x FOR PEER REVIEW 8 o 14
3
ho
2.2464
0.4287
0.1357
0.0378
0.0060
0.0047
Close examina ion o he new p edic ed empe a u es in Table 6 shows a d ama ic
imp o emen o he p edic ed alues e sus he ini ial e o . E en o he poo es he mal
pa ame e s co ela ion ( hose wi h he highes s anda d de ia ion), he new se o empe -
a u es ma ch well wi h he co ec empe a u es. Howe e , esul s when conside ing h ee
load cases ins ead o wo load cases o he co ela ion a e wo se, which is an unexpec ed
esul . The au ho s ha e no clea explana ion o his ac .
3.2. 7 Nodes Model
The p e ious 4 nodes model showed he ends and limi a ions o he co ela ion
me hod when applied o a heo e ical small he mal model. I is in e es ing o make an
equi alen s udy o a bigge model, which co esponds o a eal de ice. In his sec ion,
we s udy a educed 7 nodes model o he T ibolab ins umen , a space ibome e ha was
lown on boa d he In e na ional Space S a ion [15]. Th ee o he nodes a e sink nodes:
wo adia ion sink nodes (nodes 99,241 and 99,271) and one conduc ion sink node (node
10,000). The model consis s o ou linea conduc ances, wo adia ion conduc ances and
ou he mal ine ias, o he ansien cases, which can be seen in Figu e 3. In he ansien
cases, he calcula ions un o 86.400 s ( ha is, one day) and ime s ep Δ𝑡=600 𝑠. This
makes a o al o 144 ime s eps. The ini ial empe a u e conside ed o 𝑡=0 is 𝑇=20 °C.
Figu e 3. The 7 nodes model.
Powe s a e applied in no sink nodes (85,040, 85,041, 85,070 and 85,071) and sink em-
pe a u es a e known. Fo his 7 nodes model, bo h s eady s a e and ansien esul s ha e
been used in he pa ame e co ela ion.
3.2.1. 7 Nodes Model. Co ela ion o S eady S a e Cases
The esul s ob ained o he pa ame e s co ela ion in he s eady s a e cases a e col-
lec ed in Table 7. Two di e en si ua ions a e conside ed in hese esul s: All he empe -
a u es a e known ( i s ow) o wo empe a u es ( hose o nodes 85,040 and 85,050) a e
unknown ( ows 2 and 3). Fo he co ela ion, he empe a u es o wo load cases we e
used ( i s ow), h ee load cases (second ow) o ou load cases ( hi d ow).
Table 7. E o s ob ained o he TMM he mal pa ame e s. S eady s a e 7 nodes model.
SD = 0.01
SD =
0.001
SD =
0.0001
SD = 0.00001
SD = 0.0
Un-
knowns
No. o
Load Cases
Ini ial
E o
E o
E o
E o
E o
E o
0
2
53.64%
60.94%
2.16%
0.87%
0.05%
0.02%
2
3
53.64%
39.79%
16.11%
0.54%
0.07%
0.02%
2
4
53.64%
174.16%
3.26%
0.14%
0.03%
0.02%
Figu e 3. The 7 nodes model.
Powe s a e applied in no sink nodes (85,040, 85,041, 85,070 and 85,071) and sink
empe a u es a e known. Fo his 7 nodes model, bo h s eady s a e and ansien esul s
ha e been used in he pa ame e co ela ion.
3.2.1. 7 Nodes Model: Co ela ion o S eady S a e Cases
The esul s ob ained o he pa ame e s co ela ion in he s eady s a e cases a e col-
lec ed in Table 7. Two di e en si ua ions a e conside ed in hese esul s: All he empe -
a u es a e known ( i s ow) o wo empe a u es ( hose o nodes 85,040 and 85,050) a e
unknown ( ows 2 and 3). Fo he co ela ion, he empe a u es o wo load cases we e used
( i s ow), h ee load cases (second ow) o ou load cases ( hi d ow).
Table 7. E o s ob ained o he TMM he mal pa ame e s. S eady s a e 7 nodes model.
SD = 0.01 SD = 0.001 SD = 0.0001 SD = 0.00001 SD = 0.0
Unknowns No. o
Load Cases
Ini ial
E o E o E o E o E o E o
0 2 53.64% 60.94% 2.16% 0.87% 0.05% 0.02%
2 3 53.64% 39.79% 16.11% 0.54% 0.07% 0.02%
2 4 53.64% 174.16% 3.26% 0.14% 0.03% 0.02%
As expec ed, a smalle s anda d de ia ion in he empe a u es measu emen s implies a
be e co ela ion o he pa ame e s. The p esence o unknown empe a u es on some o he
nodes makes i mo e di icul (o e en impossible) o ob ain a co ec he mal pa ame e s
co ela ion. Howe e , he use o mo e load cases balances he added di icul y and esul s,
o he he mal pa ame e s a e qui e sa is ac o y i SD is equal o lowe han 0.001. I is also
in e es ing o no e ha he esul s o SD = 0.1 beha e in some unexpec ed way ( hey a e
wo se han he ini ial e o ) bu he use o smalle alues o SD imp o es he esul s clea ly.
Once he he mal pa ame e s ha e been ob ained om he co ela ion, all he he mal
s eady s a e cases ha e been calcula ed again using hem. Resul s a e collec ed in his case
in Table 8.
Table 8. E o s ob ained o p edic ed empe a u es o s eady s a e 7 nodes model.
SD = 0.01 SD = 0.001 SD = 0.0001 SD = 0.00001 SD = 0.0
Unknowns No. o
Load Cases
Ini ial
E o E o E o E o E o E o
0 2 4.9444 1.0961 0.0466 0.0201 0.0012 0.0000
2 3 4.9444 0.6115 0.3247 0.0102 0.0012 0.0000
2 4 4.9444 0.7695 0.0814 0.0032 0.0005 0.0000
Ae ospace 2022,9, 821 9 o 13
Once again, a d ama ic imp o emen o he empe a u es p edic ed is achie ed, e en
o big s anda d de ia ions. The p esence o unknown empe a u es is well balanced wi h
he e ec o aking in o accoun mo e load cases o he co ela ion.
3.2.2. 7 Nodes Model: Co ela ion o T ansien Cases
The simple 7 nodes case o he T ibolab ins umen is also used in he ansien e sion
o he co ela ion algo i hm. The measu ed empe a u es ha e been used in h ee di e en
si ua ions, when all he empe a u es a e known, when one empe a u e is missed ( ha o
node 85,040) and when wo empe a u es a e missed ( hose o nodes 85,040 and 85,070). Fo
each o hese si ua ions, wo load cases, wo load cases and ou load cases ha e been used,
espec i ely. E o s wi h di e en s anda d de ia ion alues we e added o he e e ence
empe a u es o each ime s ep conside ed.
The esul s ob ained o he he mal pa ame e s can be seen in Table 9.
Table 9. E o s ob ained o he he mal pa ame e s o ansien 7 nodes model.
SD = 0.01 SD =
0.001
SD =
0.0001
SD =
0.00001
Unknowns No. o
Load Cases
Ini ial
E o E o E o E o E o
0 2 49.88% 35.26% 1.10% 0.03% 0.02%
1 2 49.88% 1800.74% 218.57% 0.09% 217.82%
2 4 49.88% 22.94% 0.31% 0.06% 0.02%
Wi h 0 and 2 unknowns and using wo o ou load cases, he esul s o he co ela ion
a e good: smalle empe a u e de ia ions and enough load cases conside ed lead o a be e
co ela ion, also o his ansien case. Howe e , beha io is somehow e a ic when one
empe a u e is unknown and wo load cases a e used. The e is no a clea explana ion o
his ac . A possible human e o in he model o a non con e gence si ua ions a e possible
explana ions o he unexpec ed beha io .
Resul s o he empe a u es calcula ed wi h he he mal pa ame e s ob ained om
he co ela ion, can be seen in Table 10.
Table 10. E o s o p edic ed empe a u es o ansien 7 nodes model.
SD = 0.01 SD = 0.001 SD = 0.0001 SD = 0.00001
Unknowns No. o
Load Cases Case Ini ial
E o E o E o E o E o
0 2 cold 5.4945 1.6181 0.0262 0.0004 0.0001
0 2 TEM_cold 4.4784 0.2512 0.0162 0.0001 0.0000
0 2 ho 4.0152 1.1499 0.0190 0.0002 0.0001
1 2 cold 5.4945 0.6900 2.9599 0.0006 2.9444
1 2 TEM_cold 4.4784 2.2728 8.2142 0.0020 8.2095
1 2 ho 4.0152 3.2454 2.0224 0.0004 2.0349
2 4 cold 5.4945 0.1182 0.0051 0.0008 0.0000
2 4 TEM_cold 4.4784 0.1341 0.0059 0.0005 0.0000
2 4 ho 4.0152 0.1702 0.0075 0.0004 0.0000
The esul s o he empe a u es a e good, o ze o and wo unknowns, e en i he
he mal pa ame e s co ela ed we e no ha exac . The e o in he empe a u es is small,
e en o high s anda d de ia ions. Howe e , he one unknown case does no beha e well,
which is consis en wi h he poo esul s ob ained o he he mal pa ame e s
