The mal conduc ion in h ee-dimensional p in ed po ous samples by high
esolu ion in a ed he mog aphy
D. Mu~
noz Codo níu, J.J. Moyano, M. Belmon e, M.I. Osendi, P. Mi anzo
*
Ins i u o de Ce 
amica y Vid io, CSIC, Campus de Can oblanco, 28049, Mad id, Spain
ARTICLE INFO
Keywo ds:
3D p in ed s uc u es
Po ous ma e ials
The mal conduc i i y
In a ed he mog aphy
ABSTRACT
The he mal conduc i i y (κ) is a key pa ame e ha defines many o he echnological uses o h ee-dimensional
(3D) po ous a chi ec u es. Despi e he a ie y o me hods o de e mining κ, p oblems gene ally a ise when e-
sea che s y o apply hem o cellula ma e ials and 3D s uc u es. The p esen wo k p oposes an a o dable lab-
made de ice o analysing aniso opic hea flow in 3D po ous a chi ec u es ia high esolu ion in a ed he -
mog aphy. The me hod is alida ed using dense ma e ials o known he mal conduc i i y. Tempe a u e g adien s
measu ed o po ous specimens ha e been co ela ed o he he mal conduc i i y es ima ed om a simple esis o s
model, assessing he main ac o s ha a ec he expe imen al measu emen s. The po ous specimens o SiC, MAX-
phase and g aphene-based nanos uc u es a e in-house manu ac u ed by di ec ink w i ing ( obocas ing).
1. In oduc ion
The mal conduc i i y is he main pa ame e go e ning he hea
ans e ha de e mines many o he echnological uses o ma e ials,
especially in he mal managemen , ene gy ha es ing and he mal en-
e gy s o age (TES) applica ions. The measu emen o he he mal con-
duc i i y can be p oblema ic when dealing wi h highly po ous ma e ials,
in pa icula , o h ee-dimensional (3D) a chi ec u es manu ac u ed by
using a p in ing p ocess om compu e -aided designs (CAD). These ypes
o complex s uc u es ha e applica ion in a ious fields whe e hea
dissipa ion is a undamen al ma e [1,2], o example, in ca alysis [3–5],
ene gy s o age and p oduc ion [6,7], and hea exchange s and hea sinks
[8,9]. The e ec o he p in ing pa ame e s on he he mal p ope ies o
3D p in ed s uc u es has been ea ed in se e al s udies ocused on
polyme s and me al s uc u es p ocessed by used deposi ion modelling
(FDM) [10–15], bu only ela i ely ew wo ks ha e deal wi h 3D ce amic
s uc u es.
A g ea a ie y o me hods o de e mining he he mal conduc i i y
a e based on moni o ing empe a u e changes p oduced du ing sample
hea ing, ob aining he he mal di usi i y (
α
) and conduc i i y (κ) wi h
dis inc p ecision le els ha depend on he selec ed me hod. In he case
o po ous ma e ials, each po e ep esen s a small olume filled wi h gas
(ai ), a medium o e y low he mal conduc i i y (0.023 W m
1
K
1
unde no mal condi ions o p essu e and empe a u e [16]), which
gene a es no able he e ogenei ies ha complica e he κmeasu emen .
Mo eo e , since he ex e nal su aces o hese ma e ials a e usually e y
ough, con ac he mal esis ances become an addi ional p oblem. The
p esen wo k uses a lab-made de ice o measu ing empe a u e p ofiles
gene a ed in 3D po ous a chi ec u es placed be ween wo hea ing sou -
ces wi h an in a ed (IR) came a, examining he p os and cons o his
s aigh o wa d es ing gadge .
2. The mal conduc i i y me hods
The me hods o measu ing he he mal conduc i i y o ma e ials can
be classified acco ding o he sample empe a u e dis ibu ion as s eady-
s a e o ansien hea flow me hods. The mos common me hods a e
compa ed in some ecen e iews [17,18].
S eady-s a e hea flow me hods a e gene ally based on inducing a
cons an unidi ec ional empe a u e g adien ac oss he sample. In his
way, samples a e a anged be ween a ho sou ce and a cold sink, while
hey a e he mally insula ed om hei su oundings. This allows
simpli ying he Fou ie ’s conduc ion equa ion in pa ial de i a i es o he
ollowing exp ession:
∂
2T
∂
z2¼0(1)
whe e z coincides wi h he hea flow di ec ion. In he case o pe ec
insula ion, he empe a u e p ofiles in he sample will be linea in he
* Co esponding au ho .
E-mail add ess: pmi anzo@ic .csic.es (P. Mi anzo).
Con en s lis s a ailable a ScienceDi ec
Open Ce amics
jou nal homepage: www.edi o ialmanage .com/oce am
h ps://doi.o g/10.1016/j.oce am.2020.100028
Recei ed 21 Augus 2020; Recei ed in e ised o m 15 Oc obe 2020; Accep ed 16 Oc obe 2020
A ailable online 21 Oc obe 2020
2666-5395/©2020 The Au ho s. Published by Else ie L d on behal o Eu opean Ce amic Socie y. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Open Ce amics 4 (2020) 100028
hea flow di ec ion and cons an in he pe pendicula planes. Acco d-
ingly, he he mal conduc i i y is calcula ed om he empe a u e
g adien in he specimen and he hea flowing h ough a ce ain sec ion
o he sample. In p ac ice, hese p ocedu es a e subdi ided in o wo
ypes: absolu e and compa a i e. In absolu e me hods, he he mal powe
ans e ed o he sample is known; whe eas in compa a i e me hods, he
es sample and one o mo e e e ence specimens o known conduc i i y
a e s acked pa allel o he hea flow di ec ion. The unknown he mal
conduc i i y is, hen, ob ained by compa ing g adien s measu ed in he
es sample and he e e ence ma e ials, assuming he same hea flux
h ough he samples in con ac . The achie emen o a unidi ec ional hea
flow is one o he main issues o bo h absolu e and compa a i e
me hods. Fo ha pu pose, gua d ings a ound he empe a u e mea-
su emen zone a e usually employed, which no mally consis o he -
mally insula ing ma e ials wi h a se ies o la e al elec ical esis ances
ha ep oduce a empe a u e g adien simila o he eached in he
measu ing s acking o a oid/ educe la e al hea losses.
The mos widely used s eady-s a e me hods o po ous ma e ials a e
he gua ded ho pla e (ASTM C177-13), alid o ma e ials wi h e y low
he mal conduc i i y (κ<1Wm
1
K
1
); he gua ded-compa a i e-
longi udinal hea flow me e (ASTM-1225), limi ed o ma e ials wi h
conduc i i ies highe han ~ 1 W m
1
K
1
; and also a modifica ion o he
la e (ASTM D5470-17) ha consis s o measu ing he he mal esis-
ance o he ma e ial as a unc ion o he specimen hickness when he use
o he mocouples is no possible. These p ocedu es ha e all in common
he easiness o he undamen al equa ions, he high p ecision, and he
ad an age o a simple sample geome y (cylinde o squa e p ism);
howe e , he usually long s abiliza ion imes (up o 24 h) o ge s eady-
s a e condi ions and la ge sample size equi emen s a e some o he
limi a ions ( ypically specimens a e up o 5 cm high and 5-50 cm
diame e o cylinde s o side leng h o squa e p ism specimens). In
addi ion, he e a e some addi ional p oblems like he impe ec insu-
la ion and he o en icky a achmen o he he mocouples. The si ua-
ion ge s wo se in he case o po ous samples. On he one hand, he
su ace o a po ous ma e ial is qui e i egula , which p e en s in ima e
con ac be ween he specimen and he hea sou ces and he mocouples
and, on he o he hand, uncoun ed hea losses and imp ecise measu e-
men s o he he mal g adien s may occu . Addi ionally, o compa a i e
me hods, he con ac he mal esis ances be ween he e e ence and es
samples a e o pa amoun impo ance, qui e mo e ele an han o dense
samples o smoo he su ace. Finally, he lowe he mal conduc i i y o
po ous ma e ials u he inc eases he ime equi ed o he he mal
s abiliza ion.
T ansien me hods moni o he empo al e olu ion o he empe a u e
a e he sample hea ing o a ime pe iod in he o de o seconds. Fo his
eason, al hough he he mal insula ion is s ill necessa y, he hea losses
o he en i onmen a e less significan han in he s eady-s a e me hods.
The he mal conduc i i y calcula ion is indi ec ly ca ied ou since mos
o hese me hods usually de e mine he he mal di usi i y and calcula e
he conduc i i y om he ma e ial’s bulk densi y (
ρ
) and he specific hea
(c
p
) using he exp ession κ¼
α
⋅
ρ
⋅cp. The highe he di usi i y, he
as e he medium esponds o empe a u e changes. Some me hods use a
wi e o pla e embedded be ween wo simila specimens o he same
ma e ial o Joule e ec hea ing; he mos widely employed a e he ho
wi e (HW, ASTMC1113) and he ansien plane sou ce (TPS, ISO 22007-
2) echniques. A close con ac be ween he hea sou ces and he speci-
mens is equi ed in bo h me hods. On he o he hand, a lase pulse is used
o hea one o he specimen aces in he lase flash me hod (ASTM E-
1461); hence, a eal con ac be ween he hea sou ce and he sample is
a oided. Recen ly, he TPS me hod has been modified (MTPS, ASTM
D7984-16) by means o a fla hea e /senso and a gua d ing o use only
one specimen and, he e o e, a single in e ace, which p ac ically ensu es
a one-dimensional hea ans e .
The main ad an ages o he ansien me hods a e he educed hea
losses, he smalle sample size, and he wide ange o measu able con-
duc i i ies, om 0.02 o mo e han 2000 W m
1
K
1
depending on he
me hod, wi h easonable accu acies (1–5%) [17,18]. In gene al, he
ansien me hods allow handling samples o sizes ha a y om ~5 cm
in heigh and 100–200 cm
2
in sec ion, in he case o ho wi e, o ~2 mm
in heigh and sec ions o ~1.3 cm
2
o he lase flash echnique. Howe e ,
specific p oblems a ise ega ding hei applicabili y o cellula ma e ials
and 3D s uc u es. The ho wi e and he ansien plane sou ce me hods
p esen simila p oblems linked o he no mally poo con ac be ween he
hea sou ce and he specimen. As o he lase flash me hod, i s appli-
cabili y o hese ma e ials is ques ionable because he lase beam would
impac di ec ly he in a ed de ec o wi hou in e e ing wi h he sample
a all.
In he case o 3D p in ed s uc u es, some he mal conduc i i y
s udies can be ound o polyme s ab ica ed by FDM ha employ ei he
he TPS me hod [10,11] o simple longi udinal hea flow me e me hod
[12–15]. The la e one consis s o one hea sou ce and one hea sink, wo
me al ba s, gene ally coppe , be ween hem and he es specimen in he
cen e o he assembly. Se e al empe a u e senso s a e inse ed in he
op and bo om me al ba s o measu ing he empe a u e g adien and,
acco dingly, no only he sample bu he wo con ac esis ances a e
included in his measu emen . F om he he mal impedance (sum o he
he mal esis ance o he ma e ial and all con ac esis ances) measu e-
men s in a ious specimens o di e en hickness (be ween 4 and 10
mm), he e ec o he in e acial he mal esis ances can be elimina ed,
and, hence, he he mal conduc i i y can be es ima ed. The specimen
sec ion a ies be ween 25 25 and 40 40 mm
2
, hus, simila o hose
used in TPS. Rega ding addi i ely manu ac u ed 3D ce amic s uc u es,
o he bes o he au ho s knowledge, he e a e no expe imen al wo ks
analysing hei he mal conduc i i ies, al hough compa a i e e alua ions
o he hea dissipa ion capabili y o di e en p in ed ma e ials du ing
cooling ha e ecen ly been add essed [19–22].
3. Ma e ials desc ip ion and he mal conduc ion p ocedu e
3.1. Ma e ials
Dense and po ous ma e ials we e selec ed o he s udy. Dense cyl-
inde s o comme cial ma e ials o known he mal conduc i i y we e used
o alida e he me hod he e p oposed (see sec ion 3.2), whe eas he
po ous specimens we e in-house manu ac u ed by di ec ink w i ing
( obocas ing). Robocas ing is an addi i e manu ac u ing echnique ha
allows p in ing 3D ma e ials om highly concen a ed inks con aining
ce ain amoun o o ganic addi i es o gain con ol o e he ink iscosi y
and heology. The p ope ink is ex uded h ough a needle, ypically o
diame e be ween 250 and 800
μ
m, ollowing a p e iously compu e -
designed pa e n [23].
Re e ence s anda ds o Py ex®7740, Py oce am®9606, and alumina
(NIST SRM 720) o ce ified he mal conduc i i y we e selec ed as dense
ma e ials, al hough hei he mal conduc i i ies we e also measu ed
using he lase flash me hod. Py ex®is a low- he mal-expansion bo o-
silica e glass (SiO
2
and B
2
O
3
, wi h small amoun s o Na
2
O and Al
2
O
3
)
de eloped by Co ning Inc. wi h a ce ified κo 1.1 W m
1
K
1
a oom
empe a u e. Py oce am®9606 is a magnesium aluminium silica e glass-
ce amic wi h TiO
2
as nuclea ing agen , also de eloped by Co ning Inc.; i s
he mal conduc i i y is 4.1 W m
1
K
1
a oom empe a u e, which is
highe han κo Py ex®. The las ma e ial is a s anda d alumina o
significan ly highe he mal conduc i i y (33 W m
1
K
1
). These samples
a e discs o 12.7 mm diame e and 2 mm hick.
The specimens p in ed by obocas ing a e h ee-dimensional s uc-
u es wi h a ne like in e io o med by laye s o pa allel ods o diame e
“Ø”, wi h an in-plane sepa a ion “a”, and o hogonally s acked in z-di-
ec ion (Fig. 1a). The dis ance be ween consecu i e laye s wi h he same
od o ien a ion (h) is below wice he od diame e o assu e some od
o e lapping and a good con ac be ween ods in z-di ec ion (h/2ؼ
π
/4
in he p esen designs). The p in ed s uc u es ha e a ame ha con ou s
he sca olding p o iding addi ional suppo , as i can be clea ly seen in
Fig. 1b whe e an op ical iew o one ep esen a i e p in ed s uc u e is
D. Mu~
noz Codo níu e al. Open Ce amics 4 (2020) 100028
2
shown. The cell pa ame e s define he mac o-po osi y o he s uc u es,
π
mac o ¼1
π
⋅Ø2
2⋅a⋅hVF ame=VTo al[24]. A e p in ing, he s uc u es
we e usually hea ea ed a empe a u es in he 415–600 C ange o
emo e he o ganic addi i es o he ink and, subsequen ly, densified a
he co esponding sin e ing empe a u e o imp o e he pa icle con ac s
and he mechanical pe o mance o he ma e ial. Besides he mac o-po es
associa ed o he designed hollow cells ha a e clea ly obse ed in
Fig. 1b, addi ional po osi y may emain inside he ods ð
π
odÞ, which is
de e mined om
π
mac o and he o al po osi y, gi en by he measu ed
geome ical densi y and he heo e ical densi y o he od ma e ial. This
po osi y can be obse ed in he images (Fig. 2) aken by scanning elec-
on mic oscopy (SEM).
Th ee dis inc 3D ma e ials we e es ed, in pa icula , wo ce amics
ha co esponded o silicon ca bide (SiC) and C
2
AlC MAX-phase spec-
imens, and a composi e consis ing o educed g aphene oxide nano-
pla ele s ( GO) and a c oss-linked p ece amic polyme (Table 1). One o
he ce amic samples, labelled as SiC
50
-7, was p ocessed om SiC nano-
powde s o 50 nm o pa icle size (Nanos uc u es &Amo phous Ma e-
ials Inc., USA, poly ype 3C) con aining 5 w .% o Y
2
O
3
and 2 w .% o
Al
2
O
3
, bo h used as sin e ing addi i es. The sca old was sin e ed a 1700
C o 5 min in A a mosphe e (6 Pa o p essu e) using he Spa k Plasma
Sin e ing (SPS) echnique wi hou applying any mechanical p essu e
[19]. The MAX-phase specimen was p in ed om 98% pu e C
2
AlC MAX
lab-syn hesised powde s [25] and densified by p essu eless SPS a 1200
C o 10 min in A [20].
The so-called GO-PSZ sca old was ob ained by p in ing g aphene
oxide nanopla ele s (GO, N002-PDE Angs on Ma e ials Inc., USA, 2–3
nm hick and la e al size in he x-y plane 7
μ
m). The GO p in ed sample
was educed a 1200 C in ni ogen a mosphe e inside a g aphi e u nace
and, subsequen ly, acuum infil a ed wi h a liquid o ganic-polysilazane
(PSZ, a p ece amic polyme o Si, C, H, N) and ea ed a 200 Cina
ubula elec ic u nace unde a con inuous ni ogen flow o polyme
c osslinking [7]. The skele on consis ed o 93 and 7 w % o c osslinked
PSZ and GO, espec i ely. Fig. 2 illus a es ep esen a i e od mic o-
s uc u es o he h ee selec ed specimens.
3.2. The mal conduc ion es ing
The p oposed expe imen al se ing is based on he s eady-s a e uni-
di ec ional hea flow and is o ad an age, pa icula ly, when he mo-
couples a e di ficul o place in he sample. The scheme o he de ice is
p esen ed in Fig. 3a. I consis s o wo Pel ie cells espec i ely ac ing as
hea e and coole pla es ha allow s abilizing empe a u e di e ences o
~50 C be ween he ho (~60 C) and cold (~5 C) pla es. Two cylinde s
(15 mm in diame e and 45 mm in heigh ) o coppe wi h high he mal
conduc i i y (~400 W m
1
K
1
) we e placed in con ac o he hea e and
coole de ices o imp o e he hea di usion, while he es sample was
se in be ween. In some es s, addi ional s ainless s eel (AISI 310, labelled
as AISI) cylinde s (15 mm in diame e and 10 mm heigh ) o lowe
he mal conduc i i y (~16 W m
1
K
1
) we e in oduced be ween he
sample and he Cu cylinde s o inc ease he excessi ely small g adien s
achie ed jus wi h Cu (~0.03 C⋅pixel
1
). The me al pieces in con ac
wi h he specimen, ei he Cu o AISI, we e used as con ol samples in he
measu emen o he he mal g adien s. The p oblem sample should ha e
a sec ion simila o ha o he con ol samples. The whole measu emen
sys em (cylinde /specimen/cylinde ) was he mally insula ed o p e en
hea losses and induce uniaxial hea flow. A e ical slo (~5 mm wide)
Fig. 1. (a) Schema ic d aw o he inside sca olding design wi h indica ion o
he cha ac e is ic la ice pa ame e s ( he od diame e (Ø), in-plane sepa a ion
be ween ods (a), and dis ance be ween consecu i e laye s wi h same od
o ien a ion (h)); and (b) op ical image o a ep esen a i e eal p in ed s uc u e,
co esponding o SiC
50
-7, whe e he ame can be clea ly obse ed.
Fig. 2. SEM images o he ac u e su ace o ods co esponding o he 3D
specimens o : SiC
50
-7 (a), MAX-phase (b), and GO-PSZ (c). (a) F om Re s [19].
Table 1
Dimensions o he 3D p in ed specimens including D (squa e side in he x-y
plane) and Z (heigh ); and hei cha ac e is ic densi ies ( heo e ical –Th-, geo-
me ic –Geo- and ha o he od -Rod) and po osi ies ( o al, od and mac o).
3D-specimen dimensions
(mm)
densi y (g⋅cm
3
) po osi y (%)
D Z Th Geo Rod To al Rod Mac o
SiC
50
-7 11.6 4.6 3.3 1.1 2.7 67 23 59
MAX-phase 10.2 5.0 5.2 2.2 3.8 60 24 44
GO –PSZ 11.6 4.7 1.14 0.4 1.1 65 24 54
D. Mu~
noz Codo níu e al. Open Ce amics 4 (2020) 100028
3
was machined in he insula ion case along he he mal flow di ec ion
(Fig. 3a) o measu e he su ace empe a u e o he samples wi h a high
esolu ion in a ed came a (FLIR A325 SC, USA), as he a achmen o he
he mocouples in he p esen case was a challenging ask. The exposed
sample su ace was coa ed wi h g aphi e o elimina e e o s associa ed
wi h di e ences in he emissi i y be ween he samples and he con ol
me al cylinde s. The came a cap u es he adia ion o e he wa eleng h
ange o 8–12
μ
m and gene a es he mog aphic images om he em-
pe a u e dis ibu ion; i s esolu ion is 320 240 pixels and he accu acy
is 2%. Two ypes o lenses we e used, he s anda d lens o 18 mm (25)
and he close-up lens (IFOV 50
μ
m op ics) sui able o obse a ions a
highe magnifica ion (mac o lens). The obse a ion dis ance was 25 cm
o he s anda d lens and 2 cm o he mac o one.
Di e en fibe glass ma e ials we e used o he he mal insula ion, in
pa icula , igid iles and a flexible blanke , he la e wi h a he mal
conduc i i y o ~0.04 W m
1
K
1
a oom empe a u e, which is much
lowe han ha o he iles (~0.1 W m
1
K
1
). The he mal image shown
in Fig. 3b co esponds o a es in which insula ing fibe glass iles we e
used o he op and bo om egions o he s acking; whe eas a fibe glass
blanke was employed o he cen al measu emen a ea. I is e iden ha
while he blanke adequa ely insula es he sample, being undis inguish-
able om he en i onmen in he he mal image, he ile-co e ed egions
exhibi conside able hea losses, since hey display dis inc empe a u es
om ha o he su ounding; specifically, he uppe pa shows a em-
pe a u e ~7 C abo e he oom empe a u e and he lowe pa is ~1.5
C below. The e o e, he flexible insula ion fibe glass blanke was
p e e ed o he mally shielding he measu emen a ea.
As can be seen in Fig. 3c, he con ac he mal esis ance be ween he
di e en ma e ials led o impo an empe a u e d ops a he in e ace
(~20 C in he case o aluminium and coppe cylinde s). Acco dingly, a
high he mal conduc i i y he mal pas e (RS®, 503-357 ZP, 2.9 W m
1
K
1
, which con ains Ag pa icles) was applied o all in e aces. The use o
he he mal pas e ensu es be e hea ans e by conduc ion be ween he
wo ma e ials, dec easing he empe a u e d op o jus 5 C (75%
educ ion) in he case o aluminium and coppe ba s (Fig. 3c).
4. Resul s
4.1. Dense s anda d specimens
The unknown conduc i i y o he sample (κ
s
) can be calcula ed by
compa ing he g adien measu ed along i ( Ts) wi h ha o a e e ence
ma e ial ð T e Þo known he mal conduc i i y (κ e Þusing he
exp ession:
κs¼κ e ⋅ T e
Ts
(2)
Tempe a u e g adien s (in C⋅pixel
1
) measu ed o he Py oce am®,
Py ex®and alumina s anda ds, as well as o he con ol s ainless s eel
ba , a e collec ed in Table 2. In he absence o hea losses, he s ainless
s eel ba (κ¼16.3 W m
1
K
1
) can be used as e e ence ma e ial. The
applica ion in his case o he Eq. (2) p o ided he mal conduc i i y
alues ~ 65% lowe (κs;AISI equals o 0.4, 1.5 and 12.3 W m
1
K
1
o
Py ex®, Py oce am®and alumina, espec i ely, as shown in Table 2)
han hose ce ifica ed o he h ee e e ence ma e ials (κo 1.1, 4.1 and
33.0 W m
1
K
1
, espec i ely). The e o e, he hea losses h ough he
obse a ion slo , ine i able in he p oposed design, a e no negligible.
Ano he op ion was, hen, p oposed o de e mine he he mal conduc-
i i y o he samples in a mo e accu a e way. I consis ed in compa ing
g adien s ob ained in wo di e en es s o he specimen o unknown
he mal conduc i i y (s es ) and he e e ence ma e ial ( e es ), bo h
specimens ha ing he same dimensions. To do ha co ec ly, g adien s
should be fi s ly no malized by di iding hem by he a io “C”be ween
he g adien s along he AISI cylinde s in bo h es s, o he sample (s
subsc ip ) and e e ence ( e subsc ip ):
Fig. 3. (a) Schema ic o he he mal conduc i i y measu emen sys em showing
he s acking o he hea ing and cooling Pel ie elemen s, he con ol ba s and he
specimen (on he le ) and he he mal insula ion case wi h he obse a ion slo
(on he igh ). (b) The mal image o a es wi h wo me allic con ol cylinde s
(bo h labelled as M), using fibe glass iles (a op and bo om) and a fibe glass
blanke (cen al measu emen a ea) o he insula ing case ( empe a u e scale is
in C). (c) Tempe a u e p ofiles along he hea flow di ec ion o an aluminium/
coppe assembly wi hou (di ec con ac ) and wi h he mal pas e applied a
he con ac .
Table 2
Tempe a u e g adien s ( T) measu ed o he Py oce am®, Py ex®and alumina
dense ma e ials, as well as o he con ol me al cylinde in each es ; and he mal
conduc i i y calcula ed using he s ainless s eel (κs;AISI ) and da a o he Py o-
ce am® un (κ
s,Py oce am
) p e iously no malized using Eq. (3) as e e ences.
Ce ified he mal conduc i i y alues and a e age empe a u es (T
a e age
) a e
also included.
Tes
(ce ified κ)
Ma e ial
(T
a e age
)
T
(C⋅Pixel
1
)
κs;AISI (W
m
1
K
1
)
κ
s,Py oce am
(W
m
1
K
1
)
Py oce am®(4.1
Wm
1
K
1
)
AISI (37 C) 0.13 1.5 –
specimen
(21.5 C)
1.42
Py ex®(1.1 W
m
1
K
1
)
AISI (55 C) 0.05 0.4 1.15
specimen
(33 C)
2.07
Alumina (33.0 W
m
1
K
1
)
AISI (30 C) 0.12 12.3 33.50
specimen
(21.5 C)
0.16
D. Mu~
noz Codo níu e al. Open Ce amics 4 (2020) 100028
4
C¼ TAISI;s
TAISI; e
(3)
so ha empe a u e g adien s measu ed o he AISI ba s would be equal
in bo h es s.
The Py oce am® es was used as he e e ence o e alua e he he -
mal conduc i i y o Py ex®and alumina. In oducing he no malized
g adien s o he unknown-κsample and co ec ed da a o T e and κ e
o Py oce am®in Eq. (2), he deduced he mal conduc i i y alues
(labelled as κ
s,Py oce am
in Table 2) o Py ex®and alumina we e close o
he ce ified da a, wi h di e ences o less han 3%, which alida es his
p ocedu e.
4.2. Po ous 3D p in ed specimens
Table 3 shows da a o he po ous 3D samples es s. These s uc u es
we e analysed in wo o hogonal di ec ions, e e ed o as longi udinal
and ans e se, which co esponded o IR came a ocusing on he
pa e ned su ace (hea flowing along he x-y plane) o on one ame
la e al su ace (hea flux in z-di ec ion), espec i ely, as indica ed in
Fig. 4a. In addi ion, obse a ions we e made using bo h ypes o lens,
s anda d and mac o. A ele an esul is he aniso opy obse ed in he
hea flow as highe he mal g adien s we e measu ed o he ans e se
o ien a ion, which would indica e a lowe he mal conduc i i y in he z-
di ec ion, i.e. pe pendicula ly o he plane in which he ods a e aligned.
A compa able aniso opy has also been confi med o he elec ical
conduc i i y in simila s uc u es o di e en ma e ials, such as SiC,
g aphene nanopla ele s (GNP), and SiC-GNP composi es [24,26].
The plo o Fig. 4b shows an inc eased empe a u e g adien when
mo ing om SiC
50
-7 >MAX-phase > GO-PSZ s uc u es. I should be
no ed ha he obse ed endency canno be linked o he specimen
po osi y, nei he he mac o-po osi y no he od po osi y. In pa icula ,
al hough he h ee specimens ha e simila mac o-po osi ies, being in he
ange o 44–55%, and also he same od po osi y o ~24% (Table 1),
significan di e ences in g adien s a e e idenced, which imply ha
SiC
50
-7 and MAX-phase s uc u es would be ~4 imes mo e conduc i e
han GO-PSZ one. The a io be ween he longi udinal and ans e se
g adien s, which gi es an idea o he aniso opy, a ies om 1.1 o he
SiC
50
-7 s uc u e o ~1.7 o he GO-PSZ one.
Da a ep esen ed in Fig. 4b co esponds o he s anda d lens, since
la ge empe a u e fluc ua ions we e obse ed wi h he mac o lens.
Fig. 5a displays some ep esen a i e examples o he mal images and
empe a u e p ofiles o 3D p in ed specimens when obse ing he x-y
plane, i.e. he longi udinal o ien a ion. As seen, he empe a u e dis i-
bu ion allows pe cei ing he ods and mac o-po es in he x-y plane (IR
images in Fig. 5a), and also pe iodic fluc ua ions in he empe a u e
p ofiles induced by he mac o-po osi y inhe en o hese p in ed designs.
The he e ogenei ies associa ed wi h he p esence o hollow cells and
ods along he p ofile a e be e app ecia ed wi h he mac o lens han
wi h he s anda d ones (see he wo empe a u e p ofiles o he MAX-
phase s uc u e in Fig. 5a) due o he g ea e numbe o da a eco ded
pe cm (5 imes mo e han he s anda d lens). On he o he hand, es s
ca ied ou on a GO-PSZ s uc u e le elled by gen ly g inding showed
ha , al hough he p ofiles conside ably fla en (e en he p ofile along a
od is p ac ically linea ), local empe a u e d ops associa ed wi h he
mac o-po es we e s ill obse ed.
On he o he hand, he he mal conduc i i y o he me allic ba s also
a ec ed da a. Thus, es s ca ied ou wi h he coppe cylinde s in di ec
con ac wi h he specimen, wi hou in e cala ing s eel cylinde s, ga e
some p oblems, especially o he highly po ous samples. As i can be
seen in Fig. 5b o he GO-PSZ sample, using he coppe cylinde s
s ongly al e ed he empe a u e p ofiles pe pendicula o he hea flow,
since hey we e no fla excep in he cen al zone, changing om con ex
o conca e when mo ing away om he hea sou ce. Tha is, in he uppe
zone, highe empe a u es we e measu ed a he cen e han a he la e al
edges close o he insula ing case, while in he lowe zone, he empe -
a u es a he cen e we e lowe han a sides. Thus, i seems ha he
came a was somehow cap u ing he empe a u e o he op and bo om
me al con ols. In ac , when s eel cylinde s, wi h a much lowe con-
duc i i y han coppe (16 e sus 400 W m
1
K
1
, a oom empe a u e),
we e placed be ween he coppe and he es specimen, he cu a u e
significan ly educed, pa icula ly a he lowe pa o he sample, as
shown in Fig. 5b.
The e ec i e he mal conduc i i y wi hou conside ing he hea
Table 3
3D specimens he mally analysed indica ing he es ing o ien a ion, me al cyl-
inde s, a e age empe a u e, obse a ion lens and empe a u e g adien ( T).
G adien s a e no he same o bo h lenses because he obse a ion dis ances a e
di e en .
O ien a ion Me al
cylinde s
Specimen T
(C)
Lens T
(C⋅pixel
1
)
Longi udinal Cu GO-PSZ 25.3 S anda d 1.92
Mac o 0.12
Cu SiC
50
-7 26.8 S anda d 0.49
Mac o 0.04
AISI SiC
50
-7 25.9 S anda d 0.31
Mac o 0.06
AISI MAX-
phase
27.3 S anda d 0.64
Mac o 0.14
AISI GO-PSZ 23.3 S anda d 2.00
T ans e se Cu GO-PSZ 31.2 S anda d 3.24
Mac o 0.35
Cu SiC
50
-7 26.0 S anda d 0.57
Mac o 0.03
AISI MAX-
phase
33.3 S anda d 0.81
Mac o 0.16
Fig. 4. (a) Hea flow di ec ion in he es s o he longi udinal and ans e se
o ien a ions and (b) ba diag am showing he inc easing g adien s measu ed o
he 3D ma e ials o simila pa e ned s uc u e in he di e en es ing condi ions
shown in he legend.
D. Mu~
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5

ans e by con ec ion and adia ion o his ype o s uc u e o he wo
o ien a ions analysed, ha is, longi udinal (κ
L
), along he plane o
c issc oss ods, and ans e se (κ
T
), pe pendicula o ha plane, can be
es ima ed om he conduc i i y o he od (κ
od
) using a simple model o
he mal esis o s [24]:
κL¼∅
2aκ od (4)
κT¼h∅
a21h
2∅⋅κ od (5)
and in oducing he pa ame e s ha define he s uc u e, Ø, a and h
(Fig. 1a), which a e collec ed in Table 4 o each o he h ee es ed
ma e ials. As he ac o s ha mul iply κ od in he Eqs. (4) and (5) a e
simila o he h ee specimens (~0.17 and ~0.04, espec i ely), he
e ec i e conduc i i y would exclusi ely depend on κ od, which is
s ongly a ec ed by
π
od
and he in insic conduc i i y o he ma e ial.
F om he da a collec ed in Table 4, i is deduced ha he κ
L
/κ
T
a io was
~4, highe han he aniso opy es ima ed om he co esponding g a-
dien s a io (<2 as shown in Table 3). This can be explained by he
con ibu ion o he s uc u e ame in bo h o ien a ions, which has no
been aken in o accoun in he esis o s model. The ame con ibu ion o
he mal conduc i i y in he longi udinal and ans e se o ien a ions was
calcula ed in S1 sec ion o he supplemen a y in o ma ion. As seen, i
a ec s di e en ly o κ
L
and κ
T
, educing he aniso opy o alues anging
be ween 1.3 and 1.9, bo h close o hose deduced om he g adien
a ios.
Values o κ od in Table 4 we e es ima ed om he he mal conduc-
i i y o dense ma e ials (κ
0
) co ec ed by he od po osi y using Pabs -
G ego o a’s exponen ial ela ion [27]. In he case o he SiC
50
-7 and
MAX-phase dense specimens, κ
0
was measu ed by he lase flash me hod
(30 and 16 W m
1
K
1
, espec i ely). Fo he GO-PSZ specimen a κ
0
o 9
Wm
1
K
1
was es ima ed om he GO and PSZ olumen ac ions
conside ing bo h GO and PSZ as con inuos phases, and he mal con-
duc i i ies o 400 W m
1
K
1
o he GO (see calcula ion in S2 sec ion)
and 1 W m
1
K
1
o he c oss-linked polyme . A e κ
0
was co ec ed by
he od po osi y, κ od alues included in Table 4 we e in e ed, which
indica es ha he conduc i i y o he SiC
50
-7 s uc u e would be ~2 and
3 imes highe han ha o he MAX-phase and GO-PSZ s uc u es,
espec i ely. This ac ag ees wi h he expe imen al obse a ions when
compa ing hei g adien s unde simila condi ions (i.e. same me al
con ols). Thus, he g adien measu ed in he specimen o he
Cu/ GO-PSZ es is 4 imes la ge han ha measu ed o SiC
50
-7 and,
acco dingly, ha measu ed o he AISI/MAX-phase sys em is 3 imes
highe han ha measu ed o AISI/SiC
50
-7 (Fig. 4b and Table 3).
Al hough o ge absolu e alues o κ, a s anda d es would be necessa y
(a s uc u e o iden ical dimensions o hose es ed and o ce ified
he mal conduc i i y), hese esul s demons a e ha he p oposed
me hod o e s easonably da a in a compa a i e way ha allow p edic -
ing he he mal beha iou o he mac opo ous s uc u es. This p ocedu e
also has he c ucial ad an age ha using he mocouples is unnecessa y.
The co ela ion be ween he measu ed he mal g adien s and he he mal
conduc i i y es ima ed o he s uc u es is shown in Fig. 6, whe e da a
we e fi ed wi h T¼A⋅κ
x
being x ¼1.3.
Finally, he he mal conduc i i ies o he s uc u es es ed wi h he
AISI ba s we e es ima ed applying he p ocedu e desc ibed in sec ion 4.1,
ha is, compa ing he no malized g adien s using Eq. (3) and he es
wi h Py oce am®as a e e ence, as i was done wi h he dense ma e ials.
Thus, he conduc i i y o he MAX-phase s uc u e was es ima ed in 3.4
and 2.2 W m
1
K
1
o he longi udinal and ans e se di ec ions,
espec i ely, he a io be ween hem being 1.6; while o he SiC
50
-7 and
he GO-PSZ s uc u es, a longi udinal conduc i i y o 3.0 and 0.5 W m
1
K
1
, espec i ely, was ob ained. These alues, al hough somewha
di e ing, a e consis en wi h hose deduced om he esis o s model
conside ing he con ibu ion o he ames (Table 4). The e a e se e al
ac o s ha can explain he obse ed disc epancies, like e o s due o: i)
changes in he hea flow associa ed wi h he c oss sec ion di e ences
be ween he me al ba s (176 mm
2
) and he po ous 3D specimens (~125
and ~50 mm
2
o he on and la e al configu a ions, espec i ely, ac-
co ding o he da a in Table 1), which would induce la e al hea losses; ii)
Fig. 5. The mal images o he x-y pa e ned su ace (mac o lens da a) o he longi udinal o ien a ion and plo s o he empe a u e p ofiles along he lines depic ed in
he IR images o (a) and (b), co esponding o di ec ions aligned wi h (a) and pe pendicula o (b) he hea flow, espec i ely, and o di e en 3D ma e ials. P ofile
gene a ed wi h he s anda d lens o he MAX-phase s uc u e (dashed ed line) is also included in (a) plo o compa ison. Two p ofiles a e included o he g inded
GO-PSZ in plo (a): one h ough he mac o-po es (con inuous line) and ano he aken along a od (dashed line). Tempe a u e p ofiles in (b) we e eco ded a di e en
dis ances om he hea sou ce (ma ked on he he mog aphic images o (b)) o he Cu/ GO-PSZ and AISI/SiC
50
-7 a angemen s. The small empe a u e a ia ions a e
linked o he mac o-po es. (Fo in e p e a ion o he e e ences o colou in his figu e legend, he eade is e e ed o he Web e sion o his a icle.)
D. Mu~
noz Codo níu e al. Open Ce amics 4 (2020) 100028
6
he use o an inapp op ia e e e ence es ha co esponded o a dense
specimen o di e en geome y and size; iii) he la ge empe a u e a -
ia ions associa ed wi h he p esence o he mac o-po es and ods ha can
also a ec he measu ed g adien s; and finally, i ) a con ibu ion o he
con ec ion and adia ion hea ans e ha has no been aken in o ac-
coun in he esis o s model. I should be poin ed ou ha , in his me hod,
he he mal esis ances linked o con ac s would no ha e influence in he
measu emen o he empe a u e g adien s, as he use o he IR came a
allows pe ec ly hei disc imina ion.
5. Conclusions
The p oposed app oach o es ima e he he mal conduc i i y o
mac opo ous ma e ials based on high esolu ion in a ed he mog aphy
allows de e mining empe a u e g adien s and pe ec ly disce ning
he mal esis ances linked o con ac s while a oiding he use o he -
mocouples, as a aching he mocouples o 3D po ous s uc u es de el-
oped by di ec ink w i ing esul s qui e a challenge. Tempe a u e
fluc ua ions associa ed wi h he e icula ed s uc u e a e pe ec ly
ep oduced along he he mal p ofile eco ded when using he mac o
lens. An excellen co ela ion be ween he measu ed empe a u e g a-
dien s and he he mal conduc i i y o he 3D s uc u es, es ima ed om
a simple esis o s model, is es ablished. The mal g adien s de e mined in
wo o hogonal di ec ions, co esponding o IR obse a ions o he od
la ice (hea flowing along he x-y plane) and he la e al ame (hea flux
in z-di ec ion), suppo he esul o an aniso opic he mal conduc i i y
wi h an aniso opy ac o <2 o his ype o s uc u e.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing financial
in e es s o pe sonal ela ionships ha could ha e appea ed o influence
he wo k epo ed in his pape .
Acknowledgmen s
This wo k was suppo ed by Spanish p ojec RTI2018-095052-B-I00
(MICINN/AEI/FEDER, UE). JJM acknowledges he financial suppo o
MICINN h ough he FPI con ac e : BES-2016-077759. DMC hanks he
UPM financial aid o g adua e s uden s.
Appendix A. Supplemen a y da a
Supplemen a y da a o his a icle can be ound online a h ps
://doi.o g/10.1016/j.oce am.2020.100028.
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Table 4
Pa ame e s o he 3D s uc u es (diame e , “Ø”, in-plane dis ance be ween ods, “a”, dis ance be ween pa allel ods in z-di ec ion, “h”), es ima ed od he mal con-
duc i i y (κ od) e ec i e he mal conduc i i y in he longi udinal (κ
L
) and ans e sal (κ
T
) di ec ions deduced om he esis o s model (Eq. (4) and (5)) and co e-
sponding alues including he ame con ibu ion. The expe imen ally es ima ed (Exp) alues a e also included.
Ma e ial Ø(
μ
m) a (
μ
m) h (
μ
m) The mal conduc i i y (W m
1
K
1
)
k od kL
Eq. (4)
kT
Eq. (5)
kL
F ame
kT
F ame
kL
Exp
SiC
50
–7[19] 210 690 360 19.2 2.9 0.7 4.0 2.5 3.0
MAX-phase [20] 265 760 415 10.0 1.7 0.4 2.4 1.8 3.4
GO-PSZ [7] 345 1010 540 5.6 1.0 0.2 1.3 0.7 0.5
Fig. 6. Tempe a u e g adien s as a unc ion o he he mal conduc i i y es i-
ma ed o SiC
50
-7, MAX-phase and GO-PSZ s uc u es (da a included in
Table 4). Cu e is he da a fi wi h T¼A⋅Kx.
D. Mu~
noz Codo níu e al. Open Ce amics 4 (2020) 100028
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