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Nume ical analysis on he s a e o cha ge o an ul a-
high empe a u e la en hea ene gy s o age sys em
The case s udy o FeSiB alloy phase-change
My o Zeneli1 h ps://o cid.o g/0000-0002-7875-7025, and Alejand o Da as1 h ps://o cid.o g/0000-0001-5964-3818
1 Ins i u o de Ene gía Sola , Uni e sidad Poli écnica de Mad id, A . Complu ense s/n, 28040, Mad id, Spain
*Co espondence: My o Zeneli, [email p o ec ed]
Abs ac . Ul a-high empe a u e he mal ene gy s o age (UHTES) and con e sion is an
eme ging ield o echnology ha enables much highe ene gy densi ies (>1 MW h) and con-
e sion e iciencies han con en ional he mal ene gy s o age echnologies. Ou esea ch
g oup o Sola Ene gy Ins i u e is cu en ly de eloping a no el la en hea he mopho o ol aic
(LHTPV) ba e y ha u ilizes Si-based alloys o s o e ei he su plus enewable elec ici y o
concen a ed sunligh in he o m o la en hea a empe a u es close o 1200 ºC and con e
i back o elec ici y on demand. De e mining he S a e o Cha ge (SoC) o his ul a-high em-
pe a u e he mal ba e y is impe a i e o egula e i s eal- ime ope a ion and op imize i s pe -
o mance. Howe e , using o se e al senso s wi hin he s o age sys em –as mainly done in
low empe a u e phase change ma e ials (PCMs) o quan i y hei SoC - becomes cos ly and
challenging o his ange o ope a ing condi ions. This s udy p esen s a nume ical me hod,
which is used o ge an unde s anding o he physical p ocesses aking place du ing he LHTPV
ope a ion and cap u e comp ehensi e da a o ime a ying low a iables ha can be di icul
o eco d du ing eal- ime ope a ion. Ou esul s indica e ha we can desc ibe he sys em’s
SoC by measu ing he ime- a ying empe a u e a i s sidewalls and he inpu /ou pu hea lux
alues, wi hou he need o knowing be o ehand he he mophysical p ope ies o he used
ma e ials. Based on hese a iables we can de ine se e al indica o s ha can help us ob ain a
be e unde s anding o he equi ed physical signals o be measu ed in o de o de e mine i s
SoC, du ing eal- ime ope a ion.
Keywo ds: S a e o cha ge, The mal ene gy s o age, Nume ical simula ion
1. In oduc ion
The u gen need o deca boniza ion necessi a es a global swi ch owa ds low emission uels
o clean ene gy al e na i es. The mos po en deca boniza ion s a egy is inc easing he sha e
o Renewable Ene gy Sou ces (RES) in he ene gy mix. The mal ene gy s o age (TES) is a
game change echnology, essen ial o op imizing he use o he in e mi en RES, owing o
i s abili y o s o ing he excess elec ici y om a iable RES (sola and wind) o di ec ly s o ing
o sola ene gy in he o m o hea , which can be con e ed in o elec ici y on demand. Ul a-
high empe a u e he mal ene gy s o age (UHTES) is an eme ging ield o science and ech-
nology [1] ha enables much highe ene gy densi ies (>1 MWh h) and hea - o-elec ici y con-
e sion e iciencies han con en ional TES echnologies, i.e. mol en sal s, which ope a e a
maximum empe a u es o ~600 ºC. Among he se e al UHTES solu ions ha a e being de-
eloped, UPM is de eloping a i s p emises in Mad id a pilo -scale he mal ba e y [2] ha
s o es ei he su plus enewable elec ici y (powe - o-hea - o-powe -P2H2P) o concen a ed
sunligh (sola - o-hea - o-powe -S2H2P) in he o m o la en hea a ex eme empe a u es
Au ho las name e al. | Publica ion sho i le x (yea ) "con e ence name” (i applicable)"
(o e 1200 ºC) and con e s i back o elec ici y on demand by using ad anced he mopho o-
ol aic echnology. This La en Hea The mopho o ol aic (LHTPV) ba e y (Figu e 1) is a com-
pac (10 imes highe han concen a ed sola powe - CSP) de ice ha can s o e he mal
ene gy a low cos s (< 10 €/kWh h). The echnology uses silicon-based phase change ma e ials
(PCMs) o TES and he mopho o ol aic (TPV) gene a o s o he mal- o-elec ic ene gy con-
e sion. TPV de ices ha e simila ope a ion o sola cells, bu hey a e con e ing he mal
adia ion, ins ead o sunligh , in o elec ici y. TPVs consis o an incandescen emi e ( empe -
a u es > 1000 ºC) ha adia es pho ons owa ds TPV cells, which p oduce elec ici y h ough
he pho o ol aic e ec . This de ice has eached 40 % con e sion e iciencies [3, 4], being as
e icien as comme cial s eam u bines. The high e iciency combined wi h a simple and com-
pac design (e.g. lack o hea ans e luids o mo ing pa s) open he oad o a new ype o
echnology ha can lead in he sola ene gy ield du ing he o hcoming decades.
A undamen al challenge in his b eak h ough concep is de ining i s s a e-o -cha ge (SoC),
when ope a ing in a cycling pe o mance. Moni o ing he SoC is necessa y o he op imal,
e icien and sa e ope a ion o a TES sys em, which can become challenging when ope a ing
a ul a-high empe a u es. In la en hea he mal ene gy s o age uni s, his can be equi alen
wi h he amoun o he s o ed la en hea o he liquid ac ion (LFR: a io o liquid PCM o
o e all PCM mass) alues [5]. A possible app oach o de ine such pa ame e s is o u ilize
se e al p obes o measu e he empe a u e change ac oss he PCM olume o a he c ucible
sidewalls, whe eas addi ional da a such as he PCM he mal p ope ies a e usually equi ed,
e en o da a-d i en app oaches cu en ly a ailable in he ecen li e a u e [6]. In a p ac ical
applica ion, quan i ying accu a ely he PCM he mal p ope ies becomes o en challenging.
Thus, he e is a need o de ining he SoC o a LHTES sys em by elying only on ope a ing
pa ame e s, such as inpu /ou pu hea lux and empe a u e alues.
The p esen wo k s udies a LHTES sys em, which will be ope a ing wi h measu ing p obes on
he ex e io su aces o he c ucible, due o he demanding ope a ing condi ions. A nume ical
model is de eloped o obse e he ansien esponse o he sys em unde bo h cha ging and
discha ging phases and gain a be e unde s anding on he speci ic equi emen s ha need o
be me o ins all he necessa y equipmen in a eal ime ope a ion. This model has he capaci y
o simula ing he mel ing (cha ging) and solidi ica ion (discha ging) p ocess o he ba e y and
o cap u e comp ehensi e da a on ime a ying low a iables (bo h local and global) ha can
be di icul and expensi e o eco d du ing eal- ime ope a ion, owing o he co esponding
sophis ica ed ins umen a ion equi ed. Se e al indica o s a e quan i ied in o de o assis he
measu ing p ocess. Such analysis aims o he i s ime o ex ac gene alized esul s o a
he mal ene gy s o age sys em´s beha io and SoC, by elying only on eal- ime measu ed
ope a ing pa ame e s ( empe a u e and hea lux), decoupling, hus, he SoC de ini ion by he
ma e ial he mal p ope ies al eady used in p e ious wo ks.
Figu e 1 Schema ic o he LHTPV ba e y sys em nume ical s a egy o de ine i s SoC (TL: Ou e wall
empe a u e, TR: Inne wall empe a u e acing he TPV con e e ).
2. Nume ical model
A ansien compu a ional luid dynamics (CFD) model based on he en halpy-po osi y ap-
p oach [7], a ailable in Ansys Fluen TM ( 24.R2) pla o m is used o he simula ions. The CFD
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model applied in his wo k has been p e iously e i ied agains an analy ical model [8], de-
sc ibing he silicon solidi ica ion p ocess inside a sealed c ucible and alida ed agains expe -
imen al measu emen s [9] o pa a in wax mel ing a low empe a u es, owing o he lack o
expe imen al da a o such high empe a u es. Some o he assump ions adop ed include:
1. T ansien luid low and hea ans e mechanisms;
2. Inclusion o g a i y e ec ;
3. Solu ion o lamina low condi ions, due o low Reynolds numbe s o he cases s udied;
4. The mol en PCM is ea ed as an incomp essible New onian luid;
5. Bo h solid and liquid phases o he PCM a e homogeneous and iso opic;
6. Tempe a u e dependen he mophysical p ope ies o he PCM and c ucible;
7. The PCM densi y is cons an ;
8. Radia i e hea ans e wi hin he essel is neglec ed;
9. Con ac he mal esis ances be ween solid/liquid PCM and walls a e negligible;
10. E apo a ion o any con ained gas (p obably obse ed du ing he ini ial cycles o he
phenomenon) is no conside ed.
A mo e de ailed analysis o he en halpy po osi y equa ions can be ound in [8] and [10].
Conce ning he SoC de ini ion, we use a liquid ac ion-based app oach acco ding o which:
SoC = LFR( )
(1)
The main pa ame e s ha can be used o implici ly quan i y i s LFR and, hus, and i s SoC
include he a) Time- a ying a ea weigh ed a e age empe a u e a he ba e y sidewalls (T) b)
Inpu / Ou pu lux (cons an Q o ime a iable Q( )), and c) cha ging/discha ging ime ( ).
2.1 Geome y and mesh
The physical domain is a cylind ical c ucible made o g aphi e inside which a FeSiB alloy un-
de goes solidi ica ion-mel ing h ough a cycling pe o mance (Figu e 1). Bo h he PCM and he
c ucible a e explici ly simula ed as luid and solid domains, espec i ely. Owing o he c ucible’s
symme y ac oss he cen al axis a 2D axisymme ic case is simula ed o educe he induced
compu a ional cos . Fu he mo e, a con o mal g id o 6,348 quad ila e al elemen s is u ilized
o inc ease he le el o accu acy, nea he phase-change egion. This g id has been chosen
based on a p elimina y g id independence s udy compa ing 3 g ids o a) 6,348, b) 25,392 and
50,784, esul s o which can be ound in he Supplemen a y ma e ial.
2.2 Model se -up
Conce ning he model se -up, he mos impo an pa ame e in he en halpy-po osi y app oach,
which a ec s he PCM mel ing a e, is he mushy zone pa ame e . This pa ame e is usually
aken by de aul equal o 105. Based on he p e ious expe ience o he au ho s ega ding he
solidi ica ion-mel ing o Si alloys his pa ame e can be quan i ied by he ollowing equa ion
[11]:
Amush =180 𝜇
𝜌 𝑆𝐷𝐴𝑆2
(2)
, whe e μ is he liquid PCM iscosi y, ρ is i s densi y and SDAS he seconda y dend i es a m
spacing. By assuming an indica i e SDAS alue o 40 μm o FeSiB alloys we ge an Amush
alue o 5*105. I should be no ed ha in o de o accu a ely p edic he mel ing beha io o a
speci ic PCM an expe imen ally measu ed alue o he SDAS should be used. In he solidi i-
ca ion p ocess, which is a conduc ion domina ed p ocess his pa ame e does no a ec sig-
ni ican ly he phase-change a e. Indica i e ma e ials p ope ies can be ound in [12].
2.3 Bounda y and ope a ing condi ions
App op ia e bounda y condi ions a e se a he c ucible sidewalls. Du ing cha ging phase (mel -
ing), a cons an inpu hea lux lows in o he domain h ough he c ucible (ou e ) wall. Du ing
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discha ging phase (solidi ica ion), a cons an hea lux is ex ac ed om he TPV con e e ,
which aces he c ucible inne wall. Adiaba ic condi ions a e assumed o he uppe and bo om
c ucible walls. A ini ial condi ions, he whole domain is pa ched o 27 ºC (mel ing phase). The
solidi ica ion p ocess s a s igh a e he PCM mel ing ends.
Th ee s a es a e used du ing mel ing phase (I. sensible hea s o age: only solid phase, II. la en
hea and sensible hea s o age: phase change, III. sensible hea s o age: only liquid phase).
The same applies o he solidi ica ion phase (IV. sensible hea ex ac ion: only liquid phase,
V. la en hea and sensible hea ex ac ion, VI. sensible hea ex ac ion: only solid phase).
Du ing calcula ion, he ansien esponse o he wall empe a u e is moni o ed o hea lux
alues wi hin [5Q e -Q e /5], wi h Q e = 143 kW (i is he ene gy equi ed o phase-change o 1-
2 h) and se e al indica o s a e de i ed o desc ibe he sys em’s SοC, indica i ely desc ibed
o cha ging phase. The o e all PCM mass is equal o 717 kg, whe eas he olume is equal o
0.134 m3 o he PCM i sel and 0.221 m3 o he whole sys em.
3. Resul s and discussion
3.1 SoC indica o s
1s indica o : Time- a ying wall empe a u e
This a common indica o used o desc ibe a sys em’s SoC. Figu e 2 shows ha he ou e wall
empe a u e (TL) ollows a linea p o ile du ing he PCM p ehea ing (only solid) and hen un-
de goes a slope change, when he PCM mel ing s a s (poin 1). This s eep change signi ies
he quick ansi ion om solid o liquid phase a his a ea. As he phenomenon e ol es, he
phase-change on mo es inwa ds he domain un il i eaches he c ucible inne su ace
( ouching he TPV) and he empe a u e he e (TR) changes acco dingly (poin 2). F om his
poin and onwa ds he TR is almos cons an owing o he phase change phenomenon e olu ion
a he inne su ace. When he solidi ica ion-mel ing ends, he empe a u e a his su ace in-
c eases ab up ly (poin 3). Al hough being a he simple o moni o , i is e iden ha due o he
sys em’s complex dynamics du ing eal- ime applica ion i is di icul o unde s and he meaning
o such empe a u e changes, especially when hey a e no s eep, and, hus, ex ac accu a ely
he sys em’s SoC.
Figu e 2 Time a ia ion o he wall empe a u e alues o he LHTPV ba e y (mel ing, Q e -, Q e /2 --).
2nd indica o : Time a ying D(TL-TR)/d a e
A mo e accu a e ep esen a ion o he sys em SoC a ises om he ime e olu ion o he global
empe a u e g adien (DT/d ) a e wi h ime. No ably, in his case he empe a u e is moni o ed
no as an absolu e alue a each su ace o in e es . On he con a y, he empe a u e di e -
ence be ween he wo su aces is moni o ed, i.e. T = TL-TR is used o desc ibe his a iable.
Based on he nume ical analysis aluable esul s can be ex ac ed o he DT/d : D(TL-TR)/d
esponse du ing bo h he cha ging and discha ging phases (Figu e 3a-c).
Du ing cha ging (Figu e 3a), he PCM s a s om he sensible phase (s a e I). In his egion,
he DT/d g adien s a s a a high alue du ing he sys em s a -up and declines un il i eaches
a small alue (in he o de o ~1. e-3 K/s o Q equal o Q e ). Du ing his s a e, he sys em
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beha es as a conduc i e ma e ial and he incoming hea is s o ed in o he sys em in he o m
o sensible hea . When he phase-change s a e s a s (s a e II) an ab up change in he DT/d
g adien is obse ed. Du ing his s a e, he DT/d g adien ends o inc ease – signi ying a
decele a ion mechanism on he sys em empe a u e inc ease- owing o he high la en hea
sou ce e m, equi ed o he phase change. When he mel ing phenomenon inishes, he DT/d
g adien unde goes a apid change and hen he DT/d alue ends o s abilize in a small alue,
likewise o s a e I. A simila phenomenon is obse ed du ing he PCM discha ging, Figu e 3b.
I should be no ed ha he lowe he hea lux inpu /ou pu he lowe he change o magni ude
o he DT/d a iable du ing he ansi ion om he di e en S a es. On he o he hand, as Q
inc eases, signi ican oscilla ions o he DT/d a iable a e obse ed, du ing he ini ial minu es
o he phase change p ocess (e.g. yellow line in Figu e 3c), making i di icul o de ine he
s a ing poin o he phase change p ocess.
a)
b)
c)
Figu e 3 Time a ia ion o he a ea-weigh ed a e age empe a u e di e ence (TL-TR) g adien a e wi h
ime du ing a) mel ing and b) solidi ica ion phase o Q e , and, c) mel ing phase o all cases (blue egion:
solid s a e, pink egion: liquid s a e, whi e egion: phase- ansi ion s a e).
3 d indica o : (DT/d /Q) a ia ion wi h PCM mel ac ion (LFR)
In eal ime applica ion, he Q alue (in lux/ou lux) will be ime- a ying. Thus, i is impo an o
obse e he phase-change beha io o he sys em o di e en Q alues. As can be no iced,
almos simila p o iles a e achie ed o he (DT/d )/Q a ia ion wi h he LFR alues, Figu e 4a.
This is an impo an obse a ion, o he desc ip ion o he phase-changing p ocess as i leads
o he conclusion ha a gene al equa ion combining hese wo pa ame e s can be ex apola ed
based on he nume ical esul s. Such an equa ion – o a simila one - can be inco po a ed in o
he PLC algo i hm o desc ibe he SoC o he sys em based on only expe imen ally measu ed
pa ame e s (a) he empe a u e a he walls, T and b) he inpu /ou pu hea lux, Q).
Ano he conclusion d awn om his analysis, is ha he PCM liquid ac ion ollows and almos
linea p o ile, Figu e 4b, du ing he phenomenon e olu ion, o he speci ic ange o ope a ing
condi ions s udied. Thus, by knowing he s a ing and ending poin o he PCM phase we can
es ima e he LFR by a linea in e pola ion (assuming he hea inpu /ou pu ) is cons an o a
speci ic ime in e al. This can be ex ac ed by obse ing he changing in he DT/d g adien
wi h ime, Figu e 3. Fo his p ocess, his o ical da a a e needed o be ex ac ed du ing he
sys em eal- ime ope a ion.
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a)
b)
Figu e 4 a) (DT/d /Q) g adien wi h LFR (LFR: 0 = solid, 1 = liquid) and b) ime a ia ion o he PCM
liquid ac ion o se e al Qin alues (mel ing phase).
4 h indica o : Tempe a u e uni o mi y
A supplemen a y indica o is he empe a u e uni o mi y index, UI
1
(a dimensionless numbe
a ying wi hin 0 and 1, whe e 0 indica es poo uni o mi y and 1 indica es pe ec uni o mi y
[13]) a he c ucible sidewalls. While his pa ame e does no explici ly desc ibe he SoC o he
sys em, i is c ucial o quan i y i nume ically o de e mine he numbe and placemen o he
measu emen p obes o be used du ing he expe imen al campaigns. Du ing he cha ging
phase (Figu e 5a) as he phase change p og esses, he solid-liquid on eaches he inne
su ace o he c ucible (close o he TPV), he eby a ec ing he empe a u e ield. Owing o a
non-uni o m phase-changing on in his su ace (Figu e 2) and, hus, a non-uni o m hea
ans e mechanism (pa ly in he o m o sensible hea and pa ly as la en hea ), a non-uni o m
empe a u e p o ile is c ea ed in his egion. On he con a y, a he ou e su ace (hea ed wall)
he liquid ully PCM occupies he su ace om he s a o he phase change, esul ing in a high
empe a u e uni o mi y (wi h he uni o mi y index close o uni y).
a)
b)
Figu e 5 Uni o mi y index s. ime o Q e , Q e /2 and 2.5 Q e /5 (a: mel ing and b: solidi ica ion phase).
1
UI= 1-Σ(Ti-Ta e )/2*Ta e , wi h Ti: Local empe a u e, Ta e : A e age empe a u e in a su ace
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Du ing he discha ging phase (Figu e 5b) he e e se occu s. The solid phase o ms uni o mly
a he inne su ace o he c ucible i s . Then, he phase-changing on mo es g adually o-
wa ds he ou e c ucible su ace. In he inal s ages o he phase change, a sligh inhomoge-
nei y in he empe a u e dis ibu ion is obse ed a he ou e c ucible walls, owing o he non-
homogeneous phase-changing on o med a his a ea.
To ob ain accu a e empe a u e measu emen s, mul iple p obes a e equi ed, gi en he non-
uni o m empe a u e p o iles du ing bo h p ocesses. The highe he hea ing and/o cooling a e
he g ea e he numbe o p obes needed o p ope ly cap u e he phenomenon. No ably, o
hea ing/cooling a es exceeding Q e he minimum uni o mi y index d ops below 0.95 du ing he
phase change p ocess.
4. Conclusions
In his wo k, we in oduce a heo e ical analysis o e alua ing he s a e o cha ge (SoC) o a
la en hea he mopho o ol aic ba e y ope a ing a ul a-high empe a u es (~1200 oC). We
u ilize an ad anced ansien compu a ional luid dynamics model o in oduce and quan i y
se e al indica o s ha allow o he es ima ion o he SoC using only he ex e nal wall empe -
a u e o he PCM con aine and he inpu /ou pu hea lux da a. The nume ical esul s demon-
s a e ha moni o ing hese ope a ing pa ame e s is su icien o accu a ely acking he SoC,
a inding ha holds po en ial o applica ions whe e senso placemen wi hin he PCM is chal-
lenging, as in ul a-high empe a u e applica ions. Fu he mo e, his app oach demons a es
ha he e is no need o know a p io i he sys em’s he mophysical p ope ies, simpli ying he
SoC es ima ion du ing i s eal- ime ope a ion. In a u u e analysis, ou esea ch g oup en is-
ages o de elop a gene ic model capable o de ining he SoC o a wide ange o ope a ing
condi ions, PCMs, and c ucible dimensions by using a da a-d i en analysis. This model can
be, hen, inco po a ed in o a PLC o moni o du ing eal- ime ope a ion he sys em´s SoC.
Concluding, he analysis p o ides ini ial conclusions o a speci ic PCM and ope a ing condi-
ions, pa ing he way o a mo e gene ic model o be de eloped in u u e s udies.
Au ho con ibu ions
My o Zeneli: W i ing o iginal d a , Me hodology, Fo mal Analysis, Da a Cu a ion, Visualiza-
ion; Alejand o Da as: W i ing – e iew & edi ing, Me hodology, Supe ision
Da a a ailabili y s a emen
Da a suppo ing he esul s o he a icle can be accessed h ough excel iles in Zenodo epos-
i o y (h ps://doi.o g/10.5281/zenodo.13758813).
Unde lying and ela ed ma e ial
Simula ion esul s a e deposi ed on Zenodo eposi o y (//doi.o g/10.5281/zenodo.13758813)
Compe ing in e es s
“The au ho s decla e ha hey ha e no compe ing in e es s.”
Funding
This p ojec has ecei ed unding om he Eu opean Union’s Ho izon 2020 esea ch and in-
no a ion p og amme unde he Ma ie Skłodowska-Cu ie g an ag eemen SHINE
(GA:101145914). Pa o he wo k has been also ca ied ou wi hin THERMOBAT p ojec unde
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g an ag eemen 101057954. Views and opinions exp essed a e howe e hose o he au ho s
only and do no necessa ily e lec hose o he Eu opean Union. Nei he he Eu opean Union
no he g an ing au ho i y can be held esponsible o hem.
Re e ences
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