Resea ch Pape
A no el wo-dimensional e ec i eness-NTU me hod o high- empe a u e
cascade la en hea s o age
Hussein Alawai Ib ahim Al-Saaidi , An on Lopez-Roman , C is ina P ie o
*
Uni e si y o Se ille, Ene gy Enginee ing Depa men , Camino de los Descub imien os s/n, 41092 Se illa, Spain
ARTICLE INFO
Keywo ds:
The mal ene gy s o age sys em
Cascade phase change ma e ials
Two- dimensional
ε
-NTU me hod
CFD model
Me al wool
ABSTRACT
This s udy p esen s a no el wo-dimensional e ec i eness numbe o ans e uni (
ε
-NTU) me hod o cha ac-
e ise and op imise cascade la en hea s o age (CLHS) sys ems, inco po a ing me al wool enhancemen o
imp o ed he mal pe o mance. The p oposed model p o ides a mo e accu a e and compu a ionally e icien
al e na i e o con en ional one-dimensional app oaches o analysing hea ans e in high- empe a u e la en
hea s o age (LHS) applica ions. Cu en pe o mance e alua ion me hods s ill lack p ecision due o he eliance
on simpli ied assump ions ega ding ma e ial p ope ies and hea ans e dynamics. Mo eo e , mos enhance-
men s a egies ocus on single-PCM sys ems, necessi a ing dedica ed esea ch e o s o CLHS-speci ic solu ions.
Valida ion h ough compu a ional luid dynamics (CFD) simula ions demons a es s ong ag eemen , con i ming
he me hod’s eliabili y in p edic ing he mal beha iou du ing bo h cha ging and discha ging p ocesses. The
esul s e eal ha he cascade con igu a ion educes he mal esis ance and enhances hea ans e e iciency,
while he addi ion o me al wool inc eases he e ec i eness o he sys em by up o 54% in single la en hea
s o age (SLHS) and 20% in CLHS wi hou addi i es. These indings es ablish he wo-dimensional
ε
-NTU me hod
as a obus design and op imisa ion ool o nex -gene a ion he mal ene gy s o age sys ems in enewable ene gy
eal-wo ld applica ions.
1. In oduc ion
Sola ene gy is one o he mos p omising enewable ene gy e-
sou ces. Howe e , i s in e mi en na u e poses a signi ican challenge o
ensu e a s able ene gy supply. To add ess his p oblem, he mal ene gy
s o age (TES) sys ems ha e been de eloped as an e ec i e solu ion o
imp o e ene gy e iciency and balance supply and demand luc ua ions.
TES sys ems a e inc easingly being used o s o e excess ene gy gene a ed
om enewable sou ces, such as sola o wind, and elease i when de-
mand is high, hus imp o ing ene gy e iciency and g id s abili y [1,2].
TES can be classi ied in o h ee main ca ego ies: sensible hea s o age,
la en hea s o age (LHS), and he mochemical hea s o age. Among
hese, LHS sys ems s and ou due o hei high ene gy s o age densi y
and abili y o main ain a cons an phase change empe a u e [3,4].
Despi e hese ad an ages, a majo limi a ion o LHS sys ems is he low
he mal conduc i i y o phase change ma e ials (PCMs), which hinde s
hea ans e and leads o a slow cha ging and discha ging a e [5,6]. To
mi iga e hese d awbacks, cascaded la en hea s o age (CLHS) has
eme ged as a p omising al e na i e, o e ing imp o ed hea ans e
a es, enhanced e iciency, and a mo e uni o m ou le empe a u e o he
hea ans e luid (HTF) compa ed o single PCM sys ems [7]. Howe e ,
CLHS echnology emains la gely in he heo e ical and labo a o y
esea ch s ages, wi h limi ed eal-wo ld applica ions [8].
Se e al s udies ha e explo ed di e en CLHS con igu a ions and
hei he mal pe o mance enhancemen s. Fa id and Kanzawa [9]
in es iga ed a he mal s o age sys em wi h h ee PCMs, epo ing a 15 %
inc ease in hea ans e a e compa ed o single PCM sys ems. Simila ly,
Michels and Pi z-Paal [10] conduc ed expe imen al and nume ical as-
sessmen s o CLHS o high- empe a u e applica ions, demons a ing i s
bene i s in achie ing highe hea ans e a es and mo e s able ou le
empe a u es. Howe e , hey highligh ed ha he design complexi y and
low he mal conduc i i y o PCMs emain signi ican obs acles. Fu he
nume ical s udies ha e explo ed inno a i e CLHS con igu a ions. Rud a
Mu hy e al. [11] compa ed he hea ans e pe o mance o a ape ed
CLHS shell-and- ube sys em wi h a con en ional cylind ical design,
epo ing a 17.6 % imp o emen in mean powe ou pu and a highe
mel ing a e. Likewise, Elsanusi and Nso o [12] s udied he he mal
beha iou o mul iple PCMs in a hea exchange and ound ha a se ies
PCM a angemen educed he o al mel ing ime by 15.5 % compa ed o
a single PCM se up.
Fo he design and op imisa ion o he mal ene gy s o age sys ems,
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (C. P ie o).
Con en s lis s a ailable a ScienceDi ec
Applied The mal Enginee ing
jou nal homepage: www.else ie .com/loca e/ap he meng
h ps://doi.o g/10.1016/j.appl he maleng.2025.127452
Recei ed 22 Ap il 2025; Recei ed in e ised o m 30 June 2025; Accep ed 3 July 2025
Applied The mal Enginee ing 278 (2025) 127452
A ailable online 5 July 2025
1359-4311/© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
he one-dimensional e iciency-numbe o ans e uni s (
ε
-NTU) me hod
is widely used. Tay e al. [13] in oduced a one-dimensional
ε
-NTU
echnique o cha ac e ise PCM-based TES sys ems, allowing apid p e-
dic ion o he hea ans e a e and op imisa ion o he sys em.
Fu he mo e, [14] alida ed his me hod using a CFD-based shell-and-
ube PCM model, p edic ing bo h he mel ing and solidi ica ion p o-
cesses. Despi e ad ancemen s in he one-dimensional
ε
-NTU me hod,
his me hod assumes uni o m he mophysical p ope ies and simpli ied
hea ans e beha iou [15,16] and which makes he selec ion o PCMs
in CLHS sys ems mo e complica ed due o une en mel ing and solidi i-
ca ion ime occu ing due o he di e en he mophysical p ope ies o
hese ma e ials. These app oxima ions in oduce signi ican e o s in
eal-wo ld applica ions and limi hei applicabili y o mo e complex
CLHS con igu a ions, highligh ing he need o mo e ad anced model-
ling echniques.
The e a e ew heo e ical s udies ha ho oughly examine he op i-
misa ion and pe o mance o he CLHS sys em. Xu and Zhao [17] op i-
mised he CLHS design using en ansy heo y, demons a ing ha CLHS
con igu a ions could ex end he applicable empe a u e ange o mul i-
g ade he mal ene gy applica ions. Fu he mo e, in [18] bo h he mo-
dynamic i e e sibili y and hea ans e a e me hods based on en opy
and en ansy heo ies we e used o op imise he CLHS sys em. The e-
sul s indica ed ha he mal e iciency in en ansy op imisa ion su -
passes ha in en opy op imisa ion, whe eas exe gy e iciency is highe
in en opy op imisa ion compa ed o en ansy op imisa ion. Ne e he-
less, a challenge wi h en ansy dissipa ion heo y lies in he incomple e
de ini ion o s o ed and ans e ed hea , as no ed by Kos ic e al. [19].
Me al wool has been p oposed as an e ec i e hea ans e
enhancemen s a egy as a esul o i s high he mal conduc i i y, ease o
in eg a ion wi h PCMs, and cos -e ec i eness. P ie o e al. [20]
demons a ed ha inco po a ion o me al wool in o PCM sys ems
inc eased he e ec i e he mal conduc i i y by 300 %, signi ican ly
imp o ing hea ans e pe o mance in high- empe a u e applica ions.
Simila ly, Fa ache e al. [21] in es iga ed he use o high-conduc i i y
me allic wool as a hea ans e augmen a ion echnique in PCMs,
epo ing imp o ed hea ans e du ing solid-phase ansi ions,
al hough na u al con ec ion e ec s in he liquid phase emained
limi ed. P ie o e al. [22] explo e he easibili y o employing me al wool
as an economical me hod o imp o e he mal conduc i i y o la en
he mal ene gy s o age (TES) in sola p ocess hea applica ions. To
ensu e he s abili y o me al wool in ypical indus ial condi ions, ex-
pe imen s we e conduc ed unde high empe a u es and in an ine a -
mosphe e. These es s aimed o assess he he mal deg ada ion o me al
wool when subjec ed o ele a ed he mal cycling empe a u es. The es s
we e ca ied ou a a b oad ange o empe a u es 200 C o 500 C in an
ine a mosphe e. The esul s e eal no chemical o physical deg ada-
ion, hus con i ming he sui abili y o his echnique o he conside ed
applica ion.
Despi e hese p omising indings, key esea ch gaps emain in he
op imisa ion o he CLHS sys em. Cu en pe o mance e alua ion
me hods s ill lack p ecision due o he eliance on simpli ied assump-
ions ega ding ma e ial p ope ies and hea ans e dynamics. Mo e-
o e , mos enhancemen s a egies ocus on single-PCM sys ems,
necessi a ing dedica ed esea ch e o s o CLHS-speci ic solu ions. In
his pape , we p opose a wo-dimensional
ε
-NTU echnique o add ess
hese challenges, o e ing a mo e accu a e and compu a ionally e icien
app oach o analyse high- empe a u e CLHS sys ems. Unlike con en-
ional one-dimensional me hods, his echnique accoun s o spa ial
a ia ions in hea ans e and ma e ial p ope ies, which makes i be e
sui ed o complex TES applica ions. To alida e his app oach, a
compu a ional luid dynamics (CFD) model is implemen ed, allowing o
p ecise compa ison and pe o mance assessmen . Fu he mo e, we
e alua ed he he mal pe o mance o me al wool-enhanced CLHS
con igu a ions, examining hei impac on hea ans e e iciency,
cha ging/discha ging a es, and o e all sys em e ec i eness. The ind-
ings o his s udy con ibu e o he de elopmen o mo e obus and
Nomencla u e
A a ea o hea ans e , m
2
Cp PCM speci ic hea o he PCM
C
p HTF
speci ic hea HTF, kJ/kg K
SLHS single la en hea s o age
CLHS cascaded la en hea s o age
h
hea ans e coe icien o he HTF, W/m
2
K
h sensible en halpy, kJ/kg
H o al en halpy, kJ/kg
k
PCM
he mal conduc i i y o he PCM, W/m K
k
PCM1
he mal conduc i i y o he PCM
1
, W/m K
k
PCM2
he mal conduc i i y o he PCM
2
, W/m K
k
w
he mal conduc i i y o he ube wall, W/m K
k
e
e ec i e he mal conduc i i y, W/m K
L o al leng h o he ube, m
L
1
leng h o he i s con aine includes PCM
1
, m
L
2
leng h o he second con aine includes PCM
2
, m
LHS la en hea ene gy s o age sys em −
NTU numbe o ans e uni
N numbe o PCMs
P P ac o , −
Q
ac
ac ual s o ed ene gy wi hin a PCM sys em, kJ
Q
max
maximum ene gy s o age, kJ
i
inne adius o he ube, m
o
ou e adius o he ube, m
max
adius o PCMs, m
R
1
o al he mal esis ance o he i s pa allel hea low pa h
o HTF, ube, and PCM
1
, K/W
R
2
o al he mal esis ance o he second pa allel hea low
pa h o HTF, ube and PCM
2
, K/W
R
PCM1/PCM2/PCMn
he mal esis ance o he iso he mal hea low pa h
consis ing o PCM
1
, PCM
2
and PCM
n
, K/W
R
HTF
HTF he mal esis ance, K/W
R
HTF1
HTF he mal esis ance o he i s hea low pa h, K/W
R
HTF2
HTF he mal esis ance o he second hea low pa h, K/W
R
iso
o al he mal esis ance using iso he mal hea low, K/W
R
pa allel
o al he mal esis ance using pa allel hea low, K/W
R
PCM
PCM he mal esis ance, K/W
R
PCM1
PCM he mal esis ance o he i s hea low pa h, K/W
R
PCM2
PCM he mal esis ance o he second hea low pa h, K/W
R
T
o al he mal esis ance, K/W
R
Tube
ube he mal esis ance, K/W
R
Tube1
ube he mal esis ance o he i s hea low pa h, K/W
R
Tube2
ube he mal esis ance o he second hea low pa h, K/W
R
Tuben
ube he mal esis ance o he numbe o hea low pa h,
K/W
S e an–Bol zmann cons an , W/m
2
K
4
γphase change ac ion, −
ε
hea exchange e ec i eness, −
ε
s
emissi i y o he adia ing su ace
βcoe icien o he mal expansion, 1/K
U o e all hea ans e coe icien , W/m
2
K
˙
mmass low a e o HTF, kg/s
δpo osi y, −
λ he a io o he in e sec ion adius o diame e o he wool
ib e
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
2
scalable CLHS designs, which ul ima ely suppo s he ad ancemen o
nex -gene a ion he mal ene gy s o age solu ions o enewable ene gy
applica ions.
2. Ma hema ical o mula ion
A ma hema ical model based on one dimensional
ε
-NTU echnique
has been de eloped and expe imen ally alida ed o he ube in a PCM
sys em using a compu a ional luid dynamics (CFD) model by Tay e al.
[13]. The one-dimensional me hod canno be implemen ed o design
complica ed con igu a ions such as a CLHS and LHS wi h enhancemen
echniques because i assumes o simpli y he ma hema ical model by
making he hea low only in one di ec ion. Mo eo e , he wo-
dimensional me hod can be a mo e app op ia e ep esen a ion o he
he mal esis ance o a CLHS ha includes di e en he mophysical
p ope ies and olumes o PCMs. This app oach helps o de e mine he
ac ual a e age hea ans e a e and he phase- ansi ion ime
h oughou he phase- ansi ion p ocess, using a speci ic se o design
pa ame e s o he ube-in- ank con igu a ion. A e age e ec i eness
was desc ibed as he hea low o a hea sou ce/sink o in ini e speci ic
hea and was exp essed as he ollowing:
ε
=1−exp(− NTU) = Qac
Qmax
(1)
The NTU a any poin in ime can be de ined as [23]:
Fig. 1. (a) Pa allel and (b) iso he mal hea low o he CLHS sys em.
Fig. 2. (a) Pa allel hea low model, and (b) he mal ci cui .
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
3
NTU =UA
(˙
mCp)=1
(RT˙
mCp)(2)
The
ε
-NTU me hod depends on he cha ac e isa ion o he he mal
esis ance be ween he HTF and he phase change bounda y. An accu-
a e depic ion o he he mal esis ance in a CLHS sys em would aid in
he design and op imisa ion o such a sys em wi hou equi ing CFD
modelling. Consequen ly, i is sugges ed o u ilise CFD modelling o
calcula e he a e age e ec i eness, which will hen be used o calcula e
he equi alen o al he mal esis ance o he CLHS sys em and om his
alue he equi alen o al he mal esis ance o he CLHS sys em.
Two-dimensional hea ans e is bounded by he limi s o pa allel
hea low, cha ac e ised by an in ini e ans e se he mal esis ance, and
iso he mal hea low, whe e he ans e se esis ance is ze o. Fo a
CLHS, pa allel hea low occu s when he e is no ans e se hea low,
hus allowing hea low only in a one-dimensional di ec ion pa allel o
he ube wall, as shown in Fig. 1(a). Howe e , iso he mal hea ans e ,
cha ac e ised by he absence o la e al he mal esis ance, allows hea
low in a one-dimensional di ec ion pe pendicula o he ube wall, as
shown in Fig. 1(b). The ac ual phase change p ocess is a combina ion o
hese wo mechanisms.
Go goleski [24] de eloped he ma hema ical model o he concep
o s eel ames-b idged insula ion. To de e mine o al he mal esis-
ance, a ac o called P is used o ep esen he p opo ion o esis ance
on pa allel and iso he mal pa hs, wi h alues anging om 0 o 1.
Typically, he ac o P is assumed o be 0.5 acco ding o Belusko [25],
bu i should ideally be de e mined h ough expe imen a ion o h ee-
dimensional conduc ion modelling, such as compu a ional luid dy-
namics (CFD). Ac ual he mal esis ance, R
T
, is iden i ied by Eq. (3)
[26]. I is sugges ed o employ CFD o assess he o al he mal esis ance
and hen iden i y he sui able P ac o .
RT=P⋅Rpa allel +(1−P)⋅Riso (3)
The o al he mal esis ance o he sys em, when he hea low pa hs a e
un pa allel, is e e ed o as he pa allel esis ance, as explained in Eq.
(4). Fig. 2 illus a es he esis ance diag am ha cla i ies he iden i i-
ca ion o wo hea low pa hs. The i s pa h in ol es he HTF, ube wall,
and PCM
1
, while he second pa h consis s o he HTF, ube wall, and
PCM
2
.
Rpa allel =R1⋅R2.Rn/(R1+R2+Rn)(4)
R
1
and R
2
a e he he mal esis ances h ough he i s and second hea
low pa hs, while R
n
ep esen s he he mal esis ance o an unknown
numbe o PCMs in he cascade sys em, as shown in he equa ions. (5–7),
espec i ely.
R1=RHTF1 +RTube1 +RPCM1 (5)
R2=RHTF2 +RTube2 +RPCM2 (6)
Rn=RHTF n +RTube n +RPCM n (7)
R
HTF1
and R
HTF2
in Eqs. (8)–(9), deno e he HTF he mal esis ances o
he i s and second hea low pa hs, while R
HTFn
in Eq. (10) ep esen
he he mal esis ances o he unknown numbe o HTFs, which is
cha ac e ised by o ced in e nal con ec ion.
RHTF1 =1/(2
π
iL1h )(8)
RHTF2 =1/(2
π
iL2h )(9)
RHTFn=1/(2
π
iLnh )(10)
R
Tube1
and R
Tube2
in Eqs. (11–12) deno e he he mal esis ances o he
ube wall o he i s and second hea low pa hs, while and R
Tuben
in Eq.
(13) ep esen s he esis ance o he unknown numbe o ubes.
RTube1 =ln( o/ i)/(2
π
kwL1)(11)
RTube2 =ln( o/ i)/(2
π
kwL2)(12)
RTuben=ln( o/ i)/(2
π
kwLn)(13)
R
PCM1
and R
PCM2
in Eqs. (14)–(15) deno e he PCMs esis ances in he
i s and second hea low pa hs, while R
PCMn
in Eq. (16) ep esen s he
unknown numbe o PCMs in he cascade sys em, de ined by conduc ion
and he ele an shape ac o .
RPCM1 =ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)/2
π
L1kPCM1 (14)
RPCM2 =ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)/2
π
L2kPCM2 (15)
RPCMn=ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)/2
π
LnkPCMn(16)
Fig. 3. Schema ic o he physical model o (a) single (b) cascade la en hea s o age.
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
4
In Equa ion (17), he second esis ance, Riso, deno es he o al he mal
esis ance o he sys em, achie ed by assuming ha each componen o
he sys em (HTF, ube wall, and PCM) main ains an iso he mal s a e.
This esis ance is de i ed om he esis ance ci cui illus a ed in Fig. 2,
which indica es he p esence o h ee dis inc laye s.
Riso =RHTF +RTube +RPCM1 /PCM2/PCMn (17)
The esis ance o he i s laye (HTF) is gi en in Eq. (18).
RHTF1 =1/(2
π
iLh )(18)
The esis ance o he second laye ( ube wall) is gi en in Eq. (19).
RTube =ln( o/ i)/2
π
kwL(19)
The esis ance o he hi d laye (PCM
1
, PCM
2
, and PCM
n
) is gi en in Eq.
(20). This laye consis s o he o al numbe o PCMs. The he mal
esis ance o his sec ion is de ined by h ee esis ances in pa allel (Fig. 2
b), as e ealed in he equa ions. (20) and (21).
RPCM1 /PCM2/PCMn =1/⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
[2
π
L1kPCM1/ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)]+
[2
π
L2kPCM2/ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)]
[2
π
LnkPCMn/ln({γ( 2
max − 2
o)+ 2
o}1/2/ o)]
+⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
(20)
1/Rpcm1 /PCM2/PCMn =1/RPCM1 +1/RPCM2 +1/RPCMn (21)
3. P oblem s a emen and nume ical modelling p ocedu e
3.1. Physical model
A h ee-dimensional schema ic diag am o wo e ical cylind ical
shell and ube a angemen s ep esen ing SLHS and CLHS wi h 90 %
po osi y o ungs en s ainless s eel me al wool is shown in Fig. 3 (a) and
(b). The me al pipe is used wi h an inne diame e o 22.22 mm and a
wall hickness o 1.65 mm, while he shell, which encloses he phase
change ma e ials (PCMs), is cons uc ed om s ainless s eel wi h an
ou e diame e o 73 mm and a wall hickness o 2.11 mm. The cylin-
d ical la en hea s o age uni has a leng h o 888 mm. In he case o he
CLHS con igu a ion, he s o age shell is di ided in o wo sec ions o
equal leng hs, as men ioned by Jain e al. [27]. The minol 66 is used as
HTF, which passes h ough he inne ube om op o bo om wi h a
uni o m inle eloci y ange om 0.01 o 0.12 m/s, while he inle
empe a u e o 615 K and 519 K is implemen ed du ing he mel ing and
solidi ica ion p ocesses o all simula ions. The ini ial empe a u e o all
simula ions is se a 519 and 615 K o he mel ing and solidi ica ion
p ocesses, espec i ely. HTF discha ges om he s o age uni a a mo-
sphe ic p essu e. The adiaba ic bounda y condi ions a e implemen ed
on he aces be ween s ages and on he ex e nal aces ha a e exposed o
he en i onmen . Table 1 lis s he he mophysical cha ac e is ics o
PCM
1
, PCM
2
, and HTF employed in his s udy.
3.2. Go e ning equa ions and assump ions
The en halpy-po osi y me hod was implemen ed o pe o m a h ee-
dimensional ansien simula ion o he mel ing and solidi ica ion p o-
cesses, in which he in e acial phase change zone (mushy zone) is
ea ed as a po ous medium [28]. The bounda y condi ion used in his
s udy is summa ised in Table 2. The equa ions co e ed in he nume ical
analysis a e no ed by Mayeli e al. 2021 [29]:
Hea ans e luid domain
The con inui y equa ion o he HTF domain is as ollows:
∂ρ
∂
+∇•(
ρ
V
→)=0 (22)
The momen um conse a ion equa ion o he HTF domain as ollows:
ρ
(DV
→
D )= − ∇p−2
3∇(
μ
∇• V
→)+∇•[
μ
(∇V
→+(∇V
→)T)] (23)
The ene gy conse a ion equa ion o he HTF domain is as ollows:
ρ
cp (DT
D )= ∇•(k ∇T)(24)
Phase change ma e ials domain
Con inui y equa ion
∇.V
→=0 (25)
Whe e is he eloci y ec o , and i s componen s, u, , and w, a e
loca ed, espec i ely, in he , Θ and z di ec ions
Momen um equa ion
∂
V
→
∂
+V
→⋅∇V
→=1
ρ
(− ∇P+
μ
∇2V
→+
ρ
g
→β(T−T e ))+Sm(26)
Ene gy equa ion
∂
h
∂
+
∂
H
∂
+∇⋅(V
→h)= ∇⋅(k
ρ
Cp∇h)(27)
Whe e, h is sensible en halpy and H is o al en halpy and P,
ρ
, V and g
deno e he luid p essu e, densi y, eloci y, and accele a ion due o
g a i y, espec i ely. The a ia ion o densi y is p esen ed as he Bous-
sinesq assump ion:
ρ
=
ρ
l
/(β(T −T
l
) +1)
whe e
ρ
l
ep esen s he densi y o liquids and β is he he mal expansion.
Fu he mo e, S
m
is a sou ce o momen um ha adds o he mo-
men um equa ion and is de ined by Ol ian e al. 2020 [30]
Table 1
The mophysical p ope ies o PCMs, HTF [27], and me al wool [20].
P ope y PCMs HTF Me al wool
PCM
1
(NaNO
3
)
PCM2
(NaNO
2
)
The minol
66
Tungs en
s eel
Tempe a u e o
mel ing/
solidi ica ion (K)
579 555 − −
Densi y [kg/m
3
] 1908 1812 775 7913
La en hea o usion
[kJ/kg]
176 180.12 161 −
Speci ic hea solid [kJ/
kg K]
1.60 1.733 −0.488
Speci ic hea liquid
[kJ/kg K]
1.655 2.553 2.730 
The mal conduc i i y
solid [W/m K]
0.8 0.765 −66
The mal conduc i i y
liquid [W/m K]
0.68 0.665 0.0893 −
Dynamic iscosi y
[mPa s]
2.69 2.666 0.336 −
Coe icien o The mal
Expansion (1/K)
0.0004 0.00028 −
Table 2
Bounda y condi ions o all cases o bo h cha ge and discha ge p ocesses.
Bounda y condi ions Mel ing Solidi ica ion
Veloci y o HTF, [m/s] 0.01–0.12 0.01–0.12
HTF inle empe a u e, [K] 615 519
PCM ini ial empe a u e, [K] 519 615
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
5

Sm= − Am(1−γ)2
(γ3+Φ)( − p)(28)
Equa ion (28) es ablishes a dis inc ion be ween he liquid and solid
phases o PCM based on he en halpy-po osi y echnique, Gow eesunke
e al. [31]. A
m
, he mushy zone cons an , wi h a alue be ween 10
4
and
10
7
,
υ
p
is he solid eloci y, and Φ is a small numbe o 0.001 ha
p e en s ze o di ision. γ is he liquid ac ion ha esul s om he
ansi ion be ween he liquid and solid phase, which is used o cha ac-
e isee he deg ee o mel ing p ocess is a cons an be ween 0 and 1 and
can be desc ibed as ollows:
γ=⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
0 T <Tsolidus
T−Tsolidus
Tliquidus −Ts
Tsolidus ≤T≤Tliquidus
1 T >Tliquidus
(29)
The esul s om he sol e a e hen analysed in he pos -p ocessing s age
using he same CFD p og amme o gi e he esul s.
The heo e ical model p oposed o he e ec i e he mal conduc-
i i y o composi es, in oduced by [32] and expe imen ally alida ed in
[33], is used as ollows:
ke=
2
√
Ra+Rb
(30)
Ra=4λξ
k +[2
π
λ2ξ2
3+
π
ξ2
2(1
2λξ −3+2
2
√)](ks−k )
(31)
Rb=
2
√−4λξ
k +
2
√
π
ξ2(ks−k )(32)
ξ=54CA2−4B3+6
3
√[(27C2A2−4CB3)1
2A]1
3
3A
+
4B2{54CA2−4B3+6
3
√[(27C2A2−4CB3)1
2A](−1
3)}−B
3A
(33)
A=4
2
√
π
λ3
3−3
2
√
π
λ,B=3
π

2
√,C=1−δ(34)
The e ec i e he mal conduc i i y o he composi e was exp essed
h ough he conduc i i ies o he luid (k ) ep esen ing PCM and he
he mal conduc i i y o he solid s uc u e (ks) ep esen ing me al wool,
he po osi y o he me al wool (δ) and he a io o he adius in e sec ion
o he ligamen diame e o wool ibe (λ = /d).).). This pa ame e was
de i ed om expe imen al obse a ions. Acco ding o he da a in [33],
he alue λ =60 p o ided he bes i .
The ollowing p esump ion was conside ed when analysing p o-
cesses using he LHS sys em:
•Excep o he densi y, all he physical p ope ies o PCM a e
cons an .
•The HTF cha ac e is ic p ope ies a e cons an wi h empe a u e.
•Liquid PCM is conside ed as a lamina , New onian, and incom-
p essible luid.
•Igno ing he iscous dissipa ion o ene gy.
•In bo h p ocesses, i is assumed ha he sys em is adiaba ic.
•The liquid phase h ough he PCM is modelled using he Boussinesq
app oxima ion [34].
3.3. Nume ical model
The compu a ional p ocess can be classi ied in o h ee s eps: p e-
p ocessing, sol e (p ocessing), and pos -p ocessing. The p e-p ocessing
s age in ol es cons uc ing he compu a ional domain ha con ains he
assembly o he model, meshing, and bounda y condi ions. To ensu e
accu a e esul s wi h a easonable compu ing cos , an independen s udy
o he g id and he ime s eps needs o be conduc ed. In he cu en
s udy, h ee di e en numbe s o elemen s (196500, 697500, 813750)
and h ee di e en ime s eps (0.1, 0.5, and 1 s) we e e alua ed. Based
on analysis, numbe o elemen and ime s eps 697,500 and 1 a e aken
on he conside ed in he u he compu a ions as seen in Fig. 4 (a) and
(b). A g id wi h N =697,500 cells was selec ed o i s op imal compu-
a ional cos sa ing. Fig. 4 (a) indica es ha e ining his g id u he
does no signi ican ly a ec he nume ical solu ion. Addi ionally, as
shown in Fig. 4 (b), he ime s ep o 1 s was de e mined o be su icien o
main ain he solu ion’s independence and s abili y h oughou he
en i e phase ansi ion. The sol e s age in ol es expo ing he model o
ANSYS FLUENT 2023R2 o se up and analysis. Due o he ime
dependence p oblem, a ansien s a e was selec ed. An implici second-
o de ansien o mula ion is applied o he solu ion. The ene gy had
con e gence c i e ia o 10
-9
, while momen um in all di ec ions had a
c i e ion o 10
-6
. Acco ding o nume ous unning es s, he se o unde -
elaxa ion pa ame e s is sui able in his s udy o p e en di e gences.
The alues o momen um, ene gy, and liquid ac ion we e 0.3, 0.9, and
0.1, while hose o body o ces, densi y, and p essu e we e 1, 0.8, and
0.3. The p essu e– eloci y coupling and he Semi-Implici Me hod o
p essu e-linked equa ions (SIMPLE) echnique a e used. P essu e
Fig. 4. E ec o (a) he numbe o compu a ional elemen s and (b) ime s ep on he e alua ion o liquid ac ion o NaNO
2
wi hin compu a ional domain.
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
6
in e pola ion employed he S agge ing P essu e Op ion (PRESTO)
me hodology p essu e co ec ion [35].
4. Resul s and discussion
4.1. Valida ion o CFD model
To e i y he accu acy o he cu en nume ical model in p edic ing
he phase ansi ion p ocess, he de eloped model is compa ed wi h he
nume ical esul o SLHS and CLHS o Jain e al. [27]. They pe o med
nume ical in es iga ions on he NaNO
3
SLHS based single s age s o age
wi h he same dimensions men ioned in de ail [27], which has a shell
made o s eel (SS316) and HTF ube made o coppe . The leng h o he
s o age is conside ed 888 mm. The NaNO
3
/NaNO
2
CLHS used he same
geome y, bu i is di ided in o wo equal pa s o es ablish wo s ages
con igu a ion. The same he mophysical p ope ies and ini ial
condi ions we e used o alida ion such as inle empe a u e and e-
loci y o HTF we e 615 K and 0.04 s/m, espec i ely, as well as he ini ial
empe a u e o PCMs was 519 K. G id, ime s ep independence and
sol e se ing a e chosen as he same e e ence men ioned abo e. Fig. 5
(a) and (b) show he compa ison o p og ess in liquid ac ion o e ime
and empe a u e con ou s o wo pe iods 120 mins and 240 mins be-
ween he p esen model and he esul s by [27] o SLHS and CLHS
−based PCMs. The igu e clea ly shows ha he p esen model is capable
o accu a ely p edic ing he e alua ion phase change and empe a u e
p o ile.
4.2. Valida ion o wo-dimensional
ε
-NTU me hod
Figs. 6 and 7 show he compa ison be ween he a e age e ec i eness
ac oss di e en mass low a es o a ea om he de eloped wo-
dimensional
ε
-NTU me hod and he CFD model du ing he cha ging
Fig. 5. Compa ison o (a) Liquid ac ion and (b) Tempe a u e con ou s wi h esul s o Jain e al. [27].
Fig. 6. Valida ion o he mel ing esul s o
ε
-NTU me hod wi h CFD o single
and cascade PCMs.
Fig. 7. Valida ion o he solidi ica ion esul s o
ε
-NTU me hod wi h CFD o
single and cascade PCMs.
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
7
and discha ging p ocesses o PCM con igu a ions based on SLHS and
CLHS. I can be obse ed ha he cascade con igu a ion d ama ically
inc eases he a e age e ec i eness du ing he cha ging p ocess by 37%
and discha ging p ocess by 34% compa ed wi h single s o age and ha
in ag eemen wi h p e ious s udies [36,37]. The wo-dimensional
ε
-NTU
me hod and he CFD model showed good ag eemen ac oss all low
a es. Fu he mo e, his ag eemen was achie ed o bo h SLHS and
CLHS con igu a ions. One o he mos impo an limi a ions o he one-
dimensional
ε
-NTU me hod is ha i does no ake in o accoun he e ec
o na u al con ec ion. This is due o only conside ing he hea low
passing h ough one di ec ion and igno ing he hea low in o he di-
ec ions. Mo eo e , he na u al con ec ion is dependen on he em-
pe a u e di e ence in he sys em, while his me hod is in ended o be
empe a u e independen . Meanwhile he wo-dimensional me hod can
conside he e ec o na u al con ec ion by de e mining he hea low in
wo di ec ions om compa ing he esul s ob ained om CFD. In his
way, he e ec o na u al con ec ion can be adjus ed h ough he alue
o P ac o . The e o e, ini ially aiming o alida e he wo-dimensional
me hods wi h CFD esul s, in his sec ion he buoyancy e ec has been
igno ed in he CFD model o educe he a e age e o as much as
possible. In addi ion, an impo an limi a ion o he wo-dimensional
me hod is ha i can ake in o accoun only some he mophysical
p ope ies such as he mal conduc i i y while i igno es o he p ope ies
such as en halpy. An a e age e o was ound o be app oxima ely 3.6 %
and 5% o he mel ing and solidi ica ion p ocesses, espec i ely.
The e o e, he newly de eloped me hod, which assumed phase change,
has been alida ed o accu a ely desc ibe he cascade con igu a ion
du ing bo h cha ging and discha ging p ocesses.
4.3. Calcula ed a e age e ec i eness and he mal esis ance a io
Figs. 8 and 9 illus a e he calcula ed a e age e ec i eness ac oss
di e en mass low a es o a ea o he mel ing and solidi ica ion p o-
cesses in bo h single and cascade-based PCMs, wi h and wi hou me al
wool con igu a ions. The esul s indica e ha CLHS exhibi s a signi ican
imp o emen in he mal pe o mance o e he single LHS con igu a ion.
The addi ion o me al wool as an enhancemen echnique u he im-
p o es e ec i eness in bo h con igu a ions du ing cha ging and dis-
cha ging p ocesses. Speci ically, he a e age e ec i eness o CLHS
inc eases by 43 % and 44 % compa ed o a single PCM con igu a ion.
When me al wool is inco po a ed, he e ec i eness is enhanced by 54 %
and 54.7 % o e SLHS and by 20 % and 16 % o e CLHS-based PCMs o
he mel ing and solidi ica ion p ocesses, espec i ely.
The imp o emen in hea ans e e iciency wi h me al wool is
a ibu ed o wo key ac o s: (1) he high he mal conduc i i y o me al
wool, which acili a es apid hea ans e , and (2) he la ge hea ans e
su ace a ea c ea ed by he me al wool s uc u e, enabling be e he -
mal dis ibu ion wi hin he PCMs. This enhancemen o e comes he
inhe en low he mal conduc i i y o PCMs, allowing o mo e e icien
phase ansi ion p ocesses.
Figs. 10 and 11 p esen he a e age he mal esis ance wi hin he
PCMs, ela i e o he o al he mal esis ance, as a unc ion o HTF e-
loci y o bo h single and cascade con igu a ions wi h and wi hou me al
wool. These igu es demons a e ha he mal esis ance wi hin PCMs is
signi ican ly educed in cascade con igu a ions, indica ing ha he
Fig. 8. Compa ison he mel ing esul s o
ε
-NTU me hod o SLHS and CLHS
wi h and wi hou me al wool.
Fig. 9. Compa ison he solidi ica ion esul s o
ε
-NTU me hod o SLHS and
CLHS wi h and wi hou me al wool.
Fig. 10. Compa ison he he mal esis ance a io o he mel ing p ocess o
SLHS and CLHS wi h and wi hou me al wool.
Fig. 11. Compa ison o he he mal esis ance a io o he solidi ica ion p ocess
o SLHS and CLHS wi h and wi hou me al wool.
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
8
dominan esis ance shi s owa d HTF con ec ion esis ance (R
HTF
).
Addi ionally, a subs an ial educ ion in esis ance is obse ed wi h he
in oduc ion o me al wool in bo h SLHS and CLHS con igu a ions.
A no able ansi ion is obse ed a a HTF eloci y o 0.05 m/s, whe e
he Reynolds numbe eaches alues cha ac e is ic o u bulen low.
This ansi ion esul s in a highe Nussel numbe , he eby enhancing
con ec i e hea ans e and educing o al he mal esis ance. The
impac o cascade esis ance a ies, a e aging 15 % du ing solidi ica ion
and eaching a maximum o 80 % du ing mel ing. No ably, he he mal
esis ance o he ube and con ec i e esis ance wi hin he luid emain
signi ican con ibu o s o o e all sys em pe o mance.
These indings unde sco e he impo ance o cascade con igu a ions
and me al wool in eg a ion in imp o ing he mal managemen and e -
iciency in la en hea s o age sys ems, wi h di ec implica ions o high-
empe a u e he mal ene gy s o age applica ions.
4.4. The mal pe o mance assessmen o SLHS and CLHS-based PCMs
This sec ion e alua es he cha ging and discha ging p ocesses in bo h
single and cascade la en hea s o age (LHS) sys ems o analyse he e ec
o cascaded PCM con igu a ions. Addi ionally, he impac o me al wool
as a cos -e ec i e enhancemen echnique o high- empe a u e appli-
ca ions is examined [20,38]. The he mal pe o mance o SLHS using
NaNO
3
and NaNO
2
as s o age mediums, and CLHS inco po a ing bo h
NaNO
3
and NaNO
2
wi h and wi hou me al wool, is assessed. Bo h
con igu a ions a e a anged in a e ical o ien a ion, whe e he hea
ans e luid (HTF) lows h ough he inne ube while he PCMs a e
con ained in he su ounding ou e shell. In all cases, ho HTF en e s he
sys em a 0.08 m/s om he op and exi s a he bo om, ollowing he
di ec ion o g a i y, as illus a ed in Fig. 3.
Fig. 12 (a) p esen s he phase change in e ace and empe a u e
dis ibu ion a di e en ime s eps du ing he cha ging phase o CLHS.
Ini ially, he phase change occu s in PCM2, as i s mel ing empe a u e
(555 K) is lowe han he HTF empe a u e. Du ing he ea ly phase, hea
conduc ion go e ns he p ocess, causing PCM1 o unde go a sensible
hea ing phase be o e eaching i s highe mel ing empe a u e (579 K).
Due o he lowe empe a u e di e ence be ween PCM1 and HTF, he
ene gy ans e a e declines, leading o an ex ended cha ging ime.
As cha ging p og esses, mel ing occu s mo e apidly a he op han
a he bo om o bo h s ages. This is a ibu ed o HTF empe a u e
educ ion along he downwa d low and buoyancy e ec s, which cause
he hea ed mel o ise, slowing he mel ing p ocess a he bo om.
Fig. 12. Liquid ac ion and empe a u e dis ibu ions con ou s du ing cha ging o bo h (a) CLHS and (b) CLHS +MWcon igu a ions h ogh di e en ime s eps.
H.A. Ib ahim Al-Saaidi e al.
Applied The mal Enginee ing 278 (2025) 127452
9