Numerical Study on Solidification of Phase Change Materials Embedded with Metal Foam

Abstract

This work investigates the solidification of phase change material (PCM) embedded with metal foam (MF) in a spherical capsule which its outer layer is exposed to convective heat transfer. The one-dimensional energy equation is resolved by performing finite volume method accompanied with temperature transforming technique. Four separate scenarios are developed for different porosity value of MF in order to analyze the thermal behavior of composite PCM with MF. The numerical model is validated by experimental data taken from the literature and substantially good agreement is demonstrated. The results show that at the case where the porosity ε =0.92, the elapsed time for complete solidification is decreases by 88% compared to the case without MF (ε =1.0).

Full text

1. INTRODUCTION
Several felds of the energy system such as ndustral,
transportaton and buldngs sectors have a hgh prorty
n energy ecency researches, due to ther hghest ener-
gy savng potental n total fnal energy consumpton. Ener-
gy storage methods come to the forefront for the ecent
utlzaton of renewable and alternatve energy sources [1].
Energy can be stored n order to get more beneft from ener-
gy sources. Energy storage technologes ncrease energy ef-
fcency by dmnshng msmatch between supply and de-
mand [2]. e search for new and renewable energy sources
as well as more ecent solutons for the storage of energy
s ongong. Most of the studes on storage materals have
focused on thermal energy storage (TES) systems. Materals
that are used n TES systems are known as phase change
materals (PCMs) because of ther hgh latent heat, sutable
phase change temperature, low volume change durng pha-
se change and hgh storage capacty. ere are nformatve
revew papers whch examned the heat transfer process n
the TES system [3,4]. Furthermore, Kensarn et al. [5] com-
prehensvely revewed the heat transfer mechansm of PCM
n sphercal geometry. Pedroso and Domoto [6] analytcally
examned the soldfcaton wthn sphercal contaner by
utlzng perturbaton technque under constant wall tem-
perature. Tao [7] performed a mechansm for the evaluaton
of soldfcaton tme n cylndrcal and sphercal geometres.
e numercal soluton was based on a fxed grd method.
Blr and Ilken [8] examned the soldfcaton of a PCM en-
capsulated n a cylndrcal and sphercal geometres. ey
formulated and solved the dmensonless form of governng
equatons numercally by utlzng enthalpy approach wth
control volume method. Erek and Dncer [9] numercally n-
vestgated an encapsulated ce TES system wth a uctuated
heat transfer coecent. ey reported that the soldfca-
ton mechansm s essentally managed by the value of the
Stefan number, number of capsule row and dameter of cap-
sule. Most common PCMs materals used n the ndustry
have low thermal conductvty, whch reduces the ecency
of TES systems. erefore, these systems alone do not have
a good economc justfcaton. In order to mprove the per-
formance of TES systems, varous methods are handled, the
most mportant of whch are: ncreasng heat transfer area
by usng fns [10], PCM encapsulaton, smultaneous use of
multple PCMs [11] and fllng PCMs n shape stable stru-
ctures [12]. Narasmhan et al. [13] used an enthalpy met-
hod to analyze the soldfcaton nsde a sphercal capsule.
In ther study, the PCM s ntegrated wth macro partcles.
er results showed that the ntegraton of hgh thermal
conductvty partcles to PCMs mproves the rate of soldf-
caton properly. Varous researchers have suggested PCMs
flled n MFs n order to mprove heat transfer process. Due
to PCMs have small thermal conductvty that reduces heat
exchange, MF s suggested as a key to enhance thermal con-
Numerical Study on Solidification of Phase Change Materials
Embedded with Metal Foam
Mustafa Asker1 , Hadi Genceli2
1Aydın Adnan Menderes University, Faculty of Engineering, Department of Mechanical Engineering, Aydın, Turkey
2Avrasya University, Faculty of Engineering, Department of Mechanical Engineering, Trabzon, Turkey
Abstract
is work investigates the solidication of phase change material (PCM) embedded with metal foam (MF) in
a spherical capsule which its outer layer is exposed to convective heat transfer. e one-dimensional energy
equation is resolved by performing nite volume method accompanied with temperature transforming
technique. Four separate scenarios are developed for dierent porosity value of MF in order to analyze the
thermal behavior of composite PCM with MF. e numerical model is validated by experimental data taken
from the literature and substantially good agreement is demonstrated. e results show that at the case where
the porosity ε =0.92, the elapsed time for complete solidication is decreases by 88% compared to the case
without MF (ε =1.0).
Keywords: Control volume approach; Phase change materal; Metal foam; Soldfcaton.
e-ISSN: 2587-1110
* Corresponding authour
Email: mustafa.asker@adu.edu.tr
European Mechanical Science (2021), 5(1): 1-5
doi: https://doi.org/10.26701/ems.783892
Received: August 21, 2020
Accepted: October 15, 2020
EUROPEAN
MECHANICAL
SCIENCE
Research Paper

ductvty. An ntegraton of PCM n MF has also been recog-
nzed to be eectve for TES. erefore, t s reasonable to
utlze MFs n order to enhance the ecency of TES due to
ther hgh thermal conductvty. Varous MF materals such
as Al2O3, Copper and Nckel have been utlzed [14]. Reh-
man et al. [15] wdely revewed the mproved thermal pro-
pertes of porous/foam metallc and carbon-based materals
and heat transfer process n the LHTES of PCM. e ther-
mal behavor of PCM ncorporated wth MF durng meltng
mechansm are nvestgated analytcally and expermentally
by Zhang et al. [16]. e basc objectve of ths current rese-
arch study s to examne the thermal characterstcs of PCM
embedded n MF for TES compared to the PCM wthout
MF. In ths context, tme wse temperature dstrbuton s
performed to determne the approprate condtons for TES
system. In addton, the mpact of porosty of MF on the
complete soldfcaton of PCM s examned.
2. MATERIAL AND METHOD
e sphercal capsule model s schematcally dsplayed n
Fgure 1. e one-dmensonal model conssts of a sphercal
capsule of radus rout =0.05 m flled wth PCM embedded n
MF. e thermo-physcal propertes of the PCM and MF are
gven n Table 1. e meltng temperature of the PCM wth
Tm=0˚C s lower than ts ntal temperature Tinitial. e outer
surface of the capsule s exposed to convectve coolng wh-
ch s retaned at temperature Tf and convectve heat transfer
coecent h. e latent heat s taken as L=333400 kJ/kg.
Figure 1. Mathematcal model for the sphercal capsule embedded wth
MF.
Table 1. Thermo- physcal propertes of the PCM and MF [17, 18].
Properties Water Solid Water Liquid Aluminum
Specific heat, [kJ/kg K] 2040 4210 902
ermal conductivity, [W/m K] 1.88 0.567 218
Density, [kg/m3] 916.8 998.8 2700
e followng assumptons are conducted n the mathemat-
cal model to smplfy the problem:
• e thermal properties of the PCM and MF are
independent of temperature. However, the PCM are
comprised of two distinct regions which are solid and
liquid.
• e PCM is considered homogenous and isotropic.
e temperature transformng method descrbed by Cao
and Faghr [19] s used to smulated the heat conducton n
the sphercal capsule. s technque has consdered that the
soldfcaton process ncludes three phases namely, sold,
lqud and mushy regons. In order to smplfy the problem,
the non-dmensonal parameters are used and lsted n Table
2.
Table 2. Non-dmensonal parameters
Parameters Definition
Radius
out
r
Rr
=
Temperature
m
mf
TT
TT
θ
−
=−
Specific heat
( )
LL
c
Cc
ρ
ρ
=
ermal conductivity
L
k
Kk
=
Source term
( )
LL m f
s
S
cT T
ρ
=−
Stefan number
( )
Lm f
cT T
Ste L
−
=
Fourier number
2
L
out
t
Fo r
α
=
Biot number
out
L
hr
Bi k
=
e one-dmensonal energy equaton for sphercal capsule
expressed n the dmensonless form as
( )
2
2
1
CS
KR
Fo R R Fo
R
θθ
∂∂ ∂∂

= −

∂ ∂ ∂∂

(1)
where C, S and K varables descrbed as follows
( ) ( )
11
1
22
1
sl m
sl m m
m
m
C Solid
C C Mushy
Ste
Liquid
θ δθ
θ δθ θ δθ
δθ
θ δθ
<−


= + + − ≤≤


>

(2)
( ) ( )
11
1
22
1
sl m m
m sl m m
sl m m
C Solid
S C Mushy
Ste
C Liquid
Ste
δθ θ δθ
θ δθ δθ θ δθ
δθ θ δθ

<−


= + + − ≤≤



+>


(3)
( ) ( )( )
1
2
1
sl m
sl m
sl m m
m
m
K Solid
K
K K Mushy
Liquid
θ δθ
θ δθ
θ δθ θ δθ
δθ
θ δθ
<−

−+

= + − ≤≤


>

(4)
e ntal state for equaton (1) s
0initial
Fo
θθ
=→=
(5)
Whereas the boundary condtons can be wrtten as follows
00RR
θ
∂
=→=
∂
(6)
( )
11R K Bi
R
θθ
∂
=→ =−+
∂
(7)
Equatons 1 to 7 s resolved by applyng the control volume
approach as explaned n Versteeg and Malalasekera [20] as
2 European Mechanical Science (2021), 5(1): 1-5
doi: https://doi.org/10.26701/ems.783892
Numerical Study on Solidication of Phase Change Materials Embedded with Metal Foam

follows:
( )
2
2
1
V Fo V Fo V Fo
CS
dV dFo KR dVdFo dV dFo
Fo R R Fo
R
θθ
∂∂∂ ∂

= −

∂ ∂∂ ∂

∫∫ ∫∫ ∫∫
(8)
e dscretzed non-dmensonal energy equaton for nter-
or nodes s wrtten n general form as
( ) ( )
33 33
00 0
22
33
ew ew
PW
PP PP E P P P
ee ww
ew
RR RR
C C SS
KR K R
Fo R R Fo
θθ
θ θ θθ
−−

−
− −−
=−−

∆ ∆ ∆∆

( ) ( )
33 33
00 0
22
33
ew ew
PW
PP PP E P P P
ee ww
ew
RR RR
C C SS
KR K R
Fo R R Fo
θθ
θ θ θθ
−−

−
− −−
=−−

∆ ∆ ∆∆

(9)
At the control surface, the harmonc mean s used to cal-
culate the thermal conductvty Patankar [21]. Numercal
computatons are carred out by resolvng Equatons 1 to 8.
A self-developed computer program coded n C++ language
s used to obtan the transent temperature varatons. Due
to nonlnearty of the energy equatons, teratve soluton s
needed. Based upon the prelmnary runs, the tme step sze
and convergence crteron are decded as 0.1s and 1E-6, ac-
cordngly. Furthermore, the soldfed mass fracton (SMF)
can be calculated as follows [22].
Mass of solid
SMF Total mass
=
(10)
In ths work, Alumnum as a MF s ntegrated n PCM to
enhance the heat transfer mechansm. Bhattacharya et al.
[23] proposed a smple correlaton to evaluate the eectve
thermal conductvty as follows:
( ) ( )
( )
1
1
1
eff PCM AL
PCM AL
A
k Ak k
kk
εε ε
ε
−

= +− +


−
+


(11)
Where ε represent the porosty and A s constant (A=0.35).
Besdes, the eectve specfc heat, densty and latent heat
correlatons are obtaned from lterature [24].
3. RESULTS AND DISCUSSION
s research study performed a numercal analyss n order
to nvestgate the eect of porosty of MF combned wth
the PCM on soldfcaton process nsde sphercal capsu-
le. roughout ths work, the B number, the phase change
temperature ranges and the number of nodes s selected to
be 5, 0.01 and 100, respectvely. In order to demonstrate the
relablty of the soluton method, the numercal model that
s handled n ths study s compared wth expermental and
numercal data whch s publshed by (Ismal et al. 2003) as
shown n Fgure 2. e value of rout, Tf and Tinitail are fxed
at 0.064 m, -7.5˚C and 25.8˚C, accordngly. It can be obser-
ved that the transent varaton of center temperature of the
sphere consstence wth the expermental and numercal
method that s reported n Ismal et al. [22]. Consequently,
the present numercal model s sutable to tackle the phase
change problem n sphercal capsule.
Figure 2. Comparson of the current numercal model wth the results
obtaned from Ismal et al. [22].
In ths work, four derent cases for studes are establshed
accordng to the porosty of MF as gven n Table 3. In ad-
dton, the eectve thermal conductvty value for compo-
ste sold and lqud statuses of the PCM are also ncluded n
Table 3. Case 1 wth ε =1.0 represent baselne case n wh-
ch there s no MF contans wthn the PCM. Case 2 wth ε
=0.98, Case 3 wth ε =0.96 and Case 4 wth ε =0.92 whch
have hgher ke compared to the other cases. e eectve
thermal conductvty of PCM s calculated based on Equat-
on (11). e soldfcaton mechansm begns from the outer
layer that s subjected to the heat convecton, and advances
nward towards the center of the sphercal capsule. e sol-
dfcaton process s acheved when the lqud PCM s trans-
formed to sold phase. Hence, the thermal equlbrum of the
system has compassed wth the outsde condtons.
Table 3. Confguratons of the eectve thermal conductvty for each
case.
Cases Porosity
(ε)
ke (solid)
[W/m.K]
ke (liquid)
[W/m.K]
1 1.0 1.88 0.567
2 0.98 3.417 2.096
3 0.96 4.956 3.626
4 0.92 8.037 6.687
Fgure 3 llustrates the tme wse varaton of center tempe-
rature for derent cases. In general, the center temperature
of the sphere whch s orgnally at lqud status drops untl t
fnally becomes constant as t proceeds to the mushy regon
then abruptly reduces to ts fnal stuaton whch s the sold
phase. It can be seen that Case 4 needs remarkably lower
soldfcaton tme compared to the baselne case (Case 1)
because the heat conducton mechansm s hgh that makes
the soldfcaton process to occurs rapdly. Fgure 4 depcts
the tme wse varaton of SMF for derent cases. In ths
fgure, ntally the PCM s at lqud phase and the SMF s at
ts mnmum poston (SMF=0) and ncreases untl t even-
tually reaches to ts maxmum pont (SMF=1) whch means
that the complete soldfcaton s taken place. It can be ob-
served that the SMF trend for Case 4 wth (ε =0.92) occurs
3
European Mechanical Science (2021), 5(1): 1-5
doi: https://doi.org/10.26701/ems.783892
Mustafa Asker, Hadi Genceli

faster than the other cases because of the ke values of Case
4 s larger than the other cases.
Figure 3. Evoluton of temperature for derent MF porosty.
Figure 4. Eect of porosty of metal foam on soldfed mass fracton.
Fgure 5 demonstrates that the tme for complete soldfca-
ton for Case 1 wth (ε =1.0) can be acheved at about 6.13
hours whle for Case 4 wth (ε =0.92) the complete soldfca-
ton can be reached after about 0.72 hour. s s due to
the eectve thermal conductvty for both sold and lqud
phases n Case 4 are hgher than the other cases (see Table 3)
whch enhances the heat conducton processes.
Figure 5. Tme for complete soldfcaton for derent cases.
Fgure 6 llustrates the characterstcs of temperature profle
for surface and center temperatures of the PCM ncorpora-
ted wth MF nsde sphercal capsule. In ths fgure, Case 3
wth ε =0.96 s consdered. In ths fgure, the center porton
of the sphere temperature ntally at lqud status for about
24 mn whereas the sold regon at the center begns after
about 1.28 h. between these two duratons, the mushy phase
occurs. e abrupt drop n center temperature after 1.28 h
can be attrbuted to the extractng sensble heat from the
soldfed porton whereas phase change be formed at the
converson from mushy regon to sold status.
Figure 6. Tme wse varaton of temperatures for composte PCM wth
metal foam.
4. CONCLUSIONS
s study examned the soldfcaton of PCM ntegrated
wth MF n sphercal capsule by applyng the fnte volu-
me technque. e proposed model s valdated wth pre-
vously publshed work. e thermal characterstcs of the
center temperature for varous porosty s smulated. It can
be concluded that the eectve thermal conductvty wll be
sgnfcantly mproved by addng MF to the PCM. In addt-
on, a hgher ke results n a hgher heat conducton and as a
result, faster soldfcaton process s acheved. In ths con-
text, the use of MF s a reasonable approach for enhancng
the heat transfer n TES systems. In addton, the crtera
requred for the desgn of TES can be determned based on
these analyzes.
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5
European Mechanical Science (2021), 5(1): 1-5
doi: https://doi.org/10.26701/ems.783892
Mustafa Asker, Hadi Genceli