Applied Thermal Engineering 37 (2012) 1e9
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Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
Experimental and numerical studies on melting phase change heat transfer
in open-cell metallic foams filled with paraffin
W.Q. Li, Z.G. Qu*, Y.L. He, W.Q. Tao
MOE Key Laboratory of Thermo-Fluid Science and Engineering, Xi’an Jiaotong University, 28# XianNing Road, 710049 Xi’an, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 28 July 2011
Accepted 2 November 2011
Available online 17 November 2011
In the current study, the melting phase change heat transfer in paraffin-saturated in open-celled metallic
foams was experimentally and numerically studied. The experiments were conducted with seven highporosity copper metal foam samples (3 90%), and paraffin was applied as the phase-change material
(PCM). The wall and inner temperature distribution inside the foam were measured during the melting
process. The effects of foam morphology parameters, including porosity and pore density, on the wall
temperature and the temperature uniformity inside the foam were investigated. The melting heat
transfer is enhanced by the high thermal conductivity foam matrix, although its existence suppresses the
local natural convection. A numerical model considering the non-Darcy effect, local natural convection,
and thermal non-equilibrium was proposed. The velocity, temperature field, and evolution of the solid
eliquid interface location at various times were predicted. The numerically predicted results are in good
agreement with the experimental findings. The model as well as the feasibility and necessity of the
applied two-equation model were further validated.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
Phase-change material
Metallic foam
Melting
Non-equilibrium model
1. Introduction
Phase-change materials (PCMs) have been widely used in many
applications, such as passive cooling for electronic devices, protection systems in aircrafts, food processing, and energy conservation in
buildings, because of their high latent heat, chemical stability, suitable phase-change temperature, and reasonable price. Experimental
and analytical/numerical studies in published literatures have
focused on moving boundary problems [1e4]. Agyenim et al. [5]
summarized the various applications of PCMs with different
melting temperatures in suitable thermal energy storage systems.
The size and the shape of the PCM container were also taken into
consideration to ensure the long-term stable thermal performance of
the system. Dutil et al. [6] presented main four types of numerical
solutions dealing with the thermal behaviors of PCM in solid/liquid
systems and showed the predicted results of different configurations.
Although some organic PCMs, such as paraffin, are very popular
in energy storage applications and electronic cooling systems
because of the aforementioned advantages and their low density
compared with other kinds of heat storage materials (e.g., metal
PCMs and hydrated salts). However, organic PCMs suffer from low
conductivity (z0.1 W m1 K1), which is likely to cause failure in
* Corresponding author. Tel./fax: þ86 029 82668036.
E-mail address: [email protected] (Z.G. Qu).
1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.applthermaleng.2011.11.001
electronic devices. For example, in some electronic cooling systems,
the chip experiences a transient or periodic heat generation, which
requires a highly efficient coolant system to dissipate the heat in
case of chip exposure to extreme-temperature environments.
However, the poor conductivity of organic PCMs reduces the rate of
heat storage, thereby increasing the junction temperature of the
devices beyond the allowable range.
In order to address this unacceptable problem, thermal
conductivity enhancement techniques that increase heat transfer
rates have been developed. These enhancement techniques are
summarized in the following three methods: (1) Dispersing highconductivity particles in PCMs; Wang et al. [7] experimentally
proved that the thermal conductivity of composite PCMs is
enhanced by incorporating a b-aluminum nitride additive. (2)
Utilizing high-conductivity matrices, such as a metal or a graphite
compound, as heat delivery promoters; Kim and Drazal [8]
improved the effective thermal conductivity of paraffin by stirring
exfoliated graphite nanoplatelets (xGnP) in liquid paraffin. The
authors found that the thermal conductivity of paraffin/xGnP
composite PCMs increases as the xGnP loading content increases
without reducing the latent heat of the paraffin wax. (3) Filling
extensive surfaces such as fins into the body of PCM; Shatikian et al.
[9] numerically investigated the effect of internal fins on melting
rate, melting front profiles, and heat transfer. (4) Varying the shape
of the heat storage-heat transfer system; Banaszek et al. [10].
2
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
Nomenclature
A
asf
C
CI
cpf
cps
ctd
df
dp
dk
fl
g
hsf
K
k
ke
kf
kfe
ks
kse
L
Nu
P
PPI
Pr
q
Re
additional term in the momentum equation
interfacial surface area (m1)
constant parameter
inertial coefficient
specific heat capacity of PCM (J kg1 K1)
specific heat capacity of metal foam (J kg1 K1)
thermal dispersion coefficient
fiber diameter (m)
pore size (m)
characteristic length (m)
liquid fraction in the pore
gravitational acceleration (m s2)
interfacial heat transfer coefficient (W m2 K1)
permeability (m2)
thermal conductivity (W m1 K1)
effective thermal conductivity (W m1 K1)
thermal conductivity of fluid (W m1 K1)
effective thermal conductivity of fluid (W m1 K1)
thermal conductivity of solid (W m1 K1)
effective thermal conductivity of metal foam
(W m1 K1)
latent heat of paraffin (J kg1 K1)
Nusselt number
pressure (Pa)
pore number per inch
Prandtl number
heat flux (W m2)
Reynolds number
designed an energy conservation system using helix-shaped
channels where PCM and heat transfer fluids were separately
enclosed in every other channel. Such a design makes it possible for
the fluid to stick to the channel wall more strictly when flowing
through the channel because of centrifugal force and more efficiently take away heat from the PCM. However, a disadvantage of
such a structure for heat exchange is the increased consumption
and complexity of the cooling system.
In addition to the aforementioned methods, inserting PCMs into
a porous media appears to be an attractive choice to enhance heat
transfer. A number of experimental and theoretical investigations
on this problem have been conducted in the past two decades.
Beckermann and Viskanta [11] conducted both experimental and
numerical studies on the melting of gallium in glass beads. In their
numerical study, the one-equation energy model was adopted to
predict the temperature, taking into account the natural convective
effect. To the authors’ best knowledge, the one-equation model is
suitable only when the thermal properties of the solid matrix and
the saturated PCM are in the same order of magnitude; that is, the
macroscopic temperatures of the two materials should be close
enough so that a single temperature model can describe the whole
melting process. This equilibrium model fails when the thermal
properties of the two materials are significantly different or when
the convective effect becomes vitally important [12]. Extensive
studies are conducted to improve the accuracy of the model.
Harris et al. [13] theoretically studied the phase-change process in
porous media using a two-temperature model. However, the main
drawback of this approach is its difficulty in handling complex
cases, such as buoyancy-driven natural convection in a molten
region in porous media. Krishnan et al. [14] numerically investigated the thermal transport phenomena associated with phase
change in a rectangular cavity filled with metal foam. The
T
Tm1
Tm2
t
u,v
x,y
temperature ( C)
the lower limit of melting point
the upper limit of melting point
time (s)
velocity in x and y directions (m s)
cartesian coordinates
Greek symbols
r
density (kg m3)
d
liquid fraction (¼3 fl)
b
thermal expansion coefficient (K1)
m
dynamic viscosity (N s m2)
3
porosity
c
tortuosity coefficient
u
pore density (pore per inch, PPI)
Subscripts
amb
ambient
f
fluid (both solid and liquid paraffin)
fl
liquid paraffin
fs
solid paraffin
m
melting
p
pore
s
solid matrix
sf
surface
td
thermal dispersion
Superscripts
n
iteration number at the present time level
influences of the Rayleigh, Stefan, and Nusselt numbers on the
evolution of the solid/liquid interface were reported and discussed.
Although this technique can solve more complicated cases, the
thermal dispersion effects were not considered, thereby making
the technique unsuitable for cases with a high Rayleigh number.
Moreover, the effective thermal conductivity is only determined by
the volume fraction of each phase without considering the
geometric configuration effect of the metal structure, which is not
suitable for metal foam. Lafdi et al. experimentally investigated the
phase-change heat transfer within a PCM composite permeated
with aluminum foam [15], and then numerically studied the cooling process of the electronic device using heat sinks of solid matrix
impregnated with PCM [16]. Zhao et al. [17] experimentally and
numerically studied the paraffin melting in metallic foam.
However, in their model, the flow motion of liquid paraffin, which
has been proved to be very important, was not taken into account.
Jegadheeswaran and Plhekar [18] have summarized various techniques for enhancing the thermal performance in different heat
thermal storage systems and enumerated the pros and cons of
these enhancement techniques.
The aim of the present work was to experimentally and
numerically investigate the melting process of paraffin in copper
foams. Metal foam, a metallic porous matrix material, is a porous
media that exhibits the excellent combination of compactness, low
weight, and high thermal conductivity [19]. An experimental test
rig was built to measure the wall and internal temperatures. The
effects of porosity and pore density on heat transfer were examined
and discussed. A two-equation model was subsequently applied to
numerically study the solideliquid phase-change process in highporosity metallic foam. In the model, the natural convection of
liquid paraffin, the thermal dispersion effect, the irregular geometry configuration, and the non-Darcy effects were fully considered.
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
The numerical predictions were validated by comparing the
experimental results. The velocity field of the molten paraffin, the
temperature distributions for PCM and the metal matrix, as well as
the time-related interface variations were presented and discussed.
2. Experimental apparatus and procedure
2.1. PCM melting point and latent heat measurement
The latent heat and melting point of the paraffin were measured
using a differential scanning calorimeter (DSC; TA-Q20). The results
shown in Fig. 1 indicate that the latent heat of paraffin is 102.1 J/g,
and the melting process starts at 46.48 C and ends at 60.39 C. The
related thermophysical properties were listed in Table 1.
2.2. Experimental apparatus and procedure
As shown in Fig. 2, a plexiglass cavity was designed to cage the
paraffin-saturated metal foam based on the size of the metal foam.
However, considering the volume expansion of PCM during phase
change, a 5 mm gap between the top surface and the inner glass
surface was left for the expansion of the melted paraffin. In case of
leakage during melting, two other 30 mm thick plexiglass plates
were fabricated and fixed to the left and right sides of the cavity and
tightened at the corners with four studs. All copper foam samples
having a uniform dimension of 100 mm 100 mm 45 mm were
sintered on a 2 mm thick copper substrate at the bottom to improve
the thermal conduction to the internal space and minimize the
contact thermal resistance. A 100 mm 100 mm 0.15 mm
electric insulated film heater was glued to the left side of the metal
foam, i.e., a brazed copper substrate, to provide constant heat flux.
The other four sides (top, bottom, front, and back) were kept
thermally insulated with 30 mm thick urethane foam plates. A
constant heat flux of 4000 W/m2 was supplied by a DC voltage
stabilizer to the film heater during the experiment. For each
sample, a temperature-tracking task was completed by a Keithley
2700-multimeter/data acquisition system and data were obtained
every 5 min. A small cylindrical hole (3 mm in diameter, 45 mm in
height) erected on the sintered copper plate was drilled in the
center of the foam to better understand the time-related variation
in the inner temperatures. Five T-type thermocouples were
selected, as seen in Fig. 2(b). One was attached to the copper plate
to measure the wall temperature, and the other four were fixed
Fig. 1. DSC results for paraffin.
3
Table 1
Thermal physical properties of paraffin.
Parameters
Value
rf (at 20 C)
785.02
1.021 105
46.48
60.39
0.30
0.10
2850
3.65 103
3.085 104
L
Tm1
Tm2
kfs
kfl
cpf (at 20 C)
m (at 65 C)
b (at 65 C)
inside the hole at 3, 13, 28, and 43 mm from the left side of the
metal foam. The uncertainties were estimated based on the random
errors during temperature measurements. The uncertainty for the
T-type thermal couple temperature is 0.1 C. The uncertainties for
the D.C. power output and the film heater power output are 1.22%
and 5.0%, respectively.
The morphology of the metal foam used in the experiment is
shown in Fig. 3. Seven copper foam samples with different porosities and pore densities were used, the geometric characteristics of
which are listed in Table 2.
3. Experimental results and discussion
3.1. Effect of the metal matrix and foam porosity
Fig. 4 displays the wall temperature variation history with time
of three metal foams with different porosities (3 ¼ 0.90, 0.95, and
0.98), with the pore density fixed at 40 PPI. The wall temperature
variation of pure PCM was used as the reference. The whole melting
process can be divided into three regions: solid, mush, and pure
liquid. In the solid region, the temperatures for pure PCM and foamPCM composite both increased by absorbing explicit heat from the
wall. The foam-PCM composite wall temperature was lower than
that of pure PCM and showed better thermal management ability
because the effective PCM thermal conductivity was improved by
the copper metal matrix. As time progressed, the metal ligaments
reached a temperature higher than the PCM melting point and
started triggering a local phase change. Two factors dominate the
phase change, namely, the heat conduction and the natural
convection of the molten PCM. The final performance depends on
whichever of these two factors prevails in the melting process. For
the pure PCM, the paraffin absorbed the latent heat at the solid/
liquid melting interface dominated by natural convection, thereby
increasing the temperature of pure paraffin in a flattened manner.
However, in the foam-PCM composite, the heat conduction thermal
resistance was evidently reduced by the foam matrix, although the
liquid paraffin was constrained in the metal foam pore to some
extent and suppressed natural convection, which plays a positive
role in heat transfer by accelerating the melting rate. Consequently,
heat conduction dominated the melting process, and the wall
temperature of the foam-PCM composite was lowered and
increased almost linearly, governed by the combined effects of heat
diffusion, natural convection, and the latent heat absorption at the
moving boundary. When the melting process was fully complete,
the explicit heat exerted its role again and the temperatures for
both pure PCM and the composite continued to increase linearly.
Points A, B, C, and D represented the end of the melting process in
the mush region of the composites with 0.90, 0.95, and 0.98
porosities and pure PCM (3 ¼ 1). The total time consumption
decreased in the order of decreasing porosity because a higher
porosity indicates a higher PCM volume fraction. Higher porosity
means a lower effective thermal conductivity at a fixed pore
4
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
Fig. 2. Experimental apparatus and test section. (a) Experimental apparatus, (b) Test section.
density, and implies a smaller fiber diameter or smaller heat
transfer surface area for the prevailing heat conduction process.
Thus, higher porosity allows the sample to achieve higher wall
temperature. The effective thermal conductivity can be calculated
according to Ref. [20]. The interfacial surface area can be obtained
from Eq. (1) [21]:
asf ¼
3pdf
d2p
(1)
3.2. Effect of pore density
Fig. 5 depicts the relationship between the wall temperature
and the duration time for the copper foam filled with PCM of three
pore densities (10, 20, and 40 PPI) and pure PCM with a fixed 0.9
porosity. A similar trend was observed when the wall temperature
for the copper foam filled with PCM was lower than that of pure
PCM. The total melting duration time at various pore densities were
almost the same because the PCM volume fraction were identical at
a fixed porosity. The wall junction temperature became higher with
Fig. 3. Typical structures of tested metal foam (3 ¼ 0.90, u ¼ 40 PPI).
increasing pore density at the same time; however, the influence
was less significant for the pore density compared with the
porosity. As pore density increased, the interfacial surface area and
the effective thermal conductivity increased, thereby improving
the melting heat transfer; nevertheless, the permeability
decreased, resulting in a strong suppression of the natural
convection of molten wax. The two opposite factors competed and
strong suppression of the natural convection is comparatively
considerable, resulting in higher wall temperature for high pore
density foam and a less evident temperature difference among the
three composites.
Table 2
Geometrical parameters of the sample foams.
Sample
Porosity
Pore density (PPI)
dp (mm)
df (mm)
1
2
3
4
5
6
7
0.90
0.90
0.90
0.95
0.95
0.95
0.98
10
20
40
10
20
40
40
2.540
1.270
0.635
2.540
1.270
0.635
0.635
0.309
0.154
0.077
0.218
0.109
0.055
0.035
Fig. 4. Variations of Tw with time of PCM composites with different porosities at a pore
density of 40 PPI.
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
5
PPI foam than for the 40 PPI foam. The motion of molten paraffin wax
in the 10 PPI foam was less restricted in the matrix because of the
larger pore size, leading to a stronger natural convective effect.
Consequently, the temperature distribution was more uniform in the
horizontal direction. The effect of porosity on temperature distribution uniformity is related to pore density because the total thermal
resistance is determined by both the effective thermal conductivity
and natural convection. The natural convection was severely suppressed at the 40 PPI pore density, whereas it was considerable at the
10 PPI pore density. When the porosity varied from 0.95 to 0.9, the
effective thermal conductivity increased or the heat conduction
thermal resistance decreased. The influence of the effective thermal
conductivity increment by decreasing the porosity from 0.95 to 0.9 on
the total resistance was evident for the 40 PPI foam compared with
the 10 PPI foam. Hence, the improvement in temperature distribution
uniformity is more significant for the 40 PPI foam than for the 10 PPI
foam, as shown in Fig. 6(a)e(d).
Fig. 5. Variations of Tw with time for the PCM composites with different pore densities
(3 ¼ 0.90).
4. Numerical simulation
4.1. Physical and mathematical model
3.3. Internal temperature distributions
The two-dimensional schematic diagram of the present transient melting heat transfer is shown in Fig. 7. The copper foam
saturated with solid paraffin measured 45 mm in length and
100 mm in height. The top, bottom, and right sides can be
considered adiabatic. The initial temperature was set as the
ambient temperature at 25 C. The left side was heated at
Fig. 6 quantitatively presents the internal temperature distribution
at four different foam composite test spots at two porosities (3 ¼ 0.90
and 0.95) and pore densities (u ¼ 10 and 40 PPI). A comparison of
Fig. 6(a) and (b) or Fig. 6(c) and (d) shows that the temperature
difference between the two adjacent tested spots is smaller for the 10
a
b
c
d
Fig. 6. Internal temperature-time variation.
6
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
Continuous equation:
!
vrf
þ V$ rf h U i ¼ 0
vt
(2)
Momentum equations:
rf vhui rf !
m rf CI !
vP m
þ 2 ðh U i$VÞhui ¼ þ fl V2 hui fl þ pffiffiffiffi jh U ij hui
d vt d
K
vx d
K
þ Ahui
ð3Þ
rf vhvi rf !
mfl rf CI !
vP mfl 2
þ 2 ðh U i$VÞhvi ¼ þ V hvi þ pffiffiffiffi jh U ij hvi
d vt
d
vy
K
K
d
ð4Þ
þ rf g b Tf Tm1 þ Ahvi
Energy equation for PCM:
Fig. 7. Schematic diagram of the physical domain.
a constant heat flux of 4000 W m2, consistent with the input
power of the heating film. The following assumptions are made in
the mathematical model for the present problem: (1) the metal
foam is assumed homogeneous and isotropic; (2) the liquid PCM is
considered impressible and Newtonian, and subjected to the
Boussinesq approximation; (3) the changes of density and heat
capacity of paraffin during the whole melting process is neglected
while the other thermal physical properties of PCM are constant at
each phase but treated separately for the solid and liquid phases.
Based on the above-mentioned considerations, the volumeaveraging continuous, momentum, and two-energy equations for
PCM and the solid matrix can be written as follows:
! D E
D E
v Tf
!
dfl
3 rf cpf þ L D E
þ rf cpf ðh U i$VÞ Tf
vt
d Tf
D E
D E
¼ kfe þ ktd V2 Tf þ hsf asf hTs i Tf
(5)
Energy equation for metal matrix:
ð1 3 Þrs cps
D E
vhTs i
¼ kse V2 hTs i hsf asf hTs i Tf
vt
(6)
Where thermal properties ofrf, L,cpf,mfl,b are presented in Table 1.
The properties ofrs (8920 kg m-3), cps (380 J kg-1 K-1) and ks
(401 W m-1 K-1)are density,heat capacity and the thermal
conductivity of copper, respectively. Other parameters in the
Table 3
The employed semi-empirical correlations of metallic foam parameters.
Parameter
Permeability, K
Correlation
K ¼
3
Reference
2 d2
k
[23]
36ðc 1Þc
4p 1
þ cos1 ð23 1Þ
3
3
c ¼ 2 þ 2cos
dk ¼
Inertial coefficient, CI
c
3c
dp dp ¼
22:4 103
u
pffiffiffiffi
FI ¼ CI ð1 3 Þ0:132 ðdf =dp Þ1:63 = K , CI ¼ 2.12 103
rffiffiffiffiffiffiffiffiffiffiffi
13
dp
df ¼ 1:18
3p
( 0:76Re0:4 Pr0:37 k =d ; 1 Re 40
f
f
d
d
0:37
kf =df ; 40 Red 103
0:52Re0:5
d Pr
0:37
0:26Re0:6
kf =df ; 103 Red 2 105
d Pr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Red ¼ rf u2 þ v2 df =ð3 mfl Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ctd
r C d u2 þ v2 , Ctd ¼ 0.36
ktd ¼
1 3 f pf f
1
ke ¼ pffiffiffi
2ðRA þ RB þ RC þ RD Þ
4l
RA ¼
ð2e2 þ plð1 eÞÞks þ ð4 2e2 plð1 eÞÞkf
[26]
hsf ¼
Local heat transfer coefficient, hsf
Thermal dispersion conductivity, ktd
ðe 2lÞ2
ðe 2lÞe2 ks þ ð2e 4l ðe 2lÞe2 Þkf
pffiffiffi
ð 2 2eÞ2
RC ¼
pffiffiffi
pffiffiffi
pffiffiffi
2
2
2pl ð1 2 2eÞks þ 2ð 2 2e pl ð1 2 2eÞÞkf
2e
RD ¼ 2
e ks þ ð4 e2 Þkf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
pffiffiffi
2ð2 ð5=8Þe3 2 23 Þ
pffiffiffi
l ¼
, e ¼ 0.339
pð3 4 2e eÞ
(
kfs;e
solid region
kfe ¼ fl kfl;e þ ð1 fl Þkfs;e Mush region
kfl;e
liquid region
[24]
[25]
RB ¼
Effective thermal conductivity, ke
kse ¼ ke jkf ¼0 , kfe ¼ ke jks ¼0
[20]
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
governing equations, including K (permeability), CI (inertial coefficient), kse (the effective thermal conductivity of the solid matrix),
kfe (the effective thermal conductivity of paraffin), ktd (thermal
dispersion conductivity), and hsf (local heat transfer coefficient), are
from the semi-empirical correlations in the literature and can be
seen in Table 3, where the values of CI and Ctd shown the best fit to
the experimental data in the corresponding literatures. For interfacial heat transfer coefficienthsf, since no exact correlation is
suitable for natural convection in porous media, an empirical
correlation provided by Zukauska [24] has been introduced. The
Reynolds number is based on the cylinder diameter and Pr is the
fluid Prandtl number. In this study, the Re range was verified to be
less than 5.0 which satisfied the first term of correlation propased
by Zukauska [24] and then was applied. The bracket “hi” represents
the volume-averaging model based on the DupuiteForchheimer
relationship [22]. The second and fourth terms on the right side of
Eqs. (3) and (4) account for the extension of Darcy’s law to explain
the non-Darcy effects, and the boundary and inertial effects
proposed by Brinkman and Forchheimer, respectively. The last
terms in the momentum equations are additional source terms,
where parameter A is related to the liquid fraction in the pore
volume based on Kozeny’s equation [22], which is introduced as the
following equation:
A ¼
fl ¼
C 1 fl2
:
Tf Tm1
Table 4
Initial and boundary conditions for the governing equations.
Definite conditions
Positions
Velocity
Temperature
Initial condition
0 x L, 0 y H
x ¼ 0, 0 y H
u¼v¼0
u¼v¼0
x ¼ L, 0 y H
u¼v¼0
0 x L, y ¼ 0
u¼v¼0
0 x L, y ¼ H
u¼v¼0
Tf ¼ Ts ¼ Tamb
q ¼ qfþ qs
¼3 q þ (1 3 )q
vTf
vTs
¼
¼ 0
vx
vx
vTf
vTs
¼
¼ 0
vy
vy
vTf
vTs
¼
¼ 0
vy
vy
Boundary conditions
5. Numerical results and discussion
5.1. Code validation
The numerical codes were validated with the present experimental results by comparing the interface positions at three
different moments. Fig. 8(a) and (b) are pictures of interface locations captured by a digital camera at 3600 s and 3780 s for the 0.90
porosity and 10 PPI foam. The predicted solid/liquid interfaces
[Fig. 8(c) and (d)] obtained from the numerical predictions agree
(7)
S þ fl3
8
<
7
0
.
ðTm2 Tm1 Þ
1
d ¼ 3 $fl
Tf Tm1
Tm1 Tf Tm2
Tm2 Tf
(8)
(9)
where d is the liquid fraction considered as a function of PCM
temperature and porosity, and C and S are set as 1015 and 1010,
respectively, to fix the PCM velocity to zero before melting for
convenient numerical implementation. Therefore, the conduction
dominated equations for the solid phase and convection-diffusion
controlled equations for melted paraffin are unified by Eqs.
(3)e(5). The initial and boundary conditions for the governing
equations are given in Table 4. The initial temperatures for PCM and
metal foam are equal to the ambient temperature. The initial values
of velocity for PCM at both directions are zero and the same value is
applied to the velocity boundary condition based on the velocity
non-slip principle at the wall. At the left boundary, the cover plate
contacts with metal fiber or the fluid (solid or liquid), due to this
consideration, if ignoring the heat dissipation through the sintered
2 mm copper substrate, the heat is assumed distributed respectively to the metal foam and PCM by each representative volume
ratio at the wall surface as applied in [26]. The rest boundaries are
set adiabatic.
4.2. Numerical procedure
The governing equations were discretized by a finite volume
method in a staggered grid system of 90 80. The combined
equations are solved numerically using the IDEAL algorithm [27].
The regions of solid paraffin and melted region are determined
automatically during the iteration by the whole domain solution
strategy. The iteration at the present time layer is considered
convergent if the maximum relative residuals of Tf are less than
105, which can be expressed as ½ðTfn Tfn1 Þ=Tfn1 < 105 .
Fig. 8. Comparison of the solid/liquid interface between the photos and numerical
results. (3 ¼ 0.90，u ¼ 10 PPI) at different times.
8
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
Fig. 9. Velocity field and temperature distributions (3 ¼ 0.95, u ¼ 20 PPI) at 2700 s.
well with the experimental results. The feasibility of the present
model is verified.
whereas only diffusion existed in the solid paraffin region. A
difference in the temperature distributions between PCM and the
metal matrix was observed, especially at the liquid paraffin region.
This quantitative result validates the feasibility and necessity of the
two-temperature model.
5.2. Temperature distribution
Fig. 9 shows the velocity field and temperature contours of PCM
and the solid matrix (3 ¼ 0.95, u ¼ 20 PPI) at 2700 s. The hotter
liquid moved upward because of the natural convection of molten
paraffin stimulated by buoyancy, pushing the upper portion of the
interface faster than the bottom. For the temperature distribution,
it could be clearly observed that there exist great differences
between metal foam and the PCM. Also, at the same local position,
metal foam temperature was higher than that of PCM. In the solid
PCM region, conduction-induced heat transfer produced similar
temperature profiles for the PCM and the metallic matrix; however,
the metallic foam temperature was higher than that of the solid
PCM due to the local thermal non-equilibrium effect. In the liquid
wax region, the liquid temperature pattern differs from that of the
metallic matrix. The temperature profile of the liquid paraffin was
more uniform in the horizontal direction than that of the metallic
matrix because natural convection prevailed in the liquid region,
Fig. 10(a) and (b) compare the time-related interface position for
the 0.9 and 0.95 porosities at a fixed 20 PPI pore density. At the
beginning of the melting process, the interface tended to be vertical
because the liquid fraction was relatively small, indicating that heat
conduction dominated in the process. As melting continued, the
upper part moved faster than the lower part because of the eddy
flow, promoting the fusion rate at the top and the phase change was
accelerated because of the natural convection induced by buoyancy
in the molten PCM. Moreover, the interface movement rate for the
0.90 porosity foam was higher than that of the 0.95 porosity foam.
At 1500 s, the top phase interface for the 0.90 porosity foam had
already attached to the right boundary, whereas the interface of the
0.95 porosity foam had just crossed over the middle (X/H ¼ 0.57).
b
0.10
0.10
0.08
0.06
0.06
600s
900s
1200s
1500s
1800s
2100s
2400s
3000s
Y
0.08
Y
a
5.3. Interface variations
0.04
600s
900s
1200s
1500s
1800s
2100s
0.02
0.00
0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045
X
ε=0.90
0.04
0.02
0.00
0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045
X
ε=0.95
Fig. 10. Solid/liquid interface locations at various times (u ¼ 20 PPI).
W.Q. Li et al. / Applied Thermal Engineering 37 (2012) 1e9
The above-mentioned results are consistent with the experimental
results indicated in Fig. 4.
6. Conclusion
The melting phase-change heat transfer in paraffin PCM was
enhanced by the porous metallic foam. The mechanisms of heat
diffusion and natural convection dominated the melting phasechange heat transfer. The enhancement of heat conduction is
more prominent than the suppression of natural convection of
molten liquid by the copper matrix in the foam-PCM composite,
whereas natural convection prevailed in the pure PCM. The final
total thermal resistance was lower for the foam-PCM composite
than the pure PCM to result in the lower corresponding wall
temperature. The influence of pore density on wall temperature
was less sensitive than that of porosity. The uniformity of the
temperature distribution inside the foam-PCM composite was
augmented either by decreasing pore density to accelerate natural
convection or by decreasing porosity to improve the effective
thermal conductivity. The feasibility of the numerical model, in
which the non-Darcy effects, the natural convection in the liquid
paraffin, and the local non-equilibrium effects were considered,
was validated by the experimental visualization. The temperature
distribution profile for the local solid matrix was quite different
from that of the filled PCM and validated the feasibility and
necessity of the applied two-equation model. The solid/liquid
movement rate was higher for the low porosity foam, which is also
consistent with the experimental results.
Acknowledgements
The current study was supported by the National Key Projects of
Fundamental R/D of China (973 Project: 2011CB610306), the
National Natural Science Foundation of China (Nos. 51176149 and
50736005), the National Excellent Doctoral Dissertation Foundation of China (201041), the Doctoral Fund of the Ministry of
Education of China (200806981013).
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