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Study on solidification of phase change material in fractal porous metal foam

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The Sierpinski fractal is introduced to construct the porous metal foam. Based on this fractal description, an unsteady heat transfer model accompanied with solidification phase change in fractal porous metal foam embedded with phase change material (PCM) is developed and numerically analyzed. The heat transfer processes associated with solidification of PCM embedded in fractal structure is investigated and compared with that in single-pore structure. The results indicate that, for the solidification of phase change material in fractal porous metal foam, the PCM is dispersedly distributed in metal foam and the existence of porous metal matrix provides a fast heat flow channel both horizontally and vertically, which induces the enhancement of interstitial heat transfer between the solid matrix and PCM. The solidification performance of the PCM, which is represented by liquid fraction and solidification time, in fractal structure is superior to that in single-pore structure.
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Fractals, Vol. 23, No. 1 (2015) 1540003 (8pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0218348X15400034
STUDY ON SOLIDIFICATION OF PHASE
CHANGE MATERIAL IN FRACTAL
POROUS METAL FOAM
CHENGBIN ZHANG,LIANGYU WUand YONGPING CHEN,,
Key Laboratory of Energy Thermal Conversion
and Control of Ministry of Education
School of Energy and Environment
Southeast University
Nanjing,Jiangsu 210096,P. R. China
School of Hydraulic,
Energy and Power Engineering
Yangzhou University
Yangzhou,Jiangsu 225127,P. R. China
ypchen@seu.edu.cn
Received June 5, 2014
Accepted August 3, 2014
Published February 25, 2015
Abstract
The Sierpinski fractal is introduced to construct the porous metal foam. Based on this frac-
tal description, an unsteady heat transfer model accompanied with solidification phase change
in fractal porous metal foam embedded with phase change material (PCM) is developed and
numerically analyzed. The heat transfer processes associated with solidification of PCM embed-
ded in fractal structure is investigated and compared with that in single-pore structure. The
results indicate that, for the solidification of phase change material in fractal porous metal
foam, the PCM is dispersedly distributed in metal foam and the existence of porous metal
matrix provides a fast heat flow channel both horizontally and vertically, which induces the
enhancement of interstitial heat transfer between the solid matrix and PCM. The solidification
Corresponding author.
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C. Zhang, L. Wu & Y. Chen
performance of the PCM, which is represented by liquid fraction and solidification time, in
fractal structure is superior to that in single-pore structure.
Keywords: Solidification; Fractal; Metal Foam; Porous Media.
NOMENCLATURE
c: specific heat capacity
k: thermal conductivity
Li:lengthofith pore, see Fig. 2
L: length of metal foam
T: temperature
Tm: phase equilibrium temperature
Tc: cold surface temperature
x,y:coordinates
Greek symbols
λ: thermal conductivity
ρ:density
Γ:interface
Subscripts
i: number of level
ini : initial condition
l: liquid
ma : solid matrix
s: solid
1. INTRODUCTION
Owing to high heat storage density, small volume
requirement and moderate temperature variation,
the liquid–solid phase change heat transfer using
phase change material (PCM) is of great inter-
est in wide range of potential applications,1,2such
as smart building, compact electronics cooling and
spacecraft thermal control system. In addition, the
liquid–solid phase change heat transfer has long
been the subject of scientific investigation due to
the inherent difficulties associated with the nonlin-
earity of the interface conditions and the unknown
locations of the moving boundaries. Therefore, the
solidification phase change heat transfer is of con-
siderable importance in the development of phase
change energy storage heat transfer exchanger and
has become a topic of increasing concern and
growing scientific interest within the past several
years.38
In low temperature latent heat storage systems,
most of phase change media suffer from the common
problem of low thermal conductivity. In this case,
the porous metal foam, which possesses the advan-
tage of high thermal conductivity and low density
(i.e. light weight), have become introduced as an
attractive optional matrix to enhance the liquid–
solid phase change heat transfer. This is a new type
of functional composite composed of metal foam
saturated with a phase change material in pores.
To understand the heat transfer performance in
porous metal foam embedded with PCM, there
have been several attempts to describe and inter-
pret liquid–solid phase change process (including
solidification and melting) by the use of numerical
simulation and experimental test.914 Tong et al.11
conducted a theoretical model of solid–liquid phase
change in a vertical annulus space demonstrated the
heat transfer rate increase during solid–liquid phase
change by inserting a high-porosity metal matrix
into the PCM. Damronglerd and Zhang12 pre-
sented a modified temperature-transforming model
to solve melting phase change in porous media, tak-
ing into consideration of the heat capacity depen-
dence on the fractions of a solid and a liquid in the
mushy zone. Their investigation indicates that melt-
ing is accelerated under a higher Rayleigh number,
and the melting is dominated by conduction for a
lower Darcy number. Siahpush et al.13 performed a
detailed experimental and analytical study to eval-
uate how copper porous foam enhances the heat
transfer performance in a cylindrical solid/liquid
phase change thermal energy storage system. Farid
et al.14 laid out the understanding of latent heat
storage with a focus on the PCM materials, encap-
sulation and applications.
In summary, the available investigations provide
insight into the solid–liquid phase change heat
transfer process in porous metal foam. However,
little attention has been paid to optimize the geo-
metrical structure of porous metal foam so as
to further improve the solidification phase change
performance. Recent researches have demonstrated
that the fractal geometry has distinctly unique
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Study on Solidification of Phase Change Material
advantages in the optimization of heat and mass
transport structure. Should we construct fractal
architectures to optimize the liquid–solid phase
change system, more advanced transport perfor-
mance may be acquired. Pia and Sanna1517 utilize
the Sierpinski fractal to model the porous mate-
rials and investigated how the pore size distri-
bution and geometric microstructure parameters
affect the thermal conductivity of porous materials.
The results indicate that the pore structure and dis-
tribution is of significance in the determination of
thermal conductivity of porous materials.
It is realized that the architecture of Sierpinski
fractal is analogous to the topological structures
of metal foam. In addition, the Sierpinski fractal
metal foam is a close-celled structure, and no break
of solid matrix occurs inside. Such solid matrix
structure could provide a fast heat flow channel
for PCM solidification both horizontally and verti-
cally. Therefore, the Sierpinski fractal is utilized to
construct the geometrical structure of porous metal
foam in this paper. Based on this fractal description,
an unsteady model of solidification phase change
in porous metal foam is developed and numeri-
cally analyzed to investigate the solidification phase
change heat transfer process in a fractal structure,
in an attempt to identify the superiority of the frac-
tal architecture. In addition, the heat transfer pro-
cesses associated with solidification of PCM embed-
dedinfractalstructureiscomparedwiththatin
single-pore structure.
2. MATHEMATICAL MODEL
2.1. Sierpinski Fractal
In the current investigation, the Sierpinski carpet
is applied to construct the porous metal foam. As
shown in Fig. 1, the construction of the Sierpinski
carpet can be listed as follows:
(1) The first step (i= 1) in this construction begins
with a square, cut the square into 9 congruent
i = 1 i = 2 i = 3
Fig. 1 Generation of Sierpinski fractal.
subsquares in a 3-by-3 grid, and marks the cen-
tral subsquare as a pore that can be embedded
with PCM.
(2) In the second step (i= 2), the same procedure
is then applied recursively to the remaining 8
subsquares.
(3) This recursion is done infinite times; then
the Sierpinski carpet is obtained. The fractal
dimension of Sierpinski carpet can be deter-
mined as D=ln8/ln3 = 1.893, and the scale of
ith pore is Li=(1/3)iL0.
In order to compare the thermal performance of
solidification heat transfer, a single-pore structure
is designed with only a pore in the center of solid
matrix. The single-pore structure is of the same
porosity with fractal metal foam. In the simula-
tion, all the generated pores are filled with PCM,
and the solid matrix is assumed to be aluminum.
In this paper, a mixture of Mn(NO3)2·6H2Oand
MnCl2·4H2O is adopted as PCM for latent heat
storage, and the thermal properties are presented in
Table 1. The Sierpinski fractal with i= 3 is applied
to construct the metal foam, and the length and
width of the metal foam is of the same, which are
both equal to L0=9cm.
2.2. Solidification Phase Change in
Porous Metal Foam
2.2.1. Governing equation
The unsteady heat transfer process associated with
solidification in porous metal foam embedded with
PCM is shown in Fig. 2. In order to simplify the
model of solidification phase change in porous metal
foam, the following assumptions are applied:
(1) The pores are fully embedded with PCM, and
the solid, liquid and mushy regions exist for a
PCM in phase change heat transfer process.
(2) The natural convection and sub-cooling that
occurs for PCM is negligible.
As a latent heat storage material, the solidifica-
tion phase change of PCM occurs at a certain tem-
perature range (TmTTTm+∆T). In
the current investigation, the effective heat capac-
ity method is applied to model the solidification
phase change process of PCM in pores of metal
foam. In this method, the phase change latent heat
of PCM is regarded as a large sensible heat capac-
ity over this temperature range. Therefore, the uni-
form energy equation can be constructed for whole
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C. Zhang, L. Wu & Y. Chen
Table 1 Thermal Properties of PCM.
Density,Capacity,Conductivity,Latent Heat,
ρ/kg ·m3c/J·(kg ·K)1λ/W·(m ·K)1hfg/kJ ·kg1
Aluminum 2700 880 217
PCM 1700 2500 0.60 125.9
(a) (b)
Fig. 2 Schematic of heat transfer in metal foam: (a) fractal and (b) single-pore.
domain, and its nonlinear heat transfer problem is
directly treated instead of the domain decomposi-
tion. The governing equation for both the region of
solid matrix and PCM pores could be expressed as
ρc(T)∂T
∂t =
∂x k(T)∂T
∂x +
∂y k(T)∂T
∂y ,(1)
where Tis temperature, ρis the density, cis specific
heat capacity and kis thermal conductivity.
For the region of solid matrix, the density, specific
heat capacity and thermal conductivity is constant,
i.e. ρ=ρma,c=cma,k=kma, in which the sub-
script ma denotes the solid matrix.
For the region of pores, the density, specific heat
capacity and thermal conductivity is constant for
the PCM at the liquid region and solid region; at the
mushy region, the latent heat of PCM is regarded
as a sensible heat capacity between solidus temper-
ature and liquidus temperature of PCM, and the
thermal conductivity of PCM is assumed to be lin-
ear with temperature. Therefore, the specific heat
capacity and thermal conductivity of PCM in the
solidification phase change process can be written as
c(T)=
csT<(TmT),
hfg
2∆T+cs+cl
2(TmT)T(Tm+∆T)
clT>(Tm+∆T)
,(2)
k(T)=
ksT<(TmT),
ks+klks
2∆T[T(TmT)] (TmT)T(Tm+∆T)
klT>(Tm+∆T),
,(3)
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Study on Solidification of Phase Change Material
where Tmis the phase equilibrium temperature,
subscript land sare liquid phase and solid phase of
PCM. In the simulation, Tm=20
C, the tempera-
ture range of phase change is between solidus tem-
perature 15C and liquidus temperature 25C.
2.2.2. Boundary conditions
At the initial time, both the solid matrix and PCM
are of the same temperature, Tini.Inthesimulation,
the initial temperature is set as Tini =30
C. Once
the solidification heat transfer occurs, the heat will
be removed by the left cold surface with tempera-
ture Tc, as shown in Fig. 2. In this case, the upper,
right and lower boundary condition of porous metal
foam are assumed to be adiabatic expecting the left
boundary condition which is constant temperature
boundary condition. Note that all the initial con-
ditions and boundary conditions of the single-pore
structure are the same as the fractal structure of
the metal foam.
Initial boundary condition:
T(x, y, 0)|t=0 =Tini.(4)
The upper boundary,
y=L:λ∂T
∂y
y=L
=0.(5)
The lower boundary,
y=0:λ∂T
∂y
y=0
=0.(6)
The right boundary,
x=L:λ∂T
∂y
x=L
=0.(7)
The left boundary,
x=0:T|x=0 =Tc.(8)
Heat transfer in the porous metal foam is a con-
jugate problem which combines heat conduction in
the solid matrix and solid–liquid phase change in
PCM pores. These two heat transfer modes are cou-
pled at the solid surface of pores with the continu-
ities of temperature and heat flux,
Ts,Γ=Tl,Γ,(9)
λs∂Ts
∂n
Γ=λl∂Tl
∂n
Γ.(10)
2.2.3. Numerical simulation
In this paper, the effective heat capacity method
is applied to simulate the unsteady heat trans-
fer process accompanied with solidification phase
change for the fractal porous metal foam filled with
PCM. The sensible heat capacity method has the
advantage that it can convert the heat transfer
problem associated with solidification phase change
in two separated regions into a nonlinear heat con-
duction problem as the whole computational region
conveniently. Note that the distribution of specific
heat and heat conductivity in the heat transfer
region should be re-constructed, respectively.
A structured mesh based on rectangular grid is
applied to mesh the complex geometry structure
of fractal porous metal foam shown in Fig. 2. A
nonuniform grid arrangement with a large num-
ber of grids in pore region is arranged to resolve
the heat conduction accompanied with solidifica-
tion processes. The governing differential equation
of Eq. (1) with boundary conditions as described
in Eqs. (4)–(10) is numerically solved by the use of
control volume finite-difference technique. The uni-
form energy equation, Eq. (1), puts temperature T
as the sole parameter to be solved. The tempera-
ture in every control unit can be obtained by iter-
ation calculation, and hence the temperature dis-
tribution of the whole computational domain can
beacquired.Inaddition,thenumericalcodeisver-
ified in a number of ways to ensure the validity of
the numerical analysis. A grid independence test is
conducted using several different mesh sizes.
3. RESULTS AND DISCUSSION
3.1. Solidification Process Analysis
The existence of porous solid matrix plays an
important role in the solid–liquid phase change heat
transfer. Figure 3 presents the liquid fraction (ϕl)
of PCM during the solidification phase change heat
transfer in fractal structure and single-pore struc-
ture. As shown, at the initial time, the PCM embed-
dedinallporesareattheliquidstate.ThePCMin
the pore region near the left wall begins to solidify
first owing to cold surface at the left, and the solid-
ification region gradually proceeds toward the right
wall. Owing to interstitial heat transfer between the
PCM and solid matrix, the solidification front is
closely parallel with the solid wall of PCM pores.
The Sierpinski fractal metal foam is a close-
celled structure composed of hierarchical PCM cells,
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C. Zhang, L. Wu & Y. Chen
0min 1min 10min 20min
ϕ
l
Fig. 3 Liquid distribution of PCM during solidification process (Tc=5C).
whichprovidesmoreareaforliquidincontactwith
solid matrix when compared with single-pore struc-
ture. It is beneficial to interstitial heat transfer
between the solid matrix and PCM. During solidifi-
cation heat transfer, the PCM solidification front in
every pore moved steadily inward with time and the
solidification layer adjacent to the wall grew thicker
until the PCM is completely solidified. In such hier-
archical structure, the solidification rate of PCM
inside small pores is faster than that in large pores,
so it is observed that the PCM in the largest pore
finishes solidification at the latest time.
It is also indicated by Fig. 3 that, as time pro-
ceeds, the PCM in the pores begins to solidify, and
solidification rate of fractal structure is faster than
that of single-pore structure. At the case of cold
surface temperature 5C, the liquid is almost com-
pletely frozen at 20 min in fractal structure while
nearly 30% of PCM remain on the liquid state in
single-pore structure. In addition, the time needed
to solidify the PCM in fractal structure is shorter
than that in single-pore structure. The explanation
of this phenomenon is that: (1) unlike the phase
change of PCM alone, the solid matrix of Sierpin-
ski fractal metal foam could provide a fast heat
flow channel for PCM solidification both horizon-
tally and vertically; (2) the Sierpinski fractal metal
foam provides more area for liquid in contact with
solid matrix with respect to single-pore structure,
leading to superior heat transfer accompanied with
solidification phase change; (3) the PCM is dispers-
edly distributed in metal foam, which induces the
0.02min 1min 10min
T /K
Fig. 4 Temperature distribution during solidification process (Tc=5C).
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Study on Solidification of Phase Change Material
enhancement of interstitial heat transfer between
the solid matrix and PCM.
In order to give a clearer understanding of phase
change heat transfer in porous metal foam, Fig. 4
presents the unsteady temperature distribution of
solid matrix and PCM pores in fractal structure and
single-pore structure. Once the left wall tempera-
ture suddenly falls to Tc, the heat flows from the
PCM to the wall. The latent heat of the PCM has
an important influence on the temperature profiles
during solidification. The temperature throughout
the solidification region is maintained at the solid-
ification point, which has a distinct impact on the
temperature profiles in the PCM. During the solid-
ification process, with increasing solidified PCM,
the temperature gradient in solid matrix signifi-
cantly reduces and the major temperature differ-
ence occurs inside the PCM pores, which results in
the reduction of the heat transfer rate.
3.2. Performance Evaluation
From an engineering standpoint, the liquid fraction
of PCM and time-dependent solidification phase
change are of interest in foam-enhanced PCM units.
In order to analyze the role of metal foam on the
performance of PCM, two parameters, liquid frac-
tion, ϕl, the portion of liquid PCM to the whole
PCM, and solidification time, tsol, the duration time
to solidify, are introduced to evaluate the perfor-
mance of heat transfer accompanied with solidifica-
tion phase change.
Figure 5 compares the evolution of liquid fraction
during the solidification process of PCM between
fractal structure and single-pore structure. No mat-
ter whether the metal foam is applied, there is rapid
decrease of liquid fraction in the initial stage of
solidification phase change heat transfer process,
and the decrease trend is slower as the solidification
0 102030405060
0.0
0.2
0.4
0.6
0.8
1.0
ϕ
l
t /min
Fractal
Single-pore
Fig. 5 Evolution of liquid fraction during the solidification
process.
-5 0 5 10
20
50
80
110
140
t
sol
/min
T
c
/
Single-pore
Fractal
Fig. 6 Effect of cold surface temperature on solidification
time.
process proceeds. In the comparison, we can con-
clude that the solidification rate of PCM embedded
in fractal metal foam is faster than that in single-
pore structure. For the case considered in Fig. 6,
there are approximately 25 min for the PCM to be
completely solidified for fractal metal foam while
almost 60 min are required to solidify all the PCM
in single-pore structure.
In order to provide a further insight into the
solidification phase change heat transfer, the role
of cold surface temperature on the solidification
time is presented in Fig. 6. As expected, the lower
the cold surface temperature, the solidification time
is shorter, i.e. the faster is the heat transfer rate
occurred between the porous matrix and the PCM
in pores. In other words, increases in temperature
gradient lead to large amount of interstitial heat
transfer between the solid matrix and PCM. It
is also indicated by the figure that, for the same
heat sink condition, the presence of porous metal
foam could significantly reduce the solidification
time during phase change.
4. CONCLUSIONS
In this paper, the Sierpinski fractal is applied to
construct the geometrical structure of porous metal
foam. Based on this fractal description, an unsteady
solidification heat transfer model of PCMs in porous
metal foam is developed and numerically analyzed
to investigate the solidification phase change heat
transfer process, in an attempt to identify the supe-
riority of the fractal architecture. The heat trans-
fer processes associated with solidification of PCM
embedded in fractal structure is compared with that
in single-pore structure. The liquid fraction and
solidification time of PCM during phase change in
porous metal foam are determined and analyzed.
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C. Zhang, L. Wu & Y. Chen
The conclusions can be summarized as follows:
(1) In fractal porous metal foam, the PCM is dis-
persedly distributed in metal foam, and the
existence of solid matrix provides a fast heat
flow channel both horizontally and vertically
and introduces large area for PCM in contact
with solid surface, which induces the enhance-
ment of interstitial heat transfer between the
solid matrix and PCM.
(2) The solidification performance of the PCM in
fractal structure is superior to that in single-
pore structure. The solidification rate of the
PCM embedded in fractal structure is faster
than that of single-pore structure during solidi-
fication process. The solidification time of PCM
in fractal structure is shorter than that in
single-pore structure.
(3) During the solidification process, with increas-
ing solidified PCM, the temperature gradient in
solid matrix significantly reduces and the major
temperature difference occurs inside the PCM
pores, which results in the reduction of the heat
transfer rate. Correspondingly, there is a rapid
decrease of liquid fraction in the initial stage of
solidification process, and the decrease trend is
slower and slower as the solidification process
goes on.
ACKNOWLEDGMENTS
This work was supported by National Natural
Science Foundation of China (No. 11190015) and
Natural Science Foundation of Jiangsu Province
(No. BK20130621).
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1540003-8
... Like most PCMs, an important challenge during the ice storage process is determining how to accelerate the solidification rate in shell-tube ice storage units. To offset the efficiency losses arising from the inherent drawback of poor thermal conductivity, comprehensive numerical [8,9] and experimental [10,11] studies have been performed over the past few decades to investigate the icing and melting processes. Several thermal enhancement methods, including the use of extended surfaces [12,13], employing multiple PCMs [14,15], thermal conductivity enhancement [3,16], and the microencapsulation of PCMs [17,18] have been proposed to meet the above challenge. ...
... fine fin was recommended for more effective enhancement. Zhang et al [8] inserted the metal foam into PCM for solidification improvement of the ice storage unit. Seeniraj et al. [15] utilized the combination of fins and multiple PCMs to achieve performance enhancement of a solar dynamic LHTS module. ...
... Generally, the effect of NC is usually overlooked in investigations on the discharging process of a latent heat storage unit, because the solidification process is dominated by thermal conduction [8,19,20]. As opposed to conventional PCM, the special nature of water density reversal (i.e., the evolution of water density with temperature reverses when the temperature is larger or less than the transition temperature) leads to the buoyancy reversal during the solidification process. ...
Article
The present paper reports on a numerical study of the ice storage process in finned shell-tube ice storage (STIS) units, with a focus on the special solidification behavior using water as the phase-change material (PCM). The proposed model is experimentally verified using an energy-discharging process in an ice storage unit. The effects of natural convection and buoyancy reversal on the solidification behavior are examined and investigated. Moreover, the Taguchi method is utilized to optimize the fin geometry of STIS units. The results indicate that the natural convection and buoyancy reversal are negatively correlated with the ice storage performance. An increase of the superheat factor leads to the enhancement of the buoyancy reversal intensity, which is not conducive to the acceleration of the solidification rate. In addition, the increases in fin height, fin width, and fin number are positively correlated with ice storage performance. It is demonstrated that the fin height is the dominant factor affecting the overall ice storage performance, and it is independent of the superheat factor. From the perspective of trade-off between ice storage rate and ice storage capacity, the optimal fin parameters for the STIS unit are fin height H = 40 mm, fin number N = 10, fin width Δ = 3 mm for engineering applications.
... The improvement of the thermal response efficiency is up to 83.32% comparing with the pure paraffin. In a further study, Zhang et al. [20] introduce the Sierpinski fractal to construct the porous metal foam. The porous metal matrix constitutes an efficient heat transfer network by which the heat flow between the solid matrix and the PCM is speeded up significantly. ...
... In addition, PCMs generally have the disadvantages of poor thermal conductivity, low energy storage/release efficiency, and a heat transfer unit that is too large, which leads to a larger temperature gradient inside the PCM during the working process, increasing the energy loss of heat transfer and hindering the application of PCMs in thermal energy storage. A wide range of approaches have been applied to enhance the heat transfer performance of PCMs, including dispersing particles [12][13][14], adding fins [15,16], metal foam [17][18][19], composite methods [20,21], and more, which have proven that adding high thermal conductivity materials in various forms improves the heat transfer performance of the PCMs to a certain extent. After optimizing their performance, PCMs will have wider application prospects in the fields of building energy conservation, waste heat recovery, thermal protection of electronic devices, solar power plants, and so forth. ...
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In this study, fractal net fins were introduced to improve the melting performance of a thermal energy storage unit. A transient model for melting heat transfer for phase change material (PCM) was presented and numerically analyzed, to study the melting performance in a thermal energy storage unit using fractal net fins. The melting phase change process was modelled using the apparent heat capacity method. The evolutions of temperature and the liquid fraction in the thermal energy storage unit were investigated and discussed. The effects of the length and width ratios of the fractal net on melting performance were analyzed to obtain the optimal fin configuration. The results indicated that the fractal net fins significantly enhanced the melting heat transfer performance of the PCM in a thermal energy storage unit. The fractal net fins configuration was optimal when the length and width ratios of the fractal net were 0.5. The temperature response at the corner points of the fractal net fins was faster than that in the central points.
... Zhao et al. [7] and Zhou et al. [8] found that foam metal was more effective than expanded graphite for enhancing heat transfer performance. The experimental results of Tian and Zhang [9][10][11] also showed that a porous metal foam could effectively increase the heat transfer rate of an energy storage system. Historically, adding foam metal into a PCM has been a popular way to increase thermal conductivity of the composite material. ...
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Different foam metals combined with paraffin and other materials were analyzed to determine their effective thermal conductivity and the macroscopic thermophysical properties of the composite materials. A W-P model composed of six tetrakaidecahedrons and two irregular dodecahedrons was used to simulate the melting heat transfer process in open foam metal at pore-scale under constant temperature. The results show that the porosity and conductivity of the foam metal and the conductivity of the phase change material (PCM) have a significant influence on the effective thermal conductivity of the composite PCM, while the pore size has no obvious influence. The effective thermal conductivity of composite PCMs increased with increasing foam metal thermal conductivity, and increased more rapidly with lower foam metal porosity. The effective thermal conductivity of composite PCMs is related to the ratio of foam metal conductivity to PCM conductivity. The microstructure of the foam metal had an obvious effect on the solid-liquid phase distribution during the PCM melting process, where the heat was transferred mainly through the melted liquid PCM field. Conduction was the dominant heat transfer mechanism, and natural convection in the liquid PCM was weak for the confinement of foam metals. For heat transfer during the PCM melting process, conduction through the skeleton of the porous metal played the most important role. The PCM adjacent to the heating source and foam metal frame melted first, with the fusion zone gradually spreading to the pore center. The melting rate of the PCM increased with increasing boundary temperature and thermal conductivity of the foam metal, but decreased as foam metal porosity increased. During the melting process, the liquid phase fraction did not linearly grow with time; the melting rate was very large at the initial stage, but decreased gradually with time.
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The study of the thermal contact resistance at the liquid–solid interface is an important subject of the heat transfer of phase change materials. In this work, a fractal model for predicting the thermal contact resistance at the liquid–solid interface is established by considering the self-affine fractal geometry of the rough surface. Based on the fractal characteristic of roughness structures, topographical and mechanical analyses have been conducted to identify the position of the liquid–solid interface and determine the thermal contact resistance at the interface. The relationship between contact parameters and the thermal contact resistance are studied. Based on the analytical predictive model for thermal contact resistance at the liquid–solid interface, the three-dimensional melting process of a nanoparticle-enhanced phase change material with different thermal contact resistances was simulated by using the finite volume method, and the enthalpy-porosity model is employed. The effects of thermal contact resistance between the composite phase change material and the heat source are investigated. It is found that the augmentation of thermal contact resistance decreases the melting and heat transfer rates and the influence of thermal contact resistance becomes more pronounced with a higher volume fraction of nanoparticles.
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Porous fins with temperature-dependent internal heat generation are frequently used to improve performance in a wide range of heat transfer and porous media applications. Thermal analysis of the porous fin fractal model with temperature-dependent heat generation is generated using fractal derivatives and investigated analytically using a novel Maclaurin series method (MSM). Nonlinear temperature distribution in a porous longitudinal fin is produced by the MSM. The porous fin solution is demonstrated using the Sierpinski fractal, which is based on time-dependent heat generation. The effects of the convection parameter, porosity, internal heat production, and generation number parameter on the dimensionless temperature distribution are discussed. MSM results are graphically and tabularly compared to existing solution methods such as HPM, CM, CSCM, LWCM, and GWRM. A comparison study reveals that MSM is a very reliable, accurate, and effective addition in the field of differential equations.
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The performance enhancement of latent heat storage (LHS) units is of great consequence for the development of sustainable energy. In this article, the transient models of phase‐change heat transfer processes in LHS units with consideration of natural convection are developed and numerically analyzed, in an effort to explore the role of the fractal tree‐shaped fin in the energy charging performance. The solid‐liquid interface evolution and dynamic temperature response in the tree‐shaped finned LHS unit are presented and compared with the wheel‐shaped one. Moreover, the influence of the fin number is investigated for maximizing the energy charging performance. The results indicate that the tree‐shaped fin has merits of effective layout optimization for the point‐to‐area heat transfer and the time‐coordination of energy charging and discharging process. Compared with the wheel‐shaped fin, the melting duration time is decreased only a little for the presence of tree‐shaped fin, however, the solidification process is decreased significantly. The presence of tree‐shaped fin leads to a lower energy charging rate during the early and middle stages of the charging process due to the natural convection suppression, however, the energy charging rate is faster during the later stage due to the thermal conduction dominated interior heat transfer. For maximizing the melting performance, the appropriate fin number in practical applications is 16.
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The fractal geometry provides a new insight for the description for actual pore structure of metal foams. In this paper, the fractal grids are introduced to describe the pore structure of metal foams. A two-dimensional unsteady melting model of phase change material (PCM) in the cross-fractal metal foam is developed and numerically simulated. The transient temperature variation and melting front evolution during the charging process in the cross-fractal metal foam are analyzed. The effects of initial temperature difference, porosity and fractal dimension on the melting heat transfer in fractal metal foam are examined and analyzed. The results indicate that the temperature distribution is more uniform and the melting rate is faster in the cross-fractal metal foam compared with that in the corresponding cavity structure. Interestingly, the fractal metal foam with smaller fractal dimension provides a faster heat flow path and hence enhances the melting performance though the porosity is identical. The melting performance in fractal metal foam can be enhanced when the metal foams have a lower porosity, a smaller fractal dimension and a larger initial temperature difference.
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Several studies have shown that fractal geometry is a tool that can replicate and investigate the nature of the materials and their physical properties. The Sierpinski carpet is often utilized to simulate porous microstructures. By using this geometric figure it is possible to study the influence of pore size distribution on deterministic fractal porous media. The determination of the thermal conductivity can be carried out using the electrical analogy. So, microstructure models have been converted in electrical fractal patterns. This fractal procedure is characterized by a close relationship with the actual microstructure and prevent papers has been validate it with experimental data in a series of former papers. In this work it is possible to show how thermal conductivity changes in relation to pore size distribution and geometric microstructure parameters.
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A detailed experimental and analytical study has been performed to evaluate how copper porous foam (CPF) enhances the heat transfer performance in a cylindrical solid/liquid phase change thermal energy storage system. The CPF used in this study had a 95% porosity and the phase change material (PCM) was 99% pure eicosane. The PCM and CPF were contained in a vertical cylinder where the temperature at its radial boundary was held constant, allowing both inward freezing and melting of the PCM. Detailed quantitative time-dependent volumetric temperature distributions and melt/freeze front motion and shape data were obtained. As the material changed phase, a thermal resistance layer built up, resulting in a reduced heat transfer rate between the surface of the container and the phase change front. In the freezing analysis, we analytically determined the effective thermal conductivity of the combined PCM/CPF system and the results compared well to the experimental values. The CPF increased the effective thermal conductivity from 0.423 W/m K to 3.06 W/mK. For the melting studies, we employed a heat transfer scaling analysis to model the system and develop heat transfer correlations. The scaling analysis predictions closely matched the experimental data of the solid/liquid interface position and Nusselt number.
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Transient solid/liquid phase change occurring in metal foams impregnated with phase change material (PCM) is investigated. Natural convection in the melt is considered. Volume-averaged mass and momentum equations are employed, with the Brinkman-Forchheimer extension to the Darcy law to model the porous resistance. Owing to the difference in the thermal diffusivities between the metal foam and the PCM, local thermal equilibrium between the two is not assured. Assuming equilibrium melting at the pore scale, separate volume-averaged energy equations are written for the solid metal foam and the PCM, and are closed using an interstitial heat transfer coefficient. The enthalpy method is employed to account for phase change. The governing equations are solved implicitly using a finite volume method on a fixed grid. The influence of Rayleigh number, Stefan number, and interstitial Nusselt number on the temporal evolution of the melt front location, temperature differentials between the solid and fluid, and the melting rate is documented and discussed. The merits of incorporating metal foam for improving effective thermal conductivity of thermal storage systems are discussed.
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Experimental work was conducted to compare the thermal–hydraulic performances of cross-flow microchannel condensers using louvered fins and metal foams as extended surfaces. Three copper foam surfaces with pore densities of 10 and 20 pores per inch (PPI) and porosities of 89.3 and 94.7%, and three aluminum louvered fins with lengths of 27 and 32 mm (in the flow direction) and heights of 5 and 7.5 mm were evaluated. The experiments were carried out in a closed loop wind-tunnel calorimeter equipped with a R-600a refrigeration loop. A condensing temperature of 45 °C was used in all tests, with face velocities ranging from 2.1 to 7.7 m/s. A comparison based on the thermal conductance and air-side pumping power showed that the surfaces enhanced with louvered fins performed better than the metal foams under all conditions investigated.
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Increase in the heat transfer rate during melting and freezing of a phase change material (PCM) with a low thermal conductivity is demonstrated by inserting a high-porosity metal matrix into the PCM. A vertical annulus space (r1⩽r⩽r0, 0⩽z⩽h)homogeneously fitted with water and an aluminum matrix, is selected for this study. The Navier-Stokes equations are modified to account for Darcy's effect. For both the melting and freezing cases, the density inversion phenomenon of water is considered. The irregularity and time-varying nature of the solid and the liquid regions are accounted for by a geometric coordinate transformation. The numerical results are presented in the form of solid-liquid interface movements, isotherms, streamlines, and heat transfer rates for some representative cases. The heat transfer rates for enhanced cases show an order-of-magnitude increase over the base case, where no metal matrix is inserted.