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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 WU∗and 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.3–8

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.9–14 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 Sanna15–17 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 (Tm−∆T≤T≤Tm+∆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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Table 1 Thermal Properties of PCM.

Density,Capacity,Conductivity,Latent Heat,

ρ/kg ·m−3c/J·(kg ·K)−1λ/W·(m ·K)−1hfg/kJ ·kg−1

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<(Tm−∆T),

hfg

2∆T+cs+cl

2(Tm−∆T)≤T≤(Tm+∆T)

clT>(Tm+∆T)

,(2)

k(T)=

ksT<(Tm−∆T),

ks+kl−ks

2∆T[T−(Tm−∆T)] (Tm−∆T)≤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 15◦C and liquidus temperature 25◦C.

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=−5◦C).

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 −5◦C, 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=−5◦C).

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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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