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Thermophysical properties estimation of paraffin/graphite composite

phase change material using an inverse method

Mohamed Lachheb a,, Mustapha Karkri b, Fethi Albouchi a, Foued Mzali a, Sassi Ben Nasrallah a

a Ecole Nationale dIngnieurs de Monastir, Laboratoire dEtudes des systmes Thermiques et Energtiques (LESTE), Avenue Ibn El Jazzar, 5019 Monastir, Tunisiab Universit Paris-Est, Centre dEtude et de Recherche en Thermique, Environnement et Systmes (CERTES), France

a r t i c l e i n f o

Article history:Received 21 November 2013

Accepted 8 March 2014

Available online 31 March 2014

Keywords:

Phase change material (PCM)

Graphite

Periodic temperature method

Thermal conductivity

Thermal diffusivity

Inverse technique

a b s t r a c t

In this paper, two types of graphite were combined with paraffin in an attempt to improve thermal con-

ductivity of paraffin phase change material (PCM): Synthetic graphite (Timrex SFG75) and graphite waste

obtained from damaged Tubular graphite Heat Exchangers. These paraffin/graphite phase change mate-

rial (PCM) composites are prepared by the cold uniaxial compression technique and the thermophysical

properties were estimated using a periodic temperature method and an inverse technique. Results

showed that the thermal conductivity and thermal diffusivity are greatly influenced by the graphite

addition.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

Thermal energy storage plays an important role in an effective

use of thermal energy and it has applications in various areas, such

as building heating/cooling systems, solar energy collectors and

industrial waste heat recovery[1]. Thermal energy can be stored

as a change in internal energy of a material as thermo-chemical

reaction, sensible heat or latent heat [2].

In this work, we are interested in the latent heat storage meth-

od through phase change materials (PCM). Recent studies have

been focused on the solidliquid phase change because of its high

storage capacity and nearly isothermal heat storage/retrieval pro-

cess [36]. Phase change materials for latent heat storage can be

classified into three major categories as organic material, inorganic

material and eutectic PCMs [7,8]. Paraffin is taken as the mostpromising phase change material because it has a large latent heat,

low cost, little or no super cooling, low vapor pressure, good ther-

mal and chemical stability, nontoxic and noncorrosive [1,9,10].

However, paraffin suffers from a low thermal conductivity (0.21

0.24 W m1 K1) [11]. These drawbacks reduce the rate of heat

storage and extraction during the melting and solidification cycles

and restrict their wide applications, respectively. Consequently,

more working effort has been focus to improve the thermal con-ductivity of PCMs, by dispersing of high conducting particles with-

in the PCM [1215], impregnation of PCM into high thermal

conductivity material with porous structures[1618].

The use of graphite particles has advantages such as high ther-

mal conductivity, low density in contrast to metals and high resis-

tance to corrosion. The scope of this study is to make an

experimental investigation on the effect of graphite on thermal

conductivity, diffusivity and specific heat of paraffin/graphite

composites. Two kinds of graphite were used to enhance thermal

conductivity of the paraffin: Synthetic graphite (Timrex SFG75)

and graphite waste obtained from damaged Tubular graphite Heat

Exchangers. Paraffin/graphite phase change composites with the

masse fraction of 5%, 10%, 15% and 20% were prepared by cold

uni-axial compression method. A periodic measurement methodwas used to determine simultaneously the experimental thermo-

physical properties of paraffin/graphite composites at room tem-

perature. Composite sample is fixed between two metallic

plates, the front side of the first metallic plate is heated periodi-

cally using a sum of five sinusoidal signals and the temperature

at the front and rear sides of both plates is measured and the

experimental transfer function is calculated. The theoretical ther-

mal heat transfer function is calculated by the quadrupole meth-

od. Then, thermal conductivity and diffusivity are simultaneously

identified by comparison of experimental and theoretical heat

transfer function.

http://dx.doi.org/10.1016/j.enconman.2014.03.021

0196-8904/2014 Elsevier Ltd. All rights reserved.

Corresponding author.

E-mail address: lachheb_med@yahoo.fr(M. Lachheb).

Energy Conversion and Management 82 (2014) 229237

Contents lists available at ScienceDirect

Energy Conversion and Management

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2. Experimental investigation

2.1. Investigated materials

2.1.1. PCM and graphite selection

The PCM tested in the present work is: paraffin with melting

temperature of 5658 C and with specific density of 900 kg m3.

The thermal conductivity enhancement is obtained by addition of

conductive graphite particles. Two different kinds of graphite were

used in this study:

One type of graphite is the Timrex (SFG75) powder supplied by

Timcal Graphite & Carbon at a bulk density of 2240 kg m3. It is

a synthetic graphite with spherical shape and an average size of

75lm, it is characterized by a well-aligned crystal structure

and by a high thermal conductivity in plane[3].

The second kind that has been tested is an industrial graphite

graphite waste. It was obtained fromdamaged Tubular graph-ite Heat Exchangers. It is a form of carbon with crystalline struc-

ture; it has good thermal and mass transfer characteristics that

have led to its use for thermal conductivity enhancement. The

measured bulk density is 1936 kg m3 with an average size of

85lm. Moreover, graphite has strong resistance to corrosion

and chemical attacks which makes it compatible with most

PCM. The recycling of graphite has a lot of benefits, it can pre-

serve natural resources of graphite for future generations i.e.

recycling graphite reduces the need for raw materials; it also

uses less energy, and it have economic benefits.

2.1.2. Paraffin/graphite material elaboration

The elaborationmethod developed in the present study is based

on the cold uni-axial compression, in this method paraffin powdersand graphite particles are mixed together, then the obtained mix-

ture (paraffin + graphite) is poured into a stainless steel mould fol-

lowed by a uni-axial compression (80 bar) at ambient temperature

(Fig. 1).

This technique leads to an anisotropic composite structure

whose porosity is partially occupied by paraffin grains. Cylindrical

paraffin/graphite samples were prepared under the same manufac-

turing conditions by the cold uni-axial compression method with

mass fraction of 5%, 10%, 15%, and 20%. The thickness and the

diameter of all this specimens were 5 mm and 60 mm respectively

(Fig. 2).

Then, neatly cut of the cylindrical samples (Fig. 2b) to obtain a

parallelepiped-shape specimens (Fig. 2c), with dimensions

(42 mm42mm5 mm) for thermophysical propertymeasurements.

2.2. Thermophysical property measurements

2.2.1. Experimental set up

A periodical method was used to estimate simultaneously ther-mal conductivity, diffusivity and specific heat of paraffin/graphite

composite materials at room temperature (Fig. 3). This method is

based on the use of a small temperature modulation in a parallel-

epiped-shape sample (42 42 5mm3). The advantage of this

method is that allows estimating simultaneously the effective

thermal conductivity and diffusivity with their corresponding sta-

tistical confidence bounds[19]. It is well suited for polymers and

composite materials with a thickness between 1 and 10 mm.

In the configuration used (which is presented in Fig. 3), the

sample is sandwiched between two metallic plates. A thermal

grease of high conductivity is applied on the contact surfaces be-

tween the sample and the metallic plates to ensure good thermal

exchange between the various elements.

The front side of the first metallic plate (brass) is also fixed toheating device (thermoelectric cooler) and heated periodically

using a sum of five sinusoidal signals. The whole device is placed

in a vacuum chamber connected to a pumping system. The rear

side of the second metallic plate (copper) is in contact with air at

ambient temperature and high vacuum. The temperature is mea-

sured with thermocouples placed inside both front and rear metal-

lic plates.

2.2.2. Theoretical model

The system under study is composed of several layers with dif-

ferent thermophysical properties. A heat transfer function is de-

fined at each frequency as the ratio between the Fourier

transforms of temperature at the xr and xf point of the x-axis

(Fig. 4). We assume that the temperature of the front side of themetallic plate is modulated. The heat transfer exchanges on the

rear face are taken into account trough a global exchange coeffi-

cient, which is considered constant during the experiment and

the thermal properties of the sample are supposed constant.

By assuming a one-dimensional heat transfer in the x-direction,

the conservation of energy equation in each layer can be written

as:

@2T

@x21

a@T

@t 8t> 0 1

In the case in periodic heating, the heat transfer equation is de-

fined by[20]:

@

2~

T@x2jwa ~Ta2 ~T 2

Nomenclature

e thickness (m)D conductivity ratio between the two phasesm measured mass (kg)t time (s)f frequency (Hz)

I thermal conductivity intensificationa thermal diffusivity (m2 s1)Cp specific heat capacity (J kg

1 K1)V volume of composite (m3)b thermal effusivity (W s1/2 m2 K1)H heat transfer functionT temperature (K)~T Fourier transform of the temperature

Greek symbolsk thermal conductivity (W m1 K1)/ flux, filler fractionq density (kg m3)x pulsation

Subscriptseff effective (composite)m matrix, massf fillerc composite, contactv volume fractionexp experimentalth theoretical

230 M. Lachheb et al. / Energy Conversion and Management 82 (2014) 229237

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With;affiffiffiffiffiffiffiffiffijw

pa

;w2p f~Tis the Fourier transform of the temperature.

This system is modeled with one dimensional quadrupoles the-

ory i.e. matrices with two inputs and two outputs. It is also possible

to obtain the expression of the theoretical heat transfer function

using the quadrupoles method. Let (Tk(s),uk(s)) be the temperatureand flux at the rear side of the k-th layer of homogeneous material.

With thisk-th layer is associated a quadupole matrix (Fig. 5).

A quadrupole matrix [Q] is defined by:

Q cosh ffiffiffiffiffiffiffiffiffiffiffiskSp RKffiffiffiffiffiffiffisk sp sinh ffiffiffiffiffiffiffiffiffiffiffiskSp ffiffiffiffiffiffisk S

pRk

sinh ffiffiffiffiffiffiffiffiffiffiffiskSp cosh ffiffiffiffiffiffiffiffiffiffiffiskS

p

2

4

3

5 3

where;s e2

a: the Fourier time,R e

k: the thermal resistance.

So that:

~Tk1s~/k1s

" # Q

~Tks~/ks

" # 4

Fig. 1. Cold uni-axial compression technique.

Fig. 2. Example of paraffin/graphite composite samples: (a) paraffin, (b) paraffin/

graphite and (c) parallelepiped-shape specimen for thermophysical property

measurements.

Fig. 3. Experimental set-up of periodic method.

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This form is used for thermal modeling of plates of brass and

copper, and for the sample.

The grease layer at each interface is represented by a thermal

contact resistance.

QGrease 1 R0 1

5

The equivalent quadrupole ofNlayers in series is the product ofNquadrupole thermal relative to each layer.

Q YNi1

Qi 6

The relationship between input variables and output is written

as follow[20]:

~Tfront~/front

" # QBrass QGre QSample QGre QCopp

~T0

h~T0

" # 7

And;~Trear~/rear

" # QCopp

2

~T0

h~T0

" #

whereQBrass; QGre; QSample; QCopp; are the quadrupoles associated

to brass, grease, sample and copper respectively.

Thus, a theoretical heat transfer function can be obtained as:

Hthf; b ~Trearf~Tfrontf

8

where the parameter vector b includes two unknown parameters,

thermal conductivity and diffusivity of the sample and f is the

frequency.

The knowledge of the experimental heat transfer function isvery important to find the sample thermal parameters. The

complex experimental heat transfer function ~H is calculated at

each excitation frequency[20].

~HexpFFTTxr FFTTxf

FFTTxf2 9

where FFT(T(xr)), FFT(T(xf)) are the Fast Fourier Transfer of the tem-

perature in the rear and in the front face respectively and FFT(T(xf))

is the complex conjugate ofFFT(T(xf)).

The experimental heat transfer functions amplitude and phase

are given by the ~Hmodulus and argument for each excitation fre-

quency respectively.

2.2.3. Parameter identification procedure

In this study, a parameter estimation technique was applied to

estimate the optimal values of both thermal conductivity and dif-

fusivity. The identification procedure consists in finding the set of

parameters b that minimize the squared difference between the

theoretical and experimental heat transfer functions. The minimi-

zation function is given by:

Sbk;a XNi1

~Hrealfi Hrealfi2 ~Himagfi Himagfi

2h i

10

where (~Hreal fi) and Hrealfi are, respectively, the real parts of the

experimental and the theoretical heat transfer function, ~Himagfi

and Himagfiare, respectively the imaginary parts of the experimen-

tal and the theoretical heat transfer function and b is the vector of

the estimated parameters kc andac.

The identification of the thermo-physical parameters is a non-

linear optimization problem that is solved iteratively using the

LevenbergMarquardt method given by:

bk1 bk XkT Xk ukI

1 XkT ~HexpHtheb; k 11

The variancecovariance matrix of the estimated thermal prop-

erty vector Bcan be approximated as:

covb r2XXT1 12whereX @H

@b

represents the Jacobian matrix and XT the Jacobian

transpose and r represents the standard deviation of themeasurements.

We assume a normal distribution for the measurement errors.

The approximate statistical confidence bounds for the estimated

thermal conductivity and diffusivity valuesbk;a are at a confidence

level of 95%.

2.3. Analytical models for thermophysical properties

2.3.1. Specific density, volume fraction

The determination of the composite density is important not

only for the specific thermal capacity estimation but for checking

the quality of the samples. It is clear that if composite samples

are well processed, i.e., good homogeneity is reached without air

bubbles in the sample as well as without unfilled pores at the par-

affin/graphite interface, the specific density of the composites

should have a linear dependence upon the volume fraction, accord-

ing to the simple rule of mixtures given by:

qeffqf/f qm/m 13

where qm andqf are the densities of the paraffin matrix and thegraphite filler, respectively.

Fig. 4. Schematic view of the experimental exchange model.

Fig. 5. Model of the set-up: each of the five layers is modeled using a quadrupole

matrix.

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Then, by using the values ofqmand qf, it is possible to computethe volume fraction of each compound knowing the graphite mass

fraction /m and using the classical relationship.

/v 1= 1 qf

qm1 /m

/m

; /m1 /v 14

Two methods were adopted for measuring the densities of par-

affin/graphite composite:

The first one is the pycnometer method, in this method the

measurements are carried out with a balance and a pycnometer

for small quantities of different composites. The density of the

composite qc1 can be obtained by using the followingequation:

q m1m1m2m3 qwater 15

where m1is the sample weight,m2is the weight of the pycnom-

eter filled with water and m3 is the weight of the pycnometer

containing the sample and filled with water.

In the second method, the density measurements were

achieved using square-plate samples. A MettlerToledo

AT61 delta range balance was used to measure the mass of

the samples and the sample sizes were measured using a caliper

square. Then, the density of square-plate samples qc2 isdefined as mass divided by volume of the composites. The

uncertainty on the density u(q) measurement is obtained fromthe following equation:

Dqq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD

2m

m2 D

2V

V2

s 16

Or; DV2

V D

2L1L21

D2L2L22

D2e

e2

2.3.2. Effective thermal conductivity

Many analytical models were proposed to predict thermal con-

ductivity of composite materials. However, only few models take

into account the size of fillers and the thermal contact resistance

between fillers and matrix.

In this study, the effective thermal conductivity of paraffin filled

with graphite can be investigated using two theoretical models:

the model of Lewis and Nielsen and the model of Bruggeman.

2.3.2.1. Model of Lewis and Nielsen. This model is originally used for

the prediction of mechanical properties of composites [21]. It was

also applied to the prediction of the effective thermal conductivity

of composites [21] by taking into account different parameterssuch as the shape, distribution and orientation of fillers into the

matrix and also the thermal conductivity values of filler and ma-

trix. The main interest for the use of such model is that predictions

also consider the value of the maximum packing fraction of fillers

/max. This model is defined by[21]:

keff km1 ALNBLN/1 BLNw/ 17

With:BLN kfkm 1

=kf

kmALN

w1 1/max//2max

8