ORIGINAL

Comprehensive analysis of melting and solidification of a phase changematerial in an annulus

Rohit Kothari1 & S. K. Sahu1& S. I. Kundalwal1

Received: 24 April 2018 /Accepted: 14 August 2018# Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractA one-dimensional conduction heat transfer model has been proposed to study the melting and solidification of phase changematerial (PCM) inside an annulus. Here, the phase change process is divided into two main sub-processes such as melting andsolidification sub-process. Subsequently, each sub-process is analyzed for various temporal regimes. The temporal regimesinclude completely solid, partially molten and completely molten for melting sub-process and in reverse order for solidificationsub-process. Later on, the solution for temperature distribution for each temporal regime is obtained either by employingVariational formulation or using a method of quasi-steady state. The solution of each temporal regime is united to provide aclosed form solution for temperature distribution for the sub-process. Present model exhibits good agreement with the existingexperimental data. The results indicate that melt duration can be increased by increasing the thickness of PCM in an annulus. It isalso found observed that for any thermal storage unit there exists a particular percentage of TCE-PCMdistribution through whichmaximum melt duration can be achieved.

Nomenclaturecs, cl Solid and liquid specific heat, J/kg-Kh Outside heat transfer coefficient, W/m2-Kr1, r2 Inner and outer radii of cylinder, mks, kl Solid and liquid thermal conductivity, W/m-KLp Latent heat of PCM, J/kgq″ Heat flux, W/m2

T Temperature, 0CT∞ Ambient temperature, 0CTi Initial temperature of PCM, 0CTm Melting temperature, 0CTs, Tl Solid and liquid temperature, 0CΔT Tm − T∞,

0Ct Time, s

t0 Thermal penetration time, stm Time for start of melting, st′ Time for complete melting, st″ Time for complete solidificationts Time for start of solidificationε(t) Thermal penetration depth, mR (t) Melt interface location, mBis, Bil

hr2ks; hr2ks (Biot number)

Stes, StelcsΔTLp

; clΔTLp

(Stefan’s number)r coordinateGreek symbolsαs, αl Solid and liquid thermal diffusivity, m2/sβ r1

r2γ r2

R tð Þφs

ksΔTq′′ , m

φlklΔTq′′ , m

ρ Density of PCM, kg/m3

Subscriptss Solidl Liquidp Phase change material

1 Introduction

Removal of higher heat flux has been a great challenge in thedesign of future electronic components. So, there is a need of

Research highlights• Analytical model is proposed for melting and solidification of phasechange material (PCM) in an annulus.• The phase change process of PCM is divided into three temporal re-gimes; namely, completely solid, partially molten and completely molten.•Closed form expressions for the temperature distribution is obtained as afunction of various modeling parameters.• Present prediction exhibits good agreement with the test data.

* Rohit Kotharirohitkothari2892@gmail.com

1 Discipline of Mechanical Engineering, Indian Institute ofTechnology Indore, Simrol, Indore 453552, India

Heat and Mass Transferhttps://doi.org/10.1007/s00231-018-2453-9

efficient performance of cooling techniques. Cooling tech-niques are broadly classified as active and passive coolingtechniques. As the use of active cooling techniques is limited,effective use of passive cooling techniques is needed. Phasechange material (PCM) cooling is becoming very promisingpassive cooling technique since PCM can absorb the heatduring peak heat load and reject the heat during rest of thecycle, thus stabilizing the device surface temperature. PCMfinds application in various areas such as cooling of micro-electronics assembly, mobile communication equipment, au-tomotive electronics and battery modules for electric vehicles,subjected to transient or cyclic heat loads [1–4]. For suchdevices the cooling system needs to be optimally designedconsidering various base heat load and PCMs will cater forpeak heat load that exists for a short time.

Numerous studies have investigated the heat transfer per-formance of PCM using both experimental techniques andnumerical simulations [5–13]. Duan and Naterer [5] experi-mentally investigated two different PCM designs for thermalmanagement of electric vehicle battery modules. In the firstcase, heater was surrounded by PCM cylinder and in the sec-ond case, heater was wrapped by the jacket of another PCM.The results exhibited that in both cases PCM has been effec-tive in maintaining the temperature of heater surface within aspecified temperature range for longer duration. Ramandi etal. [6] analyzed a new type of double series PCM shell con-figuration with finite volume simulation. The double PCMshell system was found to be more efficient compared to sin-gle PCM shell in terms of exergy efficiency. Liu et al. [12, 13]experimentally investigated the melting and solidificationcharacteristics of stearic acid in an annulus with and withoutfins for different heat flux. It has been reported that fins canenhance the heat transfer rate in melting process. The effect ofgeometry of fin has also been analyzed. Some of the experi-mental studies are also summarized in Table 1.

PCM based cooling technology due to isothermal phasetransition has generated interest to analyze the melting and so-lidification behavior of PCM in a Cartesian and Cylindricalgeometry with either volumetric heat generation or boundaryheat flux condition. Several theoretical models have been pro-posed to estimate the temperature distribution of PCM duringmelting and solidification [14–23]. Lu [14] presented one di-mensional conduction heat transfer model to analyze melting ofPCM slab. This model considered the specified heat flux on toplayer of PCM, while the bottom layer was exposed to convec-tive air environment. Temperature distribution in the PCM slabwas obtained as a function of time and Biot number. The solu-tions obtained were analyzed to optimize the phase changecooling strategies for high power electronics. Chakraborty andDutta [15] developed an analytical model for cyclic melting andsolidification of PCM. They obtained temperature distributionsusing either semi-infinite or quasi steady state assumptions. Theresults were compared with those reported by Laouadi and

Lacroix [16]. Jiji and Gaye [17] analyzed the effect of volumet-ric energy generation on solidification and melting of PCM.Solution for one dimensional conduction model has been ob-tained with quasi steady approximation. Kalaieselvam et al.[18] experimentally and analytically investigated the effect ofvolumetric heat generation on melting and solidification char-acteristics of PCM inside a cylindrical enclosure. A one dimen-sional quasi steady conduction model was developed in theirstudy. Deviation between experimental and analytical resultshas been found out to be 16.11%. Saha and Dutta [19] formu-lated one dimensional heat conduction model to analyze thetemperature distributions for both melting as well as solidifica-tion inside the PCM slab. This model considered the specifiedheat flux on the bottom layer of the PCM, while the top layerhas been either insulated or exposed to convective air environ-ment. The effect of various parameters such as applied heatflux, heat transfer coefficient and height of PCM on the tem-perature distribution were analyzed. It may be noted from theabove mentioned studies that the two main sub-processes suchas melting and solidification can be analyzed by consideringdifferent temporal regimes. The temporal regime includecompletely solid, partially molten and completely molten formelting sub-process and in reverse order for solidification sub-process. Temperature distribution for partially molten stage forCartesian and Cylindrical geometries has been reported by var-ious researchers and is presented in Table 2.

It is evident from the literature that most of the models forcylindrical geometry consider the phase change material initiallyat melting/solidification temperature. In real practice, PCM willbe either at lower temperature or at higher temperature comparedto melting and solidification temperature, respectively. Therefore,there is a need to develop theoretical models during melting andsolidification of PCM in Cylindrical geometry considering vari-ous temporal regimes. In this study, an effort has been made toanalyze melting and solidification of PCM in an annulus. Here,the entire problem is divided into twomain sub-processes; name-ly, melting and solidification. Subsequently, each sub-process isanalyzed for various temporal regimes. The temporal regimesinclude completely solid, partially molten and completely moltenfor melting sub-process and in reverse order for solidificationsub-process. Later on, the solution for temperature distributionfor each temporal regime is obtained either by employingVariational formulation or using a method of quasi-steady state.A variety of boundary conditions involving constant heat flux,adiabatic condition and convective conditions are used for bothanalysis ofmelting and solidification. In all the cases, closed formexpressions for the temperature distribution have been obtainedas a function of various parameters such as heat flux, heat transfercoefficient, thermophysical properties of PCM and physical di-mension of storage system. The solution of each temporal regimeis united to provide a closed form solution for temperature distri-bution for the sub-process. Present predictions are comparedwiththe available test data and good agreement between both sets of

Heat Mass Transfer

Table1

Summaryof

Experim

entalinvestig

ation

S.

No.

Source

(Year)

Problem

Adapted

Type

ofPC

M(m

.p.in

0C)

Power

input

range(W

)Dim

ensionsof

thePCM

container

Configuratio

nsObservatio

ns

1.Duanetal.[5]

Therm

almanagem

ento

felectric

vehiclebattery

modules

with

phasechange

material

RPCM

byGlacier

TekInc.(18)

T-PC

Mby

laird

Tech

(50)

1.36

Length=101mm

Inside

diam

eter=6.35

mm

Outside

diam

eter=52

mm

OnlyPCM

andPCM

jacket

PCM

hasbeen

effectivein

maintaining

thetemperature

ofbattery

modules

forlonger

duratio

nof

time.

2.Paletal.[8]

Meltin

gof

PCM

bya

uniformly

heated

source.

n-triacotane

(65.4)

15to

60130*130*12.81

OnlyPC

MAtinitialm

eltin

gstageheat

transfer

isconductio

ndominant

butatlater

stages

ofmeltin

gis

becomes

convectio

ndominant.

3.Saha

etal.[9]

Studies

onoptim

umdistributio

nof

fins

inheatsink

filledwith

phasechange

materials

Eicosane(35)

4and8

42*42*30

mm

30,9,36,81,121pinfins

and3platefins

8%TCEvolumefractio

ngives

thebestresultwith

largenumber

ofsm

allcross

sectionfins.

4.Babyetal.[10]

Experim

entalinvestig

ationon

phasechange

materialb

ased

finned

heatsinksforelectronic

equipm

entcoolin

g

n-Eicosane(36.5)

2to

780*62*25

mm

3Nofin,3platefins,

72pinfins

Operatio

nalp

erform

ance

ofportableelectronicdevicescan

besignificantly

improved

with

PCM

basedheatsink

butthe

effectivenessdepend

onvolume

fractionof

TCE,fin

configuration,

TCEmaterial,am

ount

ofPC

Mandpower

level.

5.Gharbietal.

[11]

Experim

entalcom

parison

betweendifferentconfiguratio

nsof

PCM

basedheatsink

for

electroniccomponents.

Paraffin

(51.75)

451.5*25.5*16.5

mm

3OnlyPC

M,P

CM/Si

matrix,PCM/Graphite

matrix,PCM

with

14copper

fins,P

CM

with

6copper

fins

PCM

with

long

andwell-spaced

fins

issuitablesolutio

nforthermal

controlo

felectronicsystem

6.Liu

etal.[12]

Investigationof

meltin

gcharacteristicsof

PCM

inan

annulus,with

andwith

out

thermalconductiv

ityenhancers.

Stearicacid

(67.7)

53.81

Length=550mm

Inside

diam

eter=20

mm

Outside

diam

eter=46

mm

OnlyPCM

andwith

spiral

twistedcopper

fins

Effecto

fdifferenth

eatfluxhas

been

observed

onmeltin

gprocessandithasbeen

observed

thatfins

improveboth

conductio

nandnaturalconvection.

7.Liu

etal.[13]

Investigationof

solid

ification

characteristicsof

stearicacidic

anannulus,with

andwith

out

thermalconductiv

ityenhancers.

Stearicacid

(67.7)

53.54

Length=550mm

Inside

diam

eter=19.9

mm

Outside

diam

eter=46

mm

OnlyPCM

andwith

spiral

twistedcopper

fins

Fins

improveboth

conductio

nand

naturalconvection.Enhancement

factor

during

solid

ificationhas

been

ashigh

as250%

hasbeen

observed

with

copper

fin.

Heat Mass Transfer

Table2

Temperature

distributio

nof

PCM

forstage2(Fig.3c)

reported

byvariousresearchers

S.

No

Researcher

Problem

Adapted

Temperature

distributio

nforsolid

liquidinterfacestage

Assum

ptions

Method/

Assum

ption

Validation

1.T.

J.Lu[14]

Temperature

distributio

nin

PCM

used

forthermalmanagem

ento

fhigh

power

electronics.

Meltin

gwith

Conductionin

aslab

(Heatfluxinputo

nside;convectionon

theotherside)

Ts¼

Tm−

Tm−T

∞ð

Þ1−

XtðÞ Hþ

1 Bið

Þx−

XtðÞ

fg

Tl¼

Tm−

q″ k lx−

XtðÞ

fg

XtðÞ¼

q″t−t m

ðÞ

ρL

t0¼

t mþ

ρLH q″

Where,X

(t)=

Meltfront

positio

n,L=Latenth

eat

Quasi-steady

approxim

ation;

valid

forBi<

<1

Variatio

nal

form

ulation

coupledwith

Kantorovich

Method

No

2.Chakraborty

etal.

[15]

Solutionforheattransfer

analysis

of meltin

gandsolid

ificationof

PCM

forcyclicheatloads.

Solid

ificationwith

Conductionin

aslab

(Insulated

from

oneside;convectionon

theotherside)

Ts¼

Tm−

h effTm−T

∞ð

Þh e

ffl 1−X

tðÞf

gþk s

x−X

tðÞf

gTl¼

Tmþ

ΔT l 1

x−X

tðÞf

gþΔT

2l1X

tðÞx−X

tðÞf

g2

XtðÞ¼

X1−

t−t 1

ðÞk

lΔT

ρLl 1f1−

k sk l

l 1−X

1ð

Þl 1

þkl Biþk

s

Where

X(t)=

Interfacepositio

n,L=Latenth

eat,

l 1=Thickness

ofPC

Mlayer,ΔT

=T m

−T ∞

Sem

i-infinity

consideration,

orquasisteadystate;

Valid

forBi<

<1

Variatio

nal

form

ulation

coupledwith

Kantorovich

Method

Yes

(Experim

ental)

3.Jijietal.[17]

Effecto

funiform

volumetricheat

generatio

non

meltin

gand

solid

ificationof

PCM

ina

rectangle.

Meltin

gwith

Conductionin

aslab

(Uniform

volumetricheatgeneratio

n)

θ Lε;t

ðÞ¼

β 2ε2

þ1−

β 2ε i

2�

� ε ε i

ε i¼

ffiffiffiffiffiffiffiffi 2τ1−

βτ

qτ 0

¼1

2þβ

Quasi-steady

approxim

ation;

valid

forSte.<0.1

Directintegratio

nNo

Solid

ificationwith

Conductionin

aslab

(Uniform

volumetricheatgeneratio

n)

θ Lε;t

ðÞ¼

−β 2ε2

þ1þ

β 2ε i

2�

� ε ε i

θ Lε;t

ðÞ¼

−β 2ε i

2−ε

2ð

Þ−βε i−ε

ðÞþ

1

Where,β

¼q0 00

L2

kT

f−T

∞ð

Þ,Heatg

eneration

parameter,

q'''=

Volum

etricheatgeneratio

nrate,

L=Halfthicknessof

slab,

ε¼

x L=Dim

ensionless

distance

4.Kalaiselvam

etal.

[18]

Effecto

funiform

volumetricheat

generatio

non

meltin

gand

solid

ificationof

PCM

incylin

dricalencapsulation.

Meltin

gwith

Conductionin

acylin

der

(Uniform

volumetricheatgeneratio

n)

θ Lx;F0

ðÞ¼

lnx

lns1þ

β 4−

β 4s2

�� −

β 4þ

β 4x2

Quasi-steady

approxim

ation;

valid

forSte.<

0.1

Directintegratio

nYes (E

xperim

ent-

al)

Solid

ificationwith

Conductionin

aslab

Heat Mass Transfer

Tab

le2

(contin

ued)

S.

No

Researcher

Problem

Adapted

Temperature

distributio

nforsolid

liquidinterfacestage

Assum

ptions

Method/

Assum

ption

Validation

(Uniform

volumetricheatgeneratio

n)

θ sx;F0

ðÞ¼

lnx

4lnsβ−4

−βs2

ðÞþ

1−β 4

�� þ

β 4x2

θ Lx;F0

ðÞ¼

−β 2s2

þβ 2x2

Where,

β¼

q0 00R2

kT

f−T

0ð

Þ,Dim

ensionless

heat

generatio

nparameter,

q′′′=Volum

etricheatgeneratio

nrate,

R=Radiusof

cylin

dricalcapsule,

s¼

σ R=Dim

ensionless

positio

nof

solid

-liquidinterface,

σ=positio

nof

interfacein

spatialv

ariable

5.Saha

etal.[19]

Temperature

distributio

nin

PCM

andPerform

ance

analysisof

heatsink

with

phasechange

materialsubjected

tocyclic

heating.

Meltin

gwith

Conductionin

aslab

(Heatfluxinputo

nside;insulationon

theotherside)

T s=T m

Tl¼

Tm−

q″ 2klx−X

tðÞf

gX

tðÞ¼

q″t−t m

ðÞ

ρL

t0¼

t mþ

ρLH q″

Quasi-steady

approxim

ation;

valid

forBi<

<1

Variatio

nal

form

ulation

coupledwith

Kantorovich

Yes

(Num

erical)

Solid

ificationwith

Conductionin

aslab

(Insulationon

oneside;convectionon

theotherside)

Ts¼

Tm−

h effTm−T

∞ð

Þh e

ffH−X

tðÞf

gþk s

x−X

tðÞf

gT l=T m

XtðÞ¼

Hþ

1 h eff

k s−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2h2 effk s

Tm−T

∞ð

Þtm−t

ðÞ

ρLp

þk2 s

no

r�

�

t″¼

t″ mþ

ρLpH

2ksþ

h effH

ðÞ

2heffk s

Tm−T

∞ð

ÞWhere

X(t)=

locatio

nof

meltfront

attim

et,

H=Heighto

fPC

Mlayer

6.Kalaiselvam

etal.

[20]

Effecto

funiform

volumetricheat

generatio

non

meltin

gand

solid

ificationof

nano

particle

enhanced

PCM

inspherical

encapsulation.

Meltin

gwith

Conductionin

asphere

(Uniform

volumetricheatgeneratio

n)

θ Lx;F0

ðÞ ¼

−β−6

sþβs3

61−

sð

Þþ

6sþβ

s−βs3

61−

sð

Þxþ

βx2 6

Quasi-steady

approxim

ation;

valid

forSte.<

0.1

Directintegratio

nYes (E

xperim

ent-

al)

Solid

ificationwith

Conductionin

asphere

(Uniform

volumetricheatgeneratio

n)

θ sx;Fo

ðÞ¼

6sþβ

ss−1

ðÞ

6s−1

ðÞ

þ6þ

βs2−1

ðÞx

61−

sð

Þþ

βx2 6

θ Lx;F0

ðÞ¼

βx2−s

2ð

Þ6

Where,β

¼gR

2

kT

f−T

∞ð

Þ,Dim

ensionless

heat

generatio

nparameter,

Heat Mass Transfer

Tab

le2

(contin

ued)

S.

No

Researcher

Problem

Adapted

Temperature

distributio

nforsolid

liquidinterfacestage

Assum

ptions

Method/

Assum

ption

Validation

g=volumetricheatgeneratio

nrate

R=Radiusof

sphericalcapsule,

s¼

σ R=Dim

ensionless

positio

nof

solid

-liquid

interface,σ=positio

nof

interfacein

spatialv

ariable

7.Mosaffa

etal.[21]

PCM

solid

ificationin

rectangularcontainerwith

convectiv

eboundaries

Solid

ificationwith

Conductionin

aplate

(heattransferfluidon

thewalland

fin

does

notinfluence

solid

ificationprocess)

Tsx;t

ðÞ¼

hTm−T

∞ð

Þk s

þhX

tðÞ½xþ2

∑∞

m¼1

βmX

tðÞcosβmX

tðÞð

Þ−sinβmX

tðÞð

ÞH

sþ

β2 mþH

2 s

�� X

tðÞ�

β2 mþH

2 s

�� �

exp−α

sβ2 mt

�� si

nðβmX

tðÞ−x

ðÞ

β2 m

�

þk sTmþX

tðÞhT

∞

k sþhX

tðÞ�

�Where,

βmcot(βmX(t))=

−Hs

Where,H

s=h/k s

,and

X(t)=

Distanceof

solid

liquidinterface

inxdirection

Isotherm

alsolid

ification

Separationof

variables

Yes

(Num

erical)

Heat Mass Transfer

results is obtained. Present analytical solutions are also able topredict the melting and solidification time which helps in designof PCM based cylindrical heat storage system.

2 The physical problem

Figure 1 depicts the schematic diagram of thermal storage unitconsidered in this analysis. It consists of a cylindrical contain-er with a centrally located electric heater and PCM surround-ing to the electric heater. The heater, represents the electronicsystem, generates heat and needs to be maintained in a partic-ular temperature range for effective and safe operation.Initially, the heat generated by electronic system is absorbedby the PCM and the temperature of both PCM and electronicsystem increases till the melting point of PCM is reached.After the PCM attains the melting temperature, it absorbs la-tent heat and the temperature of electronic system and PCMremains stable during that period. As the time progresses, thecomplete melting of PCM take place and the temperature ofelectronic system increases due to sensible heating of moltenPCM. On the contrary, PCM solidified during the coolingperiod as it losses heat.

The problem depicted in Fig. 2 is categorized as the BStefanproblem^. It consists of cylinder having inner surface of radiusr1 and outer surface of radius r2. Uniform constant heat flux isapplied at the inner surface of the annulus. The length of PCMcylinder (L) is assumed to be much larger than its thickness(r2-r1), that is, (L > > r2-r1). The loss of heat from upper andlower surfaces of cylinder is assumed to be negligible and theentire heat is assumed to be conducted in radial direction.

Therefore, the problem can be modeled as one dimensional.Following assumptions have been made for the analysis andare detailed below.

(i) PCM is assumed to be pure material and its melting isconsidered as isothermal.

(ii) Heat transfer in PCM is assumed to be one dimensionaland conduction dominated process.

(iii) An equivalent thermal conductivity for liquid phase isconsidered in order to account the melt convection.

(iv) Properties of solid and liquid phases are different but areassumed to be constant within the operating range oftemperature.

(v) The change of volume of PCM after phase change isneglected in the analysis.

3 Mathematical formulation

Figure 2 depicts the temperature distribution during the phasechange process of PCM in the cylindrical container. Here, theproblem involves two different parts, namely, melting domainand solidification domain. Each domain is formulated sepa-rately in subsequent sections.

3.1 Melting analysis of PCM

The melting domain is divided into three stages. Stage 1 repre-sents the case where the entire PCM remains in solid state andheat transfer takes place due to conduction. The supplied heatenergy is absorbed in PCM and the temperature of PCM risesdue to gain in sensible heat. In stage 2, the melting of PCM takesplace with partial solid and partial liquid state of PCM. In thisstage, the melt front propagates from inner layer to the outermostlayer of PCM. After complete melting of PCM (stage 3), thetemperature of molten PCM increases due to sensible heat gain.

The one dimensional transient heat conduction equationvalid for all the three stages and can be written as follows:

Stage 1: PCM is completely solid 0 ≤ t ≤ tm:

1

r∂∂r

r∂Ts

∂r

� �¼ 1

αs

∂Ts

∂tr1≤ r≤ r2 ð1aÞ

Stage 2: PCM is partially molten tm ≤ t ≤ t’:

1

r∂∂r

r∂Tl

∂r

� �¼ 1

αl

∂Tl

∂tr1≤ r≤R tð Þ ð1bÞ

1

r∂∂r

r∂Ts

∂r

� �¼ 1

αs

∂Ts

∂tR tð Þ≤ r≤ r2 ð1cÞ

Fig. 1 A typical PCM based cylindrical thermal storage system

Heat Mass Transfer

Stage 3: PCM is completely liquid t ≥ t’:

1

r∂∂r

r∂Tl

∂r

� �¼ 1

αl

∂Tl

∂tr1≤ r≤ r2 ð1dÞ

Here R (t) denotes the position of solid/liquid interface at timet.

The initial and boundary conditions are given by:

T j r; tð Þ ¼ T∞ at t ¼ 0 ð2aÞ

−k j∂T j

∂r¼ q′′ at r ¼ r1 ð2bÞ

R tð Þ ¼ r1 for t≤ tm ð2cÞ

ks∂Ts

∂r−kl

∂Tl

∂r¼ ρLp

∂R tð Þ∂t

at r ¼ R tð Þ ð2dÞ

Tj r; tð Þ ¼ Tm at r ¼ R tð Þ ð2eÞ

In this study, the outer surface is subjected to two differentboundary conditions and are given as:

−k j∂T j

∂r¼ 0 atr ¼ r2 ð3Þ

−k j∂T j

∂r¼ hT j r; tð Þ−T∞ at r ¼ r2 ð4Þ

0 1000 2000 3000 4000 5000 6000 700020

40

60

80

100

Solidification of PCM

Stage-1

Melting of PCM

Stage-3Stage-2Stage-1Stage-3Stage-2

(er

utarep

meT

0)

C

Time (s)

(a) Substage-1(a)(b) Substage-1(b) (c) Stage-2

(d) Stage-3 (e) Temperature regimes of melting and solidification

of PCM for all the stages

Fig. 2 Typical phase change curves for melting process for (a) Substage-1(a), (b) Substage1 (b), (c) Stage-2, (d) Stage-3 and (e) Temperature regimes ofmelting and solidification of PCM for all the stages

Heat Mass Transfer

where,

k j ¼ ks; for Solidkl; for Liquid

ð5Þ

T j ¼ Ts; for SolidTl; for Liquid

ð6Þ

Here, Eqs. (3) and (4) represent the adiabatic and convec-tive boundary conditions at the outer surface of the PCM (r =r2), respectively.

In addition to the initial and boundary conditions, anotherinitial condition is required to solve the conduction equation.The details are elaborated below.

3.1.1 Outer surface subjected to convective environment

Stage 1: (0 ≤ t ≤ tm)

The governing equation, initial condition and boundaryconditions for stage 1 are valid upto the time when meltingstarts at the inner surface (tm). Stage 1 is further subdividedinto two distinct time domains as below:

Case 1: Initiation of melting after the penetration of ther-mal disturbance at the outer layer of PCM, i.e. tm ≥ t0.Case 2: Initiation of melting before the thermal distur-bance penetrates the the outer layer of PCM, i.e. tm ≤ t0.Case 1: tm ≥ t0

This stage is further divided into the following twosubstages:

(a) 0 ≤ t ≤ t0(b) t0 ≤ t ≤ tm

The solutions of above problem are reported in the form ofinfinite series or Laplace transformation. In order to solve theabove mentioned problem, one needs to assume a guess tem-perature profile and is given as:

T j r; tð Þ−T∞ ¼ aþ bln rð Þ ð7Þ

Here, a simplified variational profile, satisfying the bound-ary conditions (Eqs. 2a and 2b, for substage a and Eqs. 2b and4, for substage b) coupled with Kantorovich method [24], isemployed to obtain the solution for both the substages andexpressed below.

Ts r; tð Þ−T∞ ¼

q′′r1ks

lnε tð Þr

� �for 0≤ t≤ t0

q′′r1ks

lnr2r

�þ 1

βBis1−e

−Bisαs t−t0ð Þ

1−βð Þr22

n o8<:

9=;

24

35 for t0≤ t≤ tm

8>>>>><>>>>>:

ð8Þ

where the depth of penetration ε(t) is defined as below:

ε tð Þ ¼ r1 þ αstr1

for 0≤ t≤ t0 ð9ÞIt may be noted that the penetration time (t0) for thermal

disturbance to penetrate to the outer surface of PCM can beobtained by substituting Ts(r2, t) = T∞ in Eq. (8) as follows:

t0 ¼ β 1−βð Þr22αs

ð10Þ

Depth of penetration ε(t) and penetration time t0 are inde-pendent of amount of heat flux applied at the inner surface ofPCM cylinder.

The solutions given by Eq. (8) are valid upto t = tm.Therefore, it is needed to evaluate the value of tm. Bysubstituting Ts(r1, t) = Tm at t = tm in Eq. (8), and with ΔT =Tm − T∞, we can obtain:

tmt0

¼β

1−βð Þ eφsβr2−1

h itm≤ t0

1−1

βBisln 1−Bis

φs

r2þ βln βð Þ

�� �tm≥ t0

8>><>>: ð11Þ

It may be noted from Eq. (11) that the melting analysis ofPCM consist of two parts, one with melting time less thanpenetration time and other with melting time longer than pen-etration time. For the above case tm ≥ t0 (Eq. 11), one can

obtain the condition for the heat flux q′′≤ ksΔTβr2lnβ

.

Stage 2: (tm ≤ t ≤ t’)

The solution for this stage is obtained by employing anapproximate analytical method, termed as quasi steady method.This method assumes that during melting of PCM, temperaturevariationwith time is small. Although the temperaturemay varywith time, one can assume the temperature gradient both insolid and liquid phases to be constant. By using governingequation (Eqs. 1b and 1c) and employing the initial and bound-ary conditions (Eqs. 2b, 2c and 2e, for liquid state and Eqs. 2c,2e and 4, for solid state), the temperature distribution for bothsolid and liquid phases can be expressed as follows:

T j r; tð Þ ¼Tm þ BisΔT

1þ Bisln γð Þf g lnR tð Þr

�for solid tm≤ t≤ t

0

Tm þ βr2ΔTφl

lnR tð Þr

� �for liquid tm≤ t≤ t

0

8>><>>: ð12Þ

The melt interface location R (t) can be found by substitut-ing Eq. (12) into Eq. (2d).

αlβlStelr2 t−tmð Þφl

¼ R tð Þf g22

1þ 2e2Bis

−2φsf gE1 2ln γ−φs þ1

Bis

� � �� �for interphase tm≤ t≤ t

0

ð13Þ

Heat Mass Transfer

where,

E1 ¼ −∫e−v

vdv ð14aÞ

v ¼ 2 ln γ−φs þ1

Bis

� � �ð14bÞ

For thin PCM layer coupled with air convection Bis < < 1,thus Eq. (13) can be rewritten as:

R tð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 þ

2αlβlStelr2 t−tmð Þφl

sfor inteface tm≤ t≤ t

0

ð15Þ

Now, the temperature distribution in solid and liquid phasescan be obtained by substituting Eq. (15) in Eq. (12),respectively.

The time duration at which melt front reaches the outersurface of PCM can be obtained by substituting R (t) = r2and t = t’ in Eq. (15):

t′ ¼ tm þ 1−β2� �

φlr22αlβStel

ð16Þ

The difference between time t’ and tm gives the duration oftime in which latent heat is absorbed by the PCM and temper-ature of the inner surface of PCM, at which heat flux is ap-plied, can be made stable. It may be noted that similar closedform expressions for the temperature distribution of PCMmelting as reported in stages 1 and 2 by Lu [14]. This studywas concerned with Cartesian geometry accounting constantheat flux at bottom surface and convective air environment attop surface of PCM.

Stage 3: (t ≥ t’)

In this stage, melt interface R (t) reaches at the out-ermost layer of the PCM which is completely in theliquid state. The initial condition for this stage is t = t’.Using earlier method described in section (3.1.1, stage1), using governing equation (Eq. 1d) and employingthe initial and boundary conditions (Eqs. 2b and 4),the temperature distribution in the molten PCM for t ≥t’ can be obtained as:

Tl r; tð Þ ¼ Tm þ βr2ΔTφl

hln

r2r

�þ 1

βBil1−e

−αlBil t−t

0ð Þ1−βð Þr2

2

n o24

35 for t≥ t

0

ð17Þ

Temperature distribution profile obtained in Eq. (17) issimilar to the one reported by Saha and Dutta [19] usingCartesian geometry.

3.1.2 Outer surface of cylinder is insulated

Stage 1: (0 ≤ t ≤ tm)

It may be noted that the solution of this model is similar to theone reported in the earlier section (3.1.1). Here, the governingequations (Eqs. 1a-1d), boundary and initial conditions (Eqs. 2a-2e and 3) are used to obtain the solution. In addition, differentcases for melting (stage 1) as applicable for earlier section (3.1.1)are same for this model and also, the solution for substage (a) issimilar to the earlier section (3.1.1) as there is no effect of outsideboundary condition of cylinder in this stage of melting and hencenot presented here for the sake of brevity.

Temperature distribution for substage (b) applicablein the time domain t0 ≤ t ≤ tm, is obtained by usinggoverning Eq. (1a) and initial and boundary conditions(Eqs. 2b and 3) and is expressed below.

Ts r; tð Þ−T∞ ¼ q″r1ks

hln

r2r

�þ αs t−t0ð Þ

β 1−βð Þr22for t0 ≤ t ≤ tm ð18Þ

The solution given by Eq. (18) is valid upto t = tm.Therefore, it is needed to evaluate the value of tm. Bysubstituting Ts(r1, t) = Tm at t = tm in Eq. (18), and with ΔT =Tm − T∞, one can obtain:

tmt0

¼ 1þ φs

βr21−

βr2φs

lnβ

� �; tm ≥ t0 ð19Þ

For the above case (Eq. 19), one can obtain the condition

for the heat flux q′′≤ ksΔTβr2lnβ

while for tm ≤ t0, the expression

remains same as reported in earlier section (3.1.1, stage 2).

Stage 2: (tm ≤ t ≤ t’)

The solution for this stage is obtained by employing similarmethod as described in section (3.1.1). Using governing Eqs.(1b) and (1c), and employing the initial and boundary condi-tions (Eqs. 2b, 2c and 2e, for liquid state and Eqs. 2c, 2e and 3,for solid state), the temperature distribution for both solid andliquid phases can be obtained. It may be noted that the solutionfor liquid phase is similar to that of the earlier section (3.1.1,stage 2) and temperature distribution for solid phase isexpressed as follows:

Ts r; tð Þ ¼ Tm for solid tm≤ t≤ t0 ð20Þ

Stage 3: (t ≥ t’)

In this stage, melt interface R (t) reaches at the outermostlayer of the PCM which is completely in the liquid state. Theinitial condition for this stage is t = t’. Using earlier methoddescribed in Sec. (3.1.2, stage 1) and by using governing Eq.

Heat Mass Transfer

(1d) and employing the initial and boundary conditions (Eq.2b and 3), the temperature distribution in the molten PCM fort ≥ t’ can be obtained as:

Tl r; tð Þ ¼ Tm þ βr2ΔTφl

lnr2r

�þ αl t−t″ð Þ

β 1−βð Þr22

� �for t≥ t

0 ð21Þ

The existing study by Saha and Dutta [19] using Cartesiangeometry with constant heat flux at bottom surface and insu-lated top surface of PCM also found similar closed form ex-pressions for temperature distribution of PCM melting report-ed this section.

3.2 Solidification analysis of PCM

It is considered that the solidification of PCM begins when zeroheat flux applied at the inner wall of cylindrical annulus duringcooling. This occurs in the case of cyclic or periodic heat load-ing situation. The governing Eqs. (1a)-(1d), boundary and ini-tial conditions (Eqs. 2a, 2c-2e and 4) are same as in the case ofmelting problem discussed earlier. In addition to this, followingboundary conditions are used to obtain the solution.

The initial and boundary conditions are given by:

−k j∂T j

∂r¼ 0 at r ¼ r1 ð22Þ

Here, different stages of solidification of PCM are same asdiscussed in the melting problem. However, these stages occurin the reverse order compared to the melting problem.Different stages are elaborated below.

Stage 1: (t’ ≤ t ≤ ts)

The temperature distribution in the liquid PCM after thestart of solidification (t ≥ t’) can be expressed as:

Tl r; tð Þ−T∞ ¼ τ21þ Billnr2r

�for t ≤ t ≤ ts ð23aÞ

where,

τ2 ¼ Tl r; t′� �

−T∞ �

exp −αlm t−t′� � �

and;

m ¼ Bilr22 1−β þ Bil1þ β lnβ−1ð Þ�½

ð23bÞ

The solution given by Eq. (23a) is valid upto t = ts.Therefore, it is needed to evaluate the value of ts. By substitut-ing Tl(r2, t) = Tm at t = ts in Eq. (23a), and with ΔT = Tm − T∞,one can obtain:

ts ¼ t′ þ 1

αsmln

T r2; t′ð Þ−T∞

ΔT

� �ð24Þ

ts denotes the time at which solidification at the outer surfaceof PCM starts.

Stage 2: (ts ≤ t ≤ t^)

After the start of solidification, solid-liquid interfacemoves from outer surface to inner surface of PCM layer(frozen front) and PCM dissipates latent heat. The tem-perature distribution for both solid and liquid phases canbe expressed as:

T j r; tð Þ ¼ Tm þ BisΔT1þ Bisln γð Þf g ln

R tð Þr

�for solid ts≤ t≤ t

0 0

Tm for liquid ts≤ t≤ t0 0

8<:

ð25Þ

The frozen front location R (t) can be found by substitutingEq. (25) in Eq. (2e).

4αsStes ¼ R tð Þf g2 1þ 2ln γð Þ þ 2

Bis

�for interphase ts≤ t≤ t} ð26Þ

For thin PCM layer coupled with air convection Bis < < 1,thus Eq. (26) can be rewritten as:

R tð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir22−2αsBisStes t−tsð Þ

qfor interphase ts≤ t≤ t} ð27Þ

The time duration at which frozen front reaches the innersurface of PCM can be obtained by substituting R (t) = r1 andt = t^ in Eq. (27):

t′′ ¼ ts þr22 1−β2� �

2αsBisStesð28Þ

The difference between time t^ and ts gives the duration oftime in which latent heat is dissipated by the PCM.

Stage 3: (t ≥ t^)

This stage begins when the frozen front {R (t)} reaches atthe innermost layer of the PCM. The initial condition for thisstage is t = t^. The temperature distribution in the solid PCMcan be expressed as:

Ts r; tð Þ−T∞ ¼ τ3 1þ Bislnr2r

�n ofor t≥ t} ð29aÞ

Where,

τ2 ¼ Tl r; t′′� �

−T∞ �

exp −αsm t−t′′� � �

and;

m ¼ Bis1−β þ Bis1−β þ Bis lnβ−1ð Þ�½

ð29bÞ

Once again, we found the profiles of temperature distribu-tion of PCM solidification in this section same as that of ob-tained by Saha and Dutta [19]. In their study, zero heat flux at

Heat Mass Transfer

Table3

Temperature

distributio

nsobtained

invariousstages

Stages

Meltin

gof

PCM

Solidificationof

PCM

(Inner

surfaceisinsulatedandoutersurfaceof

cylin

derissubjectedto

convectiv

eenvironm

ent)

Outer

surfaceof

cylin

derissubjectedto

convectiv

eenvironm

ent

Outer

surfaceof

cylin

derisinsulated

Stage-1

(Meltin

g:0≤t≤

t m,

Solidification:

t’≤t≤

t s)[Fig.2(a),2(b)]

Sub

stage-1(a)

(0≤t≤

t 0)

[Fig.2(a)]

Tsr;t

ðÞ−T∞¼

q″r 1 k sln

εtðÞ rno

Where,ε

tðÞ¼

r 1þ

αst r 1

t 0¼

βr2 2

1−β

ðÞ

αs

Tsr;t

ðÞ−T∞¼

q″r 1 k sln

εtðÞ rno

Where,ε

tðÞ¼

r 1þ

αst r 1

t 0¼

βr2 2

1−β

ðÞ

αs

Tlr;t

ðÞ−T∞¼

τ 21−

Bi lln

r r 2�n

oτ 2

¼Tlr;t0

� −T

∞

hi ex

p−α

lmt−t0

�

hi

and;

m¼

Bi l

r 21−

βþBi l

1þβlnβ−1

ðÞ

fg

½�

t s¼

t0þ

1αlmln

Tr 2;t0

ðÞ−T

∞

ΔT

��

Sub

stage-1(b)

(t0≤t≤

t m)

[Fig.2(b)]

Tsr;t

ðÞ−T∞¼

q″r 1 k s

lnr 2 r��

þ1

βBi s

1−e

−Bi sαs

t−t 0

ðÞ

r 21−

βð

Þ

no

()

"#

t m t 0¼

β 1−β

ðÞ

ek sΔT

q″r 1−1

�� ;

t m≤t

0

1−1

βBi sln

1−Bi s

k sΔT

q″r 2

þβln

βðÞ

�

�� ;

t m≥t

0

8 > > < > > :

Tsr;t

ðÞ−T∞¼

q″r 1 k s

lnr 2 r��

þαst−t 0

ðÞ

r2 2β1−

βð

Þh

i

t m t 0¼

β 1−β

ðÞ

ek sΔT

q″r 1−1

�� ;

t m≤t

0

1þ

1

βBi s

ΔTþq″r 1 k sln

βðÞ

�� ;

t m≥t

0

8 > > < > > :Stage-2

(Meltin

g:t m

≤t≤

t’ ,Solidification:

t s≤t≤

t”)

[Fig.2(c)]

Tpr;t

ðÞ¼

Tmþ

Bi sΔT

1þBi sln

γðÞ

fgln

RtðÞ r��

Tmþq″r 1 k lln

RtðÞ r��

8 > > < > > :Where

Tp=Ts,T

lforsolid

andliq

uidphaserespectiv

ely.

RtðÞ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffi

r2 1þ

2q″r 1

t−t m

ðÞ

ρLp

qt0¼

t mþ

ρLpr 2

2−r

2 1ð

Þ2q

″r 1

Tpr;t

ðÞ¼

Tm

Tmþq″r 1 k lln

RtðÞ r��

8 < :Where

Tp=Ts,T

lforsolid

andliq

uidphase

respectiv

ely.

RtðÞ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffi

r 12þ

2q″r 1

t−t m

ðÞ

ρLp

qt0¼

t mþ

ρLpr 2

2−r

12

ðÞ

2q″ r

1

Tsr;t

ðÞ¼

Tmþ

Bi sΔT

1þBi sln

γðÞ

fgln

RtðÞ r�

T l(r,t)=

T mRtðÞ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffir2 2−2

αsBi sSte s

t−t s

ðÞ

pt″¼

t sþ

r2 21−

β2

ðÞ

2αsBi sSte s

Stage-3

(Meltin

g:t≥

t’ ,Solidification:

t≥t”)

[Fig.2(d)]

Tlr;t

ðÞ¼

Tmþ

q″r 1 k l

lnr 2 r��

þ1

βBi l

1−e

−αlB

i lt−t0 ðÞ

r2 21−

βð

Þ

�

8 > < > :9 > = > ;

2 6 43 7 5

Tlr;t

ðÞ¼

Tmþ

q″r 1 k lln

r 2 r��þ

q″t−t″

ðÞ

ρclΔ

rTsr;t

ðÞ−T∞¼

τ 31−

Bi sln

r r 2�n

oτ 2

¼Tsr;t″�� −T

∞

� exp−α

smt−t″��

�

where;m

¼Bi s

r2 21−

βþBi s

1þβlnβ−1

ðÞ

fg

½�

Heat Mass Transfer

the bottom surface and convective air environment at the topsurface of PCM were applied in Cartesian geometry.Temperature distributions for all domains, stages and sub-stages are summarized in Table 3.

4 Results and discussions

In this section an effort has been made to investigate the in-fluence of heat flux, heat transfer coefficient, thermophysicalproperties of PCM, thermal conductivity enhancers and phys-ical dimension of storage system on the temperature distribu-tion of PCM. Here, wax has been considered as the phasechange material and its properties are given in Table 4. An

annulus filled with PCM having respective inner or outer di-ameters 4 mm and 8 mm is considered for the analysis. Theinner cylinder is assumed to be electric vehicle battery mod-ule. The dimensions of this annular geometry are summarizedin Table 4. The length of PCM cylinder (L) is chosen as400 mm.

Initially, an effort has been made to compare the presentpredictions with the test data of Duan et al. (2010) and Liu etal. (2005) as shown in Figs. 3, 4, and 5. Here, the temperaturedistribution during melting is obtained at r = r1 (shown in

Fig. 3) for the condition q′′ ¼ ksΔTβr2lnβ

. The solid region includes

two sub stages: (i) heat is not reached to the outer surface (0 ≤t ≤ t0) and (ii) heat is reached to the outer surface with PCM inthe solid state condition (t0 ≤ t ≤ tm).

Duan and Naterer [5] carried out experiments to simulatethe battery cell by installing an electric heater at the center of acylindrical container filled with RPCM as PCM. The contain-er was made of aluminium with a thickness of 0.2 mm, diam-eter of 52 mm and height of 125 mm. The phase changetemperature, specific heat, latent heat, density and thermalconductivity of RPCM was reported as 18 °C, 2.1 kJ/kg-K,195 kJ/kg, 840 kg/m3 and 0.55 W/m-K, respectively [5]. Theelectric heater is 101 mm long and 6.35 mm in diameter. Theheater has a heating rate of 1.36 W with a power supply of14 V. Present predictions (Eqs. 8, 12, 17) exhibit good agree-ment with the test data of Duan et al. [5] and are shown in Fig.3. The temperature distribution (Fig. 3) depicts the melting ofPCM involving various stages such as solid, partially moltenand completely molten condition. It may be noted that thepresent prediction exhibits excellent agreement with the testdata during solid state (0–232.49 s). While, there is some

0 1000 2000 3000 4000 5000 6000 70000

5

10

15

20

25

30

(er

utarep

meT

0)

C

Time (s)

Experimental

(Duan et al., 2010)

PCM Type: RPCM (Glacire Tek)

m.p.: 180C

Analytical

Fig. 3 Comparison of presentprediction with the test data ofDuan and Naterer [5] duringmelting of RPCM with constantheat flux at inner surface andconvective heat transfer at outersurface

Table 4 Thermophysicalproperties of wax Property Magnitude (unit)

ks 0.29 W/m-K

kl 0.21 W/m-K

ρs 910 kg/m3

ρl 822 kg/m3

cs 1.77 kJ/kg-K

cl 1.77 kJ/kg-K

Lp 195 kJ/kg

Ti 21 °C

Tm 56 °C

r2 0.004 m

r1 0.002 m

L 0.4 m

Heat Mass Transfer

deviation between present prediction and test data during par-tial melting. This is attributed to the fact that the present anal-ysis considers a quasi-steady assumption during the partialmolten condition. The mean error is found to be less than9.36% between both sets of results.

Here, efforts have also been made to compare presentpredictions during melting of PCM for insulated outersurface with the test data of Liu et al. [12] as shown inFig. 4. In their experiments, electric heater of diameter20 mm is placed concentrically with PCM tube made ofstainless steel with inner diameter of 46 mm and length

of 550 mm. The outside surface of cylindrical tube waswell insulated with a porous polythene insulator. Stearicacid with melting temperature of 67.7 °C was used asPCM. The initial temperature of PCM is kept at 26 °Cand heat flux of 1558 W/m2 was considered during thetests. The properties of stearic acid are taken from Zalbaet al. [1]. Figure 4 depicts temperature distribution atthe heater surface (r = r1) obtained via both theoreticalmodel and experiments. Present predictions exhibit goodagreement with the test data. The error between bothsets of results is found to be less than 7.86%.

0 1000 2000 3000 4000 5000 6000

40

45

50

55

60

65

70

75

80

(er

utarep

meT

0)

C

Time (S)

Experimental

(Liu et al., 2005)

PCM type: Stearic acid

m.p.: 67.60C

Analytical

Fig. 5 Comparison of presentprediction with the test data of Liuet al. [13] during solidification ofPCM with insulated inner surfaceand convective heat transfer atouter surface

0 500 1000 1500 2000 2500 3000 3500 4000

20

30

40

50

60

70

80

90

100

110

120

(er

utarep

meT

0)

C

Time (s)

Experimental

(Liu et al.,2005)

PCM type: Stearic acid

m.p.: 560C

Analytical

Fig. 4 Comparison of presentprediction with the test data of Liuet al. [12] during melting of PCMwith constant heat flux at innersurface and insulation at outersurface

Heat Mass Transfer

The theoretical predictions obtained from the present anal-ysis during solidification of PCM with the outer surface sub-jected to convective air environment are compared with thetest data of Liu et al. [13] as shown in Fig. 5. In their exper-iments, electric heater with diameter 19.9 mm and 550 mmlong is placed concentrically with heat exchanger made of twoconcentric cylindrical pipes of stainless steel. The externaltube was 600 mm long with inner diameter of 91 mm, whileinternal PCM tube was 550 mm long having 46 mm inner

diameter and 2.5 mm wall thickness. The outside surface ofexternal cylindrical tube was well insulated with a porouspolythene insulator. Stearic acid with melting temperature of67.7 °C is used as PCM and was filled between electric heaterand internal cylindrical tube. Cooling water was circulated inthe space between internal and external cylindrical tubes. Theinitial temperature of PCM was maintained at 26 °C and heatflux of 1558 W/m2 was used during the experiment. Figure 5depicts the temperature distribution at the outer PCM surface

0 2000 4000 6000 8000 100000

20

40

60

80

100

(er

utarep

meT

0)

C

Time (s)

q'' = 250 W/m

2

q'' = 500 W/m

2

q'' = 1000 W/m

2

q'' = 2000 W/m

2

(a)

0 500 1000 1500 2000 2500 3000

20

40

60

80

100

120

(er

utarep

meT

0)

C

Time (s)

q'' = 250 W/m

2

q'' = 500 W/m

2

q'' = 1000 W/m

2

q'' = 2000 W/m

2

(b)

Fig. 6 PCM inner surfacetemperature distribution fordifferent values of heat flux (a)Melting and solidification withconvection at outer surface ofcylinder (b) Melting withinsulation at outer surface ofcylinder

Heat Mass Transfer

(r = r2) surrounded by pure stearic acid obtained through boththeoretical model and experiments. The properties of stearicacid are taken from Zalba et al. [1]. The heat transfer coeffi-cient is assumed to be 26.42 W/m2-K for all the stages ofsolidification process. The agreement between the present the-oretical prediction and test data is found to be good and themaximum error is around 10.2%.

Figure 6a and b depict the distribution of temperature withtime for various heat flux values. Figure 6a illustrates thetemperature distribution for different heat flux values for

convective boundary condition at outer surface of cylindercontaining PCM. The values of r1 and r2 are taken as 2 mmand 4 mm, respectively. The melting process completes whenthe temperature at the inside surface of PCM reaches a setpoint temperature of 100 °C. The heat transfer coefficient atthe outer surface of the PCM is assumed to be 5W/m2-K. Theheat flux values are varied between 250 W/m2 to 2000 W/m2.In earlier studies, heat flux values were found to be in thisrange for electric vehicle module [5–7]. It can be noticed thatat lower heat flux values, the temperature of the inside surface

0 200 400 600 800 1000 1200 1400 1600 180020

40

60

80

100

120

(er

utarep

meT

0)

C

Time (s)

h = 0 W/m2K

h = 5 W/m2K

h = 10 W/m2K

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000

0

20

40

60

80

100

(er

utarep

meT

0)

C

Time (s)

h = 5 W/m2-K

h = 10 W/m2-K

(b)

Fig. 7 PCM inner surfacetemperature distribution fordifferent values of heat transfercoefficient (a) Melting withinsulation and convection at outersurface of cylinder. (b) Meltingand solidification with convectionat outer surface of cylinder

Heat Mass Transfer

of the PCM cylinder is stabilized for significant duration byabsorbing the latent heat, while at higher heat flux values, thetemperature rises very fast with short duration. Sensible heatstorage is found to be significant for higher heat flux valuescompared to latent heat storage. Solidification of PCM initi-ates when heat flux at the inner surface becomes zero andouter surface of PCM is subjected to convective air environ-ment. Therefore, it can be noticed from Fig. 6a that the heat

flux values have no effect on the solidification process ofPCM. Figure 6b shows the distribution of temperature withtime during melting of PCM for various heat flux values foradiabatic condition at the outer surface of PCM cylinder.

The effect of heat transfer coefficient on inner surface tem-perature of PCM during melting with insulated outer surface(h = 0 W/m2-K) and convective outer surface (h = 5 W/m2-Kand 10 W/m2-K) is illustrated in Fig. 7a. The melting process

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

t 0/t

m

r2 (mm)

Paraffin

Erytritol

Eicosane

Stearic acid

Paraffin wax

KF.4H2O

Ga

q'' = 500 W/m

2

(a)

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

t 0/t

m

r2 (mm)

Paraffin

Erytritol

Eicosane

Stearic acid

Paraffin wax

KF.4H2O

Ga

q'' = 2000 W/m

2

(b)

Fig. 8 Dimensionless time (t0/tm)for melting and heat penetrationwith radial distance for variousPCM with T∞ = 18 °C, r1 =0.002 m and h = 5 W/m2-K. (a)q^ = 500 W/m2 (b) q^ = 2000 W/m2

Heat Mass Transfer

is assumed to be over when the temperature at the insidesurface of PCM reaches a set point temperature of 100 °C.Initially, the temperature remains same for different values ofheat transfer coefficient. However, the start of melting of PCMis delayed by certain time period with the increase in the valueof h. After certain time, as the melt front progresses, the tem-perature difference between insulated outer surface (h = 0 W/m2-K) and convective outer surface (h = 5W/m2-K, 10W/m2-K) increases as shown in Fig. 7a. This occurs because innersurface temperature of molten PCMdepends on the heat trans-fer coefficient.

0 500 1000 1500 20000.0020

0.0025

0.0030

0.0035

0.0040

)m(

noitis

op esa

hpret

ni tneis

narT

Time (s)

q'' = 250 W/m

2

q'' = 500 W/m

2

q'' = 1000 W/m

2

q'' = 2000 W/m

2

(a)

(b)

0 500 1000 1500 2000 2500 30000.0020

0.0025

0.0030

0.0035

0.0040

)m(

noitis

op esa

hpret

ni tneis

narT

Time (s)

h = 5 W/m2-K

h = 10 W/m2-K

h = 15 W/m2-K

h = 20 W/m2-K

Fig. 9 Variation of transientinterface location with time forvarious values of (a) q^ (b) h

Table 5 Thermophysical Properties of various PCMs

PCM Tm(°C)

Lp(kJ/kg)

kl(W/m-K)

ρl(kg/m3)

cl(kJ/kg-K)

Paraffin 28 244 0.15 774 2.16Paraffin wax 56 195 0.21 822 1770Erythritol 120 340 0.326 1330 2.63Eicosane 35 241 0.27 790 2.05Stearic acid 69 202.5 0.172 848 2.30KF.4H2O 18.4 246 0.48 1450 2.59Ga 29.8 80.1 33.7 6093 0.40

Heat Mass Transfer

Also, here an effort has been made to study the effect ofheat transfer coefficient on the inner surface temperature dur-ing melting and solidification of PCM with convective outersurface (h = 5 W/m2-K and 10 W/m2-K) as shown in Fig. 7b.It can be noticed from this figure that increase in the heattransfer coefficient variation does not have significant effecton melting duration. However, start of melting of PCM isdelayed by certain time period with the increase in the valueof h. Therefore, the time duration required by PCM to reach atthe set point temperature has been increased. The

solidification of PCM starts for q^ = 0 at the inner surface.Different values of heat transfer coefficient have significanteffect on solidification of PCM. As the value of heat transfercoefficient increases at the outside surface of PCM, the timeduration for inner surface to reach at ambient temperature isdecreased significantly.

In general, the analysis of melting of PCM in a cylindricalannulus of finite radii consists of two cases, one with t0 lessthan tm and other with t0 greater than tm. The former case isconsidered here. Figure 8a and b depict the variation of (t0/tm)

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

noitcarf

tleM

Time (s)

q''=250 W/m

2

q''=500 W/m

2

q''=1000 W/m

2

q''=2000 W/m

2

(a)

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

nezorf

noitcar

F

Time (s)

h = 5 W/m2-K h = 10 W/m2-K h = 15 W/m2-K h = 20 W/m2-K

(b)

Fig. 10 Variation of melt fractionwith time for various values of (a)q^ (b) h

Heat Mass Transfer

with radial distance for various PCMs and heat flux values.Here, four different categories of PCMs; namely, organic (par-affin, erythritol, eicosane, paraffin wax), fatty acid (stearicacid), salt hydrate (KF.4H2O) and metal (Gallium) are consid-ered for analysis. The thermophysical properties of PCM arereported by Zalba et al. [1] and Shamberger [22], are summa-rized in Table 5. It may be noted that for outer PCM layer of0.5 mm and q^ = 500 W/m2, among various PCM materials,only KF.4H2O starts to melt, when t = t0 as shown in Fig. 8a.On the contrary, with the increase of heat flux values to2000 W/m2, in addition to KF.4H2O, paraffin wax starts tomelt, when t = t0 as shown in Fig. 8b.

The variation of transient interface location with time forvarious heat flux values in case of melting of PCM and heattransfer coefficient in case of solidification of PCM is shownin Fig. 9a and b, respectively. For analyzing the effect of heatflux on transient solid/liquid interface position, the PCM isassumed to be initially at its melting temperature. It is ob-served that with the increase in heat flux values the melt frontspeed increases and melt front reaches the outside surface ofPCM cylinder in short duration as shown in Fig. 9a. Thisindicates that with the increase of heat flux, PCM will meltquickly and results in reducing the temperature stabilizationduration. It is observed that with the increase in the heat trans-fer coefficient, the time duration required to reach the frozenfront at inside surface of PCM cylinder decreases. This indi-cates that with the increase of heat transfer coefficient, PCMwill solidify quickly and can be considered as an importantparameter for design.

Melt fraction is defined as the ratio amount of PCMmeltedto the total volume of PCM.While, the frozen fraction denotes

the ratio of amount of PCM solidified to the total volume ofPCM. Here, the PCM is assumed to be initially at its meltingtemperature. The variation of melt fraction with time for var-ious heat flux values and variation of frozen fraction with timefor various heat transfer coefficient is shown in Fig. 10a and b,respectively. With the higher value of heat flux, the meltingoccurs quickly and the rate of melting is higher compared tolower heat flux values. It is observed that the solidification ofPCM occurs quickly at faster rate with the increase in heattransfer coefficient as shown in Fig. 10b.

Figure 10a shows the effect of heat flux on the melt fractionof PCM, whereas Fig. 10b shows the effect of heat transfercoefficient on the frozen fraction of PCM. It can be noticedfrom the Fig. 10a that PCM gets melted quickly and meltfraction increases rapidly with higher heat flux. Similarly,PCM gets solidified quickly with higher heat transfer coeffi-cient and frozen fraction rapidly approaches unity.

Figure 11 depicts the effect of radius ratio (β) on melting ofPCM. The melt duration for β = 0.5 and β = 0.33 was found tobe 1238 s and 3046 s, respectively. This indicates that meltduration can be increased by increasing the thickness of PCMover the heated battery module.

0 500 1000 1500 2000 2500 3000 3500 4000

20

40

60

80

100

(er

utarep

meT

0C

)

Time (s)

β = 0.5

β = 0.33

Fig. 11 Effect of radius ratio (β)on melting of PCM

Table 6 Properties ofTCE (Aluminium) Property Magnitude (unit)

k 202.4 W/m-K

ρ 2719 Kg/m3

c 0.87 kJ/kg-K

Tm 660.4 °C

Heat Mass Transfer

Here, an attempt has been made to analyze the effect ofthermal conductivity enhancers (TCE) on the phase changeprocess. For such investigation, aluminum is used as TCEand various volumetric percentage of aluminum (2 to 20%)is uniformly dispersed in PCM stored in the annulus of400 mm long. Thermophysical properties of aluminum aresummarized in Table 6. The TCE has been assumed to beequally dispersed throughout the length of PCM cylinder.The effective thermophysical properties of PCM and TCEare determined as follows [19].

ke f f ¼ δkTCE þ 1−δð ÞkPCM ð30aÞρe f f ¼ δρTCE þ 1−δð ÞρPCM ð30bÞρcð Þe f f ¼ δ ρcð ÞTCE þ 1−δð Þ ρcð ÞPCM ð30cÞρLp� �

e f f ¼ 1−δð Þ ρLp� �

PCM ð30dÞ

It may be noted that linear variation with volumetric frac-tion of TCE is considered to evaluate the thermophysicalproperties (Eqs. 30a–30d). Although the variation in effectivethermal conductivity with volume fraction of TCE is complex,the linear approximation is valid for various properties ofPCM such as density, specific heat and latent heat. Similarapproximations have been made by earlier researchers in theirinvestigations [19]. It is observed that with 2% TCE the timeduration to reach at the set point temperature of 100 °C in-creases. However, with further increase of TCE volume frac-tion, the time duration to reach at set point temperature de-creases as shown in Fig. 12. Also, the melt duration decreasesdue to the decrease in the amount of PCM in the thermalstorage unit. Therefore, the volume fraction of TCE needs tobe chosen wisely as there exists a particular percentage ofTCE for each configuration at which the melt duration forthermal storage unit is maximum. It is observed that TCE haveno effect on the solidification duration of PCM.

5 Conclusion

In this study, an analytical model has been developed for PCMbased cylindrical thermal storage system. In this model, theentire phase change process is divided into three stages:completely solid PCM, partially molten PCM and completelyliquid PCM. A simplified Variational profile, satisfying theboundary conditions coupled with Kantorovich method, isemployed to obtain the solution for temperature distributionof completely solid and liquid PCMs. Quasi steady approxi-mation method is employed for partially molten PCM. Thekey findings obtained from the present analysis are elaboratedbelow.

1. In all the cases, closed form expressions are obtained fortemperature distribution as a function of various model-ling parameters such as boundary heat flux, heat transfercoefficient, thermophysical properties of PCM and phys-ical dimension of thermal storage unit.

2. Present model exhibits good agreement with the test data ofDuan andNaterer [5] duringmelting of RPCMand Liu et al.[12, 13] during melting and solidification of stearic acid.

3. It is observed that increase in heat transfer coefficient haveno significant effect on the melting of PCM. However,heat transfer coefficient is an important tool for control-ling the solidification time of PCM and is also found to bea predominant factor for analyzing the transient interfaceposition and frozen fraction during solidification of PCM.

4. Decreasing the radius ratio of annulus from β = 0.5 to β =0.33 increases the melt duration from 1238 s to 3046 s.This indicates that melt duration can be increased by in-creasing the thickness of PCM over the heated batterymodule.

5. Presence of a high-conductivity material has a significanteffect on PCM temperature. It is observed that for any

0 2000 4000 6000 8000

20

40

60

80

100

Tem

per

ature

( C

)0

Time (s)

Without TCE

2% TCE volume fraction

4% TCE volume fraction

8% TCE volume fraction

12% TCE volume fraction

20% TCE volume fraction

Fig. 12 Effect of TCE on meltingand solidification of PCM

Heat Mass Transfer

thermal storage unit there exists a particular percentage ofTCE- PCM distribution through which maximum meltduration can be achieved.

Acknowledgements The financial support provided by Department ofScience and Technology (DST), India under the project grant DST/TMD/MES/2 k17/65 (G) is gratefully acknowledged. The first authoracknowledges the financial support by DST, India under DST-INSPIREFellowship program (IF170534).

Publisher’s Note Springer Nature remains neutral with regard to jurisdic-tional claims in published maps and institutional affiliations.

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Heat Mass Transfer