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International Journal of Thermal Sciences 72 (2013) 47e59

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Heat conduction in laminate multilayer bodies with applied finite heatsource

Louis Desgrosseilliers a,b, Dominic Groulx a,b,*, Mary Anne White b,c

aDepartment of Mechanical Engineering, Dalhousie University, PO Box 15000, Halifax, NS B3H 4R2, Canadab Institute for Research in Materials, Dalhousie University, PO Box 15000, Halifax, NS B3H 4R2, CanadacDepartment of Chemistry, Dalhousie University, PO Box 15000, Halifax, NS B3H 4R2, Canada

a r t i c l e i n f o

Article history:Received 1 October 2012Received in revised form29 April 2013Accepted 7 May 2013Available online 18 June 2013

Keywords:Multilayer laminateComposite heat conductionHeat spreaderNon-uniform heatingAnalytical solution

* Corresponding author. Department of MechanUniversity, PO Box 15000, Halifax, NS B3H 4R2, Cana

E-mail address: dominic.groulx@dal.ca (D. Groulx)

1290-0729/$ e see front matter � 2013 Elsevier Mashttp://dx.doi.org/10.1016/j.ijthermalsci.2013.05.001

a b s t r a c t

Multilayer laminate bodies exhibit beneficial heat spreading qualities when one or more inner layershave sufficiently high thermal conductivity to carry heat laterally under non-uniform heating conditions.The proposed model is broken up in two regions of interest: the heated region and the fin region. Thismodel predicts the behaviour experienced by laminate films when heated by a constant temperature orheat flux over only part of the domain. The one dimensional, steady-state, two-region fin model is uniquein its representation of the conduction heat transfer in the heat spreading layer only, for both Cartesianand cylindrical coordinates. Temperature uniformity is wholly attainable with applied heat fluxes, whilelaminates provide improved heat transfer rates with applied temperature sources (e.g. phase changematerials) over conventional, homogeneous, encapsulation materials. The two-region fin model predictstemperature profiles and rates of heat transfer for steady-state or pseudo steady-state analyses asvalidated by a two-dimensional finite element model under the applicable conditions.

� 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

Laminate materials (also called multilayer composites),composed of differing materials in a cohesive layered final product(heat or adhesively bonded layers), have found niche applicationsin the food and consumer electronics packaging industries. Theyare effective, lightweight barriers to the damaging effects of light,air, moisture, and microorganisms [1e5]. Laminates in these ap-plications are usually composed of outer thermoplastic layers andinner metal foil, often polyethylene (PE) film or DuPont’s Mylar�

(polyester film) and aluminium (Al) foil. Overall, the use of lami-nates is extensively dedicated to the preservation of product shelflife. In one other application, mainly stovetop cookware fabrication,the laminates in question are composed of outer layers of tough,corrosion-resistant, but typically low thermal conductivity metals(e.g. stainless steels) with inner layers (sometimes up to 9 layers intotal) of highly conductive metals, e.g. copper (Cu) or Al [6e8].Considering their heat transfer characteristics, there are otherapplications that could benefit from using laminate materials.

ical Engineering, Dalhousieda. Tel.: þ1 (902) 494 8835..

son SAS. All rights reserved.

As an illustration of laminate heat transfer, cookware is availablein an assortment of multi-ply designs with the aim of improvingthe temperature uniformity of the cooking surface on what areotherwise non-uniform heat sources. These range from ceramic coilheating elements to halogen lamps, wood fire, propane or naturalgas burners, and now induction elements. Hot spots cause food toburn or stick [8], and significant cookware overheating causesirreparable damage resulting from non-uniform thermal expansionof the various metals and surface coatings [6,7]. Temperature uni-formity is made possible through the careful combination of thewear resistant, but low thermal conductivity outer layers withhighly conducting inner layers (metal [6,7] or graphite [8]), atechnique that is primarily represented by cookware patents. Thesecookware patents suggest that while the outermost layer becomesquickly saturated by carrying heat from the heating element to thecooking surface in the direction of the surface normal, the betterconducting inner layers can convey excess heat from the source inthe transverse direction (heat spreading, or anisotropic heat con-duction for graphite cores [8]), resulting in heating more of theupper surface thanwhat was directly overtop the heat source [6,7].

On the other hand, theoretical and numerical models of heatspreading behaviour in electronics cooling devices and multilayermetal mirrors used in laser systems have been presented in theliterature. Heat transfer studies concerning composite materials

Nomenclature

Dimensional variablesA surface area (m2)_A surface area rate of growth (m2 s�1)Cp heat capacity (J kg�1)_E thermal energy accumulation (W)h convection heat transfer coefficient (W m�2 K�1)hh natural convection heat transfer coefficient in the

heated region only (W m�2 K�1)hf natural convection heat transfer coefficient in the fin

region only (W m�2 K�1)k thermal conductivity (W m�1 K�1)L heated boundary length at x, r ¼ 0 (m)_L heated boundary length rate of growth (m s�1)q00o applied finite heat flux (W m�2)Qfin rate of fin region heat transfer (W)Qlam laminate body rate of heat transfer (W)Qsimple rate of heat transfer of laminate body without heat

spreading layer (W)r radial position (m)R thermal resistance above the highly conductive metal

core (m2 K W�1)R3 thermal resistance below the highly conductive metal

core (m2 K W�1)Rh heated regionupper layer thermal resistance (m2KW�1)Rf fin region upper layer thermal resistance (m2 K W�1)t layer thickness (m)T temperature (K)Ti applied temperature heat source (K)Tinf free stream temperature (K)To boundary temperature at x, r ¼ L (K)

x Cartesian transverse position (m)

Greek symbolsa heated region constant (m�1)b heated region particular solution (K)DT variable substitution for the applied temperature

driving force (K)g fin region constant (m�1)l dimensionless positionh heat transfer effectivenessr density (kg m�3)s dimensionless temperatureUa dimensionless thermal resistance factor in the heated

regionUg dimensionless thermal resistance factor in the fin

regionUk dimensionless thermal resistance ratio of the bottom

and upper layersfo variable substitution for To in the fin region (K)qo variable substitution for To in the heated region (K)

Subscriptsave mixture average propertytransv transverse path (x or r axis) for heat conductionplanar planar normal path for heat conduction (y or z axis)sat saturated conditions in the fin region1 high conductivity metal core2 top thermally resistive layer3 bottom thermally resistive layer

Superscript0 linear flux

Fig. 1. a) Solidification of a conventional PCM; b) point heat source in seeded PCM.

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5948

have been largely motivated by two-layered heat sinks forelectronics cooling [9,10], volumetric heating and fuel simulation[11e13], and femto to nanosecond laser pulse heating of multilayerAu mirrors [14e16]. Some salient points are:

1. Applications of layered metals have been considered at thenear exclusion of polymers or other thermally resistive layers;

2. Laser pulses only apply to extreme local heat fluxes [14e16];3. Researchers have been motivated by the thermal stress and

contact resistance between dissimilar metals [10,16], less so forthe heat transfer attributes alone;

4. The solutions presented for electronics cooling are too inclusiveof all manner of non-uniform heat sources and all of the ma-terial domains. These solutions are too complex for mostcommon applications, i.e. eigenvalues must be computed forseries solutions and triple integrals solved for the source terms[9,10];

5. The most fundamental solutions only considered uniformboundary conditions and solutions spanning the planar normaldirection [11,12,17].

A new application may yet benefit from heat spreading inmultilayered bodies; conventional packaging laminates (thermo-plastic and metal foil) can add heat transfer enhancements to someof the emerging, novel heat storage technologies using supercooledsalt hydrate phase change materials (PCMs). Salt hydrate PCMs thatsupercool formmetastable, ambient temperature, liquid states thatsuccessfully retain the stored latent heat. Solidification is initiatedusing a nucleation trigger containing salt crystals in order to extract

the heat when desired. These nucleation sites grow fromwithin thebulk rather than along the heat exchange surfaces as it is for con-ventional solideliquid transitions (see Fig. 1). It is due to this so-lidification behaviour that heat spreading along heat exchangesurfaces could substantially improve the heat discharge rate fromsupercooled salt hydrate PCMs.

Lane [5,18,19] reported promising characteristics for 1.8� 10�4 mthick laminated polyethylene/Al/polyester (76/89/13 � 10�6 mlayers, respectively) pouches as salt hydrate PCM encapsulation forheat exchange with air. Schultz and Furbo [20] used 1.13 � 10�4 m

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 49

thick laminated pouches (9� 10�6m thick Al foil) for their long-termheat storage (LTHS) PCM experiments using supercooledNaCH3CO2�3H2O. Lane [19] neglected considerations of any en-hancements from the laminated film since the heat transfermedium,air, was the limiting factor and the PCMs under study were only ofthe conventional, non-supercooling kind, while Schultz and Furbo[20] made their selection based on flexibility and shelf stabilityalone.

Although Lane et al. [5] reported on the mechanical and bulkthermal and physical properties of the 1.8 � 10�4 m laminate pouch(R-2retortpouchused inmilitary rationpackaging)used inPCMheat-storage testing, there have been no studies of the potential heattransferbenefits theycouldoffer to supercooled salthydrate solutionsused inPCMLTHS systems. In fact, other PCMLTHSexperiments usingsupercooled salthydrateshaveusedeither bulkormicroencapsulatedPCM in rigid plastic [21,22] and have not sought improvements to theheat transfer rate during discharge by heat spreading.

Furthermore,models alreadypresent in the literature [9e13,15,17]are ill-suited for predicting the heat transfer behaviour from nucle-ated salt hydrate solutions and require very complex implementationfor use indesign. They leaveunsolvedparameters in integral formulas[9,10], require complex iterative approaches to achieve convergence[9e12,17], and exclude the boundary conditions and geometryneeded to model heat transfer from nucleated salt hydrate PCMs in apouch encapsulation [9e13,15,17].

To this end, a new laminate heat conduction model is proposedfor heat spreading due to non-uniform heating from supercooledsalt hydrate PCMs. The model is non-iterative and gives onlyexplicit solutions to the differential heat balance equations for easeof implementation into software and overall reduction of compu-tational effort. The solutions are derived for both Cartesian andcylindrical coordinates in order to easily adapt to the desired situ-ation. It might also find application in heat transfer modelling forcookware and electronics cooling.

This paper presents the description of the layered compositesystem separated into two domains (henceforth called ‘two-regionfin model’), mathematical derivations in Cartesian and cylindricalcoordinates for non-uniform heating (both heat flux and temper-ature) with convection heat transfer at the top surface. Dimen-sionless equations and solutions are also presented and validatedagainst finite element solutions obtained with COMSOL Multi-physics 4.2a.

2. Two-region fin heat transfer model

The system under study is presented in Fig. 2, inwhich a layeredbody has at least three layers: upper (subscript 2) and lower(subscript 3) thermally resistive layers and a high thermal

Fig. 2. Two-region fin model schematic; the heated region is left of the dashed line (xor r ¼ L), while the fin region is to the right.

conductivity core (subscript 1) that is responsible for the heatspreading component to the conduction heat transport in thesystem. The system variables presented in Fig. 2 are: layer specificthermal conductivity, k; layer specific thickness, t; heated regionlength and heated boundary x or r axis position, L; convection heattransfer coefficient, h; convection boundary free-stream tempera-ture, Tinf; constant temperature heat source, Ti; and, constant heatflux source, q00o. Subscripts 2 and 3, in Fig. 2, are ordered as such forconvenience when the bottom layer is not being considered in thecalculations. Although the PCM heat source under consideration, Ti,represents only constant temperature phase changes (congruentsystems) and most supercooled salt hydrates are incongruent sys-tems, a constant temperature boundary condition is still deemedappropriate to represent supercooled salt hydrate PCMs. Incon-gruency in the phase equilibrium of salt hydrates does not typicallyproduce very large temperature differences within the solidifyingPCM and Ti can represent the average phase change temperature.

The system in Fig. 2 can be represented equally in Cartesian orcylindrical coordinates, for which the z-axis always represents thelayer heights (thicknesses t1, t2, and t3 are along this axis) while thelayers’ lengths run along the x or r axis in Cartesian or cylindricalcoordinates, respectively.

In Fig. 2, heat is exchanged between the laminate and theenvironment in both the heated region and the unheated region(or fin region, x or r � L) and between the shared boundary of thetwo regions (x or r ¼ L). No heat is exchanged through the bottomlayer in the unheated region.

With respect to the inner metal layer, the behaviour can berepresented as a two-region fin heat transfer problem. To reducethe number of variables that must be solved in the differential heatbalance equation, the following simplifying assumptions wereapplied:

1. The heat transfer problem is symmetric at the domain origin,x or r ¼ 0;

2. Assume a planar two-dimensional laminate in Cartesian co-ordinates and axial-symmetry (rotational symmetry around z)in cylindrical coordinates;

3. Assume that temperature gradient in the z-direction of thebottom layer of the fin region is negligible when compared toTi � Tinf. Therefore, it is treated as being well insulated;

4. Assume only a transverse temperature gradient (x or r di-rections) in the high thermal conductivity metal core;

5. Assume negligible transverse heat conduction in the thermallyresistive layers;

Constant thermal conductivities and external heat transfer co-efficients, as well as negligible thermal expansion of the materialsare assumed.

The two-region fin problem reduces a two-dimensional systeminto a set of one-dimensional symmetric, steady-state heat transferequations solved only in the inner, high thermal conductivity metalcore, but in two parts: the heated region and the fin region. Solu-tions were obtained in both the Cartesian and cylindrical coordi-nate systems. While the range of application of the two-region finmodel is quite broad, consideration must be given to the implica-tions of the core assumptions in order to justify this claim.

Heat spreading behaviour can be represented as the dominanceof transverse heat conduction (x or r directions) over planar heatconduction (z-direction) in the high thermal conductivity metalcore (assumption #4), conversely that the planar thermal resis-tance dominates over the transverse thermal resistance. In an orderof magnitude approach, these are:

Rtransv ¼ L=k1 (1)

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5950

and

Rplanar ¼ t1k1

þ t2k2

þ 1h

(2)

in Cartesian coordinates only. The desired relationship for heatspreading is for

Rtransv � Rplanar: (3)

Consider, for example, a 1 � 10�4 m laminate film with ther-mally resistive thermoplastic layers (t2 ¼ t3 z 4 � 10�5 m;k2 ¼ k3 z 0.4 W m�1 K�1) and Al foil high thermal conductivitymetal core (t1 z 2 � 10�5 m and k1 z 260 W m�1 K�1), externallycooledwith air (hz 50Wm�2 K�1) and a heated length (L) equal to1 � 10�2 m. To ensure heat spreading, these parameters mustsatisfy Eq. (3). From Eq. (1),

Rtransv ¼ 0:01260

m2 KW

¼ 4� 10�5 m2KW

(4)

and from Eq. (2),

Rplanar ¼�0:00002260

þ 0:000040:4

þ 150

�m2 KW

¼ 0:02m2 KW

; (5)

satisfying Eq. (3).It would benefit to mention that the two-region fin model

could also be used in pseudo steady-state analysis of laminatesystems. Laminate bodies are typically thin, so pseudo steady-stateanalysis can often be considered. This condition can be expressedas the time derivative of the heated length, _L, being sufficientlysmall when compared to the laminate’s capacity for heat accu-mulation ð _EÞ.

In an order of magnitude approach for a constant heat flux,this is:

_E ¼_L�raveCp;aveðt1 þ t2 þ t3Þq00o

�k2

� q00o (6)

or

_L�raveCp;aveðt1 þ t2 þ t3Þ

�k2

� 1; (7)

where Cp,ave and rave are the laminate body’s mass average heatcapacity and volume average density. For example, taking the samelaminate described above and rave ¼ 900 kg m�3 andCp,ave ¼ 2� 103 J kg�1, a pseudo steady-state analysis would remainvalid if

_Lð900ð2000Þð0:0001ÞÞ0:4

sm

� 1 (8)

or

_L � 2� 10�3 ms: (9)

For an externally nucleated, supercooled, salt hydrate, thisexpansion rate could be reasonable for those with the highestlatent heat of fusion, since crystal growth becomes heat transferlimited after the initial seeding [19,23].

The assumption that the bottom thermally resistive layer can betreated as well insulated in the fin region (assumption #3),although already justified, is not the exact depiction of salt hydratesolidification. However, like all other PCMs of its kind, salt hydrates

have poor thermal conductivity, which would result in only mini-mal self-heating of the supercooled liquid PCM underneath theencapsulation’s fin region. The latter does not constitute a parasiticheat loss either, since the heat conduction path is internal to thesystem and the heat would be available for heat transfer with theenvironment nonetheless. The preferred path for conduction heattransfer in the fin region would still remain the same as the idealcase just presented.

2.1. Cartesian coordinates (applied temperature)

2.1.1. Heated regionThe Cartesian thermal energy balance equation applied only to

the high thermal conductivity metal core in the heated region(0 � x � L) with a constant applied temperature (Ti) is given by

d2Tdx2

� k3t3t1k1

ðT � TiÞ �1

Rt1k1

�T � Tinf

�¼ 0; (10)

where R ¼ t2/k2 þ 1/h. Equation (10) is a second-order, linear, non-homogeneous ODE with respect to T and can be rearranged to thegeneral form:

d2Tdx2

��1Rþ k3

t3

�T

t1k1¼ � 1

t1k1

�TinfR

þ k3t3

Ti

�: (11)

Equation (11) has to be solved using the following two boundaryconditions:

1. Tðx ¼ LÞ ¼ To ðcontinuityÞ

2.dTdx

jx¼0 ¼ 0 ðsymmetryÞ.

Applying the two boundary conditions, the temperature profilein the heated region is:

TðxÞ ¼ ðTo � bÞ coshðaxÞcoshðaLÞ þ b; (12)

where the two parameters, a and b, are

a2 ¼ 1t1k1

�1Rþ k3

t3

�; (13)

b ¼ t3Tinf þ Rk3Tit3 þ Rk3

: (14)

2.1.2. Fin regionIn the fin region, x� L, the thermal energy balance applied to the

high thermal conductivity metal core, regardless of the heat source(Ti or q00o), is shown here in the general form of a second-order linear,non-homogeneous ODE with respect to T,

d2Tdx2

� 1Rt1k1

T ¼ � 1Rt1k1

Tinf : (15)

The fin region would obey the following boundary conditions:

1. Tðx ¼ LÞ ¼ To ðcontinuityÞ2. limx/NTðxÞ ¼ Tinf ðequilibriumÞ

The fin region temperature profile then becomes:

TðxÞ ¼�To � Tinf

�e�gðx�LÞ þ Tinf ; (16)

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 51

for which

g2 ¼ 1Rt1k1

: (17)

The unknown boundary condition, To, shared by the heated and finregions at x ¼ L, implies heat continuity, so

dTdx

����x¼L

¼ aðTo � bÞtanhðaLÞ (18)

in the heated region, and

dTdx

����x¼L

¼ g�Tinf � To

�(19)

in the fin region. Equating Eqs. (18) and (19) and isolating To givesthe solution

To ¼ gTinf þ abtanhðaLÞgþ atanhðaLÞ : (20)

What remains in this exploration of the two-region fin problemare the two aspects of heat transfer: overall linear power densityentering and leaving the system, Q 0

lam, and the fin-only linear po-wer density that crosses the heated region boundary at x ¼ L, Q 0

fin.Note that the Cartesian solution heat transfer calculations mustall remain as linear power densities, W m�1, due to the two-dimensional laminate assumption in Cartesian coordinates.

Simply stated, these are:

Q 0lam ¼

ZL0

k3t3

ðTi � TÞdx ¼ k3t3

LðTi � bÞ� To � b

atanhðaLÞ

(21)

and

Q 0fin ¼ �k1t1

dTdx

����x¼L

¼�To � Tinf

� ffiffiffiffiffiffiffiffiffik1t1R

r: (22)

One final notion to discuss, applying only to constant temper-ature heat sources (e.g., PCMs), is the overall laminate heat transfereffectiveness, h, presented below.

h ¼ Q 0lam

Q 0simple

: (23)

Equation (23) is the ratio of the laminate rate of heat transferover simple encapsulation made of the same durable, but lowthermal conductivity outer layers of the laminate and of the sameoverall thickness. The simple encapsulation is assumed to onlyallow planar heat transfer.

The expression for the simple, non-laminate encapsulation asdefined by assumption #5 in Section 2, is given by

Q 0simple ¼

ZL0

1t2k2

þ t3k3

þ 1h

�Ti � Tinf

�dx ¼

L�Ti � Tinf

�t2k2

þ t3k3

þ 1h

: (24)

Since the conditions ensuring the accuracy of assumption #5 arenot essential for heat spreading, substituting Eq. (24) for Q0

simple

might underestimate the non-laminate rate of heat transfer whentwo-dimensional conduction is indeed significant and can there-fore be substituted with a more accurate solution when one isavailable.

Substituting Eqs. (21) and (24) into Eq. (23) provides a clearerexpression of h:

h ¼

�t2k2

þ t3k3

þ 1h

�k3t3

LðTi � bÞ � ðTo � bÞ

atanhðaLÞ

L�Ti � Tinf

� � 1: (25)

Contributions larger than one from the terms (t2/k2 þ t3/k3 þ 1/h)k3/t3 and �(To � b)tanh(aL)/a (To / Tinf at small values ofa), in Eq. (25), show that laminate heat transfer is usually superiorto the homogeneous kind. Themaximization of this metric, h, whilealso satisfying minimum strength requirements and encapsulationweight, is of foremost significance to designing encapsulation forsupercooled PCM, long-term heat storage.

2.2. Cylindrical coordinates (applied temperature)

2.2.1. Heated regionFor an axi-symmetric, two-dimensional laminate body, with a

constant temperature heat source, Ti, in the domain 0 � r � L, thethermal energy balance equation applied only to the high thermalconductivity metal core in this domain is

prt1k1

1rddr

�rdTdr

��pr

R

�T � Tinf

�þprk3

t3ðTi � TÞ ¼ 0; (26)

where R ¼ t2/k2 þ 1/h. Equation (26) is a non-homogeneousmodified Bessel equation of order zero with respect to T, whichcan be rearranged in the general form:

r2d2Tdr2

þ rdTdr

� a2r2T ¼ � r2

t1k1

�TinfR

þ Tik3t3

�; (27)

where

a2 ¼ 1t1k1

�1Rþ k3

t3

�: (28)

Again, the heated region would have the following two boundaryconditions:

1. Tðr ¼ LÞ ¼ To ðcontinuityÞ2.

dTdr

jr¼0 ¼ 0�symmetry

�.

The solution to the non-homogeneous modified Bessel equationof order zero gives the temperature profile obtained in the heatedregion:

TðrÞ ¼ ðTo � bÞ I0ðarÞI0ðaLÞ

þ b; (29)

for which

b ¼ t3Tinf þ Rk3Tit3 þ Rk3

: (30)

2.2.2. Fin regionIn the fin region, r� L, the thermal energy balance applied to the

high thermal conductivity metal core, regardless of the heat source(Ti or q00o), is shown here in the form of a non-homogeneousmodified Bessel equation of order zero with respect to T,

r2d2Tdr2

þ rdTdr

� g2r2T ¼ �g2r2Tinf ; (31)

Table 1Summary of the two-region fin model in Cartesian coordinates.

Parameter Ti q00o

T, 0 � x � L TðxÞ ¼ ðTo � bÞ coshðaxÞcoshðaLÞ þ b

T, x � L TðxÞ ¼ ðTo � Tinf Þe�gðx�LÞ þ Tinf

a a2 ¼ 1t1k1

�1Rþ k3

t3

�a2 ¼ 1

Rt1k1

g g2 ¼ 1Rt1k1

b b ¼ t3Tinf þ Rk3Tit3 þ Rk3

b ¼ Tinf þ Rq00o

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5952

where

g2 ¼ 1Rt1k1

: (32)

The fin region would still obey the same two boundaryconditions:

1. Tðr ¼ LÞ ¼ To ðcontinuityÞ2. limr/NTðrÞ ¼ Tinf ðequilibriumÞ

The complete fin region temperature is:

TðrÞ ¼�To � Tinf

� K0ðgrÞK0ðgLÞ

þ Tinf : (33)

The determination of the unknown boundary condition sharedby both domains at r ¼ L, To, follows the same method as was usedin Cartesian coordinates (Eqs. (18)e(20)). The heat continuity so-lution between the heated and fin regions giving To is:

To ¼ abI1ðaLÞK0ðgLÞ þ gTinfK1ðgLÞI0ðaLÞaI1ðaLÞK0ðgLÞ þ gK1ðgLÞI0ðaLÞ

: (34)

In order to evaluate the amount of heat transfer improvementfor a constant temperature heat source (e.g., PCM), the rate of heattransfer (for the representative half domain only) for the cylindricaltwo-region fin problem, Qlam, and the fin-only rate of heat transfer,Qfin, are:

Qlam ¼ZL0

k3t3

ðTi � TÞpr dr

¼ k3pLt3

L2ðTi � bÞ � ðTo � bÞ

a

I1ðaLÞI0ðaLÞ

; (35)

Qfin ¼ �k1t1pLdTdr

����r¼L

¼ k1t1pL�To � Tinf

�gK1ðgLÞK0ðgLÞ

: (36)

The resulting rate of heat transfer from the simple encapsulationmodel, Qsimple, and heat transfer effectiveness, h, in cylindricalcoordinates are:

Qsimple ¼ZL0

1t2k2

þ t3k3

þ1h

�Ti�Tinf

�pr dr ¼

pL2�Ti�Tinf

�2�t2k2

þ t3k3

þ1h

� (37)

and

h ¼2�t2k2

þ t3k3

þ1h

�k3t3

L2ðTi�bÞ� ðTo�bÞ

a

I1ðaLÞI0ðaLÞ

L�Ti� Tinf

� � 1: (38)

To To ¼ gTinf þ abtanhðaLÞgþ atanhðaLÞ

Q 0lam Q 0

lam ¼ k3t3

LðTi � bÞ � ðTo � bÞ

atanhðaLÞ

Q 0lam ¼ Lq00o

Q 0fin Q 0

fin ¼ ðTo � Tinf Þffiffiffiffiffiffiffiffiffik1t1R

r

Q 0simple Q 0

simple ¼ LðTi � Tinf Þt2k2

þ t3k3

þ 1h

Q 0simple ¼ Lq00o

h h ¼ Q 0lam

Q 0simple

� 1 h ¼ 1

2.3. Constant heat flux boundary condition

The effect of the applied heat flux boundary condition, q00o, is onlypresent in the heated region of both the Cartesian and cylindricalcoordinate models (boundary conditions to solve ODEs remain thesame). Shown in this order, their differential heat balances are:

d2Tdx2

� TRt1k1

¼ � 1t1k1

�TinfR

þ q00o

�; (39)

in Cartesian coordinates, and

r2d2Tdr2

þ rdTdr

� a2r2T ¼ � r2

t1k1

�TinfR

þ q00o

�; (40)

in cylindrical coordinates, for which a2 ¼ (Rt1k1)�1.The resulting solutions are summarized along with the equiva-

lent equations derived for the nucleated, supercooled, PCM heatsource (Ti boundary condition) in Tables 1 and 2.

2.4. Non-uniform convection

Two additional scenarios involving the two-region fin modelalso use many of the same equations from Tables 1 and 2, but someof the parameters require adjustments. In the first case, the tworegions are cooled by natural convection and their integral averageconvection coefficients must be assigned separately, simply hh andhf (also Rh and Rf) for the heated and fin regions respectively. Thesolutions presented in Tables 1 and 2 remain the same, howevereach region must have either hh or hf and Rh and Rf as substitutionsfor h and R. This method cannot include any greater accuracy of thelocally changing convection coefficient within each of the domains,since doing so would add an additional non-linear term in thedifferential equations and would be coupled to the NaviereStokesequations to solve for the exact solution of natural convection.

In the second case, consider a well-insulated top surface of theheated section only, so hh¼ 0, but the fin region remains cooled by afluid, hf s 0. For this case with the applied temperature boundarycondition (Ti), the solution in the heated domain is only affected bythe solution for b,

b ¼ Ti; (41)

while the remainder of the solutions outlined in Tables 1 and 2 isunchanged. With the applied heat flux boundary condition, how-ever, the differential heat balance equation in the heated regionmust be solved anew. The solution in Cartesian coordinates to thesecond-order, separable, non-homogeneous, differential heat

Table 3Summary of the dimensionless two-region fin model in Cartesian coordinates.

Parameter Ti q00o

s, 0 � l � 1 sðlÞ ¼ coshðl

ffiffiffiffiffiffiffiUa

pÞ

coshðffiffiffiffiffiffiffiUa

p� 1

!�foqo

��1

þ 1

s, l � 1 sðlÞ ¼ effiffiffiffiffiUg

pð1�lÞ

Ua Ua ¼ Ugð1þUkÞ Ua ¼ L2

t1k1

1R

Ug Ug ¼ L2

t1k1

1R

DT/qoDTqo

¼�1þ 1

Uk

��foqo

� 1�

DTqo

¼ 2foqo

� 1

fo/qofoqo

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Uk

ptanhð

ffiffiffiffiffiffiffiUa

pÞ fo

qo¼ �tanhð

ffiffiffiffiffiffiffiUa

pÞ

Q 0lam

Q 0lamR3qoL

¼�DTqo

�foqo

þ1�

� 1ffiffiffiffiffiffiffiUa

p tanhðffiffiffiffiffiffiffiUa

pÞ e

Q 0fin

Q 0finR3qoL

¼ 1Uk

ffiffiffiffiffiffiffiUg

p foqo

Q 0simple

Q 0simpleR3qoL

¼ 1ðUk þ 1Þ

DTqo

e

Table 4Summary of the dimensionless two-region fin model in cylindrical coordinates.

Parameter Ti q00o

s, 0 � l � 1 sðlÞ ¼ I0ðl

ffiffiffiffiffiffiffiUa

pÞ

I0ðffiffiffiffiffiffiUa

p� 1

!�foqo

��1

þ 1

s, l � 1 sðlÞ ¼ K0ðlffiffiffiffiffiffiffiUg

p ÞK0ð

ffiffiffiffiffiffiffiUg

p Þ

Ua Ua ¼ Ugð1þUkÞ Ua ¼ L2

t1k1

1R

Ug Ug ¼ L2

t1k1

1R

DT/qoDTqo

¼�1þ 1

Uk

��foqo

� 1�

DTqo

¼ 2foqo

� 1

fo/qofoqo

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Uk

p I1ðffiffiffiffiffiffiffiUa

pÞK0ð

ffiffiffiffiffiffiffiUg

p ÞK1ð

ffiffiffiffiffiffiffiUg

p ÞI0ðffiffiffiffiffiffiffiUa

pÞ

foqo

¼�I1ðffiffiffiffiffiffiffiUa

pÞK0ð

ffiffiffiffiffiffiffiUg

p ÞK1ð

ffiffiffiffiffiffiffiUg

p ÞI0ðffiffiffiffiffiffiffiUa

pÞ

QlamQlamR3qoL2

¼p2

�DTqo

�foqo

þ1�� pffiffiffiffiffiffiffi

Ua

p I1ðffiffiffiffiffiffiffiUa

pÞ

I0ðffiffiffiffiffiffiffiUa

pÞ

e

QfinQfinR3qoL2

¼ p1

Uk

ffiffiffiffiffiffiffiUg

p foqo

K1ðffiffiffiffiffiffiffiUg

p ÞK0ð

ffiffiffiffiffiffiffiUg

p Þ

QsimpleQsimpleR3

qoL2¼ p

2ðUk þ 1ÞDTqo

e

Table 2Summary of the two-region fin model in cylindrical coordinates.

Parameter Ti q00o

T, 0 � r � L TðrÞ ¼ ðTo � bÞ I0ðarÞI0ðaLÞ

þ b

T, r � L TðrÞ ¼ ðTo � Tinf ÞK0ðgrÞK0ðgLÞ

þ Tinf

a a2 ¼ 1t1k1

�1Rþ k3

t3

�a2 ¼ 1

Rt1k1

g g2 ¼ 1Rt1k1

b b ¼ t3Tinf þ Rk3Tit3 þ Rk3

b ¼ Tinf þ Rq00o

To To ¼ abI1ðaLÞK0ðgLÞ þ gTinfK1ðgLÞI0ðaLÞaI1ðaLÞK0ðgLÞ þ gK1ðgLÞI0ðaLÞ

Qlam Qlam ¼ k3pLt3

L2ðTi � bÞ � ðTo � bÞ

a

I1ðaLÞI0ðaLÞ

Qlam ¼ 1

2pL2q00o

Qfin Qfin ¼ k1t1pLðTo � Tinf Þ gK1ðgLÞK0ðgLÞ

Qsimple Qsimple ¼ pL2ðTi � Tinf Þ2�t2k2

þ t3k3

þ 1h

� Qsimple ¼ 12pL2q00o

h h ¼ QlamQsimple

� 1 h ¼ 1

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 53

balance equation in the heated region (0 � x � L) for this casebecomes:

TðxÞ ¼ q00o2t1k1

�L2 � x2

�þ To: (42)

The solution to the unknown boundary condition, To, must alsobe evaluated for the solution in Eq. (42), giving

To ¼ Tinf þL

t1k1gq00o: (43)

In cylindrical coordinates, the solutions corresponding to Eqs.(42) and (43) are:

TðrÞ ¼ q00o4t1k1

�L2 � r2

�þ To (44)

and

To ¼ Tinf þK0ðgLÞK1ðgLÞ

L2t1k1g

q00o: (45)

In both coordinate systems, with the applied heat source q00o andthe well-insulated top of the heated region, the remaining equa-tions are all identical to those in Tables 1 and 2.

2.5. Dimensionless equations

Dimensionless variables were assigned for the two-region equa-tions in Tables 1 and 2. In the heated region with the applied tem-perature boundary condition, the characteristic path length becomes,

ax ¼ lffiffiffiffiffiffiUa

p; (46)

where l ¼ x/L (substitute r for x in cylindrical coordinates) and

Ua ¼ L2

Rt1k1

�1þ k3R

t3

�: (47)

The new variables l and Ua represent the characteristic lengthand relative thermal resistance of fin heat transport in the heateddomain (indicated by subscript a), respectively.

The same approach is taken in the fin region,

gx ¼ lffiffiffiffiffiffiUg

q; (48)

where

Ug ¼ L2

Rt1k1(49)

and for which it is apparent in Eq. (47) that Ua ¼ Ug(1 þ k3R(t3)�1).The term k3R(t3)�1 being recurrent can be represented by thedimensionless variable Uk,

Uk ¼ k3Rt3

; (50)

which accounts for the relative thermal resistance of the bottomand top layers of the laminate.

Table 5Polyethylene and aluminium material properties specified in COMSOL Multiphysics4.2a.

Polyethylene [24e26] Aluminium [27]

Cp (J kg�1 K�1) 2000 900k (W m�1 K�1) 0.4 260r (kg m�3) 900 2700

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5954

Lastly, variable substitutions were also performed on theboundary conditions, the local temperature variable, and thethermal resistance in the bottom layer (all in order):

1. fo ¼ To � Tinf2. qo ¼ To � b

3. l ¼ [l � linf]/fo

4. DT ¼ Ti � Tinf or DT ¼ Rq00o � Tinf with applied heat flux5. R3 ¼ t3/k3

Tables 3 and 4 show the result of applying all of the abovevariable substitutions to obtain dimensionless equations from theequations in Tables 1 and 2.

3. Finite element model

Confidence in the two-region fin model predictions is pre-requisite to using them in design calculations for salt hydratelatent heat storage encapsulation, multi-ply cookware, and evenheat spreaders for electronics cooling. As the first means of doingso, finite element simulations were evaluated for a thin three-component film (4/2/4 � 10�5 m polyethylene/Al/polyethylene,the same as the laminate film inside Tetra Brik cartons) in Cartesianand cylindrical coordinates in order to validate the two-region finmodel for steady-state and transient, two-dimensional and axi-symmetric solutions.

Fig. 3. Close-up of the Cartesian finite element geometry and quadril

The finite element models were prepared in COMSOL Multi-physics 4.2a to represent a 0.1 m long laminate film. The funda-mental equations were added using the heat transfer in solidsmodule included in COMSOL and applying a convective coolingboundary condition on the top surface (with specified h) and eitherconstant temperature or heat flux boundary condition over aportion (0� x, r� L) of the bottom surface. The symmetry boundarycondition at x or r ¼ 0 in the heated region is satisfied by an adia-batic boundary condition, while the fin region infinite boundarycondition could only be simulated by means of giving sufficientlength to ensure complete heat dissipation in the fin region beforereaching the domain boundary. The materials models for poly-ethylene (all nominal properties) and Al are shown in Table 5. Meshconvergence was achieved in both models using 48,000 2nd orderquadrilateral mesh elements (see mesh in Fig. 3).

4. Validation

The validation exercise in this paper is designed to confirm thekey model assumptions (#4 and 5) in the context of representingtwo-dimensional heat conduction with only a one-dimensionalsolution. The validation also includes an analysis of the pseudosteady-state approach using the two-region fin model solutionssuggested in Section 2.

4.1. Validation results

A qualitative plot of the resulting two-dimensional laminateheat conduction simulation in COMSOL Multiphysics 4.2a is shownin Fig. 4. The red arrows represent the magnitude and direction ofconduction heat transfer and the dashed line represents the loca-tion of the heated region boundary. Although Fig. 4 shows the heatconduction results for only one of the finite element simulations, itconfirms the strong tendency for the transverse heat conduction in

ateral mesh in COMSOL Multiphysics 4.2a. Both axes are in mm.

Fig. 6. s vs. l constant heat flux validation of the 1D model (two-region fin model)with the two-dimensional numerical simulations using COMSOL Multiphysics 4.2a inCartesian coordinates.

Fig. 4. Arrow surface plot of two-dimensional heat conduction simulated in COMSOLMultiphysics.

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 55

the high thermal conductivity metal layer to dominate the overallheat transfer of the system, on which the two-region fin modelrelies.

Figs. 5e8 show the superimposed dimensionless temperatureprofiles, s, from the steady-state finite element simulations and thetwo-region fin model evaluated for 0.002 � L � 0.05 m, constanttemperature (Figs. 5 and 7) and constant heat flux finite heatsources (Figs. 6 and 8), for the Cartesian and cylindrical coordinatesystems. Root mean square (RMS) errors for Figs. 5 and 6 (Carte-sian) range from 8 � 10�4 to 4 � 10�3 in ascending order of L, whileRMS errors for Figs. 7 and 8 (cylindrical) range from 4 � 10�4 to3 � 10�3. While the RMS errors reported for Figs. 5e8 are small forthe profiles evaluated at L ¼ 0.05 m, the increasing trend doessuggest a very gradual departure from the fin region infiniteboundary condition with increasing L in the COMSOL simulations.Nonetheless, very good agreement is observed in all cases, thusachieving high accuracy validation for the results from the two-region fin model in the applicable conditions for assumption #4.

Validationwas also performed on the basis of heat accumulationin transient models in order to evaluate the argument presented inSection 2 for using the two-region finmodel in pseudo steady-stateanalysis. The values of the heat transfer rate in the finite elementmodel transient solution and the two-region finmodel steady-statesolution are used in Fig. 9 to identify the impact of heat accumu-lation on the two-region fin model predictions. The transient finiteelement model in Cartesian coordinates had a constant rate ofincreasing heated length ð _L ¼ 3� 10�3 m=minÞ, while the cylin-drical model had both a solution with the same imposed rate ofincreasing heated length ð _L ¼ 3� 10�3 m=minÞ and another witha constant rate of increasing heated area ð _A ¼ 1:5� 10�4 m2=minÞ.

It is only during the first interval, in either pane of Fig. 9, that adiscernable difference is observed between the steady-state heattransfer rate of the two-region fin model and the transient

Fig. 5. s vs. l constant temperature validation of the 1D model (two-region fin model)with the two-dimensional numerical simulations using COMSOL Multiphysics 4.2a inCartesian coordinates.

solutions in COMSOL, for which the logical reason is the need forheat accumulation for the system to depart from the initial condi-tions. For the results in cylindrical coordinates (Fig. 9b), however,the difference between the initial heat accumulation of the twosimulations (fixed _L and _A) and the absence thereof in the steady-state two-region fin model produces no noticeable effect. This islikely due to the small scale of the simulated heat transfer in thecylindrical case, whereas the Cartesian solution appears morepronounced because it is represented as a linear heat flux ratherthan the rate of heat exchange. Thus, high accuracy validation wasachieved for the valid use of the two-region fin model in pseudosteady-state analysis.

5. Results and discussion

The principal effect that sets apart the solutions in the two co-ordinate systems is the area for heat transfer in the fin region,which is far better enhanced in cylindrical coordinates compared toCartesian coordinates. This supposition is further proven in theanalyses of the limits of Q 0

fin (Eq. (22)) and Qfin (Eq. (36)) as theheated length, L, approaches infinity. In Cartesian coordinates, thelimit becomes

Fig. 7. s vs. l constant temperature validation of the 1D model (two-region fin model)with the two-dimensional numerical simulations using COMSOL Multiphysics 4.2a incylindrical coordinates.

Fig. 8. s vs. l constant heat flux validation of the 1D model (two-region fin model)with the two-dimensional numerical simulations using COMSOL Multiphysics 4.2a incylindrical coordinates.

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5956

Q 0fin;sat ¼ limL/NQ 0

fin ¼�gTinf þ abtanhðaNÞ

gþ atanhðaNÞ � Tinf

� ffiffiffiffiffiffiffiffiffik1t1R

r

(51)

and reduces to the following explicit expression

Q 0fin;sat ¼

�gTinf þ ab

gþ a� Tinf

� ffiffiffiffiffiffiffiffiffik1t1R

r: (52)

As for the limit in cylindrical coordinates,

Fig. 9. Validation of the two-region fin model with respect to the two-dimensionaltransient simulations obtained in COMSOL Multiphysics 4.2a and an applied con-stant temperature. a) Solutions in Cartesian coordinates and prescribed transient ofconstant _L ¼ 3� 10�3 m min�1; b) solutions in cylindrical coordinates and prescribedtransients _L ¼ 3� 10�3 m min�1 and _A ¼ 1:5� 10�4 m2 min�1.

Qfin;sat ¼ limL/NQfin

¼ k1t1pN�abI1ðNÞK0ðNÞ þ gTinfK1ðNÞI0ðNÞ

aI1ðNÞK0ðNÞ þ gK1ðNÞI0ðNÞ � Tinf

�

� gK1ðNÞK0ðNÞ ;

(53)

where it is known that I0(N) ¼ I1(N) and K0(N) ¼ K1(N), so thefinal expression can be given as

Qfin;sat ¼ k1t1pN�abþ gTinf

aþ g� Tinf

�g/N: (54)

In Eqs. (52) and (54), the subscript sat denotes the maximumtheoretical values. The value of Q 0

fin suffers diminishing returnswith every incremental increase in L since the limit exists (hencesat, meaning saturation), while Qfin shows no halt in its progression(due to the Lmultiplier in front). That is not to say that the value ofQfin does not also suffer diminishing returns with increasing L(constant first derivative with respect to L as L/N in Eq. (54)), butthe limit, Qfin;sat, does not exist. In either of the coordinate systems,the value of h does approach 1 with increasing L, as can be seen bythe curves representing h in the two coordinate systems for a fixedlaminate geometry (1 � 10�4 m thick: two 4 � 10�5 m thick poly-ethylene layers, and one 2 � 10�5 m thick Al foil layer) and a con-vection coefficient, h, equal to 25 W m�2 K�1 in Fig. 10. Thislaminate film is nominally representative of the laminate filmsfound inside Tetra Brik aseptic cartons. The axis variables in Fig. 10,excluding h, are normalized with respect to the values computed atL0 ¼ 1 � 10�3 m, being the first value in the data sets. Most inter-estingly, the fact that the curve representing Q 0

fin (shown as thenormalized ratio QfinðLÞ=QfinðL0Þ on the right axis in Fig. 10) in theupper panel shows a distinct plateau, which is the true manifes-tation of Q 0

fin;sat, while the lower panel confirms the linear increasein Qfin with increasing values of L (also normalized on the sameaxis) in cylindrical coordinates.

The non-dimensional temperature profiles obtained from theequations for constant temperature and constant heat flux sourcesin Cartesian coordinates in Table 3, with varying L, are each shownin Fig. 11. The non-dimensional temperature profiles, s, in Fig. 11represent the same laminate film composition and conditions asfor Fig. 10.

The solutions with smaller heated lengths, L, in Fig. 11a and bshow a greater extent of heat penetration into the fin regions

Fig. 10. Laminate heat transfer effectiveness for applied temperature heat source only:a) Cartesian coordinates and b) cylindrical coordinates. Lo is equal to 1 � 10�3 m.

Fig. 11. Non-dimensional temperature profiles, s vs. l, in Cartesian coordinates with varying L: a) constant temperature sources and b) constant heat flux sources.

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 57

relative to L, therefore emphasizing the observation from Fig. 10that the relative contribution from Q ’

fin to the overall rate of lami-nate heat transfer (h) diminishes at the larger values of L. Also notethat the indirect link between the Al temperature and constant heatflux sources is apparent in Fig. 11b in that the value of s in theheated region increases from l ¼ 1 to l ¼ 0. The farther away theleft edge of the heat source from the heated region boundary, the

Fig. 12. Non-dimensional temperature profiles, s vs. l, in cylindrical coordinates wit

more inaccessible the local heat flux at the far left edge becomes tothe fin region heat transfer. This condition begins to dominate theheated region when the s profile can be seen to reach a plateauapproaching l ¼ 0, as it does for L ¼ 5 � 10�2 m in Fig. 11b.

The non-dimensional temperature profiles for cylindrical co-ordinates from the equations in Table 4 for constant temperatureand constant heat flux sources, with varying L, are each shown in

h varying L: a) constant temperature sources and b) constant heat flux sources.

Fig. 13. Effectiveness, h, domain maps in dimensionless coordinates Ug and Uk forCartesian coordinates.

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e5958

Fig. 12. The laminate films represented in Fig. 12 have identicalcomposition and conditions to those Fig. 11.

The non-dimensional temperature profiles for cylindrical co-ordinates in Fig. 12 also show the same features related to theextent of heat penetration into the fin region at lower values of L aswell as the diminishing contribution of Qfin to the overall rate oflaminate heat transfer (h). The profiles in Fig. 12 also show that thecylindrical two-region fin model predicts a larger fin region influ-ence on the respective heated regions compared to the equivalentCartesian coordinate profiles in Fig. 11. Plateaux for s in the cylin-drical coordinate heated regions are largely suppressed, imposingstrong curvature instead, even in the profiles for constant tem-perature heat sources. This is largely due to the absence of a satu-ration limit for Qfin in cylindrical coordinates (Eq. (54)).

Fig. 14. Effectiveness, h, domain maps in dimensionless coordinates Ug and Uk forcylindrical coordinates.

Note that in the two types of profiles, and the respective coor-dinate systems in Figs. 11 and 12, the objective with the q00o heatsource is temperature uniformity while that of the Ti heat source isa greater rate of heat transfer. Temperature uniformity appearseasiest to achievewith rectangular heat sources rather than circularones (Fig.11b has a greater extension of s in the fin region, l� 1, andlower peak values in the heated region, 0 � l � 1 than doesFig. 12b). An increased rate of heat transfer is most easily accom-plished with circular heat sources since larger magnitudes of slopeat l ¼ 1 in cylindrical coordinates than those in Cartesian co-ordinates for the same values of L indicate greater capacities forheat transfer in the fin region. Specifically, the profile slopes at theinflection point, l ¼ 1, are equivalent to their magnitudes of dT/dxor dT/dr in dimensional coordinates, therefore representing theirmagnitudes of Q ’

fin and Qfin from Eqs. (22) and (36), respectively.In design, however, parameter selection is usually dictated by

the limiting cases. So, good long-term heat storage PCM encapsu-lation would have to improve heat transfer rates for rectangularexpanding crystal fronts (always performing better for circularexpanding crystal fronts) and a multi-ply cook-pot would have toimprove temperature uniformity for circular wound heating ele-ments (the element portions in the middle having the strongestcurvature while the outside portion would begin to resemblerectangular elements).

In the cases where the applied finite heat source is a heat fluxdevice, obtaining uniform dimensionless temperature profiles inthe heated region and far reaching profiles in the fin region are theobjectives and plots such as in Figs. 11 and 12 should be studied.In the case of salt hydrate encapsulation, the design metric is h,which should be maximized to a practical extent using the plots inFigs. 13 and 14 that cover large sections of the possible solutiondomain in dimensionless variables (Ug and Uk) in order to select adesign basis. Design constraints would either restrict the value ofUg (Eq. (49) relating the path length of heat spreading, mostimportantly the selection of L, t1, and k1) or Uk (Eq. (50) relating tothe thermally resistive layers and convective cooling, so R, t3, andk3), for which a suitable range of values of the unconstrained var-iable would be selected to yield the desired values of h, thenobtaining the exact parameters from the variable definitions in Eqs.(49) and (50). For instance, using the Tetra Brik laminate film thatwas used in the finite element model, but with h ¼ 50 W m�2 K�1

(Uk ¼ 210), one could expect h between 10.6 and 1.19 in Cartesiancoordinates and 58 and 1.42 in cylindrical coordinates, respectively,both for 1 � 10�3 m � L � 50 � 10�3 m (0.0093 � Ug � 23).

Pertaining only to the results presented for q00o finite heat sour-ces, one can consider applying the two-region fin model to thedesign of laminate systems of several layers of metals, or polymersand metals. The model would remain valid so long as there is asingle inner layer, or a contiguous group of inner layers, withsignificantly higher thermal conductivity than all the others suchthat assumption #4 still holds. All adjacent layers that are non-heatspreading (Rplanar � Rtransv) can be combined into a lumped resis-tance (R) for the top layer and a lumped conductivity for the bottomlayer (k3).

6. Conclusion

Validation of the governing principles of the two-region finmodel was achieved and demonstrated for use in the design cal-culations of salt hydrate encapsulation for long-term heat storage,multi-ply cookware, and even preliminary heat transfer calcula-tions for electronics cooling devices. The two-region fin modelrepresents the unique heat transfer attributes that are manifestedby planar systems composed of laminated dissimilar materials(with great emphasis on widely dissimilar thermal conductivities)

L. Desgrosseilliers et al. / International Journal of Thermal Sciences 72 (2013) 47e59 59

with applied finite heat sources (either temperature or heat flux)over one of the planar surfaces and convective cooling on theremaining surface. The model equations are all given explicitly, oneneed only solve algebraic equations in either their dimensional ordimensionless forms.

The two-region fin model allows engineers and scientists toassess the benefit of heat spreading laminate materials (e.g., poly-ethylene/aluminium or polyester/aluminium pouches for salt hy-drate PCM encapsulation) on the performance enhancement oflong-term heat storage salt hydrate PCMs, multiply cookware, orelectronics cooling devices. For the latter two, only the temperatureprofiles (dimensional or non-dimensional) are needed to deter-mine the degree of temperature uniformity beyond the heatedregion. For the former, the parameter, h, introduced in this paperprovides the necessary metric to determine the degree of heattransfer enhancement. The range of h is greatest with decreasingthermal resistance to heat spreading to the fin region andincreasing resistance to planar thermal conductivity. The limitingcase is always the design constraint, meaning that curved heatsources promote less temperature uniformity than do rectangularones, and the opposite it true for h.

The two-region fin model has demonstrated that the poly-ethylene/aluminium laminate film found in Tetra Brik aseptic car-tons could still promote between 960% and 19% greater heattransfer for rectangular heat sources and between 5700% and 42%for circular heat sources, both for 1 � 10�3 m � L � 50 � 10�3 m.

Acknowledgements

The authors are thankful for financial contributions from theNatural Sciences and Engineering Research Council of Canada(NSERC), Resource Recovery Fund Board (RRFB) of Nova Scotia,Dalhousie Research in Energy, Advanced Materials and Sustain-ability (DREAMS), and the Canada Foundation for Innovation (CFI).

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