ORIGINAL

Thermographic validation of a novel, laminate body, analyticalheat conduction model

Louis Desgrosseilliers • Dominic Groulx •

Mary Anne White

Received: 19 March 2013 / Accepted: 14 January 2014 / Published online: 4 February 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract The two-region fin model captures the heat

spreading behaviour in multilayered composite bodies (i.e.,

laminates), heated only over a small part of their domains

(finite heat source), where there is an inner layer that has a

substantial capacity for heat conduction parallel to the heat

exchange surface (convection cooling). This resulting heat

conduction behaviour improves the overall heat transfer

process when compared to heat conduction in homoge-

neous bodies. Long-term heat storage using supercooling

salt hydrate phase change materials, stovetop cookware,

and electronics cooling applications could all benefit from

this kind of heat-spreading in laminates. Experiments using

laminate films reclaimed from post-consumer Tetra Brik

cartons were conducted with thin rectangular and circular

heaters to confirm the laminate body, steady-state, heat

conduction behaviour predicted by the two-region fin

model. Medium to high accuracy experimental validation

of the two-region fin model was achieved in Cartesian and

cylindrical coordinates for forced external convection and

natural convection, the latter for Cartesian only. These

were conducted using constant heat flux finite heat source

temperature profiles that were measured by infrared

thermography. This validation is also deemed valid for

constant temperature heat sources.

List of symbols

Dimensional variables

h Convection heat transfer coefficient (W m-2 K-1)

k Thermal conductivity (W m-1 K-1)

L Heated boundary length and boundary-edge position

from x, r = 0 (m)

Q Total rate of heat transfer (W)

q00o Applied finite heat flux (W m-2)

r Radial position (m)

R Thermal resistance above the highly conductive

metal core (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 = L (K)

x Cartesian axial position (m)

z Vertical axis position (m)

Greek symbols

a Heated region constant (m-1)

b Heated region particular solution (K)

DT Temperature uncertainty 95 % confidence limit (K)

c Fin region constant (m-1)

Subscripts

1 High conductivity metal core

2 Top thermally resistive layer

3 Bottom thermally resistive layer

L. Desgrosseilliers � D. Groulx (&)

Department of Mechanical Engineering, Dalhousie University,

PO Box 15000, Halifax, NS B3H 4R2, Canada

e-mail: dominic.groulx@dal.ca

L. Desgrosseilliers � D. Groulx � M. A. White

Institute for Research in Materials, Dalhousie University,

PO Box 15000, Halifax, NS B3H 4R2, Canada

M. A. White

Department of Chemistry, Dalhousie University,

PO Box 15000, Halifax, NS B3H 4R2, Canada

123

Heat Mass Transfer (2014) 50:895–905

DOI 10.1007/s00231-014-1295-3

f Fin region only

h Heated region only

meas Based on measurement data

model Based on two-region fin model results

1 Introduction

Advances are currently needed in all areas of fundamental

phase change material (PCM) sciences with regards to

supercooling salt hydrates in order to support the design

and eventual commercialization of long-term thermal

energy storage systems. The liquid phase of many common

salt hydrates (e.g., sodium acetate trihydrate—NaCH3-

CO2�3H2O) have been found to persist below their equi-

librium solidification temperature in a metastable state,

called supercooled salt hydrate solution, and allows for the

practical storage of much of the latent heat of transition at

ambient temperatures [1–4]. Solidification triggering

mechanisms enable the heat withdrawal on demand at

which point the heat discharge process proceeds at the rate

of external heat transfer, but with each individual solidifi-

cation site generating an isolated heat source within the

bulk (i.e., non-uniform heat sources) [1, 2, 5].

While there are no standing heat losses in the super-

cooled state at ambient temperatures, heat losses are nee-

ded to achieve supercooling from the higher temperature

liquid phase. Unlike other PCMs, the accessible amount of

heat storage in supercooling salt hydrates is directly

affected by their enthalpy histories [2, 6]. The thermody-

namic and physical properties of some important super-

cooling salt hydrates have been examined for more than

100 years [2, 3, 7–11].

The encapsulation technologies and heat transfer

improvements belonging to conventional, non-supercool-

ing, PCM heat storage technologies are ill-equipped for the

emerging use of supercooling salt hydrate PCMs [12, 13].

Laminate film encapsulations, typically composed of thin

layers of polyethylene or polyester (Mylar�) enclosing a

middle layer of Al foil, are one such encapsulation type

well-suited to supercooling salt hydrate PCMs. They are

lightweight (minimal thermal mass), flexible, provide

excellent barriers to light, moisture and oxygen, and can

work well to incorporate electromechanical solidification

triggers [12, 14–16].

Lane outlines the fundamental criteria for PCM encap-

sulation [15] and discusses macro-encapsulation options

for inorganic (salt hydrates) and organic (paraffins, fatty

acids), non-supercooling, PCMs [14, 16]. Lane has exam-

ined the properties of commercial polymeric bottles, lam-

inate pouches, and metal cans and tested their use with a

variety of non-supercooling PCMs [13, 15]. A suitable

PCM encapsulation must address mechanical durability

and integrity, adequate heat transfer rates, vapour imper-

meability, and chemical compatibility [14, 15].

Supercooling PCMs require an additional encapsulation

criterion to those for non-supercooling PCMs, whereby the

selected encapsulation technology must accommodate a

solidification triggering mechanism to act on the PCM

contained within [12]. The inherent flexibility of laminate

films enables external actuation of a solidification trigger-

ing device as well as leak-tightness around any necessary

perforations for solidification actuation. A heat transfer

improvement aiding the solidification of supercooling salt

hydrate PCMs (e.g., non-uniform heat sources) that is made

possible when using laminate film encapsulation has been

identified in [12, 13].

Desgrosseilliers et al. [13] have proposed a simple to

use, one-dimensional, steady-state, laminate heat conduc-

tion model to predict the two-dimensional, non-uniformly

heated, heat conduction behaviour that would arise from

using laminate film encapsulation for supercooling PCMs

in order to improve their heat discharge rates during

solidification. Desgrosseilliers et al. [13] developed a new

laminate heat conduction model (called ‘two-region fin

model’) since other models in the literature, as discussed in

their paper, were found to be ill-suited for the geometry,

composition, boundary conditions, and objectives of lam-

inate film, supercooling PCM encapsulation heat conduc-

tion and would be too impractical to adapt to the desired

application. The handbook by Kraus, Aziz, and Welty

discusses the effect of cladding and fouling on high thermal

conductivity metal cores to longitudinal fins and spines, but

these used only heating at the base (excluding the effect of

non-uniform heating transverse to the fin) and their con-

clusions relate only to large temperature difference envi-

ronments occurring in gas turbine systems and heat pumps

(DT [ 100 �C) [17].

The two-region fin model for laminate film heat con-

duction describes the qualities enabling heat transfer

improvements to the supercooling salt hydrate PCM

solidification process (Fig. 1) [13]. The two-region fin

model derivation focuses on the lateral heat conduction

(planar heat conduction) component in the inner layer of Al

foil that dominates the majority of the system’s heat

transport from a finite heat source to an outside heat sink

occupying the entire heat-exchange surface. The resulting

one-dimensional, steady-state, laminate heat conduction

model uses only explicit inputs and provides, straightfor-

ward, non-iterative solutions to the laminate body’s tem-

perature and heat flux profiles in both the heated (0 B x,

r B L in Fig. 1) and fin regions (x, r C L in Fig. 1).

The two-region fin model derivation in Cartesian and

cylindrical coordinates, for both constant temperature (e.g.,

PCM) and constant heat flux (e.g., stovetop cookware and

electronics cooling) finite heat sources is described in [13].

896 Heat Mass Transfer (2014) 50:895–905

123

The numerical validations of the one-dimensional heat

conduction and pseudo-steady-state approximations using a

transient, two-dimensional, laminate body heat conduction

finite element model are included in [13] in both Cartesian

and cylindrical coordinates with constant temperature and

constant heat flux finite heat sources.

This paper reports on the experimental, steady-state,

validation of the two-region fin model performed using

infrared (IR) thermography and constant heat flux finite

heat sources. These validation experiments evaluated the

heat transfer mechanisms in laminate films recovered by

hydropulping post-consumer Tetra Brik cartons. Experi-

ments were conducted under conditions of natural con-

vection and forced external convection for cooling the top

surface of the laminate while also heating only a fraction of

the bottom surface with either thin rectangular heaters or

thin circular heaters. Constant temperature validation was

not deemed necessary since such finite heat sources pro-

duce temperature profiles in the heated region far less rich

in defining features than their constant heat flux counter-

parts [13]. Furthermore, the fin region temperature profiles

are independent of the nature of the heat source [13].

2 Analytical model

The two-region fin model equations under evaluation in

both Cartesian and cylindrical coordinates are listed in

Table 1. The variables T, Tinf, x, r, L, k, and h refer to the

middle layer Al foil temperature, free-stream temperature,

axial positions (x and r), heated boundary length, thermal

conductivities of the individual laminate body layers, and

convection heat transfer coefficient in Fig. 1. The remain-

ing variables in Table 1, To, R, a, c, and b, represent the

heated boundary temperature T (x = L), combined thermal

resistance above the metallic foil layer, heated and fin

region space constants, and the hypothetical temperature at

the origin (x or r = 0), respectively. I and K represent

modified Bessel functions of order zero or one used for the

solutions of second order differential equations in cylin-

drical coordinates.

Under conditions of non-uniform convection between

the heated and fin regions, the values of h differ in the

calculations of R in the heated and fin regions, and sub-

sequently a and c [13]. In these cases, h becomes hh and hf

in the heated and fin regions, as well, R becomes Rh and Rf,

respectively.

3 Experimental methods

Constant heat flux finite heat sources (q00o in Table 1) were

simulated in steady-state experiments using rectangular,

flexible silicone rubber, fibreglass insulated heaters and

circular, thin Kapton� heaters from Omega Engineering

Inc. (controlled by a STACO 0–140VAC VARIAC,

±0.5 V), resting plainly behind a flattened laminate film

secured to a sheet of 1’’ thick polystyrene foam insulation

(backing and border clamped thereto). Figure 2 shows this

arrangement for mounting laminate films for the constant

heat flux experiments.

The laminate film was put in place on the foam sheet

first using electrical tape along its edges (trapping only a

very thin layer of air underneath the laminate film), then

clamped in place with the foam border (the laminate film

slightly exceeding the inner dimensions of the foam border,

but not protruding from the mounting frame). The exposed

laminate surface was allowed to exchange heat through

either natural or external forced convection.

Adhesive T-type thermocouples (nominal 25.4 lm

diameter wire) affixed to the laminate film’s surface

recorded the surface temperatures via a NI CompactDAQ

and LabView 2011 in order to calibrate IR thermographic

still images from an Indigo Merlin IR camera. Image

intensities (converted from RGB to grayscale using the

Image Processing Toolbox in MatLab R2011a) along cut

lines were used to construct temperature profiles after

calibration (see Fig. 3). Each IR thermographic still image

used for analysis required individual calibration with the

corresponding thermocouple data.

Thermocouples adhered to the surface of the laminate

distorted local temperature fields (adding thermal resistance

greater than that of the film), so they were kept along the

outer edges of the film. Figure 4 shows the arrangement of

thermocouples and their impact on the local temperature

fields. Experimental validation was performed for rectan-

gular heat sources of constant width equal to 20.3 cm (see

Fig. 4), heated lengths (L, vertical length in Fig. 4) 25.4,

Fig. 1 Two-region fin model schematic; the heated region is left of

the dashed line (x or r = L), while the fin region is to the right.

Reproduced with permission from [13] � Elsevier Masson SAS. All

rights reserved

Heat Mass Transfer (2014) 50:895–905 897

123

50.8, and 76.2 mm and rated power density of 0.39 W cm-2.

Circular heat source validation was performed using 50.8

and 76.2 mm diameter (2L) heaters and rated power densi-

ties of 1.6 and 0.39 W cm-2, respectively.

Images recorded from the IR camera (see Fig. 4)

revealed the temperature profile of the inner layer of Al foil

since the top layer of low-density polyethylene is invisible

to IR radiation. This combination of optical properties

allowed IR thermographic data to relate directly to the

steady-state temperature profiles predicted by the two-

region fin model equations in Table 1. Al foils have low

emissivity and high reflectivity, so measurements were

conducted under reduced ambient light and with no heat or

light source directly in front of the laminate film under

evaluation.

Only one rectangular heater experiment and all the cir-

cular heater experiments were performed using external,

impinging forced convection. Natural convection could not

be used in the evaluation of circular heaters since radial

uniformity of the system could not be ensured in this

condition (one-dimensional conduction condition). Airflow

for forced cooling was supplied by a duct fan positioned

nearly perpendicular to the heated film to avoid obstructing

the IR camera.

The laminate films under examination were all sourced

from used aseptic cartons, in this case 1 and 2 L Tetra Brik,

which were separated from the outer paperboard and

polyethylene layers by water immersion and light agitation/

rubbing at room temperature for 35 min. This process

mimics high consistency hydropulping carried out in paper

mills to recycle the paperboard from cartons. Other sam-

ples were obtained from the high consistency hydropulper

at Klabin’s paper mill in Piracicaba, Brazil. Only laminate

films obtained by in-house hydropulping were used in the

validation experiments, but both these and the films pro-

cessed at Klabin’s mill in Brazil were used to measure the

mean thermal conductivity of post-consumer aseptic carton

laminate films.

The thermal conductivity of the laminate films measured

at 30 �C in a Mathis Instruments TC30 was

0.41 ± 0.05 W m-1 K-1. Laminate films from Tetra Brik

cartons are composed of two outer layers of 40 lm poly-

ethylene (0.33–0.46 W m-1 K-1 [18]) and a 20 lm layer of

Al foil (260 W m-1 K-1 [19] ) [12, 13], for which the

calculated, series, thermal conductivity is

0.41–0.57 W m-1 K-1, therefore in agreement with the

measurements done by TC30. The TC30 determines the

thermal conductivity of film or block shaped samples at

ambient pressure and 30 �C using a modified hot-wire

Table 1 Two-region fin model

equations in Cartesian and

cylindrical coordinates [13]

Parameter Cartesian Cylindrical

T, 0 B x, r B LTðxÞ ¼ To � bð Þ cosh axð Þ

cosh aLð Þ þ b TðrÞ ¼ To � bð Þ I0 arð ÞI0 aLð Þ þ b

T, x, r C L TðxÞ ¼ To � Tinf

� �e�c x�Lð Þ þ Tinf TðrÞ ¼ To � Tinf

� � K0 crð ÞK0 cLð Þ þ Tinf

ToTo ¼

cTinf þ abtanh aLð Þcþ atanh aLð Þ To ¼

abI1 aLð ÞK0 cLð Þ þ cTinf K1 cLð ÞI0 aLð ÞaI1 aLð ÞK0 cLð Þ þ cK1 cLð ÞI0 aLð Þ

RR ¼ t2

k2

þ 1

h

aa2 ¼ 1

Rt1k1

cc2 ¼ 1

Rt1k1

b b ¼ Tinf þ Rq00o

Fig. 2 Laminate film test mount schematics: a mounted laminate

with rectangular heater; b mounted laminate with circular heater

898 Heat Mass Transfer (2014) 50:895–905

123

technique. This is accomplished by measuring the interfacial

temperature rise and comparing these results to calibrations

from a series of standards.

For simplicity and to account for temperature variations

occurring in the laminate heat conduction validation

experiments, an intermediate thermal conductivity value of

0.4 W m-1 K-1 was chosen to represent the layers of

polyethylene in the two-region fin model calculations.

4 Results and discussion

The experimental validation in this paper compared one-

dimensional temperature profiles extracted from steady-

state, calibrated, IR-thermographic images to those pre-

dicted by the two-region fin model in Sect. 2, Table 1. The

laminate heat conduction experiments (constant heat flux/

finite heat source) were performed under conditions of

natural and forced convection, using rectangular and cir-

cular heaters.

The model predictions are obtained by adjusting the

value of the heat transfer coefficient in both the heated

and unheated regions, hh and hf, separately (see Sect. 2),

in a least-squares method of best fit. Although hh and hf

are not themselves predicted, since they vary in the sys-

tem due to the local intensity of natural or forced con-

vection, the model assumes hh and hf are uniform over

each of their respective regions and makes no attempt to

determine them from equations. Obtaining hh and hf by

best fit still satisfies the energy balance and avoids further

error from convection correlations, therefore providing a

better evaluation of temperature, as well as compensating

for inaccuracies in the determination of heat flux.

Fig. 3 Calibration plot of IR

thermographic grayscale

intensity (pixel intensity) and

thermocouple temperature

measurements. This calibration

corresponds to measurements

for an L = 76.2 mm naturally

cooled rectangular heater

operated at 0.059 W cm-2 and

gave an R2 value of 0.9974

Fig. 4 IR thermographic image showing thermocouples on Tetra Pak

laminate film using an L = 50.8 mm rectangular heater. The cross-

hairs and numbers at the top of the image were from internally

calibrated profiles unrelated to the experiment

Fig. 5 Experimental validation of the two-region fin model in

Cartesian coordinates and subject only to natural convection. The

corresponding free stream temperatures, Tinf, are 22.74, 22.58, and

24.28 �C for the L = 25.4, 50.8, and 76.2 mm rectangular heaters,

respectively, for which all heaters were operated at 0.059 W cm-2

Heat Mass Transfer (2014) 50:895–905 899

123

Undervaluation of the measured heat flux would cause the

least-squares determination of hh and hf to predict lower

values of the convection coefficients in order to satisfy

the imposed energy balance in the two-region fin model,

while the opposite would occur due to overvaluation of

the measured heat flux.

4.1 Rectangular heat sources (natural convection)

The temperature profiles measured by IR thermography for

the Al foil of the laminate pouches, naturally cooled and

with heaters operated at 0.059 W cm-2, are shown in

Fig. 5. Predictions from the two-region fin model are also

shown in Fig. 5 for the same experimental conditions. Note

that sudden spikes in any of the temperature profiles

measured by IR thermography are due to the presence of

residual paperboard fibers in the cut-lines from the images,

which have significantly different optical properties (higher

emissivity) than does the Al foil being measured.

The samples in Fig. 5 were mounted vertically, with the

heat source on the bottom, therefore natural convection

favoured the lowest regions, promoting lower temperatures

near x = 0 and higher temperatures in the farther reaches

of the heated region. Consequently, this also imposed low

convective cooling in the unheated region.

As is seen in the case of the rectangular heater with

L = 76.2 mm in Fig. 5, and to a lesser extent for the heater

with L = 50.8 mm, it appears the vertical laminate sur-

faces are insufficiently long to cool completely to their

corresponding ambient temperatures at the end of the fin

regions (see values of Tinf in the Fig. 5 caption). These

conditions indicate a departure from the two-region fin

model’s assumption that the fin region temperature

approaches Tinf far from the heated region. Instead, they

reveal that the laminate surfaces were subject to an

increasingly important insulated boundary condition at the

end of their fin regions (greater for L = 76.2 mm than

L = 50.8 mm). Nonetheless, the two experimental profiles

shown in Fig. 5 do not deviate too sharply from the con-

ditions anticipated by the two-region fin model and are

therefore still deemed valid for the present validation

exercises. Insulated boundary conditions in the fin regions

are the topic of continued work to expand the two-region

fin model’s analytical repertoire as well as further numer-

ical and experimental validations. It is expected that the

evaluation of the two-region fin model with an insulated

boundary condition in the fin region would merely result in

a larger value of hf and a marginally better fit.

Table 2 shows the values of the natural convection heat

transfer coefficients for the model results in Fig. 5. The

highest values are naturally in the heated regions, hh’s, and

the highest value overall was found for the heater with the

smallest heated length, L, which corresponds well with

natural convection theory for vertical cooled surfaces with

constant heat flux whereby h is inversely proportional to

the characteristic length of the system. As expected, the

overall rate of heat transfer increases with the size of the

system (Q ¼ L� 20:3 cm� 0:059 W cm�2), despite

decreasing values of h. The natural convection heat

transfer coefficients listed in Table 2 are also in the correct

order of magnitude for natural convection in air at ambient

conditions.

The temperature measurements in the heated regions in

Fig. 5 (x B L) were subject to the calibration uncertainties

(DTmeas) listed in Table 3. IR calibrations in Table 3

improved at the cooler temperatures of the unheated region

(x C L). Error propagations from the two-region fin model

(DTmodel), due to position, AC voltage and temperature

uncertainties are also found in Table 3 for the entire one-

dimensional domain. As shown for the rectangular heater

Table 2 Two-region fin model natural convection heat transfer

coefficients for the profiles in Fig. 5

L (mm) hh (W m-2 K-1) hf (W m-2 K-1)

25.4 52.9 2.72

50.8 35.6 1.22

76.2 43.2 1.03

Table 3 Temperature measurement uncertainties for the IR thermo-

graphic profiles (95 % confidence intervals) and calculated error

propagation for the two-region fin model profiles in Fig. 5

L (mm) DTmeas (�C) DTmodel (�C)

x B L x C L x C 0

25.4 1.9–2.0 1.8–1.9 0.51–0.66

50.8 1.5–1.8 1.0–1.8 0.60–0.83

76.2 1.3–2.0 0.5–1.3 0.57–0.75

Fig. 6 Plot of experimental uncertainties and error propagation for

the L = 50.8 mm rectangular heater in Fig. 5

900 Heat Mass Transfer (2014) 50:895–905

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with L = 50.8 mm in Fig. 6, the predicted temperature

profiles matched closely with those from IR measurements

and predictions were contained well within the measured

temperature profile uncertainties (typical for all cases,

including forced convection).

Overall validation of the Cartesian coordinates two-

region model with natural convection was achieved with

medium to high accuracy with respect to the appearance of

good two-region fin model fit to the features of the

experimental data, obtaining values of hh and hf in the

correct order of magnitude, and the width of the overlap-

ping model error propagations by measurement uncertain-

ties. Specifically, high accuracy validation (i.e., good

appearance of fit as well as high accuracy of temperature

predictions) was achieved for the two-region fin model

predictions of the fin region temperatures, while medium

accuracy (i.e., good appearance of fit but lacking accuracy

of temperature predictions) was achieved for the heated

region two-region fin model predictions. These validation

criteria formed the bases for all of the subsequent valida-

tion exercises in this paper.

4.2 Rectangular heat sources (forced convection)

Laminate heat conduction with a rectangular heater and

cooled by forced convection was evaluated for the 76.2 mm

heater only. The result, shown in Fig. 7, was intended to

produce a result more closely related to the conditions of

channel flow as would be encountered in cases of long-term

PCM heat storage. However, the method of exerting external

forced convection normal to the laminate’s surface could not

ensure complete homogeneity of the convection coefficient

so the model fit for the two-region fin model was still

performed using hh and hf. The values determined by

least-squares were hh = 65.8 W m-2 K-1 and hf =

14.0 W m-2 K-1, clearly showing that convective cooling

was more effective under external forced convection than

natural convection (see Table 2).

The most apparent feature, or lack thereof, of the one-

dimensional temperature profile with forced convection in

Fig. 7 is the improved uniformity in the heated region.

Forced convection induces greater confinement of the

boundary layer flow next to the polystyrene frame pro-

truding around the sample, thus abating the profile inho-

mogeneity always present near x = 0 for natural

convection (see Figs. 5 and 7). Consequently, the two-

region fin model predictions in Fig. 7 produce a better fit in

the heated region under forced convection than under

natural convection. The fit to the descending slope in the

fin region also appears to have improved due to forced

convection, but the lower plateau of the forced convection

profile diverges and remains at a higher temperature than

the ambient temperature measured directly by thermocou-

ple (see Tinf in Fig. 7 caption). It was noted, however, that

this disagreement could be eliminated by substituting the

far-end temperature from the measured profile with forced

convection for the ambient temperature recorded separately

by thermocouple.

It is likely the latter was adversely affected by the lower

quality of the IR-thermocouple calibration, being subject to

large uncertainties, around 2.8 �C over the entire forced

convection profile domain in Fig. 7. A principal reason was

the sharp decline in thermocouple temperatures from the

heated region to the fin region, such that intermediate

temperatures could not be recorded and used in the cali-

bration. Instead, the calibration (Fig. 8) relied most heavily

on the domain limit values measured under forced con-

vection. The two-region fin model error propagation was

minor compared to the IR-thermocouple calibration

uncertainty, only ranging between 0.50 and 0.58 �C.

The model prediction therefore remained within the

region of measurement uncertainty. Two-region fin model

validation for forced convection in Cartesian coordinates

was achieved with medium to high accuracy, although

having greater accuracy in the heated region than was

achieved for natural convection.

4.3 Cylindrical heat sources

Temperature profiles for the circular heaters

(2L = 50.8 and 76.2 mm) with forced external convection

necessitated judgments to correctly ascertain the true

diameter of the heater behind the laminate film in the IR

thermographs. The direction of the one-dimensional heat

conduction in the case of rectangular heaters could be

easily inferred from the heater orientation to the vertical,

but could not be ascertained for the circular heaters (see

Fig. 2). The location of the heaters’ diameter at 0̊ from the

Fig. 7 Experimental validation of the two-region fin model in

Cartesian coordinates for the L = 76.2 mm heater only, operated at

0.059 W cm-2, and with either natural (Tinf = 24.28 �C) or external

forced convection (Tinf = 21.05 �C)

Heat Mass Transfer (2014) 50:895–905 901

123

horizontal needed to be determined manually since IR

thermography does not reveal physical features very well.

Figure 9 shows thermographs of the two sizes of heaters

used for validation, and drawing the largest possible

horizontal line across each heated circle indicated the

closest diameter that could be extracted from the images.

The profiles were fit to the models using the radii, so only

half the cut line could be used to extract a profile.

Fig. 8 Calibration plot of IR

thermographic grayscale

intensity (pixel intensity) and

thermocouple temperature

measurements. This calibration

corresponds to measurements

for an L = 76.2 mm rectangular

heater operated at

0.059 W cm-2 under external

forced convection and gave an

R2 value of 0.9974

Fig. 9 IR thermographic

images of the 2L = 50.8 mm

(left) and 76.2 mm (right)

circular heaters with external

forced convection

Fig. 10 Experimental

validation of the two-region fin

model in cylindrical coordinates

for 2L = 50.8 mm heater using

two adjustable heat transfer

coefficients to achieve best fit in

conditions of forced convection.

Heaters were operated at 0.11

and 0.19 W cm-2, as indicated,

for which Tinf = 22.20 and

22.04 �C, respectively

902 Heat Mass Transfer (2014) 50:895–905

123

The results of the temperature profiles that were mea-

sured for the 2L = 50.8 mm circular heaters at 0.11 and

0.19 W cm-2 as well as the model predictions obtained by

the method of least squares, still using the separate values

of hh and hf, are shown in Fig. 10.

Table 4 shows the values of hh and hf for the two-region fin

model in Fig. 10. Their values are much greater than they were

for natural convection in Table 2 and still in the correct order

of magnitude for external forced convection with air at

ambient conditions. The convection coefficients in Table 4

are also larger than what was obtained for forced convection in

Sect. 4.2, which was simply the result of rectangular heater

misalignment with the impinging fan stream made difficult by

the proximity of the polystyrene boarder to the heater.

Note, however, that the values listed for the heater

operated at 0.19 W cm-2 in Table 4 appear counter intui-

tive, since hf is larger than hh, but they are nonetheless in

the correct order of magnitude overall. This apparent error

in the determination of hf and hh for the heater operated at

0.19 W cm-2 was likely caused by a minor overestimation

of the heated region boundary position, r = L input to the

model. This effect is seen in Fig. 10 from the slight over-

prediction of the two-region fin model temperatures in the

heated region approaching r = L and under-predictions in

the fin region just past r = L. Another reason for the dis-

crepancy could be the fact that the temperature profile for

the heater operated at 0.19 W cm-2 in Fig. 10 was prone to

greater measurement uncertainty than for the heater profile

at 0.11 W cm-2.

The circular heaters in Fig. 10 were all placed in the

center of the sample laminate film mounted on polystyrene

foam, so flow characteristics arising from edges next to the

heated areas did not interfere with their thermographic

measurements. Just as for the rectangular heater measured

under external forced convection, forced convection resulted

in uniformly smooth temperature profiles. However, the use

of circular heaters increased the magnitude of slope of the

temperature profiles in the heated region compared to those

from rectangular heaters—resembling the shape of domes

rather than plateaux. Clearly, the radially increasing Al foil

cross-section has a measurable impact on the fin region heat

transfer, affecting the heated region acutely.

The quality of the fit in Fig. 10 was noticeably better for

the heater operated at 0.19 W cm-2 since the higher tem-

peratures gave increased resolution of the temperature

profile over the measurement uncertainty, making the slope

easier to fit with the model. Both the heater powered at 0.11

and 0.19 W cm-2 had model error propagations that were

bound by their measurement uncertainties (Table 5).

The temperature profiles measured from 2L = 76.2 mm

circular heaters at 0.12, 0.14, and 0.19 W cm-2 are shown

in Fig. 11.

Table 6 shows the values of hh and hf for the two-region fin

model in Fig. 11. Once again, the values of hh and hf are in the

correct order of magnitude for external forced convection with

air at ambient conditions; in fact, they are very close to the

values listed in Table 4 under similar conditions.

The high quality of fit to each measured temperature

profile in Fig. 11 was uniformly achieved, adapting very

Table 4 Two-region fin model external forced convection (imping-

ing flow) heat transfer coefficients for the model profiles in Fig. 10

P (W cm-2) hh (W m-2 K-1) hf (W m-2 K-1)

0.11 138 69.8

0.19 96.3 187

Table 5 Measurement uncertainties for the temperature profiles

measured by IR thermography (95 % confidence intervals) and cal-

culated error propagation for the two-region fin model profiles in

Fig. 10

P (W cm-2) DTmeas (�C) DTmodel (�C)

r B L r C L r C 0

0.11 1.4–2.9 0.50–0.62

0.19 2.6–5.2 1.9–2.6 0.50–1.2

Fig. 11 Experimental validation of the two-region fin model in

cylindrical coordinates for 2L = 76.2 mm heater using two adjustable

heat transfer coefficients to achieve best fit in conditions of forced

convection. Heaters were operated at 0.12, 0.14 and 0.19 W cm-2, as

indicated, for which Tinf = 21.73, 21.82 and 21.92 �C, respectively

Table 6 Two-region fin model external forced convection (imping-

ing flow) heat transfer coefficients for the model profiles in Fig. 11

P (W cm-2) hh (W m-2 K-1) hf (W m-2 K-1)

0.12 126 85.3

0.14 123 27.7

0.19 151 61.3

Heat Mass Transfer (2014) 50:895–905 903

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well to the changing features with increasing heater power.

The domes of the heated regions retained much of the

curvature that was noted in Fig. 10, but these were all

flatter at the peaks, obviously an important feature of the

larger circular heaters, but still indicative of the importance

of radially increasing cross-sectional area for conduction

heat transfer. To the contrary, when comparing the tem-

perature profiles of the forced convection rectangular

heater experiment (Fig. 7) to that of the circular heaters

(Figs. 10 and 11), it becomes apparent that heat penetration

in the fin region is always more acute with rectangular

heaters. This confirms the finding of [13] that temperature

uniformity in a laminate body can be more easily achieved

using rectangular heat sources than with circular ones,

while circular heat sources achieve more efficient heat

dissipation.

Note that the complete profile measured at

0.14 W cm-2 was difficult to ascertain; hence there is an

absence of measurements between 0 and 5 mm in Fig. 11.

Nonetheless, this did not diminish the confidence to assess

validation.

Measurement uncertainties and error propagations listed

in Table 7 show that the model profiles were bound by the

experimental profiles once again. The overall validation of

the two-region fin model in cylindrical coordinates was

observed as medium to high accuracy and equally applies

to cases of constant temperature, as was stated in Sect. 1.

5 Conclusion

Non-uniform heating experiments were conducted using

thin rectangular heaters and thin circular heaters applied to

thin laminate films composed of low-density polyethylene

and Al. The observed steady-state heat transfer, with

temperature profiles measured by IR thermography con-

firmed the anticipated behaviour from the two-region fin

model in both the Cartesian and cylindrical coordinate

systems. The two-region fin model equations all performed

as desired in capturing the measured temperature profiles

and the two-region fin model profiles were always bound

by the experimental uncertainty for the one-dimensional

measured temperature profiles. In fact, all validations were

judged medium to high accuracy, supporting the use of the

two-region fin model in design calculations. Medium

accuracy validations were those that represented the fea-

tures of the temperature profiles well, but lacked accuracy

on the temperature predictions themselves, whereas high

accuracy validations had both good representation of the

features and good temperature prediction accuracy.

Circular heat sources benefit greatly from the capacity to

dissipate heat in the fin region with increased effectiveness

as the active surface area for heat transfer and the cross-

section area for heat conduction in the Al foil increase

proportionally to r2. Rectangular heaters, on the other hand,

show a greater degree of heat penetration in the fin region,

therefore supporting arguments for better temperature

uniformity using rectangular finite heat sources.

It was also noted in the experiments carried out with

only natural convection for cooling that insufficiently

available length in the fin region increases the importance

of modeling an insulated boundary condition in the fin

region rather than an infinite one. Nonetheless, the two-

region fin model compensated for the measured behaviour

by estimating hf smaller than it would likely be with the

correct boundary condition. This mainly affected the two-

region fin model agreement with the experimental tem-

perature profiles at the end of the fin regions, as all other

aspects remained in good agreement.

The interaction of adjacently heated boundaries, as

encountered in stovetop cooking and likely to arise in the

development of laminate film encapsulated supercooling

salt hydrate PCM heat storage pouches, indicate the

importance of improving the repertoire of the two-region

fin model for real-world applications. The implementation

of an insulated boundary condition to the two-region fin

model rather than the infinite boundary condition currently

used is expected to capture this behaviour and provide

high-quality design input for supercooling salt hydrate

PCM encapsulation studies in particular.

Acknowledgments The authors are thankful for financial contri-

butions from the Natural Sciences and Engineering Research Council

of Canada (NSERC), Resource Recovery Fund Board (RRFB) of

Nova Scotia, Dalhousie Research in Energy, Advanced Materials and

Sustainability (DREAMS), and the Canada Foundation for Innovation

(CFI). Technical expertise and use of equipment was assisted by

Michel Johnson of the Institute for Research in Materials (IRM) at

Dalhousie University, Klabin, Tetra Pak Brazil, EET Brasil (member

of TSL Ambiental), and Mokhtar Mohamed at Dalhousie University.

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