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Polymer Testing 250142-9418/$ - see front matter q 2005 Elsevier Ltd. All rights reservKeywords: Functional analysis; Superposition; Bench-scale test; Heat release rate

1. Introduction

The heat release rate in a room fire has to be

understood [1,2] in hazard assessment. This will give

key information on the size of the fire; the rate of fire

growth and, consequently, the release of smoke and

toxic gases; the time available for escape or fire

suppression and the type of suppressive action that is

likely to be effective. Other attributes that define the fire

hazard, such as the possibility of having a flashover fire,

can be estimated.

Different combustibles, including both movable and

fixed fuel load [3], are stored in a building. The rate of heat

release in burning these materials together should be

estimated. Mostly likely, only the heat release rates of

individual materials or single items are available. How the

curves can be combined to estimate the resultant heat

release rate curve [4,5] should be understood.

The heat release rate per burning surface area of a

material can be measured by a cone calorimeter [4,6].

Models based on the cone calorimeter results have beenAbstract: In fire hazard assessment, the resultant heat release rate of burning different combustibles has to be known. The principle

of superposition is commonly applied to estimate the total heat release rate from the individual curves measured for single items.

Accuracy of such an approach will be studied with bench-scale tests in this paper.

The heat release rate curves of burning each sample cube of polymethylmethacrylate (PMMA), polyvinyl chloride (PVC),

polycarbonate (PC) and wood were first measured individually by a cone calorimeter. Radiative heat fluxes of 50 and 70 kW mK2

were applied. After that, a PMMA cube was burnt with a cube of another material under the same heat flux. The resultant heat

release rate curves of burning these two cubes were measured. Heat release rate curves of burning the single cube were used to

estimate the resultant curves. The technique of fundamental analysis will be applied for comparing the predicted curves with the

experiments. Importance of the parameter s for estimating the secant inner product cosine specifying the data points intervals will

also be discussed.

For the samples tested, it is observed that superposition gives good estimations of the total heat released curve if those for

individual items were measured under the same radiative heat fluxes. However, the results will not be so good where the curves

for each combustible were measured at different heat fluxes. This point is very important in estimating the possible heat release rate

for a design fire.

q 2005 Elsevier Ltd. All rights reserved.Property

Superposition of he

for combustibles w

W.K. Chow

Areas of Strength: Fire Safety Engineering, Research Centre for Fire

Received 29 June 2005;delling

release rate curves

bench-scale tests

, S.S. Han

neering, The Hong Kong Polytechnic University, Hong Kong, China

pted 2 September 2005

(2006) 7582

www.elsevier.com/locate/polytestdifferent applications such as for train compartments

[79].

ed.

doi:10.1016/j.polymertesting.2005.09.016

* Corresponding author. Tel.: C86 852 2765 7198.E-mail address: [email protected] (W.K. Chow).developed to predict the heat release rate for burning

those materials in bigger rooms [5], and applied for

THR Z

0

Q0ctdt (2)

Test A: Two sample cubes at two sides of the tray

with arrangements:

A1: PMMA and PC at 70 kW mK2, 25 mmunder the cone

A2: PMMA and PVC at 70 kW mK2, 25 mmunder the cone



A5: PMMA and wood at 50 kW mK2, 50 mmunder the cone

Test B: PMMA cube only at one side of the tray with

arrangements:

B1: PMMA at 70 kW mK2, 25 mm under thecone



Test C: PC cube only at one side of the tray with only

Schematic view of heat fluxes

Rfaff

Fig. 1. Cone calorimeter tests.

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 758276In this paper, how the resultant heat release rate

of two different polymeric materials can be

combined is explored. Polymethylmethacrylate

(PMMA), unplasticized polyvinyl chloride (PVC),

polycarbonate (PC) and oak wood widely used in

the market as consumer products and construction

materials are taken as examples. Samples of those

materials were selected and cut into small cubes of

size 20 mm. The samples were exposed to the same

conditions in a cone calorimeter for measuring the

heat release rates under heat fluxes of 50 and

70 kW mK2. The samples were tested individually

by themselves first. A sample PMMA cube was

then burnt with another sample for comparing with

the calculated heat release rate using Eq. (1).

2. Samples tested

Experimental measurements on one or two sample

cubes of PMMA, PVC, PC and wood were conducted

in a cone calorimeter. The samples were placed at the

side of the cone tray as shown in Fig. 1. The radiative

heat flux of the cone was set at 70 or 50 kW mK2. In

following the procedures in the standard tests [13],

samples were placed at 25 mm below the cone. In this

paper, some samples were also tested by moving

down to 50 mm below the cone [14]. This will give

different ventilation conditions as in real fire

scenarios. Different heat release rate curves can then

be achieved.

The testing arrangements as shown in Table 1 areThe combined heat release rate Q0T for burning twosamples A and B with heat release rates Q0A and Q0B issuggested to be [10]:

Q0T Z Q0A CQ

0B (1)

This was tested before with a cone calorimeter

[11,12].

The heat release rate per unit area Q0ct curves ofdifferent materials measured with bench-scale tests in a

cone calorimeter, time to ignition (TTI, in s), peak heat

release rate per unit area (pkRHR, in kW mK2) and

total heat released per unit area (THR, in MJ mK2) are

estimated. Note that THR is given by:

Nsummarized in the following:Test A

Tests B, C, D, E

100 mm20 mm cube

100 mm

Cone tray

Thermal radiation

R

Rfc

(a)

(b)

(c)one test:

Table 1

Functional analysis on the superposition results

Curves

tested

Para-

meters

A1: PMMACPC A2: PMMACPVC A3: PMMACPVC A4: PMMACPVC A5: PMMACwood

Same

heat flux

Under different

heat fluxes

Same

heat flux

Under different

heat fluxes

Different

heat flux

Same

heat flux

Different

heat flux

Under different

heat fluxes

Same

heat flux

Under different

heat fluxes

Same

heat flux

B1CC1 B2CC1 B3CC1 B1CD1 B2CD1 B3CD1 B1CD2 B2CD2 B3CD2 B1CD3 B2CD3 B3CD3 B1CE1 B2CE1 B3CE1

Q0ct Norm 0.13 0.21 0.46 0.17 0.31 0.55 0.19 0.23 0.51 0.70 0.5 0.22 0.82 0.52 0.15Cosine

(sZ1)0.86 0.83 0.56 0.83 0.53 0.16 0.44 0.72 0.14 K0.07 K0.02 0.16 0.11 0.16 0.60

Cosine

(sZ2)0.90 0.88 0.61 0.89 0.59 0.18 0.49 0.78 0.16 K0.07 K0.01 0.18 0.12 0.19 0.65

Cosine

(sZ3)0.91 0.91 0.63 0.91 0.62 0.20 0.54 0.80 0.17 K0.07 K0.01 0.20 0.12 0.21 0.70

Cosine

(sZ4)0.92 0.93 0.65 0.93 0.65 0.21 0.59 0.82 0.19 K0.07 0.01 0.22 0.13 0.23 0.75

Cosine

(sZ5)0.93 0.93 0.66 0.94 0.68 0.23 0.63 0.83 0.20 K0.06 0.02 0.25 0.15 0.25 0.80

THR Norm 0.08 0.16 0.30 0.16 0.25 0.39 0.06 0.14 0.31 0.05 0.51 0.04 0.61 0.41 0.11

Cosine

(sZ1)0.99 0.99 0.90 0.99 0.97 0.84 0.98 0.97 0.86 0.06 0.07 0.97 0.77 0.89 0.99

W.K

.C

ho

w,

S.S

.H

an

/P

olym

erT

esting

25

(20

06

)7

5

82

77

C1: PC at 70 kW mK2, 25 mm under the coneTest D: PVC cube only at one side of the tray with

arrangements:

D1: PVC at 70 kW mK2, 25 mm under thecone



Test E: Wood cube only at one side of the tray with

only one test:

E1: Wood at 50 kW mK2, 50 mm under thecone.

Each testing arrangement was tested several times to

check the repeatability [15]. Only one typical set of

results was used to study the superposition. Results of

the heat release rate per unit area Q0ct measured in thecone and the total heat released per unit area THR for

each test are shown in Figs. 26.

In a real fire, combustibles placed together are

exposed to different radiative heat fluxes. Therefore,

different external radiative heat fluxes and separation

distances among them should be assessed. For the two

samples as tested in this paper, there was a constant heat

flux emitted from the cone Rfc, a heat flux from the

adjacent burning sample Rfa and a heat flux feedback

from the flames Rff acting on the burning surface of the

sample, as shown in Fig. 1c. The distances between the

two samples might become shorter than 6 cm (even

2 cm for PVC) due to melting and swelling upon

burning. There might be stronger interaction between

the two combustibles due to the shorter distance.

Although Rfa might be higher than Rfc, both heat fluxes

are likely to be less than Rff . Except at very high values,

external heat fluxes such as Rfc would only be important

in ignition. Effects of these couplings should be further

studied quantitatively but are not included in this paper.

PVC samples were difficult to ignite under lower

1000

1500

2000

2500 A1 PMMA+PCB1 PMMA at 70 kWm2C1 PC at 70 kWm2S Calculated

B1

C1

S

ase

rate

per

uni

t are

a Q'

c / k

Wm

2

A1

(a)

0 100 200 300 4000

100

200

300To

tal h

eat r

elea

sed

per u

nit

Time / s

D1

Total heat released per unit area

Fig. 3. Test A2.

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 7582780 100 200 300 4000

500

Hea

t rel

e

Time / sHeat release rate per unit area

0 100 200 300 4000

100

200

300

400

500A1

B1

C1

S

Tota

l hea

t rel

ease

d pe

r uni

t are

a TH

R /

MJm

2

Time / sTotal heat released per unit area

(b)Fig. 2. Test A1.0 100 200 300 4000

500

1000

1500

2000

2500 A2 PMMA+PVCB1 PMMA at 70 kWm2D1 PVC at 70 kWm2S Calculated

Hea

t rel

ease

rate

per

uni

t are

a Q c

' / k

Wm

2

Time / s

D1

B1

S

A2

Heat release rate per unit area

400

500

are

a TH

R /

MJm

2

B1

S

A2

(a)

(b)heat fluxes. Therefore, higher heat fluxes of 50 and

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 7582 790 100 200 300 4000

500

1000

1500

2000 A3 PMMA+PVCB2 PMMA at 70 kWm2D2 PVC at 70 kWm2S Calculated

B2

A3

S

Hea

t rel

ease

rate

per

uni

t are

a Q'

c / k

Wm

2

Time / s

D2

Heat release rate per unit area

400

500

TH

R /

MJm

2

B2

A3

(a)

(b)70 kW mK2 were used. As observed in the tests, PVC

samples melted quickly and evaporated into fuel

vapour. A large quantity of smoke with irritating

smell was liberated upon exposure to the heat fluxes.

Among the four samples tested, PVC was the most

difficult to burn with the longest ignition time. Further,

PVC cubes did not burn steadily with charring. Taking

test A4 as an example, the heat release rate oscillated

with several peaks.

There was smouldering at first in burning the wood

samples. Char was formed later at the burning stage.

PMMA was ignited easily and burnt completely with a

steady rate. The PC samples were also difficult to burn.

However, once ignited, they burnt vigorously with

smoke liberated. Except for PMMA, some residues

were left after burning.

Rate of heat transfer inside the sample from the

external heat flux depends on the effective exposure

area and the distance away from the conical heater.

0 100 200 300 4000

100

200

300

Tota

l hea

t rel

ease

d pe

r uni

t are

a

Time / s

S

D2


Fig. 4. Test A3.0 200 400 6000

500

1000

1500

2000B3

S

A4 PMMA+PVCB3 PMMA at 50 kWm2D3 PVC at 50 kWm2S Calculated

D3

A4

Hea

t rel

ease

rate

per

uni

t are

a Q'

c / k

Wm

2


300

400

500

nit a

rea T

HR

/ M

Jm2

B3

S

A4

(b)

(a)The heat flux might be reduced by 515 kW mK2 when

the distance of the sample from the heater was moved

from 25 to 50 mm when set at 50 and 70 kW mK2.

There were differences in the exposure areas for the

different samples upon burning. The melted PVC is an

obvious example. Results of heat release rate per unit

area deduced from the cone would be affected. The

exposure areas for all sample cubes were taken to be the

same upper surface area of 4 cm2. The accuracy of the

heat release rate would be affected by the above factors.

However, those effects should be the same to all

samples, giving very little deviations in judging the

superposition principle.

3. Superposition of heat release rate curves

Whether the individual heat release rate curves of

two different materials can be added together (i.e.

superposition) to give the resultant heat release rate

while burning both of them will be assessed by the

measured results [2,1012,16,17].

0 200 400 6000

100

200

Tota

l hea

t rel

ease

d pe

r u

Time / s

D3


Fig. 5. Test A4.

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 7582800 200 400 6000

500

1000

1500

2000 A5 PMMA+WoodB3 PMMA at 50 kWm2E1 Wood at 50 kWm2S Calculated

B3

S

E1

A5

Hea

t rel

ease

rate

Q' c /

kW

m2


400

500

area

TH

R /

MJm

2

B3

(a)

(b)From the heat release rate per unit area curves Q0CAand Q0CB of two samples A and B, the transient heatrelease rate per unit area estimated Q0CTE is:

Q0CTEt Z 12

Q0CAt CQ0CBt

(3)

Experimental curve on the heat release rate per unit

area for burning the samples A and B Q0CTt under thesame heat flux and the calculated curve Q0CTEt fromEq. (3) are compared in Figs. 26.

Adding the heat release rate curves of two

combustibles by superposition is useful in predicting

real fire scenarios when the combustibles are not placed

too close to each other. The results can at least be taken

as a minimum estimation.

There might be some other effects which are more

obvious for bigger fires. A correction factor might be

required in using superposition. Assuming these effects

can be neglected under a specified standard external

heat flux higher than normal heat fluxes, the net heat

calculated:

0 200 400 6000

100

200

300

Tota

l hea

t rel

ease

d pe

r uni

t

Time / s

S

E1

A5


Fig. 6. Test A5. The parameter norm is a measure of the relativedifference in magnitude of the two curves.

The parameter inner product or cosine describes theangular difference between the resultant vectors to

provide a quantitative measure on the similarity of

the curve shape.

For good agreement between the experimental and

predicted curves, the value of the norm is expected to be

close to zero, and the value of the cosine is expected to

be close to one.

Following the recommendation by Peacock et al.

[18], the Euclidean norm is calculated by the ith

experimental and model values Ei and mi at the ith timerelease rate for one sample by burning it alone will be

the same as that from burning it together with other

combustible under the same conditions. Therefore, the

upper limits of predicted results can be determined by

applying the results tested under the same standard

external heat fluxes. Test results of each sample

exposed to the same conditions could be added, though

more tests should be carried out to confirm this.

Another method is to calculate the increase in heat

flux by the adjacent burning item. The geometries of the

two combustibles and the relevant flames, the variable

heat flux and other potential factors should be

considered. These aspects will be further reported

separately later.

4. Functional analysis

Instead of comparing the estimated results from

superposition with the experiments in qualitative terms

such as good, satisfactory or bad, functional

analysis proposed for evaluating fire models by

Peacock et al. [18] is used. Fire model predictions

have been compared with test data by Friday et al. [19]

with such approach.

As both experimental and predicted data can be

described by transient curves, functional analysis

would quantify the difference between two curves in

terms of magnitude and shape. The data points within

each curve are described by vectors, summing them up

would give a resultant single vector for each curve. The

distance between the resultant vectors for the predicted

and measured curves is the error. This error can be

normalized to provide a relative difference, or norm,

between the curves. The following parameters will beincrement ti with s data points to be considered at each

Norms and inner product cosines were calculated for

the curves of Q0 t and THR for each case under the

indicating very good predictions on curve magnitude.

The shapes of the curves are in fact very close, as shown

in the figures.

For s equal to 1, the computed values of cosine at

higher heat fluxes are higher than 0.72. However, the

values of cosine for Q0ct are 0.16 for test A4 and 0.60

experiment.

7. Conclusions

Samples of PMMA, PVC, PC and wood in different

From the above study, the heat release rate of

burning two material samples together can be estimated

tiK1miKs2=s2tiKtiK1

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 7582 81(with wood at 50 kW mK2) for test A5. A possible

reason might be because wood and PVC samples did

not burn steadily under the lower heat flux of

50 kW mK2.

Values of cosine would be higher for higher values

of s. For example, the values of cosine for tests A4 and

A5 would be changed to 0.25 and 0.80, respectively, by

taking s as 5.

Results on comparing the curves with functionalc

same radiative heat fluxes. The values were computed

over the burning duration period of 400 and 600 s for

radiative heat fluxes of 70 and 50 kW mK2,

respectively.

Functional analysis results of the point-to-point

comparison are presented in Table 1. It is observed

that the values of the norm are lower than 0.23,increment inside as:

jEKmjjEj Z

PniZ1

EiKmi2s

PniZ1

Ei2s (4)

The secant inner product cosine is:

The parameter s R1 would smoothe the results togive better estimates of large-scale differences. Higher

value of s might not overcome the effects of small-scale

noise between dense data, depending on the shapes of

the curves. Values of s will be varied as 1, 2, 3, 4 and 5

in this paper to investigate its effect on the secant inner

product cosine.

5. Superposition of curves under the same radiative

heat fluxes

E;mh ijEj,jmj Z

PniZsC1

EiKEiKsmi KmiKs=s2ti KPniZsC1

Ei KEiKs2=s2tiKtiK1,Pn

iZsC1mi K

sanalysis suggested that superposition is better for testsby simple addition, i.e. by superposition of the curves

measured for each sample under the same heat flux.arrangements were tested with a cone calorimeter under

different heat fluxes and distances from the conical

heater. Several tests were repeated for each testing

arrangement to ensure its repeatability. One typical set

of results was used to study the superposition.with the distance and exposure area. The result of one

sample tested in other conditions can be taken as a

reference curve, which might be varied under the same

initial conditions. Deviation of the calculated results by

superposition can be estimated by functional analysis.

If the curves under different heat fluxes are added

together, say B3 (under 50 kW mK2) with C1 (under

70 kW mK2) instead of B1 with C1, both the norm and

cosine compared with test A1 deviated from the

matching value of 0 and 1.0. The values of norm and

cosine are 0.46 and 0.56 respectively for B3 with C1

when s is 1. The value of cosine only increased to 0.66

when s is 5.

For curves under the same heat flux but at different

distances below the cone, say combining B2 of

70 kW mK2 for 50 mm and C1 for 25 mm, the values

of norm and cosine are 0.21 and 0.83, respectively. The

value of cosine increased up to 0.93 when s is 5.

Therefore, combining the curves measured under the

same heat flux but at different distances below the cone

would not give results deviating so much from theunder higher heat fluxes, under which the combustion

was more complete.

6. Curves under different radiative heat fluxes

For real fire scenarios, the heat release rate of burning a

combustible will be affected by the total heat feedback

and the total external heat fluxes, which might be varied

(5)Values of norm and cosine gave better agreement for

exposed to higher external heat fluxes, such as a

W.K. Chow, S.S. Han / Polymer Testing 25 (2006) 758282external thermal radiation might give more complete

combustion. The results can be taken as a minimum

estimation. The coupling effects between the combus-

tibles, external heat and ambient conditions should also

be considered.

Many more tests, especially full-scale burning

tests [16,20,21] should be carried out on those

samples to support the application of the super-

position principle.

Acknowledgements

This paper is supported by the RGC project

Determination of the concentration needed for extin-

guishing fires with clean agent heptafluoropropane

(FM200) under Grant No. B-Q669.

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Superposition of heat release rate curves for combustibles with bench-scale testsIntroductionSamples testedSuperposition of heat release rate curvesFunctional analysisSuperposition of curves under the same radiative heat fluxesCurves under different radiative heat fluxesConclusionsAcknowledgementsReferences

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