Radiative ignition of combustible materials I. Polymeric materials undergoing nonflaming thermal degradation—the critical storage problem

MATHEMATICAL COMPUTER MODELLING

PERGAMON

Mathematical

and Computer

Modelling

30 (1999)

177-195 www.elsevier.nl/locate/n-cm

Radiative Ignition of Combustible Materials I. Polymeric Materials Undergoing Nonflaming Thermal DegradationThe Critical Storage Problem E. BALAKRISHNAN Department of Mathematics and Statistics Sultan Qaboos University: Sultanate of Oman M. I. NELSON* Department of Fuel and Energy, The University of Leeds Leeds LS2 9JT, England G. (3. WAKE Department of Mathematics and Statistics University of Canterbury Private Bag 4800, Christchurch. n’ew Zealand (Received

December

1998;

accepted

January

1999)

Abstract-The critical storage problem for a self-heating combustible material exposed t,o all external radiative heat source is investigated. The reaction kinetics are defined in terms of a characteristic temperature and an activation energy, linking small-scale experimental techniques to mediumscale Aarnmability tests. The mode1 is nondimensionalised in such a way that material properties of fundamental interest are retained as distinct continuation parameters. @ 1999 Elsevier Science I,td All rights reserved. Keywords--Cone

calorimeter,

Nonflaming combustion

NOMENCLATURE A

pre-exponential

E

activation

E”

nondimensionalised

7-1

ramping

(b-1)

factor

(.I Illol-- ’)

energy activation

energy, E” = E/RT,

rate used in a thermogravimetric

t

)

(KS_‘)

experiment

L

the length of the test sample

L?

a typical sample length

(111)

strength

(Js-’

.c

of external

(III)

irradiance

rl-2)

*Author to whom all correspondence should be addressed. This work was carried out whilst MIN was supported by a fellowship from the Royal Society of London to work with GCW in New Zealand. In addition support from a Marsden fund grant from the Royal Society c#f New Zealand (UOA 515) is acknowledged. 0895-7177/1999/S - see front matter @ 1999 Elsevier PII: SO895-7177(99)00207-l

Science

Ltd.

All rights reserved.

Typeset

by &&‘I$$

178

et al.

E. BALAKRISHNAN L CT

the critical heat flux

&r

IL=j

the critical heat flux of a sample of length j(m)

(Js-’

me2)

Ix+

the critical heat flux of a sample when the heat-transfer coefficient on the insulated boundary is j

(Js-’

mV2K-‘)

CR

the relative length number, CR = (CC,

(-)

Lr

(Js-l

number, Cc, = (L,,

IL=O.OI

-Cc,

Jxz=e -&

IL=o.o~/&

IL=~.o~)

IXz=o.rX1/LCcr Ix,=o)

rne2)

Lx

the noninsulation

Cc’

nondimensionalised

G,

the nondimensionalised

Ml

the combustible

Q

reaction exothermicity

(JW’)

R

the ideal gas constant

(JK-l

T

temperature

(K)

Tt

the effective ambient temperature when an external irradiance is combined with a ‘hot’ atmospheric temperature

(K)

T*

reduced temperature

(-)

TCl

ambient temperature

TC

characteristic

TE

nondimensionalised

characteristic

temperature,

T,’ = T,/T,

Y*

nondimensionalised

sample length, Y* = L/L,,

Y* = L/L,

a0

half-width of the test sample, used in the Frank-Kamenetskii

c

heat capacity

(J K-’ kg-‘)

k

thermal conductivity

(J s-l

irradiance strength, L’ = L/Tax1

(-) (-)

critical heat flux

(-)

material

(kgme3)

scale, T* = T/Ta

mol-‘)

(K)

temperature

measured in a thermogravimetric

thermal conductivity,

experiment

(K) (-) (-)

variables

k”

nondimensionalised

kd

the rate of thermal degradation

qout

nondimensionalised

t

time

t*

nondimensionalised

2

position within the test sample, 0 5 CC2 L

XV

reduced length scale, x* = x/L

9

nondimensionalised

Q

absorptivity,

6

the Frank-Kamenetskii

t

reduced activation

e

nondimensionalised temperature rise over ambient in the FrankKamenetskii variables, 0 = (E/RTz) (T - Ta)

k* = k/x1 L,

(m)

K-‘)

(-) (kgm

radiative heat loss coefficient,

m-l

qout = aTi/x~

-3 s-1

)

c-1 (4

time, t* = (k/pcLz) t

reaction exothermicity,

(-1 (4 c-1

@ = QpHLT/xlTz

c-1

0 _< QI5 1

(-_)

parameter, 6 = (pQa$Aexp

energy in the Frank-Kamenetskii

[-E/RTa])

/kRTz

variables, c = RT,/E

(-_) c-1 c-1

P

density

c7

Stefan-Boltzmann

X1

the heat transfer coefficient between the sample and the surrounding

(Js-l

mm2 K-l)

x2

the heat transfer coefficient between the sample and its insulating background material

(J s-l

mW2 K-l)

X;

the nondimensional&d

(-)

ary, x5 =x2/x1

(km-3)

constant

(J s-l air

heat-transfer coefficient on the insulated bound-

m-’ Ke4)

Unless otherwise specified, we take the following typical parameter values: E = 80 x lo3 J mol-‘, Ff = 1/60Ks- l, L = O.Olm, L, = 0.01 m, Q = 20 x lo6 J kg, T, = 298K, T, = 580K, c = 1000Jkg-lK-l, k = lJs-lm-lK-l, a = 1, p = 2000kgm-3, x1 = 30Js-1m-2K-1, 33 = 0Js-1m-2K-1.

Radiative

The following

range of parameter

Ignition

values can be considered

80 x lo3 5 E (JmolY’) 580 < T,(K)

5 240

0.01 5 ,+ (Js-’ 0 < x2 < appropriate

10-"

values

for physical

‘typical’: x

103,

I 780,

4 x 10’ 5 Qp (Jne3)

The

179

5 4 x lOlo,

m-2K-1)

5 1.0,

O.lYl.

constants

are:

R = 8.31431 JK-’

mol-’

and r = 5.67

x

J s-l m-2 K4.

1. INTRODUCTION The thermal a classic cessful

runaway

problem

of a self-heating

in combustion

in explaining

fire-risk

theory hazards

combustible

material

[l]. The associated of practical

concern

placed

in a hot environment

mathematical

theory

from a wide range

is

has been suc-

of industries,

in

manufacturing processes, storage, and transportation [a]. Two distinct fire-hazard problems arise in practice. The critical storage problem is the highest temperature at which a material can be safely stored. This problem is defined by the ste.sdystate equations, is independent of the initial conditions, and is characterised by a limit-point, bifurcation. The ctitical assembly problem relates to the fact that a material that is assembled at t.oo high an initial temperature may ignite, even though the storage condit,ions were known to t)e subcritical as far as the steady-state theory is concerned. The practical significancac> 0’ the latter problem has only been realised in recent years. We model the fire-hazard created by the radia,tive heating of a self-heating combustible material. This models a fire scenario in which a a combustible material is heated 11~ a nearby fire. Furthermore, test methods have been developed in recent years to evaluate the flammability

of polymeric

materials

wherein

the test sample

is radiatively

heated

on one side.

are rr$acing older methods, which invariable involve heating by a bunsen flame, as they more reproducible results and are more amenable to mathernat,ical modelling. In particular, mat,hematical

model considered

here incorporates

the testing

of materials

undergoing

l’hese give t,hc

nonflaming

combustion in the cone calorimeter [3-51, which is becoming a standard test method in indllstry. Our primary motivation for this work is to provide the foundation for a future investig,ation into t,he effectiveness of fire-retardants in reducing flammability. The result,s presented here should, therefore, be considered as defining the flamrnability of nonfire-retarded material:;. In this paper, we address the critical storage problem, paying particular attention to how criticality is influenced by t,he physical and chemical parameters in t,he model. The critical assembly prc~blem is cbonsidered elsewhere [6]. 1.1. Nondimensionalising

the

Model

For many years combustion problems, regardless of their source of origin, were nondlmensionalised using the variables popularised by Frank-Kamenetskii [l]. This uses a dimensionless temperature rise over the ambient given by

The simplest spatially nonuniform combustion problem t,hen contains two nondimensionalised parameters, a reduced activation energy (t) and the Frank-Kamenetskii parameter (fi), often called the Dank6hler Number in the chemical engineering literature, defined bl RT, c=---, E 6

=

pQ@ k RT,T

(2)

180

E. BALAKRISHNAN

The power provided

by this choice of variables

et al.

is that many problems

have an analytic

when the pre-exponential approximation (c = 0) is made [a]. In the mid 1980s it was realised that the Frank-Kamenetskii nondimensionalisation the role played by the ambient

temperature

in defining

the combustion

problem;

solution

complicates

in many cases this

is the most important experimental control parameter. Notice that it appears in both the reduced activation energy and the Frank-Kamenetskii parameter. Consequently, varying one of these parameters, whilst keeping the other fixed, does not reflect experimental ambient temperature is varied as the bifurcation parameter. Therefore, variables

are inappropriate

when ambient

temperature

methodology; whereby the Frank-Kamenetskii

is viewed as the main

bifurcation

param-

eter. A further complicating factor is that ambient temperature also appears in the definition of the nondimensionalised temperature-scale. am. which a nondimensionalised Accordingly a nondimensionalisation was introduced [7-91 m bient temperature is retained as a distinguished bifurcation parameter. This is discussed in more detail elsewhere [8-lo]. In particular, it was shown that the use of the exponential approximation (E = 0) leads to large errors the classical nondimensionalisation behaviour

shown when different

in the calculation of critical initial temperatures [8] and that is sufficiently nonlinear to conceal some of the differences in bifurcation

detail in Section 4.1. In this paper, we are interested rial effect its flammability. parameters

of interest

are varied

in how the physical

We, therefore,

are retained

parameters

[lo]. This is discussed

and chemical

properties

choose a nondimensionalisation

as distinct

continuation

2. DESCRIPTION

in more

of the test mate-

scheme

in which the

parameters.

OF THE

MODEL

In Section 2.1, we describe the model chemistry and discuss the formulation of the reaction kinetics using the concept of a characteristic temperature and an activation energy, rather than the traditional definition of a pre-exponential factor and an activation energy. we discuss the physics of the model. In Section 2.3, we discuss the classification results 2.1.

as either

subcritical

Description

of Model

The self-heating rhenius Kinetics,

reaction

In Section 2.2, of experimental

or supercritical. Chemistry is modelled

by a single, exothermic,

first-order

reaction

obeying

MI --t Mz.

Ar(4)

Depletion of the combustible material (MI) is assumed to be negligible. The rate constant for such a reaction is traditionally written in the form, kd = -MlAexp

[ 1. g

For polymeric materials undergoing flaming combustion proven useful to rewrite the rate constant in the form, kd = -Ml%

[ll],

(5) and references

therein,

it has

(6)

exp c

In equation

(6) the parameter

T,, the characteristic

temperature,

is routinely

measured

in

thep’mogravimetric experiments [12]. The parameter X is the heating rate used in the thermogravimetric experiment. Note that these definitions of the rate constant are equivalent with the pre-exponential factor in equation (5) defined by A=gexp[&]. c

c

(7)

Radiat,ive Ignition Formulating

the reaction

kinetics

using equation

LX1

(6) is advantageous

because,

from the victw-

point of an experimentalist, it is very natural to consider how the flammability of a class of materials depends upon their characteristic temperature. Similarly, it is useful to ruse the activation energy, at fixed characteristic temperature, as a secondary bifurcation parameter. In the latter

E ---) cc is of theoretical

case, the limit

kinet,ics in a way that is well understood, in thermogravimetric experiments. 2.2.

Description

of Model

The combustible in an environment boundary, radiative

material

heat. transfer

because

it motives perspective,

high-energy

activation

in terms of bchavirnn

Physics is assumed

at ambient

is is heated

interest

from an experimental

to be a one dimensional

temperature

by an external

(T,).

irradiance

The front

slab of length

L which is placed

of the slab (1: = 0). t,hc exposed

(C), in addition

to undergoing

convective

.md

with its surroundings,

The cone calorimeter has been designed so that the rear end of the sample (X = L). the insulated face, is perfectly insulated. We usually assume that this holds. For simplicity, when considering nonperfect

insulation

we model the heat-transfer

between

the sample

and the insulating

surface

as a convective heat-transfer process. A more detailed model would explicitly include an iriert substrate underneath the test sample and model heat-transfer across the interface by coupling two partial differential equations, one for the combustible material and one for the insulating material,

at the rear boundary.

We assume that the irradiance is uniform across the exposed face and that heat-transfer dimensional. The cone-calorimeter has been designed so that these requirements hold. A schematic diagram showing the model geometry is provided in Figure 1. The cone calorimeter

methodology

specifies

no standard

paratus allows a maximum length of five centimeters. sample is one centimeter, a typical sample length. For simplicity, we assume that material properties intention ticularly 2.3.

to investigate for porous

Classification

the effect of having

materials,

at a latter

of Model

sample

We usually

length, assume

are independent

a temperature-dependent

the design that

of the ap-

the lengt,h of the

of temperature. thermal

is one

It. is om

conductivitjy,

par-

date.

Behaviour

In the cone calorimeter, and other radiative test methods, the strength of the irradiating accordingly we use it as the primary bifurcation source (C) is the main control parameter; parameter. When considering our model from the perspective of a potential fire hazard. the irradiance (C) reflects the severity of the fire. Two types; of behaviour are observed experimentally in cone-calorimeter tests: if the irradiance is suheritical the temperature of the sample undergoes a small increase; if the irmdiance is su~ercritical the sample undergoes thermal runaway. The dividing line between subcritical and as the critical heat fluz (G,,). supercritical behaviour is known in the fire-engineerin, (r literature The calculat,ion of the critical heat flux solves the critical storage problem. Experimental reports of critical heat, flux include [2], from materials undergoing self-heating, and [13-151. for the cone calorimeter.

3. MODEL 3.1.

Dimensionalised

The system

that

EQUATIONS

Equations we study

Reaction-Diffusion

is the following. Equation. L-g

= peg

- Qp$$

exp c

(8)

E. BALAKRISHNANet al.

182

I

HOT ATMOSPHERE TEMPERATURE = Ta

x = 0 (exposed boundary)

x = L (insulated boundary)

lNwLATlNG

MATERIAL

Figure 1. Model geometry showing convective (C) and radiative (R) heat-exchange between the hot atmosphere (temperature, T,) at the exposed boundary of the sample (2 = 0). The exposed boundary is also heated by an external irradiance (I). The sides of the sample are assumed to be perfectly insulated, so that heat-transfer is one dimensional. At the insulated boundary (z = L) the sample may be either perfectly or imperfectly insulated. Standard values for the nondimensionalised parameters are: nondimensionalised activation energy, E* = 32.288; nondimensionalised characteristic temperature, T: = 1.946; nondimensionalised reaction exothermicity, @ = 2.502; nondimensionalised sample length, Y’ = 1; nondimensionalised thermal conductivity, Ic* = 3.333.

Boundary Condition on z = 0.

-kg =cd

+ Xl(Ta

- T) +

a0

(T, - T4).

(9)

Boundary Condition on x = L. -kz

m

= x2(T - T,).

Initial Condition at time t = 0. T(z, 0) = T,.

(11)

The classic combustion problem assumes that there is no external irradiance (C = 0) and redefines the convective heat-transfer coefficient so as to include the effect of radiative heatexchange. Ambient temperature is used as the primary bifurcation parameter. Assume that the rear-boundary is perfectly insulated (~2 = 0) and that convection is the only heat-loss mechanism. The boundary condition on z = 0, equation (9), is -kg

= CYC+ Xl(Ta - T), =X’({Ta+$}-T),

= Xl (T+ -

T),

where T+ can be regarded as an ‘effective ambient temperature’.

(12)

Radiative

Hence,

there is no distinction

between

purely

1x3

Ignition

‘hot atmosphere’

experiments,

purely

‘radiative’

experiments and experiments combining a ‘hot atmosphere’ and an external irradiance: each of these experiments can be defined in terms of an effective ambient temperature. Now consider the case where both convective and radiative heat-loss mechanisms are included. First observe that since all the parameters are positive we can always find a unique of T+ such that aC + xlTa + wT; == xlT+ + aaT+‘. The boundary

condition

on cx = 0, equation

-kg =aL = d

(9), can then

+ x1(7; -T) + xlT,

be written

posit,ive value (15)

as

+ acr (T; - T”) .

(1G)

- XI?’ - ucrT”.

117)

+ aoT;

= xlT+ + aaT +“ - xlT

- CWT“,

ilS)

= x1 (T+ - T) + cto (T+” - T”) . and the same conclusion

3.2.

is T+ is an “effective

holds, that

Nondimensionalised

ambient

1119) temperature”

Equations

In \-iew of the discussion

in Section

3.1, our nondimensionalisation

scheme

does not retain

ambient temperature as a distinguished bifurcation parameter. In nondimensionalking equations (8)-( 11). we introduce a nondimensionalised temperature, T*, and a nondimensionalised time, t*. Equations (8)-( 11)) can then be written as follows.

nondimensionalised t,he Nomenclature.

Reaction-Diffusion

These

length,

CI*L,a

are defined

in

Equation. a2T* i3T* a2+2=df*-

Boundary

condition

on z* = 0. aT* --_=-. ax*

Boundary

condition

Y* k”

cd*

+ (1 - T*) + aq,,t

(1 _ T*‘)}

(21)

on x* = 1. --

aT*

zz F

(22)

(T* - 1).

8X*

Initial

Condition

at time t* = 0. T*(z*,O)

where

the nondimensionalised

variables

(23)

= 1,

are defined

in the Nomenclature.

Each

parameter

of

interest has been retain as a separate distinguished bifurcation parameter. The properties of the solution set for this system of equations for the case a = 0 was first discussed by Burnell et al. [9]‘. The parameter Y*, which represents the nondimensionalised external sample length (Y* = L/L,), could be eliminated from these equations by redefining the nondimensionalised thermal conductivity and nondimensionalised reaction exothermicity

1Although Burnell

et al. use a different

k+ = k*Y*,

(24)

a+ = QY”.

(25)

nondimensionalisation

the two models

are topologically

equivalent.

184

E. BALAKRISHNAN

Since the sample

length

is a natural

control

guished continuation parameter. Effectively, ‘internal’ length scale, representing position length state

scale, representing calculations

the length

parameter,

experimentally

we retain

it as a distin-

we have two nondimensionalised length-scales: an within the sample (0 5 Z* < l), and an ‘external’

of the sample

and not to transient

et al.

calculations

(Y*).

The latter

because

is relevant

the sample-length

only to steady-

also appears

in the

time-scale. 3.3.

Numerics

The steady-state using Auto

solutions

of equation

94 [16] by rewriting

(20) with boundary

the initial-value

problem

conditions as a boundary

(21),(22)

was investigated

value problem.

4. RESULTS Our general specific material,

approach

is to fix the thermophysical

and to use the nondimensionalised

and chemical irradiance

parameters,

thus modelling

(C”), the experimentally

a

controlled

parameter, as the primary bifurcation parameter. We represent the response of the material to the ignition source by plotting the L2 solution norm against the nondimensionalised irradiance. In Section 4.1, we investigate the steady-state structure of the model, showing the existence of response diagrams containing one, three, and five steady-state solutions. When the response diagram does not have a unique solution for all values of the primary bifurcation parameter there are two cases to consider, depending upon if the steady-state diagram has multiple solutions when the primary bifurcation parameter is zero. The implications of such behaviour is discussed in Section 4.2. When the response diagram contains three, or five, steady-state solutions there is a critical of the nondimensionalised irradiance at which the ‘low-valued’ solution branch, corresponding noncombusting

material,

loses stability

irradiance slightly higher than representing thermal runaway.

at a limit point,

the ignition

limit point.

value to a

For values of the

this critical value the solution evolves onto a high-norm solution, The ignition limit point corresponds to the nondimensionalised

critical heat flux, (,C&). In Sections 4.3 and 4.4, we investigate how the nondimensionalised critical heat flux depends upon the thermophysical and chemical properties of the test material and upon experimental conditions. Although it is standard to assume that the rear boundary is perfectly insulated, anomalous experimental results, for materials undergoing flaming combustion, have been proscribed to nonideality in this boundary condition. Accordingly, in Section 4.4, we pay particular attention to the impact that heat-transfer on the insulated boundary has upon the nondimensionalised critical heat flux. Recall that there is a one-to-one relationship between our nondimensionalised continuation parameters and their dimensionalised counterparts. Hence we often write, for example, ‘the critical heat flux’ rather than ‘the nondimensionalised critical heat flux’. 4.1.

Physically Disjoint, Nonphysically and ‘Intrinsically Nonflammable’

Disjoint, Solutions

When the exponential approximation (E = 0) is not made the classical combustion problem in the slab exhibits two types of response curve: either there is a unique solution for every value of the control parameter or the response curve is the archetypal ‘S’-shaped combustion curve [9]. The former case represents ‘intrinsically nonflammable’ materials because small changes in the primary bifurcation parameter cannot produce sudden increase in the solution norm. The latter case models ‘combustible’ materials as in the vicinity of the ignition limit point a small change in the primary bifurcation parameter can have a large effect on the solution norm. When a nondimensionalisation scheme is used that accurately reflects experimental methodology there are two distinct types of ‘S’-shaped response curve. If the extinction limit point is

Radiative

in the left-half

plane

the steady-state

diagram

185

Ignition

is said to be physically-disjoint.

If it is in the

right-half plane the steady-state diagram is said to be nonphysically-disjoint. This classificat,ion arises because negative values of the primary bifurcation parameter do not make physically sense. This

phenomenon

was discovered,

though

explained

far from clearly,

practical significance was discussed by Gray et al. [lo]. It is significant that the phenomena of physically disjoint when the Frank-Kamenetskii

variables

are used.

response curve has an important practical consequence and is discussed further in Section 4.2. , W: showed in Section 3.1 that the radiative ignition to its combustion in a hot environment. investigate and the classic combustion exposed exhibited

boundary.

Consequently,

in our model,

over a range of parameter maximum

multiplicity

and the ‘new variables’ introduced

by radiative

as shown in Figure

heat-exchange

in predicting

of a combustible

material

The extra

(a) Steady-state diagrams showing: (1) physically disjoint solution, (2) nonphysically disjoint solution, (3) “intrinsically nonflammable” solution.

wa.s equivalent t,he problem we heat loss on the t,ypes are

2b shows a fourth type of solution

t,hat has,

This is a surprising

result because

in both t,he Frank-Kamenetskii

multiplicity

on the exposed

of a system

solution

solutions.

has been found

is not apparent

the flammability

that the three fundamental

2a. Figure

values, five steady-st,ate [7-g] is three.

branches

191. Its

bet)ween two types of ‘S’-shaped

Therefore, the main difference between problem is the incorporation of radiative

it is unsurprising

for a slab that

solution

The distinction

et al.

by Burnell

the

variables

is due to the ext,ra nonlinearity

boundary.

(b) Steady-state diagrams showing potential dering combustion’ behaviour.

‘smoul-

Figure 2. (a) Parameter values: nondimensionalised characteristic temperature. (1) T,’ = 1.946, (2) T,* = 5.940, (3) T,’ = 6.040. (b) Parameter values: nondimensionalised characteristic temperature, T,‘ = 3.188; nondimensionalised reaction exothermicity, + = 2.502 x 10-l. Parameter values not specified are defined in the caption for Figure 1.

4.2.

The

Transition

and Nonphysically

Between

Physically

Disjoint

Disjoint

Solutions

In Section 4.1, we discussed the distinction between physically disjoint and nonphysically disjoint solutions, an example of each is shown in Figure 2. The combustion of a material that

E. BALAKRISHNANet al.

186 190

,

1

,

I

,

,

,

,

,

100

I

I

110

,/’ _

-

90

10060

(A) Physically disjoint steady-state solutions

,/’ ,:’

,/’

70

,/’

,/’ ,: ,/ ,/’ ,,,’ ,/’

60

,/

,/’ 50

,/’ ,/ ,:’ ,:’

50 -

40 40 -

(C) “fntrinsicalfy non-flammable matenafs”

(C) “lntrinseally non-flammable materials”

30

1 3

(a) Nondimensionalised Q, = 2.502.

reaction exothermicity,

4 5 Non-dlmensionaleed

(b) Nondimensionalised + = 2.502 x 10W2.

6 7 6 characterlst!c tempearture

9

reaction exothermicity,

Figure 3. Regions of parameter space exhibiting (A) physically disjoint solutions, (B) nonphysically disjoint solutions, (C) “intrinsically nonflammable” solutions. Region (B) occurs between the solid and dotted lines. Parameter values not specified are defined in the caption for Figure 1.

exhibits a physically disjoint steady-state diagram is not extinguished if the irradisnce is decreased sufficiently slowly towards zero. The fate of if the irradiance is changed more rapidly, perhaps even turned off instantaneously, depends upon the basins of attraction of any coexisting attractors. For the model considered in this paper, we believe that combustion cannot be extinguished by lowering the irradiance to any physically attainable self-sustaining if the irradiance source is removed. Although materials

exhibiting

level, i.e., combustion

is

physically disjoint response diagrams can continue to burn in

the absence of an external irradiance, this is not true for materials with nonphysically disjoint response diagrams: in the absence of an external irradiance the only attractor is a ‘low-valued’ solution; combustion, is therefore self-extinguishing. Furthermore, combustion is self-extinguishing if the strength of the ignition source is reduced past a critical value, given by extinction limit point. The transition from a physically disjoint response curve to a nonphysically disjoint curve, occurring when the extinction limit point passes through the line L* = 0, is therefore important. The transition from nonphysically disjoint to ‘intrinsically inflammable’ is also of interest. The regions of parameter space in which these three fundamental solution types may be found are shown in Figures 3 and 4. We refer to the value of the characteristic temperature at the boundaries between these regions as the critical transition (characteristic) temperatures. We emphasis that the distinction between materials having physically disjoint and nonphysically disjoint steady-state solutions does not arise if the Frank-Kamenetskii variables are used. Figure 3 shows that, for a fixed value of the activation energy, the critical transition temperatures decrease with decreasing reaction exothermicity, as expected intuitively; this point is also illustrated in Figure 4b. Notice that as the activation energy increases so does the critical char-

Radiative

Ignition

187

(C)“lntrmskally

2.5 ^^^.

“.Wl

“.I

Non-dimensionalised thermal condktivity

(a) Variation with nondimensionalised conductivity.

thermal

non-flammable materials””

^ ^_ ^ IL”1 0.1 Non-d!mensionallsed reaction exotherkty

(b) Variation with nondimensionalised exothermicityl.

. . ...A 1”

reaction

Figure 4. Regions of parameter space exhibiting (A) physically disjoint solutions. (8) nonphysically disjoint solutions, (C) “intrinsically nonflammable” solutions. Region (B) occurs between the solid and the dotted lines. Parameter values not specified are defined in the caption for Figure 1.

acteristic temperatures, from this perspective high-activation energy materials pose a greater fire hazard. In Figure 4, we consider the ‘best case’ by taking the activation energy to be the lowest permissible value. Figure 4a shows that as the thermal conductivity decreases the critical transition temperatures increase. This is unsurprising because as the thermal conductivity decreases heat is increasingly concentrated near the exposed boundary, increasing the chance of thermal runaway. It should be noted that the lower values of the thermal conductivity

in Figure 4s are more

realistic than the higher ones. In view of the typical parameter values given in the Nomencla,ture, and of the results shown in Figure 2, we conclude that the type of materials considered in this paper will invariably exhibit physically disjoint, steady-state curves. This has an immediate consequence: materials of practical interest may ignite in the absence of an external irradiance, provided that there initial temperature profile is above some watershed. This is the critical assembly problem, which is investigated elsewhere [6]. 4.3.

The Dependence

of the Critical

Heat Flux Upon

Material

Properties

Figures 5-7 show how the critical heat flux, identified with the ignition limit point, depends upon the physical and chemical properties of the material. As our nondimensionalisation keeps physical and chemical parameters of interest as distinct bifurcation parameters, we investigate this dependence by continuing the ignition limit point with the appropriate secondary bifurcation parameter. Figure 5a shows that, for fixed activation energy, the critical heat flux increases with characteristic temperature. In Figure 5b, provided the activation energy is sufficiently large, the critical heat flux increases with activation energy; for lower values of the reaction exothermicity the crit-

188

E. BALAKRISHNANet al. 1.8

:’ ,:’ ,:’ ,:’ ,:’

,.I’ ,:’ ,:’ ,:’

,/’ ,,/

,;’ ,:’ ,,.’ ,:’ ,I’

1.Y

2

2.1

2.2

2.3

3

,:

,:

,’

,/’

,..’

,:’

---

,I

,/’

,/’

,/’

2.4

2.5

2.6

2.7

30

Non-dimensionalisedcharacterislictemperature

(a) The variation of nondimensionalised critical heat flux (LE,) with nondimensionalised characteristic temperature (T:).

40

50

60

70

80

90

100

Non-dimensionalisedachvat~onenergy

(b) The variation of nondimensionalised critical heat flux (L;,) with nondimensionalised activation energy (E*).

Figure 5. Ignition limit point bifurcation diagrams. (a) Parameter values: nondimensionalised activation energy, (1) E* = 32.288, (2) E* = 64.576, (3) E” = 96.864. (b) Parameter values: nondimensionafised reaction exothermicity, (1) @ = 2.502, (2) @ = 3.5 x 10F2, (3) + = 2 x 10W2, (4) + = 9 X 10W3. Parameter values not specified are defined in the caption for Figure 1.

ical

heat

flux initially

decrease with increasing activation energy. If the reaction exothermicity

is

sufficiently low the critical heat, flux decreases monotonically over the range of activation energies considered, as suggested by Figure 6a. Figure 6 shows the critical heat flux increasing with decreasing exothermicity, this is more noticeable for low activation energy materials, Figure 6a. The intersections of the three curves in Figure 6a reflects the range in behaviour shown in Figure 5b: in decreasing the exothermicity, we move from a regime where the critical heat flux increases monotonically with activation energy to one where it decreases monotonically. Figure 6b shows that at low exothermicity there is a substantial increase in the critical heat flux as the characteristic temperature increases. Figure 7 shows that the critical heat flux decreases with decreasing thermal conductivity. The trends with increasing activation energy and characteristic temperature reflect properties already noted. 4.4.

The Dependence

of the Critical Heat Flux Upon Experimental

Conditions

In Figures 2-7, we assumed that the sample length was 10 mm and that the rear boundary was perfectly insulated. In Figures 8-11, we show how the critical heat flux depends upon the length of the sample, Figures 8a, 9, and llb, and the degree of heat-transfer at the insulated boundary, Figures 8a, 10, and 11. It is usually assumed that the rear boundary is perfectly insulated, the latter figures show the effect of nonideality upon experimental results. Figure 8 shows the percentage change in the critical heat flux as either the length of the sample, (a), or the heat-transfer coefficient, (b), is increased from their default values. The critical

Radiative

189

Ignition

r

0.01 Nondimenslonaked

0.1 reacbon exafhermicily

(a) Variation with activation characteristic temperature.

1

energy for a fixed

0.01 Non-dlmenslonalised

0.1 reacnon exotherm~~~

(b) Variation with characteristic fixed activation energy.

1

temperature

for a

Figure 6. Ignition limit point bifurcation diagram: the variation of nondimensionalised critical heat flux (Lc,*,) with nondimensionalised reaction exothermicity (9). (a) Parameter values: nondimensionalised activation energy, (1) E* = 32.288, (2) E’ = 64.576, (3) E’ = 96.864. (b) Parameter values: nondimensionalised characteristic temperature, (1) T,’ = 1.9463, (2) TT; = 2.282, (3) T,’ = 2.617. Parameter values not specified are defined in the caption for Figure 1.

heat flux decreases monotonically with increasing sample length and increases monotonically with the degree of nonideality in the experiment. In view of the trend in Figure 8a we consider the ‘worst case’ scenario in Figure 9 and calculated the percentage difference between a sample of length 50 mm (Y’ = 5) and of length 10 mm (Y” = 1): we call this number the relative length number (CR). A length of 50mm was chosen because this is the maximum sample size that can be used in the cone calorimeter. The :sensitivity of the critical heat flux to sample length increases with decreasing activation energy and decrea@ng thermal conductivity. In Figures 10 and 11, we compare the critical heat flux when the heat-transfer coefficient on the rear boundary is 10% of that on the exposed boundary (~2 = 0.1~1) to the case when the rear boundary is perfectly insulated (~2 = 0.0): we call this the noninsulation number (C,). The case x2 = 0.1~1 was chosen as representin,0 the maximum heat-transfer coefficient that can be expected in an experiment that has been designed to be, ideally, perfectly insulated. The noninsulation number, therefore, represents the error induced by nonperfect insulation. Figure 10 shows the noninsulation number increases with decreasing characteristic temperature and decreasing activation energy for materials with high thermal conductivity. The maximum value of the noninsulation number, occurring for a reaction with a low characteristic temperature and low activation energy, is approximately 11%. This variation is less than the typical experimental uncertainty in determining the critical heat-flux under assumed conditions of perfect insulation [17]. Considering that the figure of 11% represents the worst case we, therefore, conclude that for materials with a high-conductivity the effect of nonperfect insulation on the rear boundary is relatively unimportant in the experimental determination of critical he& flux.

190

E. BALAKRISHNANet al.

“‘1

0.1

1

Non-dlmensionalised thermal conductiwty (a) Variation with activation characteristic temperature.

energy for a fixed

(b) Variation with characteristic fixed.

temperature

for a

Figure 7. Ignition limit point bifurcation diagram: the variation of nondimensionalised critical heat flux (L&) with nondimensionalised thermal conductivity (AZ*). (a) Parameter values: nondimensionalised activation energy, (1) E” = 32.288, (2) E” = 64.576, (3) E’ = 96.864. (b) P arameter values: nondimensionalised characteristic temperature, (I) Tz = 1.9463, (2) Tc = 2.282, (3) T,* = 2.617. Parameter values not specified are defined in the caption for Figure 1.

Figure

11 shows the effect of varying

the noninsulation

number.

the thermal

In view of the trends

conductivity

in Figure

and the sample

10, we consider

length

upon

the worst case sce-

nario of a reaction with low characteristic temperature (T,* = 1.946) and low activation energy (E* = 32.388). F’1g ure 11, therefore, represent an upper bound on experimental results. Figure lla shows that the noninsulation number, whilst relatively insensitive to the reaction exothermicity, increases significantly with decreasing thermal conductivity. Figure llb shows that the noninsulation number increases with increasing sample length. Whilst the increase is minor for a material with a high conductivity, line 1, it is considerable for materials with lower thermal conductivities, lines 2 and 3. In fact for lines 2 and 3 the increase in the critical heat flux due to nonperfect insulation is significant at our standard sample size (Y* = 1). We, therefore, conclude that nonperfect insulation can lead to significant variation amongst experimental results for materials with low thermal conductivity.

5. DISCUSSION 5.1. Physically Disjoint and Nonphysically

Disjoint Solutions

Figure 2a shows a physically disjoint solution, curve (l), a nonphysically disjoint solution, curve (a), and an ‘intrinsically nonflammable solutions’, curve (3). The implications of these structure for the risk assessment of combustible materials were discussed in Sections 4.1 and 4.2. It was emphasised that solution curve (3) is important because such materials cannot undergo large changes in their solution norm in response to small changes in the irradiance, they are not ignitable. However, Figure 2a shows that if the irradiance is sufficiently high the solution

Radiative

TC’=l.946

_

TC’&.282 TC’=M,,

----~-. .-

..

Ignition

90

35 -

30 -

1

1.5

4.5 Non~lmens~~Hlised3,,l*m,~b5ampl,

5

;I,ngth

(a) The percentage decrease in the critical heat flux (C,,) with nondimensionalised external sample length (Y*).

0 0 0.2 Nand~menswnalwd

0.4 0.6 heat-transfer cc&iiient

al? on the msulated bun&

(b) The percentage increase in the critical heat flux (ICC,*,)with the nondimensionalised heat-transfer coefficient on the insulated boundary (x;).

Figure 8. Nondimensionalised characteristic temperature, (1) 2’: = 1.946, solid line, (2) TT = 2.282, lower dashed line, (3) T: = 2.617. upper dashed line. Parameter values not specified are defined in the caption for Figure 1.

norm for an ‘intrinsically nonflammable’ material may be higher than that for a flammable material. Although the material represented by curve (3) has not undergone any recognisable ignition event it must be considered as undergoing combustion for sufficiently large values of the irrsdiance. This may raises doubts about the accuracy of classifying risk assessment based purely upon the structure of the steady-state curves. In Figure 2a, the value of the nondimensionalised irradiance at which curves (1) & (3’) inAs a room tersect is C* - 190. In dimensional terms this corresponds to L N 1700 kWm-*. fire burning near its maximum rate can produce irradiances to walls and contents of approximately 150 k W m-* the intersection of curves (1) and (3) is not of practical importance and the distinction between the three types of steady-state curve retains its importance. In view of the calculation in the previous paragraph consider curve (2) in Figure 2s. The ignition limit point occurs at C& N 150, which in dimensionless units is C,, - 1340 k W m-*. Although the steady-state diagram is nonphysically disjoint, so that there is a potential hazard, we conclude that in practice such a material does not pose a critical storage risk. The critical heat flux for mat,erial (2) shows that assessments of risk based purely upon the steady-state structure, Figures 3 and 4, may over-estimate the potential fire hazard. If the maximum irradianct? in a given fire scenario can be calculated then it can be used to produce a more practical classification of materials; into ‘flammable’ and ‘potentially nonflammable’. For example, it has been suggested that samples in the cone calorimeter should be tested at heat fluxes of C = 35 k W mV2 [18] (L* = 3.915), representative of typical peak heat fluxes from limited ignition sources [lQ], and 75 kW me2 [18] (C” = 8.389), rep resenting heat fluxes in a flashed-over room. One might then define the boundary between ‘flammable’ and ‘potentially nonflammable’ as critical heat fluxes of either C,, = 40 k W in-* (C& = 4.474) or C,, = 83 k W mP2 (C& = 9.284), depending upon the particular fire scenario being considered.

E. BALAKRISHNANet al.

35

40

50 63 70 80 Non-dimensionalised actttation energy

90

30 0.01

100

0.1

1

Non-dlmensionalised thermal conductivtty

(4

(b)

Figure 9. The percentage decrease in the relative length number (Lx) with: (a) nondimensionalised activation energy (E*); (b) nondimensionalised thermal conductivity (/c*). In (a), lines (1) and (2) cross at E* N 40. (a) Parameter values: nondimensionalised characteristic temperature, (1) TE; = 1.946, (2) T,* = 3.591, (3) T: = 3.624. (b) Parameter values: nondimensionalised reaction exothermicity, (1) @ = 2.502, (2) Q = 2.502 x lo-‘, (3) @ = 2.502 x 10e2. Parameter values not specified are defined in the caption for Figure 1.

5.2.

The

Dependence

of the

In view of the discussion

Critical

in Section

Heat

Flux

Upon

5.1 it is instructive

Material

Properties

to view Figures

5-7 with

respective

to a classification of materials into ‘flammable’ and ‘potentially nonflammable’. Considering the lower criterion, LE, = 4.474, the only materials that pass this test are found in Figure 6b. From this perspective, the majority of the ‘materials’ represented in Figures 5-7 would be classified as flammable. Figure 5a, 6b, and 7b show that the characteristic temperature is a good indication of fire hazard since increasing characteristic temperature corresponds to decreasing flammability, this trend also occurs in Figure 3. The dependence of the critical heat flux upon characteristic temperature is particular pronounced at low exothermicity, Figure 6b. However, at fixed characteristic temperature, the activation energy is not a good predictor of flammability. Although the critical transition temperatures increase with increasing activation energy, Figure 3, the critical heat flux may either increase or decrease with increasing activation energy, Figures 5b and 6a. Similar behaviour has been exhibited in a critical mass flux model for the ignition of thermally thin thermoplastics [20]. The reaction exothermicity for any material is a constant, determined purely by its chemistry. However, the ‘effective’ reaction exothermicity can be reduced by preparing test samples in which the combustible material is diluted with varying amounts of an inert substance-a process that is used frequently to reduce the flammability of combustible materials. Consequently, the use of reaction exothermicity as a secondary bifurcation parameter to unfold the ignition limit point can be seen as a crude model of the effect of dilution. Figure 6 then has the interpretation that

Radiative

Ignition

193

\

\

(1)

\

\

\

(2)

\

(3)

81.9

2 2.1 22 Nond~mensionalmd

2.3 2.4 charactenstii

2.5 2.6 temperature

2.7

40

50 60 Non-d~mensionalnsed

(a)

70 actvatm

80 energy

90

-

(b)

Figure 10. The percentage increase in the noninsulation number (C,) with: (a) the nondimensionalised characteristic temperature (7”); (b) the nondimensionalised activation energy (E’). (a) Parameter values: nondimensionalised activation energy, (1) E* = 32.288, (2) E* = 64.576, (3) E’ = 96.864; (b) Parameter values: nondimensionalised characteristic temperature, (1) T,* = 1.946. (2) T,* = 2.282, (3) T,” = 2.617. Parameter values not specified are defined in the caption for Figure 1.

the relative flammability ranking of a group of combustible materials may change as they are diluted. In particular, low activation energy materials become more flammable, relative to high activation energy materials, at low exothermicities. 5.3.

The Dependence

In the classic combustion

of the Critical

problem

Heat

an insulated

Flux Upon

boundary

Experimental

condition

Conditions

is used for the slab because

the solution is symmetric about its centre [l]. In Figures 2-7, we assumed that the rear-boundary of the sample was perfectly insulated because this is the aim of the experimental methodologies that we are considering. In Figures 8b, 10, and 11, we modelled nonperfect insulation by a simple heat-transfer coefficient. Figure 11 shows that nonideality in the boundary condition has a very pronounced effect on the critical heat flux for materials with a low thermal-conductivit,y. An alternative formulation of the boundary condition at the rear-end of the sample is to heat-transfer. This combine fully convective heat-transfer (~2 = x1, x; = 1) with radiative models an experiment, or a specific fire scenario, in which the rear-end of the sample is exposed to the same atmosphere as the front-end, with the external irradiance heating only the front-end. We do not address the issue of whether it is better to investigate flammability experimentally though eit,her an insulated rear-boundary or an open rear-boundary. To some extent this d.epends upon the eventual application of the material. When the rear-boundary is perfectly insulated the maximum temperature of the steady-state solution is found at CE*= 1. When the rear-boundary is imperfectly insulated the location of the maximum temperature moves into the interior of the sample. We have not shown the locality of the maximum as it is not important for our work.

E. BALAKRISHNANet al

194

S .E

.E t .-e

z L !?

40 -

EGO-

%a B g @ d

60 -

xlp”

20 -

0 0.01

0

i

1

0.1 .’ Non-dimensionalsed thermal condwtwty

2 2.5 3.5 sarrpletngth 1.5 Nondimensionalised3external

4.5

5

PI

(a)

Figure 11. The percentage increase in the noninsulation number (13,) with: (a) the nondimensional&d thermal conductivity (k’); (b) the nondimensionalised external sample length (Y”). (a) P arameter values: nondimensionalised reaction exothermicity, (1) @ = 2.502, (2) @ = 2.502~ 10-l, (3) @ = 2.502 X 10e2. (b) Parameter values: nondimensionalised thermal conductivity, (1) k’ = 3.333, (2) k* = 3.333 x lo-‘, (3) k” = 3.333 x 10d2. Parameter values not specified are defined in the caption for Figure 1.

6. CONCLUSION We have considered is exposed

the critical

to an external

radiation

storage source,

problem viewing

for a self-heating the irradiance

combustible

material

as the primary

that

bifurcation

parameter. We have shown that such experiments are equivalent to the classical combustion problem where ambient temperature is regarded as the primary bifurcation parameter. In nondimensionalising

our problem,

we have retained

physical

and chemical

constants

terest as distinct bifurcation parameters. Consequently, we have been able to directly the effect of material properties upon the critical heat flux.

of in-

investigate

Steady-state curves have been found which exhibit no two or four limit points. The first of Response curves with two or four these corresponds to an ‘intrinsically nonflammable material’. limit point are classified as being either physically disjoint or nonphysically disjoint depending upon if the extinction limit point is in the left-half or right-half plane. The three solution types, ‘intrinsically nonflammable’, physically disjoint and nonphysically disjoint, correspond to materials with fundamentally different flammability characteristics. It has been shown that the critical heat flux for materials with low thermal conductivity is particularly sensitive to nonperfect insulation on the rear-boundary.

REFERENCES 1. D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics, First Edition, Princeton versity Press, (1955). 2. P.C. Bowes, Self-Heating: Evaluating and Controlling the Hazards, Elsevier, Amsterdam, (1984).

Uni-

Radiative

Ignition

195

3. V. 13abrauskas. Development of Cone Calorimeter-A Benchscale Heat Release Rate Apparatus Bused on the Oxygen Consumption, NBSIR 82-2611, National Bureau of Standards, Washington, DC. (1982). 4. V. Babrauskas, Development of the cone calorimeter--A bench scale heat release rate apparatus based on oxygen consumption, Fire and Materials 8 (2), 81-95, (1984). 5. \‘. Babrauskas and W.J. Parker, Ignitabzlity Measurements ~sth tire Cone Calorimeter. NBSIR X6-3445. Nat ional Bureau of Standards, Gathersburg, MD. (1986). 6. 1’. Babrauskas and W.J. Parker, Ignitability meaSurements with the cone calorimeter. Pirc crnd A,fate?~ls 11. 31-43, (1987). 7. 1’. Balakrishnan, M.I. Nelson and G.C. Wake. Radiative ignition of combustible materials II. Ptrlynrrric tnaterials undergoing nonflaming thermal degradation--The critical assembly problem, (In preparat,ion). 8. t3.1,‘.Gray and M.J. Roberts, Analysis of chemical kinetic systems over the entire parameler space 1. The Sal’nikov thermokinetic oscillator, Proceedings of the Royal Society A 416, 391&402. (1988) 9. 13.1:. Gray and G.C. Wake, On the determination of critical ambient temperatures and critial init.ial ten perattires for thermal ignition, Combustion and Flame 71, 101-104, (1988). 10. .I.(:. Burnell, J.G. Graham-Eagle. B.F. Gray and GC. Wake. Determination of critical ambient tcrnpcr;~turc for thermal ignition. IMA Journal of Applred Mathematics 42, 147- 154. (1989). 11. B.F. Gray. J.H. Merkin and G.C. Wake, Disjoint bifurcation diagrams in combustion systems. Mnthl. C,‘on~p~~f. Modelling 15 (II). 25-33, (1991). 12. 11.1. Nelson, Ignition mechanisms of thermally thin thermoplastics in the cone ralorimctci. Ploceedangs of (hi Royal Society A 454, 789-814, (1998). 13. ‘l’. IIatakeyama and F.X. Quinn: Thermal Analyszs: Fundamentals und Appl~uztions to l’olyrrsw- Sczence. .lohn Wiley & Sons, (1994). 14. AF. Breazeale. Wire and cable performance as determined by a cone calorimeter. In Proceedings of the 37”’ ~nternatianal Wire and Cable Symposium, pp. 536-542, (November 1988). 15. E. htikkola and I.S. WYchman, On the thermal ignition of combustible materials. FVY nrad Al~itersals 14. 87-96, (1989). 16. B A.L. Ostman and L.D. Tsantaridis, Ignitability in the cone calorimeter and the IS0 Ignitability ‘l&t. In Proceedin.gs of Interj%m 90, pp. 175-182. (1990). 17. E. Doedel, ?(. Wang and T. Fairgrieve. AUTO 94 Software for continuation and bifurcation proble,ns in ordinary differemial equations, In Applied Mathematrcs Report. California Institute of Technology. Pasadena. CA, (1994). 18 R.H. Whiteley, Some comments on the measurement of ignitability and on the calculation of ‘critical heat Hux’, Fire Sufaty Journal 21, 177-183, (1993). 19 \s’. Babrauskas and W.J. Parker, Ignitability measurements with the cone calorimeter. F’c~c (end &fn:eria/s 11. 31-43, (1987). Nl3S Monograph, \‘olume 173, 20. \‘. Babrauskas and J.F. Krasny, Fire Behawzow of Upholstered Fwnitwe. (US) National Nureau of Standards, (1985). 21. hI.1. Nelson, .I. Brindley and A.C. McIntosh. The dependence of critical heat flux on fuel and additive propertics: A critical mass flux model, I;‘ire Safety Journal 24 (2). 107- 130, (1995).