Numerical modelling of early flame spread in warehouse fires

ELSEVIER

Fire Safety Journal 14 (1995) 247-278 ~) 1995 Elsevier Science Limited

Printed in Northern Ireland. All rights reserved 0379-7112/95/$09-50

0 3 7 9 - 7 1 1 2 ( 9 5 ) 0 0 0 2 2 - 4

Numerical Modelling of Early Flame Spread in Warehouse Fires

George Grant & Dougal Drysdale

Department of Civil and Environmental Engineering, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JN, UK

(Received 12 December 1992; revised version received 28 April 1995; accepted 5 May 1995)

ABSTRACT

An important European research initiative 'Major Hazards Arising from Fires in Warehouses and Chemical Stores' has recently concluded; Edinburgh University has contributed both experimental and theoretical determinations o f the rate o f upward flame spread on corrugated fibreboard packaging. The theory presented here is a development o f a recently published thermal model o f upward flame spread; the model is in the form o f a Volterra integral equation which is solved for the time-dependent flame spread velocity. The main advance over the previous theory lies in the modelling o f burnout behind the pyrolysis region. The original theory, with no burnout, always predicts self- extinguishment o f the flame front when a realistic, empirical, non-linear flame height correlation is adopted; the inclusion of burnout in the new model overcomes this restriction and allows the flame spread to proceed indefinitely at a steady-state. The numerical solution algorithm of the model is particularly attractive since it permits heat release data from cone calorimeter tests to be used directly as input. The FORTRAN coding of the model includes routines which perform the necessary I / 0 sequences automatically; some minor post-processing of the cone calorimeter data is occasionally required prior to running the model but this too has the potential for future automation.

K

0

NOTATION

empirical constant irradiance (kW) heat release rate (kW)

247

248

t

T V, V(t) x

A

~b

Subscripts b e f i ig O

P 1 ,2 , . . .

G. Grant, D. Drysdale

time (s), independent variable temperature (°C) time-dependent flame spread velocity (m/s) Cartesian co-ordinate (m) increment ignition delay time (s) burnout time (s)

burnout, burner effective value flame ith value in series ignition initial value pyrolysis front sequential values

Superscripts n exponent ' per unit width (m -1) " per unit area (m -2)

INTRODUCTION

If a large warehouse is involved in a serious fire, the detrimental effects are felt both in economic and environmental terms. From the economic perspective, the owners of the warehouse and the stored goods (not necessarily one and the same) have some of their business neutralised while insurance claims are settled. In addition, when insurance com- panies pay out large claims, premiums are inevitably increased, placing an additional burden on industry. The environmental hazard is discernible in two forms; the airborne fire products carried downwind, and the water runoff from fire fighting operations which, if not contained, may find its way into local watercourses. The environmental hazard is particularly serious when the warehouse contains industrial chemicals, especially fertilisers and pesticides for agricultural use. The potential for environmental damage has been recognised by the Commission of the European Communities (CEC) and has been the subject of the 'STEP' (Science and Technology for Environmental Protection) project CT-90- 0096: Major Hazards Arising from Fires in Warehouses and Chemical Stores funded by the European Community.

The potential for serious conflagrations in warehouses is increased

Numerical modelling of early flame spread in warehouse fires 249

by the twin factors of dense storage methods combined with the presence of large quantities of flammable packaging materials. Up to 90% of the available floorspace can be in use for storage and the trend for high-rack storage of palleted loads creates horizontal and vertical flues making for very efficient fire spread. The early involvement of packaging materials (such as cardboard boxes, corrugated paper, paper sacks, polystyrene and polyurethane foam, polyethylene bottles or shrink-wrap plastic) is frequently mentioned in warehouse fire investigation reports. It is therefore desirable to be able to assess the level of environmental threat posed by a fire occurring in a proposed, or existing, warehouse development and it has been recognised that the development of quantitative risk assessment tools are central in achiev- ing this goal. One of the key goals of the STEP project was the development of a flame spread model that could be used to predict the early growth of a warehouse fire under 'typical' conditions. Since the range of possible packaging materials is so diverse, it was decided to implement a flame spread model applicable to a single common packaging material in the first instance. The material selected was corrugated fibreboard, used extensively in a variety of forms by the packaging industry. The analysis was limited to the most critical scenario, that of flame spread in the upward direction.

The spread of fire, whether on individual 'fuel elements' or between various 'fuel elements' (such as furniture) in enclosures, has always occupied a large proportion of the research effort in fire science. Historically, this research effort has been aimed at ranking the fire hazard posed by various materials in various configurations; i.e. developing a rationale for quantitative risk assessment. Despite the existence of numerous 'standard' fire tests, there frequently arises a conflict between the conclusions of the different tests as to the acceptability of a given material.

The need for a rational basis in the assessment of a material's flammability (its potential fire hazard) has long been recognised and the development of mathematical theories of flame spread on individual fuel elements has received much attention. Experimental techniques have also been improved, and the introduction of the cone calorimeter as a new standard bench-scale test method has been of particular significance. Many of the theoretical models require input data which characterise the burning rate of the fuel in question, the cone calorimeter was designed primarily to obtain these transient heat release rate data from small-scale specimens tested either in the horizontal or vertical orientation. The model described here is a numerical implementation of a thermal theory which uses cone

250 G. Grant, D. Drysdale

calorimeter test data to predict the rate of vertical flame spread on corrugated fibreboard.

FLAME SPREAD MODELLING

Friedman, 1 in a review article on flame spread over solid and liquid surfaces written in 1968, observed that, 'no firmly established flame- spread theory in terms of known parameters exists even for the simplest imaginable geometry'. Figure 1 below illustrates the complex heat and mass transfer interactions occurring between solid and gas phases in the vicinity of a pyrolysis front propagating across a solid fuel bed. The particular case shown is that of concurrent, or 'wind-assisted', flame spread since this configuration is the analogue for upward flame spread on a vertical surface; in the latter the 'wind' is driven by the locally strong buoyancy forces.

Friedman ~ identified several elements whose inclusion in a given flame spread theory was deemed to be indicative of the theory's 'completeness'; these included the fuel geometry, heat transfer to the fuel, flame aerodynamics, and chemical phenomena. Much research

Conduction

Edge them layer

Dif

Pyrolysis front i Gas velocity

Fig. L

Heated layer

Schematic of wind-assisted flame spread (after Fernandez-Pello2).


effort continues to be expended on the flame spread problem with numerous theories, experimental data and review articles appearing periodically over the years. 2-2° Critical appraisals of existing theories have been published, 2-4,7,12 and new models continue to be proposed. 5"6'8-H'~3-2° Williams' review of the mechanisms of fire spread 7 identified the flame spread rate Vp as the principal quantity sought by investigators. A strong dependence of Vp upon the fuel thickness was observed and two broad r6gimes were identified. 7

In the case of an infinitely thick fuel, upward spread was predicted to be perpetually acceleratory in nature, while a large but constant spread rate was approached for materials of finite thickness. Williams 7 also identified the time-dependent flame height as an important component in theoretical models of upward flame spread; the significance of this observation is apparent from Fig. 2, where the region of forward heat transfer (pre-heating of the fuel) is defined by the zone xp < x <xf. However it was acknowledged that this simplified heat transfer model would only become practicable with the availability of reliable empirical flame height correlations for the specific case of flames produced by burning walls.

In 1984 Fernandez-Pello 2 reviewed various published flame spread theories and concluded that the relevant controlling physico-chemical mechanisms were well established; the form of the fundamental equations necessary for a rigorous analysis of the phenomenon were

:H

i !! i T x~

Fig. 2.

x:

i::::; ~ /

H

• H:~I: ~

L I x , ,

xj~ t ~

\ m

Upward flame spread with no burnout.

i x,,


likewise well known. Despite this knowledge however, the complexity of the problem was such that a general mathematical solution encom- passing realistic, arbitrary conditions (i.e. incorporating all the material and environmental factors listed by Friedman 1) remained intractable. Progress towards this goal was deemed to be dependent on mathematical and/or phenomenological breakthroughs (e.g. reacting turbulent flows, flame radiation, etc.); in the absence of such advances, it was considered that predictions of the flame spread process would be limited to three forms: 2

• Explicit expressions of the rate of spread, applicable only to limited conditions

• Numerical analyses, providing parametric or case-by-case predictions of the spread process under relatively complicated ge- ometries or environments

• Approximate engineering analyses, making use of experimental correlations or test data to determine the unknown parameters of the problem.

A recent paper by Karlsson 19 exemplifies the engineering approxima- tion method by combining a thermal theory of concurrent flame spread 15 with empirical flame height correlations 2~ and test data obtained from the cone calorimeter. 22 The following sections describe how this technique has been modified for the specific problem of modelling the upward spread of flame over cellulosic packaging materials commonly found in warehouses.

THE KARLSSON MODEL: NO BURNOUT

The main assumptions of Karlsson's model 19'23 are:

• The material is thermally thick, homogeneous and its thermal properties are independent of temperature.

• Chemical kinetics are not considered. • The flame length, xf, is proportional to the rate of convective heat

release per unit width of the flame front, Q', raised to some power.

• Heat transfer from the flame occurs only in the region Xp < x < xf, where a constant heat flux q" obtains (Fig. 2, left).

• Once ignited, all fuel surfaces continue to burn for the duration

Numerical modelling o f early flame spread in warehouse fires 253

of the simulation (i.e. 'burnout ' at the rear of the pyrolysis region is not considered).

Figure 2 above, shows the situation at three consecutive infinitesimal time steps to (ignition), tl, and rE. The dimensions of neighbouring pyrolysis regions (shaded) are exaggerated to indicate schematically the progressive involvement of subsequent fuel elements in the direction of flame spread; the ensuing regression of each pyrolysis region into the fuel bed is in accordance with the 'no burnout' condition.

The upward linear time-dependent velocity of the flame front, V ( t ) (m/s) is written as: 19

where

xf(t) -xo ( t ) d(xp) V ( t ) - - - - (1)

T dt

[ 4(q~ ]-' T = [ z r k p c ( T ~ g - To) 2 (2)

is the characteristic 'ignition delay time' (s) for the material when exposed to a net total heat flux of q" (kW/m 2) from the flame. 24

The time-dependent flame height and pyrolysis front location are given by:

and

xf(t) = K 0 ; ( t ) + xpoQ"(t) + (~"(t - tp )V( tp ) dtp (3)

Xp(t) = Xpo + V(tp) dtp (4)

respectively. Substitution of the latter two expressions into eqn (1) results in an overall expression for the velocity of the pyrolysis front as a function of time,

V ( t ) = l K t) + xpoQ ( t ) + Q"( t - tp )V( tp ) dtp}"

(5)

where tp is the 'dummy variable of integration'. The time-dependent flame spread velocity is found by solving eqn (5), which is an integral

254 G. Grant, D. Drysdale equation of the Volterra type. z~-27 Karlsson 19 showed that it was possible to solve this equation analytically if the cone calorimeter heat release rate data were first approximated by a suitable mathematical function and the flame height correlation was linearised by setting the value of n as unity. While this approach has merits, the numerical solution method to be described permits the direct use of the raw empirical heat release data and the incorporation of a more realistic, non linear (n # 1) flame height relationship. 21

In any event, eqn (5) requires some slight modification before it can be applied to the present problem. In particular, the term Q~(t) denotes the energy release rate (per unit width) from a gas burner which was present during the Swedish testsJ 9"23 Flame spread experiments performed in Edinburgh on corrugated fibreboard did not employ such a device; the 'ignition source' was simply the lowest 0.02 m of the test sample which had methanol applied to it prior to ignition.

Thus the burner contribution is not relevant here and a constant value for xpo of 0.02 m has been adopted for all numerical simulations reported hereinafter. The modified general expression for the flame spread velocity in the absence of the gas burner is, therefore,

V(t)--7;I [K{ xp°(2"(t)+ fo (~"(t-tp)V(tp)dtp) n- (Xpo+ fo V(tp)dtp)]

(6)

which has been solved by numerical methods 27-28 (results are presented in a later section).

FLAME SPREAD MODEL WITH BURNOUT

The flame spread equation (6) does not accurately represent cases where the timescale of pyrolysis is significantly less than the timescale of the flame spread over the material (i.e. when 'burnout' is significant). Thomas 2° introduced an expression for the velocity of the burnout front which is analogous to the progression of the pyrolysis front in the flame spread equation (1). In the present notation, Thomas' expression may be written thus,

Vb(t) -- xo(t) -- Xb(t) -- d(xb____~) (7) Tb dt

x~

Numerical modelling of early flame spread in warehouse fires

:ili!iiii!iiiiiii!ii!iii

~q, "1@

I!ii !i Fig. 3.

x,

A

: : , : H

ii~-:i ' i i i i i l ; i

iiiiiiiii!iiii :!

?? : . L , , I , . .

: : : , : ' : ' .

I _g_ x., xh21

Upward flame spread with burnout.

255

I

Xp2

where Zb is a characteristic 'burnout time' for the material. Figure 3, above, shows how the additional presence of burnout affects the flame spread process depicted previously in Fig. 2. Again, the shaded pyrolysis regions indicate schematically the progressive involvement of subsequent 'fuel elements' in the direction of flame spread; the consecutive stages of spread shown are each separated by a time period % and again the dimensions of the active (and burned-out) fuel elements have been greatly magnified for the sake of clarity. The regression of the pyrolysis region into the fuel bed, however, is now limited to a constant depth by the burnout condition; it can also be seen that the lower limit of the flaming region is no longer indefinitely located at the base of the fuel, but is dynamic in nature, corresponding to the instantaneous position of the lower limit of the pyrolysing region. The correct incorporation of this new condition in particular represents a significant enhancement of the model described by eqn (6) above.

Now with reference to eqn (1) and Fig. 3, it is evident that for upward flame spread with a constant burnout time,

xb(t) = Xp(t - Zb) (8)

defines the time-dependent location of the burnout front. It is also clear that for to <- t < Zb, prior to burnout of the ignition region Xpo, eqn (6) applies without modification and can be used to predict V ( t ) for all intermediate states between the leftmost and centre diagrams in Fig. 3.

2 5 6 G. G r a n t , D . D r y s d a l e

In the longer term however, for t - zb, a new description of the process is required.

From Fig. 3 and the previous discussion of flame height correlations, we may write,

xf(t) - Xb(t) = K(Q'( t ) ) ~ (9)

or by substitution from eqn (8) above,

x~(t) - Xp(t - zb) : K(Q_'(t)) ~ (10)

as the modified general expression relating flame height and heat output in the model for t - Zb.

NOW, making xf(t) the subject of eqn (1) and substituting for Xp(t) from eqn (4) gives,

L' xf(t) = rV(t) + Xpo + V(tp) dtp (11)

and by a similar substitution for xf(t) into eqn (10),

Xp(t) - xp(t - zb) + zV(t) = K((2'(t))". (12)

Introducing an integral equivalent for Xp(t)- Xp(t- 'rb) in terms of the unknown flame spread velocity and rearranging, we obtain

V ( t ) - I [K{Q'(t)}" - f ' V(tp)dtp] (13)

flame spread equation for the case which includes as the modified burnout, where

~l t Q'(t) = (~"(t - tp)V(to) dtp (14) - Z b

is the instantaneous heat release rate per unit width of the burning wall. From the above discussion, it is clear that even when the effects of

burnout are present, a general flame spread model must initially solve the expression,

V( t )= ~[K{xpoQ"(t )+ fo 'Q"(t- tp)V(tp) dtp}"- (xpo+ Ji'

when t < ~:b, and thereafter the modified form,

v tp, tpl] (6)

V(t) - 1 K (2"(t - tp)V(tp) dtp - V(tp) dtp - - "C _ ~, _ ~

for all t -> ~'b.

(15)


Once the time-dependent velocity of the flame front is known, it is a relatively simple task to perform back-substitution into the relevant integral terms in order to calculate the other dependent properties such as the locations of pyrolysis and burnout fronts and the flame height, if required; it is also feasible to calculate the total transient heat release rates per unit width of the burning surface and thus quantify the fire growth.

Press et al. 27 note that the numerical solution of integral equations is often regarded as an 'extremely arcane topic' by people who are 'otherwise numerically knowledgeable'. They proceed to give a com- prehensive discussion of the implementation of a solution method for linear Volterra equations, based on numerical quadrature. If the value of the power n in the flame height correlation is assumed to be unity (as in Karlsson's analytical solution 19) then this approach is acceptable. It has been found however, 21 that the best representation of experimental flame height data is obtained by a correlation in which n = 3z; the presence of this exponent in the flame spread equations introduces a marked non-linearity. The solution of such non-linear cases requires some additional effort on the part of the modeller in order to couple a non-linear root finding scheme with the basic numerical integration algorithm used to solve linear Volterra equationsY -2a Once the flame spread velocity has been determined as a function of time, it can be substituted back into the original equations in order to calculate the position of the pyrolysis front (and hence the rate of area increase of the fire). Also, by combining V(t) with the original heat release curve Q"(t) and integrating time-wise over the surface area sequentially involved in the fire, the fire growth in terms of heat output Q'(t) can also be calculated.

A flame spread program has been written (in FORTRAN) which performs these calculations using the heat release data obtained from cone calorimeter tests; the experimental data files are interrogated automatically at the start of the program in order to determine the ignition delay time r, the timestep value (At) and the time-dependent heat release data. The program incorporates a non-linear root finding scheme within the basic solution algorithm for linear Volterra equations; 27-28 this allows the inclusion of the n = ~ empirical flame height correlation derived by Tu and Quintiere. 21 It is not the intention of the present paper to provide a detailed tutorial on the implementation of the numerical scheme; this will be the subject of a follow-up article in the near future. The crucial point is that integral terms featuring Q " ( t - tp) are computed from actual neighbouring values of Q"(t) on the heat release rate curve (e.g. Fig. 9); thus all the transient features


of the data are preserved since functional approximations to the curve are not necessary. It is, however, instructive to illustrate how the initial flame spread velocity V(0) is calculated, since this is simple to do and is of particular relevance to some of the later discussion. It has already been demonstrated that regardless of burnout, the early stages of upward flame spread are characterised by eqn (6); in particular, when t = 0 the initial flame spread velocity is defined explicitly by

V(O) = 1 [K{xpoQ"(O)}" - xpo] (16) T

where V(0) is readily calculated by hand, in contrast to the values of V ( t ) at later timesteps--the extensive nature of the integral computations is better suited to machine calculation. The following section describes how the various empirical input data were obtained from cone calorimeter tests; an additional series of medium-scale experiments was also conducted whose objective was to measure the actual upward flame spread velocity attained in practice. This work was conducted in parallel with the model development; the virtue of this approach was that the theoretical predictions could be immediately compared with a database of physical observations of the phenomenon (i.e. the model could be validated).

(Kokkala and Baroudi 29 have also implemented a numerical solution of Karlsson's model, 19 using a conditionally stable forward Euler method; their model considered only the no-burnout case and retained the idealised flame height exponent of unity. These assumptions enabled the numerical procedure to be initially verified by comparison with Karlsson's analytical solution; ~9 the subsequent predictions of this numerical model were found to be in generally good agreement with experimentally determined heat release rates obtained from burning vertical walls of various wood products.)

FLAME SPREAD MODELLING: EXPERIMENTAL MEASUREMENTS

The experimental data which were gathered fell into three distinct categories: cone calorimeter (HRR) data, flame spread velocity valida- tion data and characteristic heat flux data.

Cone calorimeter experiments

It was indicated previously that the current investigation has been limited to the study of flame spread on corrugated fibreboard, but even this 'simple' material is supplied to industry in many forms, to suit


,VAVAVAVAVAVAVAVAVAVAVAVAVAVAVAVAVAVAV~I ~ 'E' Flute &T&VAW&T~kVAT

k V A V A V A V A V A V A V A V A V A V A V A V A

I AA/YAAAAAA

~ V A V A V A V A V A V A V A V A V A V A V A V ~ AkVAVAVAVAVAVAVAVAVA

Fig. 4.

'B'(F) Flute

'C' Flute

' F C ° Flute

Corrugated fibreboard typical cross-sections.

259

particular packaging requirements. Figure 4 above illustrates three common forms of single wall construction and one double wall combination. Single wall types are defined by the height of the 'flute' (or corrugation) and the number of flutes per metre width of the board; the corresponding nominal values for the E, F and C flute boards shown above are: 2mm/315, 3mm/160 and 4mm/135, respectively. The finished single wall board consists of a sheet of fluting fixed between two outer layers of 'liner' material using an adhesive whose principal ingredient is starch (obtained from maize and wheat).

In double wall packing cases (e.g. constructed from the FC flute board shown in Fig. 4), it is conventional for the thinner board to form the external surface; it is most important that the presence of both the internal and macro-scale anisotropy is appreciated during the execution of cone calorimeter test work and that samples be prepared accord- ingly. Figure 5 below illustrates the concept of the flame spread modelling/cone calorimeter testing adopted during the present study;

Fig. 5. Cone calorimeter test methodology for upward flame spread.


it should be noted that the test sample is in the vertical orientation, with the corrugations also running vertically. It can be seen that the test sample depicted is of the single wall type; in the case of double wall specimens the thinner board was exposed to the external heat flux q". All tests were performed in accordance with ISO 5660, 22 three replicate samples being exposed to a range of incident heat fluxes while supported in the vertical sample holder.

Of the standard reaction-to-fire measurements made, the time to ignition r and the heat release rate history were the most important since these data appear directly in the flame spread equations (6) and (15). The data were sampled at 1 s intervals to provide a high temporal resolution and minimise the timestep At in the numerical integration scheme. Supplementary measurements were also made during the standard tests to measure the burnout time ~b and the surface temperature at ignition, T~ r The former was determined simply by manual timing of the duration of flaming after the point of ignition. Surface temperatures were monitored automatically using a fine wire (0.05 mm) chromel/alumel thermocouple located just under the surface, at the centre of the sample face and connected to data logging equipment. Using this method a mean value for T~g was established as 345 °C.

Figure 9 illustrates the general form of a typical heat release rate curve obtained for FC flute corrugated fibreboard (in this case, exposed to an irradiance of 20 kW/m2). Table 1 and Fig. 6 below present the ignition delay data obtained from the cone calorimeter tests and confirm a thermally thick behaviour for the material; 24 therefore the use of Karlsson's theory 19 is justified for this material.

Upward (concurrent-flow) flame spread velocity measurements

Experimental studies of concurrent-flow flame spread are difficult; 2 in general the dynamic behaviour of the pyrolysis front is obscured by the flames themselves, making direct visual observations difficult. The experimental arrangement shown in Fig. 7 was used to estimate the flame spread rate for medium-scale samples (1300mm x 230mm) of double wall FC flute corrugated fibreboard. Since direct measurements were impossible, the method involved inferring a flame spread velocity from the time delays measured between the successive heating of thermocouples located at the fuel surface.

Initially a sample of the fibreboard was mounted on a Kaowool back- ing board as shown, held in place at the edges by two 30 mm wide steel strips (not shown) running the full length of the sample. Holes were


1 ~(s"O

0.413

0.35

0.313

0.25

0.20

0.15

0.10

0.05

0.00

0 I I I i I I I i 5 10 15 20 25 30 35 40

4" (kw/m2)

Fig. 6. Thermal ly thick ignitability correlation for FC flute corrugated.

then drilled through both materials on the vertical centreline at 80 mm intervals to accept the thermocouples; a total of 16 type K thermocouples were installed so that the welded tips were just flush with the front surface of the fibreboard. The lowest approximately 2 cm of the sample was dabbed with methanol and a match was applied to ignite the sample along this strip.

The time taken from ignition until flaming ceased at the top of the sample was, in general, 4½ minutes. The observed burning pattern was as shown in Fig. 7 where the charred and smouldering vertical

T A B L E 1 Ignitability Data for FC Flute Corrugated

~!" (kW/m2) • (s) 1/V~z (s -1~)

40 8-0 0-354 35 9-7 0"321 30 15-0 0"258 25 23"0 0"209 20 42.0 0" 154 15 115"3 0"093


Steel support frame

Corrug. ~

pwool board

Thermocouple output (I 6 No.) to data logger

Fig. 7. Upward flame spread experimental arrangement.

corrugations were visible behind the burnout front and took no further part in flaming combustion. At the end of each experiment, plots of temperature against time were generated for each of the 16 thermocouples; these were then carefully analysed together with the timed visual observations and the following conclusions were reached.

• In the majority of cases the highest recorded temperature was associated with the arrival of the pyrolysis front.

• The maximum (pyrolysis front) temperature was usually the second peak in the temperature- t ime curve for an individual thermocouple, due to the earlier presence of upward spreading flames adjacent to the unburned fuel surface.

• The maximum temperature associated with the pyrolysis front was in the range 345 °C to 600 °C.

It was assumed that the arrival of the pyrolysis front was associated


with a thermocouple reading of 345°C (equal to the mean T~g established during cone calorimeter tests). The data were analysed and the time delay between successive thermocouples attaining 345 °C was measured; knowing the separation of the probes enabled a flame spread velocity to be inferred. The data were fitted by a least-squares regression and estimates of upward flame spread velocity for this material were found to be consistently around 0.006 to 0.009 m/s.

C h a r a c t e r i s t i c h e a t f lux da ta

The final phase of the experimental study was designed to quantify the forward heat flux to the surface of the unburned fuel ahead of the pyrolysis front [4" in eqn (2)]. The apparatus was similar to that in Fig. 7, with a single Gardon-type total heat flux meter installed flush with the front surface of the sample. The output from the heat flux meter was recorded from the moment of arrival of the flame tip until the arrival of the pyrolysis front, when logging ceased. The results of five tests are shown in Fig. 8 below and from these a representative value of 20 kW/m 2 was adopted for the flame spread calculations to be described in the following section. This agrees with Hasemi's measurements of imposed heat flux in the region of solid flaming 14 although Delichatsios et al. 17 favoured the higher value of 30 kW/m 2.

q"(kW/m 2)

50"

45.

40.

35.

3o2

25.

20.

15-

10-

5-

0 0

Total heat flux meter

~, 20 kW/m' ~ - ~ - - "~.- - - ' ~ - - ' ~ - 7 ~- . . . . . . .

.' .. :;":'.-, _V ' : / : , - - -

i i i | i | i

10 20 30 40 50 60 70

time (s)

Fig. 8. Dynamic heat flux experimental data.


N U M E R I C A L F L A M E SPREAD MODEL: RESULTS AND DISCUSSION

The first cone calorimeter data to be used in the numerical model were from a test with cy' = 20 kW/m 2, corresponding to the total heat flux condition observed ahead of the flame front during the medium scale upward flame spread tests (Fig. 8). Figure 9 below shows the HRR data used, obtained from a cone calorimeter test (vertical orientation) performed on 'FC flute' corrugated fibreboard. The inset figure shows the form of the entire heat release rate curve for the test (which lasted 225 s). The shaded region of the curve, from 0 to 50 s has been enlarged to form the lower area of the diagram. The moment of ignition as recorded during the test was 40 s; however the numerical model failed to run correctly when the raw data were used, producing a negative value for the initial spread velocity V(0). The reason for this problem is apparent from eqn (16) where a finite, positive value of V(0) is produced only when the difference expression within the square brackets is also positive. Two possibilities suggest themselves to remedy the situation; either adjust Xpo or alter the observed ignition time to increase the value of Q"(o) used in eqn (16). The latter strategy has been adopted here, in all cases where this problem arose; the value of the ignition delay time z was increased in i s steps until the flame spread model produced a positive value of V(0) for Xpo = 0-02 m.

Q"(t) (kW/m 2)

120 ~40

12o

8O

60 20

40 0 0 2(

|19.9

/ in flame spcelgl model) /

20 40 60 80 100 120 140 160 180 200 220

FCV014: HRR ~ for fc-flule co m l g ~ tested at an irradiar.ce o f 20 kW/m'

20 16.1

0 5 10 15 20 25 30 35 40 45 50

time (s)

Fig. 9. FC flute corrugated fibreboard HRR curve (c]" = 20 kW/m2).


v(o (m/s)

0.050

0.045

0 . 0 4 0

0 . 0 3 5 •

0 . 0 3 0 -

0 . 0 2 5 •

0 . 0 2 0 •

0 . 0 1 5 •

0 . 0 1 0 .

0.005 J

0.000

-- - cq. (6): n=l [No Btmaout]

.... ©q. (6): n'=2/3 [No Btanout]

eqs. (6) & (15): n=2/3 [WilJa Btmaout]

I I ! I I I I I

2 0 4 0 6 0 80 I 0 0 120 140 160

Fig. 10.

time (s)

Comparison of flame spread velocity predictions for FCV014 data.

265

Thus, from Fig. 9, the modified value of ~" = 43 s was used, corresponding to a (2"(0) value of 16.1 kW/m2; the heat release rate at the recorded ignition time of 40 s was much lower (2.6 kW/m2), resulting in a negative V(0). It is evident that the ignition conditions are critical, however once a positive initial velocity is obtained, the progression of the flame front is thereafter calculated from eqns (6) and (15) with K = 0.0666 and n = ~ as given in Tu et al. 21 Figure 10 above compares some general predictions of the current model with those of Karlsson's model 19 (discussed recently by Thomas2°).

Equation (6) has been solved for two different values of n; when n = 1 (the case solved for analytically by Karlsson19), the flame spread velocity increases rapidly with time and the magnitude of the velocity soon becomes infeasible (in this case, a value of 1.8 m/s is predicted at 60 s, rising to 10.3 m/s at 75 s after ignition). In contrast, with n = ~, the magnitudes of the predicted velocities are much more credible, with a maximum of around 0.0072 m/s (or 7.2 mm/s) being predicted after a steady rise in velocity. Unfortunately, this model subsequently predicts (erroneously) deceleration of the flame front velocity, leading ultimately to self-extinction in less than 3 min. Experimental determinations of V(t) for this material were consistently around 0.006 to 0.009m/s (and fairly constant), with no self-extinction observed during a 4 min test.

Despite the obvious conflicts with the observed flame spread behaviour, the predictions of the current numerical implementation of


Q"(t) (kW/m z) 180-

160-

120-

Z o i

1 4 0 . .o .

100- ' . l ,-~

80. I I

60. I f

4 0 . ! ! ! 20.~ ~:

0 0 20 40 60

Cone ~ HRR d u t FCV002:q,'~40 kW/m 2

. . . . FCV007:q;=)0 kW/m"

-- -- FCV014:q,~20 kW/m z

" . . . ,

80 100 120 140 160 180 200

time (s)

Fill. 1L Comparison of various HRR data used for modelling.

eqn (6) are in agreement with Thomas' discussion of the analytical solution, 2° including the tendency of the model to predict self- extinction. The inclusion of burnout [equation (15)] has a dramatic effect, in this case an initial acceleration is predicted, leading to a steady-state constant spread velocity of around 6-6 mm/s; this modified behaviour is also in accordance with Thomas' expectations for a model incorporating burnout.

Following this encouraging initial trial, the numerical model was used to predict the upward flame spread rate for scenarios where a significant level of external irradiance was imposed on the fibreboard packaging (i.e. thermal radiation from adjacent hypothetical burning vertical surfaces). Figure 11 above compares the three HRR curves selected, corresponding to typical cone calorimeter data obtained at q" values of 20, 30 and 40 kW/m 2. As expected, the ignition delay time is reduced at higher heat fluxes. Also, a double heat release peak feature becomes more pronounced although the maximum magnitude of Q"(t) remains consistently in the range 140-170 kW/m 2. The (modified) ignition point necessary to yield a finite and positive V(0) is indicated by a horizontal arrow head at the left of each curve and the corresponding burnout point is shown by the inclined arrow at the right hand side; Table 2 includes some specific data from Fig. 11.

The predictions of flame spread velocity as a function of time are shown in Fig. 12 below; the time of burnout of the ignition region Xpo is also indicated on each curve.

Three observations can be made immediately.


TABLE 2 Ignitability Data for FC Flute Corrugated

267

Cone test q: (kW/m 2) • (s) Q"(0) z~ (s) (modified) (kW/m 2)

FCV014 20 43 16.1 86 FCV007 30 19 19.3 73 FCV002 40 10 15.4 65

• All curves show an initial acceleratory spread phase followed (in the longer term) by a steady-state phase; higher imposed heat fluxes result in a greater magnitude of early flame spread acceleration and also of the final steady-state flame spread velocity.

• All curves exhibit a slight, but discernible, discontinuity in the V(t) function at the moment of burnout of the ignition region Xpo; however the overall behaviour is not affected significantly and the predictions do not appear to be seriously disturbed in the longer term.

• An oscillatory (or transition) phase is seen to occur between the point of maximum V(t) and the attainment of the final steady- state value; the magnitude of the oscillations appears to be correlated with the magnitude of the imposed heat flux q".

Notwithstanding the above, the flame spread velocity data were

v(t) (m/s)

0.06.

0.05,

0.04,

0.03

0 . 0 2

0.01,

0.00 0

eqs. (6) & 0 5 ) : ,=2/3 [With B u r n o u t ]

- - FCV002:q,'=40 kW/m2

. . . . FCV007:q~=30 kW/m2

- - FCV014:q.%20 kW/m2

7~=73s / ..... . /

;° -~ / c

/ ~=86s

I ! I I ! !

1 ~ 2 ~ 3O0 4OO 5 ~ 6OO

time (s)

Fig. 12. Flame spread velocity predictions for various HRR data.


x.(t) (m)

30

25

20

15

10

5

0

©qs. (6) & (15): n=2/3 [With Burnout] FCVO02:,~,=40 kW/m~

. . . . FCV007:q~,=30 kW/m2

1 O0 200 300 400 500 600

time (s)

Fig. 13. Pyrolysis front progression plot using V(t) data.

back-substituted into the appropriate integral expressions in order to calculate the time-dependent variations in pyrolysis front location (height above base of fuel) and the total heat release per metre width of burning fuel. The results of these supplementary computations are shown in Figs 13 and 14 respectively.

Since the pyrolysis front position is strongly correlated with the functional behaviour of V(t), a region of unsteady progression is

Q'(t) (kW/m)

400

350

300

250

200

150

100

50

0

/I---ii;oV iiii:' iiw

!

l~ 2~ 3~ 4~ 5~ 6OO

time (s)

Fig. 14. Heat release rates per unit width using V(t) data.


observed initially; in the longer term the rate of progress attains a constant value which is greater for higher imposed heat fluxes. Figure 13 also serves as a useful indicator of the rate of upward flame spread in the high-rack storage environment; e.g. for an imposed heat flux of 40 kW/m 2, the pyrolysis height reaches 10 m above ground level after approximately 4 min following ignition at low level (Xpo = 0.02 m). The heat release curves in Fig. 14 are broadly similar in appearance to the V(t) curves in Fig. 12, but there appears to be a 'phase-lag' in the peaks and troughs during the oscillatory phase. In the longer term, the highest steady-state Q'(t) values are associated with the highest imposed heat fluxes.

The results presented above gave confidence that the numerical solution method was robust enough to utilise disparate experimental heat release data obtained from cone calorimeter tests, albeit that in some cases there was a requirement to refine the ignition conditions. Moreover, it was confirmed that the inclusion of burnout overcame the self-extinction restriction of the original model and thus permitted long-term predictions of flame spread to be made. It still remained to explain the transients present in the V(t) curves of Fig. 12 however, and the additional V(t) curves shown in Fig. 15 were generated in an effort to assist in this task.

Figure 15 is essentially a parametric study of the major influences on flame spread velocity as identified in the preceding sections. A 'base case' has been assumed, corresponding to the solid curve in the centre

v(t) (m/s)

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ W / m ~ : z=lOs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I I I I I I

100 200 300 400 500 600

time (s)

Fig. 15. Parametric comparison of V(t) predictions for various data.


of the diagram; this is identical to the uppermost curve in Fig. 12, where the V(t) function is calculated from the solid H R R curve in Fig. 11 (cone calorimeter test FCV002, q~ = 40 kW/m 2) and where the oscillations are particularly pronounced. The ignition delay time in this case is 10 s, with a corresponding burnout time of 65 s; the much flatter dashed line directly above this curve was generated from the same input data except that the burnout time was decreased to 35 s. The shorter burnout time was selected in order to study the effect of omitting the second peak in the FCV002 heat release data (again see Fig. 11). The uppermost pair of curves in Fig. 15 were generated using a hypothetical H R R curve with a constant heat output rate of 140 kW/m2; this value was adopted since it approximated the value of the first peak in the base case (FCV002) H R R data.

Again, the solid curve was obtained with ~'b = 65 s and the dashed line with ~'b = 35 s; however both curves retained the previous ignition delay time (r) of 10 s. Finally, the lower two curves were also generated using the conjectural H R R curve as the input, but with an ignition delay time of 43 s (chosen to reflect the characteristics of the FCV014 data in Fig. 9); as in the other cases the different line styles correspond to the two different values of "rb.

It had been supposed initially that the oscillations were in some way associated with the double peak feature noted in the heat release curves of Fig. 11, and this influenced the choice of input data used for the construction of Fig. 15. As explained above, the central 'base case' solid line has been taken from Fig. 12; the dashed line immediately above shows the effect of reducing the burnout time from 65 to 35 s. It can be seen that the steady-state is attained more quickly and the undulations, though still present, are very much reduced; the steady-state velocity is also somewhat greater than for the base case. These results were initially thought to support the belief that undulations in the HRR curve were influencing the predictions of flame spread velocity in the short term. However, the upper pair of curves in Fig. 15 are seen to exhibit identical characteristics to the central pair of curves just described (albeit with greater steady-state velocities), despite being obtained with a 'flat' (i.e. constant) H R R curve.

The latter observation effectively disproves the notion that 'waviness' in the V(t) curve is correlated with a similar attribute in the H R R data used as input to the model. The lower pair of curves (obtained using the same constant H R R curve but with increased ignition delay time) contain no oscillations; both curves show a gradually accelerating flame front which smoothly reaches a steady-state velocity (of relatively low magnitude) after some 100 s.


Given the range of steady-state velocity values manifest in Fig. 15, coupled with the presence (or otherwise) of short-term oscillatory effects, the results were re-appraised with the intention of developing a more subtle explanation for the nature of the solutions obtained. In the first instance, it is instructive to consider the mechanics of the steady-state situation, where in all cases V(t) attains a constant value for large t. In the steady-state when V(t) is constant, the pyrolysis front advances a fixed distance in unit time; consequently, all 'fuel elements' are of equal length and once ignited each burns for a fixed interval "cb. The combination of a fixed rate of advance of the flame front coupled with a constant burnout time results in a constant length of pyrolysis zone [xp(t)-xb(t)]. At any instant, each fuel element (i) within this zone contributes an incremental amount AQ~ to the heat output per unit width of the flame; this incremental quantity is defined as the product of the element length times the appropriate value of QT, obtained from the cone calorimeter HRR curve. For typical HRR signatures (see Fig. 11), Q;' is strongly time-dependent and hence the position of the element within the pyrolysis zone is significant, since the discrete burning time is shorter nearer the pyrolysis front and longer closer to the 'burnout front' at the base of the pyrolysis zone. Therefore in general, despite their equality of size, each element contributes a different AQ~; nevertheless, the steady-state sum of the individual contributions [Q'(t)] remains a constant (Fig. 14).

The oscillations observed in Fig. 12 clearly represent the successive acceleration and deceleration of the flame front and these unsteady effects can also be explained in terms of the above rationale. It is apparent from eqn (1) that acceleration of the flame front occurs when the value of the numerator increases with time; conversely, a decelerat- ing flame front occurs when there is a timewise reduction in x f ( t ) - xp(t). The most conspicuous r6gime of acceleratory spread is observed during the initial phase of the simulation immediately after ignition. At ignition, following the calculation of V(0) from eqn (16), the pyrolysis front advances at this velocity for At = 1 s, involving the next fuel element as described above; the latter then releases heat at the rate Q"(O), corresponding to the ignition point on the cone calorimeter HRR curve. Meanwhile the rate of heat release from the original ignition zone (Xpo) is now correlated with the value on the HRR curve at 1 s after the ignition point (since it has been burning for 1 s). The HRR curves in Fig. 11 all exhibit a rapid increase in heat output immediately after ignition and consequently the instantaneous magnitude of Q" associated with each fuel element will also increase rapidly during the first 20 s or so after ignition.


The cumulative effect of increasing the number of burning fuel elements coupled with the monotonic increase in HRR is to increase the total instantaneous heat release rate from the upward spreading flame. This in turn results in successive increases in the calculated flame height xf which tends to increase x f ( t ) - xp(t) and therefore V(t). As successive V(t) values are increased, so too are the lengths of successive fuel elements which become involved in the fire; this additional effect increases the burning area of the element and further enhances the AQ/ contributions to the total flame HRR. The constant element lengths shown in Figs 2 and 3 are therefore indicative only of the steady-state process and are not an accurate representation of the early stages of flame spread. Another consequence of the accelerating pyrolysis front is that Xp increases rapidly with time; whilst this tends to reduce the magnitude of xf(t)- xp(t), it is apparent that the timewise increase in flame height dominates the early stages of flame spread and necessitates an accelerating pyrolysis front.

During the initial acceleration, the position of the pyrolysis front becomes increasingly more significant, as xp advances, the rate of increase of V(t) (i.e. the acceleration of the pyrolysis front) is reduced. The difference xf(t)-xp(t) is still increasing with time, but the percentage increase over the previous value is continuously being diminished; this is demonstrated well by the gradual change in slope of the curves in Fig. 12 as the maximum value of V(t) is approached. At the point of zero gradient, xf(t)- Xp(t) reaches a maximum value and consequently V(t) also peaks at this time. The ensuing deceleration phase is characterised by the dominant effect of the advancing pyrolysis front, here the rapid decrease in xf(t)-xp(t) is equated with a correspondingly rapid attenuation of V(t).

During this phase the flame height increases much more slowly, for a number of reasons:

• The 'no-burnout' condition of eqn (6) forces the base of the flame to remain coincident with the base of the fuel (Fig. 2), so that xf is always measured from this point. When the dimension of the pyrolysis zone becomes comparable with the flame height, predictions of the latter will tend to become unreliable, since the xf correlation is based upon point or line-source plume theory; 21 this may cause xf to be underestimated.

• The discrete energy contributions AQ/of the fuel elements in the lower region of the pyrolysis zone will tend to diminish with time as their instantaneous Q" values become associated with intermediate and decaying r6gimes on the HRR curve (Fig. 11).


The newly-involved fuel elements in the upper region of the pyrolysis zone are of successively smaller lengths as a consequence of the gradual reduction in V(t); therefore their energy contributions AQ~ also tend to be reduced despite being correlated with the early fire growth region of the HRR curve where Q" values are on the increase.

The retardation in V(t) continues until the 'no burnout' model [eqn (6)] ultimately predicts self-extinguishment at the point when xf( t ) - Xr,(t ) = 0 (see Fig. 10). In the case where burnout is allowed however, a discontinuity occurs in the V(t) function at t = ~'b; this is well seen in the uppermost curve in Fig. 12. This discontinuity defines a point of inflexion and inspection of the curve shows that the gradient thereafter becomes less steep as time progresses, indicating a gradual reduction in the rate of deceleration of the pyrolysis front. During this phase the rate of reduction of x~(t)-xp(t) decreases due to the additional presence of burnout and in particular its effect on the behaviour of xf. Although the flame front continues to advance, sequential burnout of the lower fuel elements results in the continual re-positioning of the flame base to the bottom of the lowermost burning element (Fig. 3); this modification overcomes the limitation of the 'no-burnout' model and allows xf to be determined more realistically.

The increasing importance of this effect is due to the burnout of increasingly longer fuel elements generated during the initial acceleration of the flame front; eventually another point of inflexion is reached, where xf( t)- xp(t) reaches a minimum value. Subsequently the curve displays another acceleratory phase in V(t), caused by burnout of increasingly larger elements which temporarily re-establishes the domi- nance of xf in eqn (1); once again the knock-on effect is to generate burning elements of disproportionate length.

However, the acceleration is checked once more as the burnout front approaches the shorter elements produced during the initial deceleration, prior to the first maximum point; once again the relative importance of changes in xf and xp is reversed and a second maximum (of lower magnitude) follows. The deceleration-acceleration cycle is then repeated, but the tendency is for all the burning elements to attain a characteristic equilibrium length since the hereditary nature of the process effectively 'damps out' the fluctuations in V(t).

Notwithstanding the above, it is notable that the oscillations are more severe for high heat release rates, and short ignition times; i.e. conditions which maximise the early acceleration of V(t) and therefore the fuel element lengths. Under such conditions the subsequent


transient effects caused by burnout are also amplified. It is the opinion of the authors that the source of the oscillations lies in the numerical prescription of the problem rather than in physical reality; in this case a smoother transition phase would be obtained with a shorter integration step. A definitive test of this assumption requires modifications to both the FORTRAN coding of the flame spread solution and the cone calorimeter HRR data; this work is currently being pursued.

CONCLUSIONS

Williams 7 noted that 'Theoreticians often attempt to include all potentially important phenomena in their models of fire spread, believing that they cannot properly describe the process if something that contributes is neglected. Nevertheless, there is merit to the opposing view which holds that the best avenue for developing understanding is to neglect all but essential phenomena and to study thoroughly limiting cases in which different phenomena are controlling.' The attraction of 'fundamental' theories which encompass small scale fluid dynamics, chemistry, diffusion, etc. is that they are deemed to be 'rigorous'; the attraction of simpler (e.g. thermal) theories is that they require less empirical input data, are more accessible and are thus more likely to find broad usage. Nearly thirty years since Friedman's review of flame spread, 1 there is still 'no firmly established flame spread theory. . . ' . Given the often daunting appearance of journal articles on flame spread, it is perhaps not surprising that theoretical models have been applied only in the research environment; consequently it is extremely unlikely that a given model has ever been solved outside its place of origin. The present study has been applied to a practical fire hazard problem; the main achievements of this study are summarised below.

A state-of-the-art flame spread model, employed during the EUREFIC (EUropean REaction t o Fire Classification) research programme, has been modified for use as a quantitative risk assessment tool applicable to the particular case of warehouse fires. The application of the published model had previously been restricted to conceptual scenarios characterised by idealised cone calorimeter heat release data; this approach enabled an analytical solution to be obtained, but at the expense of an unrealistic (i.e. linear) flame height sub-model.


• The basic flame spread algorithm was re-written for the case of a burning vertical wall (i.e. a packing case), ignited at low level over a small area and including the effect of 'burnout' of the flame after a specified time following ignition. The problem was solved numerically so that the raw heat release data obtained from cone calorimeter reaction-to-fire tests could be used directly as the input; no 'idealisation' of the data was necessary (the development of functional approximations to the data being necessary only for the analytical solution). The numerical form of the solution also enabled a realistic, empirical, non-linear flame height correlation to be included.

• Experimental data were collected from the cone calorimeter (mostly for corrugated fibreboard packaging material), and from a medium scale test rig to provide some estimates of the upward flame spread velocity for this material (inferring a pyrolysis front velocity from the 'phase-lag' of transient temperature data from fixed thermocouples).

• A comparison of the model predictions with the empirical flame spread measurements showed good agreement. The previous, and erroneous, prediction of self-extinguishment given by the analytical model was confirmed by the equivalent numerical solution (i.e. the 'no burnout' case); when burnout was included in the more advanced model, indefinite flame spread at a steady-state was predicted in the longer term.

• The advanced model was also used to predict the expected flame spread behaviour when the burning surface is exposed to a significant external heat flux; this corresponds to the case of cross-radiation from adjacent high-rack storage arrays in warehouses. It was found that the steady-state prediction of the flame spread velocity was proportional to the magnitude of imposed heat flux. Some short term acceleration-deceleration cycles were also predicted in some cases, these were again most noticeable at higher imposed heat fluxes; it is suggested that these pulses are probably due to spurious numerical effects rather than physical reality.

• The initial conditions (i.e. time of ignition and corresponding rate of heat release) were found to be critical in order to produce a positive and finite value of the initial flame spread velocity, V(0). This required the recorded 'ignition delay time' to be increased in some cases, to increase the initial heat release rate; it should be noted that while this is tantamount to overriding the cone calorimeter operator's judgement of the reaction-to-fire


test, it is a consequence of the flame height correlation used in the model. It is suspected that the erroneous prediction of self- extinguishment in the original 'no burnout ' model is associated with the form of the flame height sub-model. The original correlations used in the sub-model were derived from plume theory applied to point and line sources; in the present model, the 'source' is a vertically oriented pyrolysis zone with dimensions comparable to the flame height. Under these conditions, some loss of accuracy in the prediction of x~ is to be expected and therefore this represents a general weakness in the model, regardless of whether burnout is included or not. Therefore there is a requirement for flame height correlations which take account of the finite vertical pyrolysis zone; this problem is the subject of a forthcoming research project at Edinburgh University (funded by the UK EPSRC).

A C K N O W L E D G E M E N T S

This work has been funded by the CECs STEP (Science and Technol- ogy for Environmental Protection) initiative under the project CT-90- 0096 Major Hazards Arising from Warehouse and Chemical Fires. The authors gratefully acknowledge this support. The efforts of Mr Les Russell and Mr Andrew Jackson were invaluable during the experimental investigation. Finally, thanks are also due to Dr John Martin of the Department of Mathematics and Statistics for several helpful discussions on the subject of Volterra integral equations.

R E FE R E NC ES

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6. Orloff, L., de Ris, J. & Markstein, G. H., Upward turbulent fire spread and burning of fuel surface. In Fifteenth Symposium (International) on Combustion. The Combustion Institute, 1974, pp. 183-92.

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Numerical modelling of early flame spread in warehouse fires

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