dynamic modeling of fire spread in building

Fire Safety Journal 46 (2011) 211–224

Contents lists available at ScienceDirect

Fire Safety Journal

0379-71

doi:10.1

n Corr

Carleton

E-m

ChengH

journal homepage: www.elsevier.com/locate/firesaf

Dynamic modeling of fire spread in building

Hao Cheng a,b,n, George V. Hadjisophocleous a

a Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada K1S 5B6b Fire Protection Program, Human Resources & Skills Development Canada—Labour, 165 Hotel de Ville Street, Gatineau, QC, Canada K1A 0J2

a r t i c l e i n f o

Article history:

Received 17 December 2009

Received in revised form

11 February 2011

Accepted 11 February 2011Available online 5 March 2011

Keywords:

Fire spread

Dynamic model

Bayesian network

Time effect

Probability

Building

12/$ - see front matter & 2011 Elsevier Ltd. A

016/j.firesaf.2011.02.003

esponding author at: Department of Civil and

University, 1125 Colonel By Drive, Ottawa,

ail addresses: [email protected],

[email protected] (H. Cheng).

a b s t r a c t

Modeling fire spread in a building is a key factor of a fire risk analysis used for fire safety designs of large

buildings. In this paper, a dynamic model of fire spread considering fire spread in both horizontal and

vertical directions is described. The algorithms for simulating the fire spread process in buildings and

calculating dynamic probability of fire spread for each compartment at each time step of simulation are

proposed. The formulae used in calculating the input data for the dynamic fire spread model are derived.

The dynamic fire spread model can easily be applied for any building including high-rise buildings.

A detailed example of calculation of fire spread in a two-storey office building is described.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Fires in buildings pose a significant risk to building occupantsand cause property damage. Traditionally, prescriptive-based build-ing codes have been used in fire safety design of buildings. Suchcodes prescribe in detail what is required for a fire-safe building.Because of the limitations inherent in prescriptive-based buildingcodes, such as not being flexible and inhibiting innovation, perfor-mance-based building codes are being adopted and used in moreand more countries. To achieve the goals required by performance-based building codes, it is necessary to undertake a fire-riskassessment and fire safety evaluation, especially for large buildings.

A lot of research has been conducted in recent decades tounderstand the mechanisms of fire spread in large buildings.Because of the difficulty of including all factors affecting firegrowth and fire spread in the input data for fire-spread models,these studies have often been carried out using a probabilisticapproach. Ramachanandran [1,2] summarized the earlier studiesof stochastic fire-spread modeling from recent decades.

In the earlier studies, epidemic theory [3,4], random walktheory [5,6], Markov processes [7–9], percolation processes[10,11] and probabilistic networks [12,13] have been used tomodel fire spread. These models can successfully describe certainaspects of fire spread in buildings. But there are some

ll rights reserved.

Environmental Engineering,

ON, Canada K1S 5B6.

disadvantages when simulating the fire spread process usingthese models. The epidemic theory cannot model fire spread,due to radiation, combustible materials or compartments that arefar away from the fire origin and unable to be directly reached byflames. The random walk theory [5,6] and percolation pro-cesses [10,11] can simulate fire spread from a fire compartmentto one of its adjacent compartments, and then from this firecompartment to another adjacent compartment. But they are notgood at simulating scenarios whereby fire may spread from a firecompartment to multiple adjacent compartments or fires frommultiple fire compartments spreading to adjacent compartments.The transition probability in the Markov process [7–9] is notreally the probability of fire spread from the fire compartment toadjacent compartments, which are affected by the fire severityand time. It only represents the relative probability that firewould spread from the fire compartment to an adjacent compart-ment, i.e. if there are two similar compartments on each sideof a fire compartment, the transition probability of fire spread toeach compartment will always be equal to 50%, regardless of theseverity of the fire and how long the fire lasts in the compartment.In addition, the fire spread process from one compartment tomultiple compartments or from multiple compartments to theiradjacent compartments at the same time cannot be described bythe Markov process.

Ling and Williamson [12] first presented a probabilistic net-work approach to study room-to-room fire spread. An exampleof modeling a network for fire spread in a building floor wasillustrated. This model did not consider barrier failure due toradiation. This network approach was complicated, each time the

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Nomenclature

Af surface area of fuelAF floor surface area of a compartmentAo area of the ventilation openingAo

ffiffiffiffiffiffiHo

pventilation factor

AT total area of fire compartment enclosing surfacec specific heat of the boundary materialD the depth of a compartmentdt simulation time stepg the acceleration due to gravityH height of a compartmentHch heat of combustion of fuelHo height of the ventilation openingk thermal conductivity of boundary materialL a large number_mv burning rate of fuel

n total time steps of simulationP(a9a0) probability of fire growth from ignition to a fully

developed fireP(a9b) probability of fire spread from compartment B to

compartment AP(a09b) probability of fire resistance failure of the barrier

between compartment B and compartment AP(b) the probability of fully developed fire occurred in

compartment Bpbf the density function of the probability of barrier

failurePbf the cumulative probability of barrier failurepfd the density function of the probability of fire growth

to fully developed firePfd the cumulative probability of fire growth to fully

developed firePig probability of fire ignitionQ total heat released by fuels_Q heat release rate_Q fo critical value of heat release rate of flashover

R a random number generatedT total simulation timet0 simulation starting timeT0 ambient temperature

tfo flashover time in the compartmenttig ignition time in the compartmenttn total time of simulationW width of a compartmentwf fuel density in a compartmentWf total mass of equivalent wood in a compartmentWo Width of the ventilation openingz flame height above opening

Greek

a growth coefficient of t2 fireb the parameter of Japanese Parametric modelmbf the mean time for the fire resistance of a barrier to failuf mean of fuel load density in compartmentmfo mean duration for fire to grow from ignition to

flashovermFRR Mean of duration of FRR to the standard ISO 834 firema mean of the growth coefficient for t2 firetbf duration of fire resistance rating to the fully

developed fireteq equivalent duration of the standard fire test in

severitytfd duration of fully developed fire phasetfo duration of fire growth phasetFRR duration of fire resistance rating to the standard ISO

834 fire of assemblytmax

gr the maximum duration of fire growth phaser0 density of airr density of materialj specific surface of fuelsbf standard deviation of barrier fire resistance failuresf standard deviation of fuel load density in

compartmentsfo standard deviation of duration from ignition to

flashoversFRR standard deviation of the duration of fire resistance

ratings to the standard ISO 834 for assemblysa standard deviation of the growth coefficient for a

t2 fire

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 46 (2011) 211–224212

compartment of fire origin was changed, a new network had to bedeveloped. Platt et al. [13] developed a simple and clear model inwhich an event tree was used to determine the probability of firespreading from the fire compartment to other compartments. Thismodel was very good in expressing the fire spread process forsmall buildings, but it is hard to develop a fire-spread event tree forlarge buildings. If the initial fire compartment was changed, a newevent tree would have to be developed, even for the same building,which makes the model difficult to program. The digraph (directedgraph) approach was used as the fire-spread sub-model of the firerisk evaluation and cost assessment model (FiRECAM) [14]. Firespread from the fire compartment to a compartment on the floorabove through the pathway of window to window due to externalflames out of a window was not included in this model. To simplifythe problem, all compartments of the same type, such as rooms,corridors, stairwells on one floor were combined as a single node ofthe network. The developed algorithm searched all possible path-ways for fire to spread from one node to another.

In this paper, a dynamic model of fire spread in buildings ispresented. It was developed based on the static model of firespread in buildings using the Bayesian Network theory proposedby Cheng and Hadjisophocleous [15], which can overcome the

disadvantages of the earlier models. Some basic concepts intro-duced in [15] are not repeated in this paper. The dynamic fire-spread model considers both horizontal and vertical fire spread ina building. In this model, the algorithms of simulating the firespread process have been developed and corresponding codeshave been written. Therefore, the probability of fire spread fromthe compartment of fire origin to any other compartment, thetime of ignition and the time to flashover in each compartmentcan be calculated. In addition, the formulae calculating the inputdata for the dynamic fire spread model were derived. Thedynamic fire spread model can easily be applied to any building,including high-rise buildings.

2. Fundamentals of fire development in compartment and firespread in building

In order to build a fire spread dynamic model simulatingfire spread in buildings, it is important to review some of thefundamentals of fire development in a compartment and firespread from the fire compartment to its adjacent compartmentsin buildings.

H. Cheng, G.V. Hadjisophocleous / Fire Safety Journal 46 (2011) 211–224 213

2.1. Fundamental of fire development in compartment

If a fire occurs in a compartment, the compartment fire mayundergo the phases of growth, development and decay as shown inFig. 1:

(1)

Tablefire g

Fire

Slo

Me

Fas

Ult

Dormant phaseThe phase during which no fire or ignition occurs in thecompartment.

(2)
Fire growth phaseOnce ignition occurs in the compartment, fire might grow upto a fully sustained fire. Usually the t2 fire [19] is used tocharacterized the fire in the fire growth phase
_Q ¼ aðt�tigÞ2

ð1Þ

where _Q is the heat release rate (kW); tig the ignition time inthe compartment (s); a the growth coefficient for t2 fire (kW/s2).The value of fire growth parameter is shown in Table 1.The fire in the growth phase in the compartment mighttransition to flashover. One criterion for flashover occurring ina compartment is that the heat release rate of the fire must reacha critical value [20].

_Q fo ¼ 750Ao


pð2Þ

where _Q fo if the critical value of heat release rate of fire forflashover (kW), Ao


pthe ventilation factor (m5/2), Ao the

area of the ventilation opening (m2), and Ho the height of theventilation opening (m).

a. Fire can flashover in a compartment

The fire growth phase ends when heat released rate(HRR) of fire in fire growth phase reaches the criticalvalue for flashover, which depends on the ventilationcondition; therefore,

_Q fo ¼ aðtfo�tigÞ2¼ aðtfoÞ

2¼ 750Ao


pð3Þ

where tfo is the duration of fire growth phase fromignition to flashover (s) and tfo the flashover time inthe compartment (s).

Tem

pera

ture

/HR

R

1rowth

grow

w

dium

t

ra fas

t (time)

FlashoverIgnition

Growth Fully developed fire DecayDormant

Decay

tig tfo tde

Extinguishment

tex

τfdτfo τde

Fig. 1. The phases of enclosure fire development.

parameter.

th rate Fire growth

parameter a (kW/s2)

Time (s) (when_Q ¼1 MW)

0.0029 600

0.012 300

0.047 150

t 0.188 75

Therefore the duration from ignition to flashover tfo canbe calculated by

tfo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi750Ao


p

a

sð4Þ

b. Criterion for flashover in a compartment

If the heat release rate (or temperature) in a compart-ment during the fire growth phase cannot reach thecritical value for flashover, this fire would not causeflashover to occur. Assume all fuels in the compartmentwould burn out during the fire growth phase. The totalheat that can be released by fuels in a compartment canbe calculated.

Q ¼wf AFHch ¼

Z tigþtmaxgr

tig

_Q ðtÞ dt

¼

Z tigþtmaxgr

tig

aðt�tigÞ2dt¼

Z tmaxgr

0aðt�tigÞ

2dt¼1

3aðtmax

gr Þ3

ð5Þ

where Q is the total heat released of fuels (kJ), wf the fueldensity of equivalent wood in compartment (kg/m2), AF

the floor area of the compartment (m2), Hch the heat ofcombustion of fuel (kJ/kg), tmax

gr the maximum durationof fire growth phase when all fuel would be burnt in firegrowth phase (s).Therefore, the maximum duration of fire growth phasecould be calculated by

tmaxgr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3wf AFHch

a3

rð6Þ

Criterion to check whether flashover can occur in a

compartment

� If tmaxgr rtfo, there is no flashover in a compartment.

� If tmaxgr 4tfo, flashover can occur in a compartment.

(3)
Fully developed fire phaseImmediately after flashover, the compartment fire becomes afully developed fire. During the fully developed fire phase,the heat release rate of the fire reaches its greatest value. Thefully developed fire might be ventilation controlled or fuelcontrolled.The criterion used to distinguish the two regimes of post-flashover compartment fire proposed by Harmathy [21,22]are as follows.Ventilation controlled fire
r0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigAo


p

Af

so0:263 ð7aÞ

Fuel surface controlled fire

r0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigAo


p

Af

sZ0:263 ð7bÞ

where r0 is the density of air (kg/m3), g the accelerationdue to gravity (m/s2), and Af the surface area of the fuel (m2).Harmathy [23] suggested that the surface area of the fuel canbe expressed as

Af ¼jWF ¼jwf AF ð8Þ

where Wf is the total mass of equivalent wood in a compart-ment (kg), j the specific surface of fuel, 0:1ojo0:4 ðm2=kgÞ [21], wf the fuel density of equivalent wood ina compartment (kg/m2), and AF the floor surface area of acompartment (m2).


Substituting r0 ¼ 1:2kg=m3,g ¼ 9:81m=s2 and Eq. (8) intoEq. (7a) and (7b), then the criterion of two regimes of post-flashover compartment fire becomesVentilation controlled fire

A0

ffiffiffiffiffiffiH0

pAf

¼A0


pjwAF

o0:07m�1=2 ð9aÞ


A0


pAf

¼A0


pjwAF

Z0:07m�1=2 ð9bÞ

For a wide range of conventional furniture j¼ 0:13ðm2=kgÞ [23]a. Duration of fully developed fire phase

Harmathy [23] presented following equations to calculatethe duration of fully developed fire phase tfd

Ventilation controlled fire

tfd ¼Wf

r0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigA0


pq ¼ 10:6Wf

A0


p ðsÞ ð10aÞ


tfd ¼151

j ðsÞ ð10bÞ

For a ventilation controlled fire, its duration depends onboth the amount of fuel in the compartment and the sizeand shape of the opening of the compartment. For a fuelsurface controlled fire, its duration is independent of thefuel load in the compartment and only depends on thespecific surface of fuel:(a) For a fuel surface controlled fire, the fire will be

contained in the compartment and flames will notcome out of the compartment openings.

(b) If a fire is ventilation controlled, flames will projectoutside of the compartment through openings such aswindows or open doors. The flame height above thesoffit of the opening is given by [24]

z¼ 12:8_mv

Wo

� �2=3

�Ho ð11Þ

where z is the flame height above the soffit of the opening(m), _mv the fuel burning rate during fully developed firephase for ventilation controlled fire (kg/s), and Wo thewidth of the ventilation opening (m).For the ventilation controlled fire, the burning rate at fullydeveloped fire phase [25] is

_mv ¼ 0:18ð1�e�0:036OÞA0


pffiffiffiffiffiffiffiffiffiffiffiD=W

p ð12Þ

where D is the depth of the fire compartment (m) and W

the width of the fire compartment (m)

O¼ Coefficient, O¼AT

Ao


p

where AT is the total area of fire compartment enclosingsurface (m2).

(4)
Decay phaseDecay occurs as fuels are consumed by the fire and the HRRbegins to decline. During this phase, the fire in the compart-ment changes from a ventilation controlled fire to a fuelsurface controlled fire.
2.2. Fundamentals of fire spread in a building

The main reason for fire spread from the compartment of fireorigin to its adjacent compartments is that the barriers betweenthe compartments fail to contain the fire. The heat in the firecompartment penetrates the barrier and may ignite the combus-tible materials in the adjacent compartments. After ignition, thefire in the adjacent compartments may grow and develop to afully developed fire. The failure of a barrier to contain the firedepends on three factors:

�
The fire severity. � Fire duration. � Fire resistance of the barrier.
Fire severity is usually expressed in terms of the heat releaserate or temperature in the fire compartment. As shown in Fig. 1,the heat release rate or temperature during the fire growth phaseis much smaller than that of the fully developed phase; therefore,it can be assumed that fire spread can only occur after flashover,during the fully developed fire phase.

The duration of a fully developed fire mainly depends on theamount of fire load in the compartment and the fire burning rate.

Fire spread from the fire compartment to its adjacent com-partments always occurs via the weakest parts of the barriers,such as doors, windows or other openings. In buildings, fire mayspread from the fire compartment to adjacent compartments bothin the horizontal direction and in the vertical direction

2.2.1. Fire spread in the Horizontal direction

The possible pathways for fire to spread from the fire compart-ment to its adjacent compartments in a horizontal direction are:

�
through a wall connecting two compartments, � through a closed door connecting two compartments, � through an open door connecting two compartments, � through a window connecting two compartments, and � from one compartment to another compartment separated by
a corridor (4 different door opening scenarios).

The mechanisms of fire spread between two compartments inthe horizontal direction are:

�
Conduction: the heat in the fire compartment is conductedthrough walls or closed doors separating the two compart-ments, causing an increase of the temperature on the unex-posed side and igniting combustible materials in the adjacentcompartment. � Convection: hot gases or flames flow through openings such as
open doors, windows or cracks to the adjacent compartmentand ignite combustible materials in it.
� Radiation: radiative heat flux from the fire compartment
transfers to compartments across the corridor and ignitescombustible materials in these compartments.

2.2.2. Fire spread in the vertical direction

The possible pathways for fire to spread from the fire compart-ment to compartments on the floor above:

�
through a ceiling connecting two compartments, � through an opening such as stairwell connecting two compart-
ments, and
� by outside flames projecting out of windows in the fire
compartment entering windows in the compartment above.


The mechanisms causing the fire to spread from the firecompartment to compartments above in vertical direction are:

�
Conduction: conduction of heat from the fire compartmentthrough the ceiling or floor, causing an increase of thetemperature on the unexposed side and igniting combustiblematerials in the compartment at the upper or lower floor. � Convection: immediately after flashover in the fire compart-
ment, hot gases flow to the compartment above throughopenings and ignite combustible materials inside.
� Radiation and convection: heat flux from the flames projecting out
of the windows of the fire compartment could break glasswindows of the compartments on the upper floor, penetratethe windows, and ignite combustible materials in the uppercompartment.

3. Fundamentals of the dynamic Bayesian network [16,17,18]

Bayesian networks cannot model temporal relationshipsamong variables since Bayesian networks do not provide amechanism for representing temporal dependencies. Therefore, itcannot represent how the value of some variables may be relatedto their values and the values of other variables at a previous pointin time. However, fire spread in a building is a dynamic process.Time is a critical factor in modeling the fire spread process in abuilding because the fire severity in the fire room and the fireresistance of the barriers are functions of time. To capture thedynamic aspects of the problem, a dynamic Bayesian network(DBN) is used to build the dynamic fire spread model for buildings.

Let X¼{X1,X2,y, Xm}denote the variables in a Bayesian net-work. Assume that the variables in X are ordered ancestrally, or ina topological ordering. That is, for every node XiAX; the index ofits ancestor Xj has the property jo i. A dynamic Bayesian networkis a Bayesian network with explicitly represented temporalvariables. A DBN structure can be treated as having two

t0 =0 t2t1

X(t0) X(t1) X(t2)

X1(t)

X2 (t)

Xm (t)

X1 (t0)

X2 (t0)

Xm (t0)

X1 (t2)

X2 (t2)

Xm (t2)

X1 (t1)

X2 (t1)

Xm (t1)

321

Fig. 2. A DBN with first-or

dimensions: the time line and the variable line, represented inFig. 2 on the horizontal axis and vertical axis, respectively. On thetime line, the time mission T (from t0¼0 to tn¼T) is divided into n

intervals. Each time step (t0, t1, t2, y tn) is called a time-slice.The relationships between variables in the same time sliceare represented by the intra-slice arcs. The edges representingthe relationships between variables in the different time slicesare called inter-slice arcs or temporal arcs. Each variable on thetime line forms a sequence Xi(t)¼{Xi(t0), Xi(t1), y , Xi(tn) } called atemporal variable sequence. The temporal variables measuredat the same time slice form a contemporaneous variable setX(ti)¼{X1(ti), X2(ti), y , Xm(ti)}.

The joint probability distribution of DBN over all variables canbe stated as

PðXðt0Þ,Xðt1Þ,Xðt2Þ,. . .,XðtnÞÞ ¼ PðXðt0ÞÞUPðXðt1Þ9Xðt0ÞÞ

UPðXðt2Þ9Xðt0Þ,Xðt1ÞÞ. . .PðXðtnÞ9Xðt0Þ,Xðt1Þ,Xðt2Þ,. . .,Xðtn�1ÞÞ ð13Þ

It is assumed that the time slices are chosen so that the states ofthe DBN satisfy the Markov condition; that is, the state of a DBNat time ti depends only on its immediate past state (the state attime ti�1). Therefore, the state of a variable of a DBN at time ti

depends on its previous state at time ti�1 and the current states ofits parent’s variables of the DBN at time ti. Then the jointprobability distribution of DBN over all variables can be written as

PðXðt0Þ,Xðt1Þ,Xðt2Þ,. . .,XðtnÞÞ ¼ PðXðt0ÞÞXn

i ¼ 1

PðXðtiÞ9Xðti�1ÞÞ ð14Þ

In Fig. 3, there is a transition network G-containing thevariables in X(ti)[X(ti + 1). Its transition probability distribution is

P-ðXðtjþ1Þ9XðtjÞÞ ¼Ym�1

j ¼ 1

P-ðXjðtjþ1Þ9paðXjðtjþ1ÞÞÞ ð15Þ

where pa(xj(ti + 1)) denotes the value of the parent node setof xj(ti+ 1).

tn=Ttime

n

X1 (tn)

X2 (t n)

Xm (tn)

X (tn)

der Markov property.

X1 (t0)

X2 (t0)

Xm (t0)

X1 (ti+1)

X2 (ti+1)

Xm (ti+1)

X1 (ti)

X2 (ti)

Xm (ti)

The transition network G →The prior network

Fig. 3. The prior network and transition network. (a) The prior network. (b) The

transition network G-.


4. Dynamic modeling of fire spread in buildings

The dynamic modeling of fire spread in buildings is based onthe static modeling of fire spread in buildings using the Bayesiannetworks derived by Cheng and Hadjisophocleous [15]. To sim-plify the problem, following assumptions have been made.

4.1. Basic assumptions for dynamic fire spread model

�
Fire spread in the building is subject to the Markov condition.That is, the state of a variable of a DBN at time ti depends on itsprevious state at time ti�1 and the current states of its parents’variables of the DBN at time ti. � If a fully developed fire has occurred in a compartment, fire
cannot return to the compartment that has already burnt outsince there are no combustibles remaining in this compart-ment to support fire ignition and fire growth.
� Once ignition occurs in a compartment, fire will develop in this
compartment independently; that is, the fire development in acompartment will not be subjected to the influence of the fireconditions in its adjacent compartments.
� Fire in a compartment can only spread to its adjacent compart-
ments in the horizontal direction and adjacent compartmentson the upper floor; it is assumed that fire spread to compart-ments on the lower floor can be ignored.
� The probability of fire spread from a corridor to adjacent
compartments can be ignored since the fuel density in thecorridor is low. Therefore the duration of a fully developed fireis not long enough for heat to penetrate the barrier to anadjacent compartment and ignite the combustible inside.
� When two compartments are separated just by an open door,
open window or there is no barrier between them, if flashoveroccurs in one of the compartments, the fire will immediatelyspread to the other compartment. For a large compartmentsuch as a hall in a building, the large compartment can beconsidered consisting of several virtual compartments withoutboundary barriers between them.
� When the doors of two compartments are separated by a corridor,
the fire can spread from the fire compartment to the other

compartment opposite the corridor through the doors of thesetwo compartments by radiation heat flux from the hot gases inthe corridor or in the room. If the door of a compartment is faraway from the position perpendicular to the door of fire compart-ment, the radiant heat flux from the fire compartment could betoo small to ignite the combustible materials in other compart-ment since the two doors could be diagonally far away from eachother across the corridor.
� If flashover could occur in a compartment such as stairwell,
elevator shaft, or duct, the fire would immediately spread to itsupper compartment.
� The common glass of windows in a compartment is assumed
to break and fall off once flashover occurs in the compartment.

4.2. The probability of fire spread

Fire spread from the fire compartment to its adjacent com-partment includes two processes:

(a)
Heat overcomes the fire resistance of the barrier between thetwo compartments, is transferred to the adjacent compart-ment, and ignites combustible material inside it.
(b)
After ignition, the fire in the adjacent compartment couldgrow to a fully developed fire.
Therefore, the probability of fire spread from compartment Bto compartment A could be written as

Pða9bÞ ¼ Pða9a0ÞPigða09bÞ ð16aÞ

Pigða09bÞ ¼ Pða09bÞPðbÞ ð16bÞ

where P(a9b) is the probability of fire spread from compartment Bto compartment A, P(a9a0) the probability of fire growth fromignition to a fully developed fire in compartment A, Pig(a09b) theprobability of fire ignition in compartment A due to heat transferfrom the fire compartment B, P(a09b) the probability of barrierfailure indicating the probability that heat is transferred from thefire compartment B to the adjacent compartment A and ignitesthe combustible materials in compartment A, P(b) the probabilityof fully developed fire occurred in compartment B.

In this paper, it is assumed that both the probability of barrierfailure and the probability of fire growth to a fully developed firefollow the properties of a normal distribution (m,s). The purpose ofthis assumption is only to show one kind of calculation of thecumulative probability of barrier failure and the probability of firegrowth to a fully developed fire, both of which are needed in the firespread dynamic model. If the probability of barrier failure and theprobability of fire growth to a fully developed fire follow another kindof probability distribution, the results of the cumulative probability ofbarrier failure and the probability of fire growth to a fully developedfire could also be used as inputs in the dynamic fire spread model.

The means and standard deviations of the probability ofbarrier failure and the probability of fire growth could changeaccording to different properties of buildings and other factorssuch as the presence of a fire suppression system.

�
The probabilities of barrier failure mainly depend on the severityof fire and structures, materials and geometry of the building. � The probabilities of fire growth to a fully developed fire mainly
depend on fuel load, ventilation, duration of the fully devel-oped fire phase and on whether fire suppression systems areinstalled or not.

If a room has several adjacent fire compartments, heat couldbe transferred to this room simultaneously from all adjacent fire


compartments, which will increase the probability of ignition ofthe combustible materials in the room considering the interactionof heat transfer to this room from more than one barrier. Assumethe compartment A has two adjacent fire compartments B and C.The probability of fire spread to compartment A due to firecompartments B and C is

Pða9b,cÞ ¼ Pða9a0ÞPigða09b,cÞ ð17aÞ

Pigða09b,cÞ ¼ Pða09bÞPðbÞþPða09cÞPðcÞ�Pða09bÞPða09cÞPðbÞPðcÞ ð17bÞ

where P(a9b,c) is the probability of fire spread from the firecompartments B and C to compartment A, Pig(a

0

9b,c) the prob-ability of fire ignition in compartment A due to heat transfer fromthe fire compartments B and C.

4.2.1. Probability of barrier failure

There are four principal ways by which fire spreads betweencompartments:

�
Radiation through a window. If two windows are alignedvertically and the flame projecting out of the lower windowis high enough to reach to the window of the compartment onthe upper floor, fire may spread to the upper compartment. � Conduction through a wall, ceiling or closed door. � Convection through an open door, or window. For example, if the
door connecting two rooms is open, the fire would spread tothe other room immediately after flashover occurs in one of therooms.
� Radiation and convection between rooms connected by a
corridor.

The probability density function of barrier failure at the time t

can be written as

pbf ðtÞ ¼1

sbf

ffiffiffiffiffiffi2pp expð�½ðt�tfoÞ�mbf �

2=2s2bf Þ ð18Þ

The cumulative probability of barrier failure at time t is

Pbf ðtÞ ¼

Z t

tfo

1

sbf

ffiffiffiffiffiffi2pp expð�½ðt�tfoÞ�mbf �

2=2s2bf Þdtð0rt�tfortfdÞ

ð19Þ

where tfo is the time of flashover in the fire compartment, mbf themean of duration of a barrier failure linking the fire compartmentto an adjacent compartment, sbf the standard deviation of dura-tion of a barrier failure linking the fire compartment to anadjacent compartment; tfd the duration of the fully developedfire phase in the fire compartment.

The calculation of the time of barrier failure

Mehaffey [26] used a Japanese fire model to assess the fireresistance of assemblies exposed to a post-flashover fire byexpressing the severity of the fire in terms of the standard ISO834 fire. The duration of the standard ISO 834 fire that isequivalent in severity to the duration of the fully developedventilation controlled fire is

teq ¼b

230

� �3=2

tfd ð20Þ

where teq is the equivalent duration of the standard fire test inseverity, tfd the duration of the fully developed ventilationcontrolled fire, b the parameter of the Japanese Parametric modelfor compartment boundaries (Ks�1/6).

The parameter b of the Japanese Parametric model could bewritten as

b¼ 3:0T0Ao


p

AT

ffiffiffiffiffiffiffiffikrc

p !1=3

ð21Þ

AT


q¼X

i

AT ,i

ffiffiffiffiffiffiffiffiffiffiffiffikirici

qð22Þ

Ao


p¼X

i

Ao,i

ffiffiffiffiffiffiffiffiHo,i

qð23Þ

where ki is the thermal conductivity of boundary material i,ðkW=ðmKÞÞ, rithe density of the boundary material i (kg/m3), ci

the specific heat of the boundary material i ðkJ=ðkgKÞÞ T0 theambient temperature (K).

In fire safety design, the fire resistance ratings of assembliestested to the standard ISO 834 fire are usually prescribed by thebuilding codes. From this, the fire resistance of the assembly tofully developed fire can be calculated by

tbf ,i ¼tFRR,i

ðb=230Þ3=2ð24Þ

where tFRR,i is the duration of fire-resistance ratings to thestandard ISO 834 fire for assembly i, tbf,i the duration of fire-resistance to a fully developed fire for assembly i.

Based on Eq. (24), the mean and standard deviation of fireresistance to a fully developed compartment fire for an assemblycan be calculated

mbf ,i ¼mFRR,i

ðb=230Þ3=2ð25aÞ

sbf ,i ¼sFRR,i

ðb=230Þ3=2: ð25bÞ

where mFRR,i is the mean of the duration of fire resistance ratingsto the standard ISO 834 fire for assembly i, sFRR,i the standarddeviation of the duration of fire resistance ratings to the standardISO 834 for assembly i, mbf,i the mean of the duration of fireresistance to a fully developed compartment fire for assembly i,and sbf,i the standard deviation of the duration of fire resistance toa fully developed compartment fire for assembly i.

4.2.2. Probability of fire growth to a fully developed fire

Once ignition occurs in a compartment, the fire may grow upto a fully developed fire. The probability of fire growth to a fullydeveloped fire depends on the following factors:

(a)
The fuel load: fuel amount and fuel types in the compartment. (b) The geometry of the compartment and its ventilation
conditions.
(c) The availability of a fire suppression system.
If the maximum heat release rate of the fire in a compartmentduring the fire growth phase is less than the critical value forflashover in the compartment; that is aðtmax

gr Þ2 o750Ao


p,

flashover would not be expected to occur in the compartment.Therefore, the probability of fire growth to a fully developed firein the compartment is zero. Otherwise, flashover can occur in thecompartment. The density function of the probability of firegrowth to a fully developed fire in the compartment at time t is

pfdðtÞ ¼1

sfo

ffiffiffiffiffiffi2pp expð�½ðt�tigÞ�mfo�

2=2s2foÞ ð26Þ

The probability of fire growth to a fully developed fire at time t

can be calculated by Eq. (27a)–(27d)


When flashover does not occur in the compartment

PfdðtÞ ¼

Z t

tig

1

sfo


2=2s2foÞdtð0rt�tig rtmax

gr Þ

ð27aÞ

PfdðtÞ ¼ 0 otherwise ð27bÞ

When flashover occurs in the compartment

PfdðtÞ ¼

Z t

tig

1

sfo


2=2s2foÞdtð0rt�tig rtfoþtfdÞ

ð27cÞ

PfdðtÞ ¼ 0 otherwise ð27dÞ

where tig is the ignition time in the compartment, mfo the meanof duration for fire to develop from ignition to flashover inthe compartment, sfo the standard deviation of duration for fireto develop from ignition to flashover, and tmax

gr the maximumduration of fire growth phase in the compartment.

The calculation of the mean time and standard deviation forfire to develop from ignition to flashover in the compartment

mfo ¼



p

ma

sð28aÞ

S2

R10R8

S1

R11 R9

E

R6R4

R5 R7R1

R2

R3

C

Fig. 4. The floor plan of an office building.

R1 R3 R5

R2 R4 R6

S1 C

R5

R6

R3

R4

R1

R2

S1

Fig. 5. (a) The general fire spread network. (b)

sfo ¼



p

m2a

ssa ð28bÞ

where ma is the mean of the fire growth coefficient for a t2 fireand sa the standard deviation of the fire growth coefficient fora t2 fire.

4.3. Dynamic modeling of fire spread in buildings

In large buildings, fire may spread between compartmentssimultaneously in both horizontal and vertical directions. Tosimplify the fire spread model, fire spread model in the horizontaldirection and fire spread model in the vertical direction aredescribed separately.

4.3.1. Modeling of fire spread in the horizontal direction

on a building floor

Fig. 4 shows the floor plan of an office building considered. Thefloor has eleven rooms, one corridor, two stairwells, and oneelevator shaft. We can transform this floor plan into a general firespread network as shown in Fig. 5(a). Each node represents onecompartment such as a room, stairwell, elevator shaft, corridor orduct, and edges represent the possible paths and directions of firespread. The arrows in both directions in the network mean thatthe general fire spread model can deal with all the possible firespread situations for any fire scenario. Ignoring the influence ofthe corridor on fire spread by assuming there is little fuel in thecorridor, the general fire spread model of the building floor can besimplified as shown in Fig. 5(b). Once the compartment of fireorigin is given, the general fire spread network can be immedi-ately converted into a specific fire spread Bayesian network whichcan be used to calculate the probability of fire spread from theinitial fire compartment to a destination compartment on thatfloor. A specific example of a fire spread model based on Bayesiannetwork is described below.

To calculate the probability of fire spread from the compartmentof fire origin (R1) to the destination compartment (R6), all possiblepathways by which fire can spread from R1 to R6 need to be found.The fire spread paths can be found using the directed acyclic graph(DAG) of the fire spread Bayesian network as shown in Fig. 6.

S2

R8R7 R10

R11R9E

R8

R9

R7

E R11

R10

S2

The simplified general fire spread network.

R5

R6

R8

R9

R7

E R11

R10

S2

R3

R4

R1

R2

S1

Fig. 6. The DAG of Bayesian network for fire to spread from Room 1 to Room 6.

1st floor

2nd floor

3rd floor

4th floor

F4

F3

F2

F1

Fig. 7. (a) A building with four storeys (b) The DAG of vertical fire spread network

of the building.

Vertical fire spread

Horizontal fire spread

Fig. 8. Fire spreads to a compartment horizontally and vertically.


4.3.2. Modeling of fire spread in vertical direction in a building

Assume that there is a four-storey building with only onecompartment on each floor as shown in Fig. 7(a). If a fire starts ina room on the first floor, the network of fire spread in verticaldirection can be expressed as in Fig. 7(b).

4.3.3. Modeling of fire spread in buildings in both horizontal and

vertical directions

A fire could spread to a compartment simultaneously in bothhorizontal and vertical directions as shown in Fig. 8. The calculationof the probability of fire spread to a compartment must consider theinteraction among the probabilities of barrier failure when heat istransferred to the compartment horizontally from adjacent com-partments and vertically from the compartment on the lower floorthrough the ceiling or window. The probability of fire spread of acompartment due to two adjacent fire compartments can becalculated by Eq. (17).

The flow chart describing the dynamic model of fire spread inbuildings is shown in Fig. 9. The detailed algorithms of dynamicmodel of fire spread in buildings, used to calculate dynamicprobability of fire spread for each compartment at each time stepof simulation, are described Table 2.

5. Example

Consider a two-floor office building with only one window ineach office room, whose floor plan is shown in Fig. 4. Assume thatthe heat release rate of fire during the growth phase follows a

slow t2 fire. If a fire starts in Room 1 on the first floor, what is theprobability of fire spreading to Room 6 on the second floor?

Case 1 all doors of the compartment are closed during the firespread process;Case 2 the doors of Room 5 and 8 are open and all other doorsare closed during the fire spread process.

Steps of calculation:

(1)
First a compartment number is assigned to each compartmenton each floor of the building.
(2)
Read input data for the simulationTotal simulation time: 480 minEach simulation step time: 1 minAmbient temperature: 20 1C
(3)
Read the input data of the compartments� Office room:
Dimension: 4 m�5 m�3 m (W�D�H)The fuel density: mean¼24.8 kg/m2; standard devia-tion¼8.6 kg/m2 [23].Heat of combustion of fuel Hch¼12.4 KJ/g [27]Fire growth parameter: a¼0.0029 kW/s2 (slow)� Window:

Dimension: 1.0 m�1.0 m (W�H),Material: common glass;Fire resistance rate to ISO 834 fire: mean¼2 min, standarddeviation¼0.3 min (assumed).� Door:

Dimension: 1.0 m�2.0 m (W�H),Material: wood;Fire resistance rate to ISO 834 fire: mean¼10 min, stan-dard deviation¼1.5 min (assumed).� Wall and ceiling:

Material: gypsum boardffiffiffiffiffiffiffiffikrc

p¼ 0:742 KJ=m2s0:5K [23]

Fire resistance rating to ISO 834 fire: mean¼60 min,standard deviation¼9 min (assumed).� Floor:

Material: normal weight concreteffiffiffiffiffiffiffiffikrc

p¼ 2:192 KJ=m2s0:5K

[25]Fire resistance rating to ISO 834 fire: mean¼90 min, standarddeviation¼13.5 min (assumed).� Stairwell:

Dimension: 5�3�3 m (W�D�H)The fuel density: mean¼0 kg/m2, standard deviation¼0 kg/m2.

(4)

(5)

Yes

No

Read the input data of each compartment

Stop

Calculate duration of each fire phase for a compartment

Set the initial conditions of a compartment, simulation duration and time step

Ti =1, Tn

Read input data of a building and assign number to each compartment

Start

j=1,Nfloor and i=1,Nroom

Calculate the probability of fire spread


Calculate the probability of ignition


Calculate the probability of fire growth

j=1, Nfloor and i=1, Nroom

Calculate the probability of barrier failure

Yes

Yes

Yes

No

No

No

No

Yes

Fig. 9. Flow char of dynamic Bayesian network.


� Elevator/Duct:Dimension: 4�5�3 m (W�D�H)The fuel density: mean¼0 kg/m2, standard deviation ¼0 kg/m2.

The output of compartment fire model used as input data forfire spread model:(a) Fire growth phase:

Maximum duration of fire growth phase (assume all fuelwill burn during the fire growth phase without flashover)

umaxgr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3uf AFHch

a3

r¼ 1853s¼ 30:88min

smaxgr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3AFHch

aðuf Þ2

3

ssf

3¼ 214s¼ 3:57min

The duration from ignition to flashover in the room

Mean : mfo ¼



p

a

s¼ 508s¼ 8:48min

Standard deviation: sfo¼0.15mfo¼1.27 min (assumed).

(b) The fully developed fire phaseCheck the fire regime during fully developed fire phase:ðA0


p=jwAF Þ ¼ 0:0155o0.07 m�1/2 (the fully devel-

oped fire is ventilation controlled)Duration of fully developed fire phase:Mean: mfd ¼ 10:6ðAfmf =A0


pÞ ¼ 5258ðsÞ ¼ 87:6min

Standard deviation: sfd ¼ 10:6ðAfsf =A0


pÞ ¼1823 (s)¼

30.0 minThe mass burning rate for the ventilation controlled fire:

_mv ¼ 0:18ð1�e�0:036OÞA0


pffiffiffiffiffiffiffiffiffiffiffiD=W

p ¼ 0:349kg=s

Flame height above soffit of window:z¼ 12:8ð _mv=WoÞ

2=3�Ho ¼ 2:70m, therefore the fire can

spread to the room on the floor above by both window andfloor.

(c) The Japanese Parametric model:The parameter b¼ 3:0T0ðAo


p=AT


pÞ1=3¼ 190:57

Ks�1=6

The mean and standard deviation of fire resistance to fullydeveloped fire can be calculated by Eq. (25a) and (25b)� Window: Mean¼2.65 min, standard deviation 0.40 min;� Door: Mean¼13.26 min, standard deviation¼1.99 min;� Wall and ceiling: Mean¼79.56 min, standard devia-

tion¼11.93 min;� Floor: Mean¼119.33 min, standard deviation¼17.90 min.

The output of two cases based on the results of dynamical firespread model:(a) Case 1: all doors are closed during fire spread process:� Based on the calculation of dynamic fire spread model, the

ignition time and flashover time in each compartment arelisted in Table 3.� The graphs of probability of fire spread during the fire

growth phase for three adjacent compartments of Room1 on the first floor and the graph of the probability of firespread during the fire growth phase of Room 6 on thesecond floor are plotted in Fig. 10.

Analysis of the simulation results of fire spread in a building byassuming that a flashover fire occurs in Room 1 on the first floor:

i.
Fire spread to Room 6 on the second floor:� The ignition time is 179 min and flashover time is 188 min
in Room 6 on the second floor after a flashover fire starts inRoom 1 on the first floor (the time of simulation starting).� The probability of fire spread to Room 6 on the second

floor is zero before 179 min.� The cumulative probability of fire spread to Room 6 on the

second floor is one after 194 min of the simulation time.
ii. Fire spread from Room 1 on the first floor to its adjacent
rooms:� Fire spread from the Room 1 on the first floor to Room 2 on

the first floor is due to barrier failure of two closed doorsacross the corridor. The ignition time in Room 2 is 26 minand flashover time is 34 min after a flashover fire started inRoom 1.� Fire spread from Room 1 on the first floor to Room 3 on the

first floor is due to barrier failure of the wall connecting thetwo rooms. The ignition time in Room 3 is 70 min and theflashover time is 78 min after a flashover fire started inRoom 1.� Fire spread from Room 1 on the first floor to Room 1 on the

second floor is due to barrier failure of window glass due toexterior flames. The combustible material in Room 1 onthe second floor is ignited by radiation and convection heat

Table 2Algorithms for dynamic fire spread model.

1. Input data of dynamic fire spread model such as simulation information, initial conditions of each compartment of a building and durations of fire development phase

in each compartment

Assume the total simulation time tn¼T, each simulation time step dt¼T/n

Set a large amount of time L¼2T

(1) Assigned a room number to each compartment

(2) Initial conditions for all compartments

Pig(i,j)¼0, tig(i,j)¼L; Pfd(i,j)¼0, tfo(i,j)¼L ( in dormant phase)

(3) Initial condition of the compartment of fire origin

i. Pig(i,j)¼1, tig(i,j)¼0; Pfd(i,j)¼1, tfo(i,j)¼0 (Assume flashover occurs in the compartment)

ii. Pig(i,j)¼1, tig(i,j)¼0; Pfd(i,j)¼0, tfo(i,j)¼L (Assume ignition occurs in the compartment but flashover does not occur inside)

(4) Calculating the duration of fire burning for each compartment

i. duration of fire growth phase

ii. duration of fully developed fire phase

2. Simulation at the time slice ti (t0rtirtn)

2.1. Calculate the probability of barrier failure from fire compartment to its adjacent compartments

(1) Loop through all floors: j¼1, Nfloor (Nfloor¼total floors in a building)

(2) Loop through all compartments on the floor j: i¼1, Nroom (Nroom¼total compartments on a floor)

(3) Check whether fully developed fire did occur in compartment (i, j);

(a) if tfo4ti, flashover did not occur in compartment (i, j), go to next i;

(b) if ti�tfo4tfd, decay did occur in compartment (i, j), go to next i;

(4) if 0oti�tfootfd, calculate the cumulative probability of barrier failure of a barrier linking this compartment (i, j) and adjacent compartments at

time ti in horizontal direction (same floor j)

(a) Loop through all compartments on the floor j: k¼1, Nroom

(b) Check whether there is a barrier linking between compartment (i, j) and adjacent compartment (k, j)

(c) If no, next k

(d) If yes; check whether decay did occur in compartment (k, j), ti�tfo(k,j)4tfd(k,j), If yes, next k

(e) If no; check what kind of barrier linking between compartments (i, j) and (k, j)

(f) Calculate the cumulative probability of barrier failure Pbf(k,i,j) due to fire spread from compartment i to compartment k by Eq. (19)

(g) go to next barrier k

(5) Calculate Pbf(0,i,j+1), (fire spread to compartment (i, j+1) due to heat transfer from the fire compartment (i, j) in vertical direction)

(a) Check whether the jth floor is the top floor of the building (j¼Nfloor?)

(b) If yes, next time slice

(c) If no, check whether a flashover did occur in the compartment (i, j+1)

(d) If yes, next i

(e) If no; check what kind of barrier linking between two compartments (i, j) and (i, j+1)

(f) If there is no barrier or a hole on ceiling such as stairwell, fire can spread to the compartment (i, j+1) immediately. Pbf(0,i,j+1)¼1, next i

(g) Calculate Pbf(0,i,j+1) using Eq. (19) (heat transfer through ceiling)

(h) Check whether fire can spread to the compartment (i, j+1) by projecting flames out of windows of the compartment (i, j).

(i) If no, next i

(j) If yes, check whether (the height of projecting flame above window)o(separation distance between two windows in vertical distance); If

yes, next i

(k) If no, that is (the height of projecting flame above window)4(separation distance between two windows in vertical distance); fire can

spread to the compartment (i, j+1) by projecting flames out of windows of the compartment (i, j).

(l) Check whether the window of the compartment (i, j+1) is closed.

(m) If yes, calculate Pbf using Equation (19) due to heat transfer to above compartment through closed window; then calculate probability of

barrier failure considering the interaction of heat transfer to the compartment (i, j+1) through both ceiling and closed window

Pbf(0,i,j+1)¼Pbf(0,i,j+1)+Pbf�Pbf(0,i,j+1)� Pbf , go to next i

(n) If no (the window is open), let Pbf(0,i,j+1)¼1(heat transfer to above compartment through open window).

(6) Next i

(7) Next j

(2.2) Calculate the probability of fire ignitionPig(i,j)

(1) Loop through all floors: j¼1, Nfloor

(2) Loop through all compartments on the floor j: i¼1, Nroom

(3) Check whether flashover did occur in (i, j) (ti4tfo(i,j))

(4) If yes, let Pig(i,j)¼1 go to next compartment i

(5) If no,

(a) When j¼1, let Pig(i,j)¼0;

(b) When j41, check whether flashover did occur in (i, j-1). If no, Pig(i,j)¼0; if yes, let Pig(i,j)¼Pbf(0,i,j)�Pfd(i,j�1)

(6) Consider the heat transfer to the compartments (i,j) form fire compartments in horizontal direction

(a) Loop through all compartments on the floor j: k¼1, Nroom

(b) Check whether a flashover did occur in the adjacent compartment (k, j), if no, next k

(c) If yes, calculate Pig(i,j)¼Pig(i,j)+Pbf(i,k,j)�Pfd(k,j)�Pig(i,j)� Pbf(i,k,j)� Pfd(k,j), next k

(7) If tig(i,j)oti , ignition did occurred in this compartment, go to next i

(8) If tig(i,j)4ti generate a random number R


Table 2. (continued )

(9) if R4Pig(i,j), ignition will not occur in compartment (i, j), go to next i

(10) if RoPig(i,j), ignition will occur, set tig(i,j)¼ti

(11) Next i

(12) Next j

(2.3) Calculate Pfd(i,j),the probability of fire growth to fully developed fire in compartment (i, j)



If tig(i,j)4ti, compartment (i, j) was not ignited yet, go to next i

If tig(i,j)oti , ignition occurred in compartment (i, j), then

(a) if tfo(i,j)oti, flashover occurred in this compartment, calculate Pfd(i,j) using Eq. (27c or 27d), go to next i;

(b) if tfo(i,j)4ti, flashover did not occur in this compartment, calculate Pfd(i,j) using Equation (27a or 27b), then generate a random number R;

(c) if R4Pfd(i,j), flashover will not occur in compartment (i, j), go to next i

(d) if RoPfd(i,j), flashover will occur in this compartment, set tfo(i,j)¼ti.

(3) Next compartment i

(4) Next floor j

(2.4) Calculate Pfd(i,j), probability of fire spread to compartment (i, j)



(3) Calculate Pfd(i,j) in compartment (i, j) using Eq. (16a)

(4) Next i

(5) Next j

3. Go to next time slice

Table 3The ignition and flashover time in each compartment for case 1.

Floor no. Compartment Ignition time

tig (min)

Flashover time

tfo (min)


tig (min)

Flashover

time tfo (min)

1 R1 0 0 2 R1 3 12

1 R2 26 34 2 R2 36 44

1 R3 70 78 2 R3 82 92

1 R4 100 107 2 R4 111 120

1 R5 135 142 2 R5 146 155

1 R6 167 176 2 R6 179 188

1 R7 208 217 2 R7 219 227

1 R8 278 285 2 R8 289 297

1 R9 312 320 2 R9 318 328

1 R10 341 350 2 R10 353 362

1 R11 376 385 2 R11 384 393

1 E 231 NF 2 E 248 NF

1 S1 69 NF 2 S1 88 NF

1 S2 416 NF 2 S2 431 NF


flux from the external flame projecting out of the windowof Room 1 on the first floor. The ignition time in Room 1 onthe second floor is 3 min and flashover time is 12 min aftera flashover fire started in Room 1 on the first floor.� Comparing the three different kinds of pathways of fire

spread in this example shows that fire spread to a roomdirectly above through windows is much faster than otherpathways for this building.

(a)
Case 2: the doors of Room 5 and 8 are open and all other doorsare closed. � Based on the results of the fire spread model, the ignition time
and flashover time in each compartment in case 2 are listed inTable 4.
� The graphs of the probability of fire spread during the fire
growth phase of Room 6 on the second floor for both cases areplotted in Fig. 11.
� Comparing Tables 3 and 4 as well as Figs. 10 and 11, the
time needed for fire to spread from Room 1 on the first floor

to Room 6 on the second floor in Case 2 is less than that inCase 1. The opening of a door has an influence on the firespread process.

6. Conclusion

Modeling fire spread in a building is a key factor of a firerisk analysis used for fire safety designs of large buildings. Theprobability of fire spread from the compartment of fire origin toother compartments in the building, in conjunction with smokeconditions in the building, is required to calculate the expectedrisk to life and expected losses in a building during a fire. Theresults of modeling of fire spread can also be used to design fireprotection strategies for buildings. This paper proposes a dynamicmodeling of fire spread in buildings using a Bayesian network.The model can consider fire spread in both horizontal and vertical

Fig. 10. Probability of fire spread in the compartment during fire growth phase.

Table 4The ignition and flashover time in each compartment for case 2.


tig (min)

Flashover time

tfo (min)


tig (min)

Flashover

time tfo (min)

1 R1 0 0 2 R1 3 11

1 R2 26 34 2 R2 33 43

1 R3 60 70 2 R3 73 82

1 R4 94 104 2 R4 105 114

1 R5 137 146 2 R5 149 157

1 R6 158 167 2 R6 169 178

1 R7 206 214 2 R7 217 227

1 R8 267 275 2 R8 279 288

1 R9 288 296 2 R9 299 308

1 R10 341 349 2 R10 347 356

1 R11 354 363 2 R11 367 375

1 E 223 NF 2 E 242 NF

1 S1 59 NF 2 S1 70 NF

1 S2 418 NF 2 S2 425 NF

Fig. 11. Probability of fire spread during fire growth phase.


directions. The algorithms for simulating the fire spread processin buildings and calculating dynamic probability of fire spread foreach compartment at each time step of the simulation have been

proposed. The formulae calculating the input data for thedynamic fire spread model have been derived. The dynamic fire-spread model can easily be applied for any building includinghigh-rise buildings. A detailed example of calculation of firespread in a two-storey office building is described.

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dynamic modeling of fire spread in building

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