NATL INST OF STAND & TECH

A11107 310MM6

NBS

PUBLICATIONS

NBSIR 83-2765

Calculations of Wall Fire Spread

in an Enclosure

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, D.C. 20234

November 1983

-QC—

p

IGU

.U56

ci3-2765

1983

V c. 2

Sponsored in part by:

Armstrong World Industries

Lancaster, PA 17604

and

U.S. Department of Health and

Human Services

Washington, D.C. 20201

NATIONAi BUREAU or BTANDASOa UBRAAT

CALCULATIONS OF WALL FIRE SPREAD

IN AN ENCLOSURE

NBSIR 83-2765

/\/o.

Kenneth D. Steckler

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, D.C. 20234

November 1983

Sponsored in part by:

Armstrong World Industries

Lancaster, PA 17604

and

U.S. Department of Health and

Human Services

Washington, D.C. 20201

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

LIST OF FIGURES v

LIST OF TABLES vii

NOMENCLATURE vili

Abstract 1

1. INTRODUCTION 1

2. APPROACH 4

3. DESCRIPTION OF THE QUASI-STEADY MODEL 5

4. DEVELOPMENT OF GAS LAYER AND WALL CONDUCTION EQUATIONS FOR

A TRANSIENT TWO-LAYER MODEL 9

4.1 Transient Equations for Conservation of Mass and

Energy in a Gas Layer 9

4.2 Transient Wall Conduction Equations 11

5. TRANSIENT MODEL FOR CRIBS BURNING IN AN ENCLOSURE 13

6. RESULTS OF CALCULATIONS FOR WOOD CRIBS BURNING IN AN

ENCLOSURE 14

7. DEVELOPMENT OF AN ENCLOSURE CORNER WALL FIRE MODEL 15

7.1 Burning and Flame Spread Rates 15

7.2 Wall Grid 18

7.3 Local Radiant Flux to Wall 18

7.3.1 Radiation from Enclosure Surfaces and Upper

Gas Layer 19

7.3.2 Radiation from Burning Wall Areas 20

7. 3. 2.1 Radiation from Burning Wall to

Adjacent Burning Wall 20

7. 3. 2.

2

Radiation from Burning Wall to

Same Wall 21

7.3.3 Total Incident Radiation 21

7.4 Wall Fire Entrainment 21

7 .5 Summary of Wall Fire Model Equations 22

8. RESULTS OF WALL FIRE CALCULATIONS FOR A HYPOTHETICAL WALL

MATERIAL 23

9. CONCLUSIONS 25

10. ACKNOWLEDGMENTS 26

11. REFERENCES 26

iii
Page

Appendix A - Subsidiary Equations 28

Appendix B - Radiation from Upper Gas Layer and Non-burning

Enclosure Surfaces 31

Appendix C - Radiation Path Length Through Upper Layer 34

Iv
LIST OF FIGURES

Page

Figure 1-1 Wall spread phenomena 28

Figure 2-1 Approach to solving wall spread problem 29

Figure 3-1 Two-layer zone model 30

Figure 4-1 Equation of continuity applied to uniform gas

layer 31

Figure 4-2 Finite difference slabs 32

Figure 6-1 Room geometry used in calculations 33

Figure 6-2 Free-burn mass loss rate for four wood cribs 34

Figure 6-3 Effect of layer transient terms on wood crib

calculations 35

Figure 6-4 Comparison of experimental and theoretical

enclosure temperatures - four wood cribs 36

Figure 6-5 Comparison of experimental and theoretical mass

loss rates - four wood cribs 37

Figure 6-6 Comparison of experimental and theoretical oxygen

concentrations - four wood cribs 38

Figure 7-1 Grid for wall spread calculation 39

Figure 7-2 Radiation from non-burning enclosure surfaces and

upper gas layer to dA 40

Figure 7-3 Geometrical factors for radiation from non-burning

enclosure surfaces and upper gas layer 41

Figure 7-4 Radiation from burning elements on adjacent wall 42

Figure 7-5 Initial area of involvement for wall spread

calculations 43

Figure 7-6 Theoretical wall spread results at t = 100 seconds ... 44

Figure 7-7 Theoretical wall spread results at t = 240 seconds ... 45

Figure 7-8 Theoretical wall spread results at t = 245 seconds ... 46

Figure 7-9 Theoretical wall spread results at t = 250 seconds ... 47

Figure 7-10 Summary of theoretical wall spread results:

initial burning area on each wall = 0.5mxH 48

V
Page

Figure 7-11 Theoretical upper gas temperatures from corner wall

spread calculations: initial burning area on each

wall -

5^ X H 49

Figure 7-12 Summary of theoretical wall spread results;

initial burning area on each wall = 0.5 m x H/2 50

Figure 7-13 Summary of theoretical wall spread results:

initial burning area on each wall = 0.8 m x H/2 51

Figure B-1 Radiation target element in lower layer 59

Figure B-2 Radiation target element in upper layer 60

vi
LIST OF TABLES

Table 8-1 Thermophys Leal and fire properties of hypothetical

wall material

Page

53
NOMENCLATURE

"b

A'x

ou

^inv

py

PP

'pw

D

E"

wall properties parameter (equation 4-11), s~^

2 total burning wall area, m

2 - burning area on adjacent wall, m

2 - area of crib exposed to external radiation, m

2 - area of floor, m

internal area of crib(s), m

2 - area of room opening in upper layer, m

2 - enclosure surface area bounding upper layer, m

2 - enclosure surface area bounding lower layer, m

2 - area of involvement, (A^ + Ap^) , m

2 - pyrolyzing wall area, m

{

empirical material constant (equations 7-7 and 7-8), m^^^ s^^^/kW

radial function in equation A-lla

- opening flow coefficient (inflow)

- opening flow coefficient (outflow)

specific heat of air, J/g-K

- specific heat of pyrolysis gases, J/g-K

- specific heat of enclosure walls, ceiling, and floor, J/g-K

- distance between area elements on adjacent walls (Appendix C) , m

2 - energy release rate per unit area, kW/m

rate of energy transfer by heat transport, kW

- rate of energy transfer by mass transport, kW

- energy release rate, kW

- fraction of total energy release radiated by flame

«

J

viii
dA,i

r.s

g

H

HoAH

m

w

m

•vap

m

m.

• It "^b

governing equations In model

radiative exchange geometrical factors

gravitational acceleration, m/s

room height, m

opening heights, m

heat of combustion, kj/kg

convective heat transfer coefficient, kW/m -K

number of horizontal strips Into which wall Is divided

(figure 7-1)

flame attenuation coefficient, m~^

upper layer attenuation coefficient, m“^

mixing coefficient

thermal conductivity of walls, celling, and floor, kW/m-K

room length, m

mean beam length based on entire upper layer volume, m

mean beam length based on portion of upper layer volume, m

effective heat of vaporization of fuel, kJ/kg

flame thickness, m

path length through upper layer (Appendix C) , m

flame spread parameter (equations 7-8)

burning rate, kg/s

p burning rate per unit area, kg/m -s

lx
“b;w,k,i

^,crit

m

m.

in

out

m

m

c

•m

•m

I

•m

\

• • m

\

• • m

m

m'

\ N

P

Q

v;w,k,i

VO

VO

qV*

q"**

1

‘i,ig

O local burning rate per unit area (figure 7-1), kg/m -s

critical fuel supply rate per unit area necessary to sustain

burning, kg/m^-s

mixing mass flow rate from upper to lower gas layer, kg/s

mass flow rate through opening into room, kg/s

mass flow rate through opening from room, kg/s

mass flow rate to layer, kg/s

mass flow rate from layer, kg/s

pyrolysis rate, kg/s

entrainment rate of gas into flame and plume, kg/s

fuel pyrolysis rate, kg/s

O fuel pyrolysis rate per unit area, kg/m -s

local pyrolysis rate per unit area (figure 7-1), kg/m -s

free-burn pyrolysis rate, kg/s

free-burn fuel pyrolysis rate per unit area, kg/m^-s

number of slabs in finite-difference wall

pressure, N/m

rate of energy release by fire (equation 7-17), kW

dimensionless rate of energy release

incident radiant flux from flaming areas of adjacent wall, kW/m^

radiant flux emitted per unit area of flaming material

(equation 7-14)

2 incident radiant flux, kW/m

minimum external radiant flux necessary to sustain burning at

reference state, kW/m^

2 minimum external flux to cause mass loss, kW/m

2 minimum flux necessary for piloted ignition, kW/m

2 heat flux convected to surface, kW/m^

X
• •• qgw

rc

• M

^rl

Ku

• 9t %

• It

At

*

t

T

T,

'gl

gu

w

incident radiant flux from enclosure surfaces and upper gas

layer, kW/m^

heat flux conducted into surface, kW/m^

external radiant flux incident upon crib, kW/m

net flux radiated through layer interface from upper to lower

spaces, kW/m^

net flux radiated to lower surfaces of enclosure, kW/m^

net flux radiated from upper layer through room opening, kW/m^

net flux radiated to upper surfaces of enclosure, kW/m^

O net flux to exposed enclosure surface, kW/m^

net flux from unexposed enclosure surface, kW/m

stoichiometric air-to-fuel mass ratio

characteristic radius (equation A-lla), m

time, s

time increment, s

characteristic burn time, s

temperature of gas or wall, °C or K

temperature of exposed surface of finite difference slab, °C

or K

temperature of unexposed surface of finite difference slab, or K

ambient temperature, °C or K

flame temperature, K

lower gas layer temperature, °C or K

upper gas layer temperature, °C or K

finite difference wall slab temperature, ®C or K

temperature of surface behind flames, (equations 7-15

and A- 6) ,K

temperature of enclosure exposed surface, °C or K

reference temperature for ideal gas calculation, ®C or K

xi
T . wj - either or (equation 7-17), °C or K

- flame spread rate, m/s

- local horizontal flame spread rate (figure 7-1), m/s

V

00 limiting flame spread rate, m/s

w - room width, m

Wo opening width, m

X - position coordinate (Appendices B and C) , m

^,k,l - horizontal coordinate used in wall spread calculation

(figure 7-1), m

1

i - depth coordinate within wall (equation 4-7) , m

[ - dummy variable (equations B-6 and B-7a)

- ratio of excess air after combustion to air required by

combustion

AX - wall thickness increment, m

y position coordinate (Appendix C) , m

Y - dummy variable (Appendix C)

Yox - oxygen mass fraction

^0X1 - oxygen mass fraction in lower layer

^oxu oxygen mass fraction in upper layer

YoXjCrit

* Yox

- critical oxygen concentration below which burning ceases

- oxygen mass fraction; reference state

Z - height coordinate, m

^w,k,i - vertical coordinate used in wall spread calculation

(figure 7-1), m

AZ - height increment, m

Zd - thermal discontinuity height, m

- entrainment height, m

Zn neutral plane height, m

xii
Zj^ - vertical wall coordinate (Appendix C) , m

Z

2 - vertical wall coordinate (Appendix C), m

6^ - horizontal dimension of initial flaming area, m

6 - wall thickness, m w

e - average emissivity of upper gas layer

S

e - average emissivity of gas within a portion of upper layer

(equation 7-9)

- emissivity of surface behind flame (equation 7-14)

5 - burning-rate parameter (equation 7-2) , kg/m -s

- area ratio (equation A-17a)

Y - area ratio (equation A-8a)

u

O p - density of gas, kg/m'^

O p - density of air at ambient temperature, kg/m"^

q p - density of walls, ceiling and floor, kg/m"^ w

q p^ - density of air at reference temperature, T^, kg/m"^

a - Stefan-Boltzman constant (5.67 x 10”^^ kW/m^-K^)

xiil
(Jtl “

%

A

'*%,# I

'*

v’*‘*i

v.» ^'

"

",?* .* '

\ ^*»,l

1
Calculations of Wall Fire Spread in an Enclosure

Kenneth D. Steckler

Center for Fire Research

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

Abstract

Mathematical models of fire growth in enclosures offer a potential for assessing the risk, of materials with respect to a hazard such as flashover. The development of a model for fire

spread over wall lining materials is presented. The work covers

the development of a transient two-layer zone model, initially

formulated for crib fires in a room, then adapted to simulate a spreading corner wall fire. The local wall flame spread rate and

burning rate per unit area are expressed in the model as functions

of the external radiation incident upon the burning surface and

the oxygen concentration in the adjacent gas layer (zone). Flame

spread is limited to the horizontal direction. Results for a fictitious wall lining material are presented. Major elements of

the results are shown to be in qualitative agreement with

experience. Finally, areas for improvement and the direction of

, future efforts are noted.

Key words: burning rate; fire models; flame spread; flashover;

hazard assessment; interior finishes; room fires; walls.

1 . INTRODUCTION

A common room-fire scenario involves the ignition of an object such as a wastebasket positioned near a combustible wall surface, ignition of the wall

lining material, and subsequent fire spread on the wall lining. In many cases

the energy released by the wastebasket alone would be insufficient to produce

a dangerous environment within the room. However, when the wastebasket is

located near a wall in a room, whether the fire continues to grow to a

1
hazardous condition such as flashover, or merely burns out, depends on the

fire characteristics of the lining material and the extent to which the

enclosure dissipates the energy and combustion products released by the

burning material. With knowledge of a characteristic energy release rate per

unit area, E", and the flame spread rate, v^, of the burning wall lining

material, an evaluation of the potential hazard of flashover should be

possible. If an evaluation of potential hazard due to development of a toxic

atmosphere is desired, then knowledge of the rates of production of hazardous

species per unit area would also be required.

In a qualitative sense, one would expect a flashover "hazard index" for a lining material to include some combination of v^ and E" to reflect the total

energy release rate of the material. Unfortunately, "combining" v^ and E" is

complicated by the fact that both parameters tend to be functions (at least)

of exposure conditions such as incident radiant flux and oxygen concentration

[1-6]. When a material burns within an enclosure, portions of the energy and

combustion products released by the fire accumulate within the enclosure and

portions feed back to the burning material. Therefore, E" and v^ are coupled

through the properties of the overall enclosure. A more detailed analysis of

the interactions between the burning lining material and enclosure is required

to assess the roles of E" and v^ in contributing to the hazard of flashover.

The required analysis is mathematical modeling.

Mathematical models of fire growth in enclosures represent assemblages of

mathematical expressions which describe in varying degrees the complex physical interactions between a burning material and its environment. One class of

models, known as zone or control volume models [7-12], treats the developing

room fire in terms of several volume and boundary elements which intercommunicate energy and/or mass. Conservation equations are written for each zone and

solved using a numerical technique. Applying such models leads to solutions

for the state variables of each zone and the various interzonal mass and

energy transfers.

Several zone models treat fires which grow in area. Emmons and Hitler

[8] and Rockett [13] modeled fire spread on a horizontal (mattress) surface

within an enclosure while McArthur and Meyer [10] treated the case of fire

2
spread on seats and walls in an aircraft cabin. Tanaka [14] and Smith [11]

modeled the developing wall fire in a room. The current work follows an approach suggested by Quintiere [15]. It differs from the earlier work in

that the different oxygen concentrations in the upper and lower gas layers

within the enclosure are taken into account in predicting the burning and

flame spread rates in these layers.

The objective is to fonnulate and execute a "first-cut" mathematical zone model which handles enough of the phenomena depicted in figure 1-1 so as to

1) produce results which are qualitatively reasonable and 2) provide a basis

for a more complete model which will provide an answer to the question: If a given wall lining is ignited, will the room reach flashover conditions? It was judged that a model having the following limited features would lead to

this objective and be tractable within the time allocated for the project.

a. The wall lining is the only combustible material in the room.

b. Only the fire spread, not ignition, is considered. This assumes that

a portion of wall-corner area is already burning and that the

ignition source is either no longer present or is insignificant.

c. Flames are restricted to the wall areas; that is, there are no flame

extensions on the ceiling.

d. As a first approximation flame spread is restricted to the horizontal

direction.

e. Burning rates per unit area and flame spread rates are expressed as functions of the oxygen concentration of the gas layer (upper or

lower) in which the area element is located and of the local incident

external radiant flux. This approach is motivated by the existence

of experimental results and theoretical analyses for steady or mean

time burning and flame spread rates as a function of external oxygen

concentration and heat flux. Therefore local burning and flame

spread rates will be assumed to instantaneously take on steady values

consistent with the instantaneous values of the time-dependent

3
external heat flux and oxygen concentration. This leads to a conser- vative estimate of these rates and avoids the problem of calculating

local wall temperature histories. As a further simplification, it

will be assumed that the external heat flux which drives the burning

and flame spread rates is solely radiative: that is, these rates are independent of convective effects from the gas zone or layer in which

the material is immersed.

f. The flame spread model assumes countercurrent gas flow in all

locations. The one place where this clearly does not bear out is the

ceiling jet region.

g. The room conditions and interactions between the room and burning

material are formulated as a transient version of the quasi-steady

room fire model of Quintiere and McCaffrey [12]

.

2

.

APPROACH

The approach used to develop the corner wall spread model is diagrammed

in figure 2-1. Transient features such as gas layer transients, wall heat

A

conduction , and time-dependent energy release of the fire source were added

to the quasi-steady model for crib fires in an enclosure [12]. Actually two

transient enclosure fire models are developed in the following sections; one

for a time-dependent crib source and the other for a spreading wall fire

source. The transient crib model provided a convenient means for evaluating

the transient conduction and gas layer sub-models before proceeding with the

more complicated wall-spread source.

A

With respect to heat transfer by conduction, the ceiling and floor are treated as extensions of the upper and lower wall surfaces, respectively.

Therefore, "wall conduction" refers to either wall and ceiling or wall and

floor heat conduction.

4
3. DESCRIPTION OF QUASI-STEADY MODEL

The quasi-steady model, which served as the starting point for this

project, will be briefly described in this section. Basically the model

predicts the quasi-steady conditions within an enclosure, including the

burning rate of the crib(s), given the free-burn rate of the crlb(s) and other

properties of the enclosure. A more detailed description can be found in the

literature [12].

The model casts the enclosure fire in terms of control volumes or zones,

and the physical boundaries of these zones (figure 3-1). Mass and energy

balances are written for each zone and boundary as functions of seven independent variables; namely.

Z

Z

n

d

T

T

T

T

gu

gl

wu

wlm

V

height of neutral plane in the opening,

height of thermal discontinuity within the room,

temperature of gas in upper layer,

temperature of gas in lower layer,

temperature of wall and ceiling bounding upper gas layer,

temperature of wall and floor bounding lower gas layer, and

fuel pyrolysis rate.

This produces seven simultaneous non-linear algebraic equations of the

form

d* gu*

T 1 . glT , T , , m ) wu wl V

L n

* ^d*

T , T T, m ) gu* gl* V

d’ gu* V’ T , T T , m ) wu wl V

f

4^^n*

Z-, T , T .)

d* gu gl

d’ gu* V’ T , T 1 , m ) wu wl* V

d* gu* V* T , T . , m ) wu wl V

d* gu* V* T , T , , m ) wu wl V

0

0

0

0

0

0

0

(3-1)

5
which must be solved by numerical methods.

Specifically, the seven equations describe the following quasi-steady

processes

a. Mass balance for the enclosure:

f =m -m -m = 0 (3-2)

1 o i V

where m is the mass flow rate outward through the room

o opening (equation A-1 listed in Appendix A),

is the inward mass flow rate through the opening

(equation A-2)

,

m^ is the fuel pjrrolysis rate (see equation 3-8).

b. Mass balance for lower gas layer:

f-=m -m. - m = 0 (3-3)

2 p 1 e

where m is entrainment rate into the crib and plume

P

(equation A-3)

,

m^ is the mass flow rate from the upper to lower layer

due to entrainment (mixing) at the opening

(equation A-4).

c. Upper gas layer energy balance:

= E_ - E

Ru Lu ^u = 0 (3-4)

where

'Ru is the rate at which energy is released in the layer

(equation A-5) , (note: in figure 3-1 the crib is

located in the upper layer).

6
is the net rate of energy loss from the layer by

heat transfer (equation A-7)

E„ is the net rate of energy loss due to mass transfer

Mu

(equation A-6)

d. Lower layer energy balance:

< 3-5)

where E^^^ is the net rate of energy loss by mass transfer

(equation A-12),

Eli rate of energy loss by heat transfer when

the gas is not considered to be absorbing or emitting (equation A-13).

e. Energy balance at upper gas-wall interface:

^5 ’

• »• qku

" q^., “ q^.. = 0 ru cu

(3-6)

where q" is the heat flxix conducted from the surface into the

^ku

upper wall (equation A-14),

q" is the net radiative flux to the upper wall surface

^ru

(equation A-8)

,

q^^ is the flux convected from the gas to the upper wall

surface (equation A-11).

f. Energy balance at lower gas-wall interface:

At! • •• • •• h ' "kl -

"’rl %1 ° (3-7)

7
where q" is the heat flux conducted from the surface Into the

kl

lower wall (equation A-15)

,

q", is the net radiative flux to the lower wall surface

^rl

(equation A-17),

q^^ is the convected flux from the gas to the lower wall

surface (equation A-16).

g. Fuel pyrolysis rate

f, = m - m" A,(Y ,/0.23) - iq" A /L ) = 0 (3-8)

7 V VO i 0x1 ^rc c vap

where m^ is the fuel pyrolysis rate.

m”^ is the peak free-burn pyrolysis rate per unit area.

A^ is the internal area of the crib,

A^ is the area of crib exposed to external radiation.

q^^ is the external radiant flux incident upon crib,

Yoxi is the mass fraction of oxygen in the lower layer

(equation A-18),

Lvap is the fuel's effective heat of vaporization.

The burning rate, la^, is defined in equation A-3 as the smaller of m^

(fuel controlled) and m^/r (ventilation controlled), where r is the

stoichiometric air-to-( volatilized) fuel mass ratio. The model also computes

the mass fraction of oxygen in the upper layer, Y^^ (equation A-19).

8
4. DEVELOPMENT OF GAS LAYER AND WALL CONDUCTION EQUATIONS

FOR A TRANSIENT TWO-LAYER MODEL

The equations presented in section 3 for the conservation of mass and

energy in the two gas layers (equations 3-2 to 3-5) are based on the assumption that the time derivative or transient terms of the more complete equations are negligible with respect to the retained terms. This is a good

assumption for near-steady conditions, but could lead to significant errors if

the early stages of fire growth were modeled as a sequence of quasi-steady

states. Similarly, the quasi-steady wall conduction terms,

qj^^

and in

gas-wall interface equations 3-6 and 3-7, could produce conqsarable errors in

such an analysis. Therefore, more complete equations for mass and energy

conservation are needed to address the spreading wall fire problem.

4.1 Transient Equations for Conservation of Mass

and Energy in a Gas Layer

Following the analysis of Quintiere [15] , the equation of continuity for

a generic uniform gas layer shown in figure 4-1 is

m, = 0. in (4-1)

The first term accounts for changes in the layer thickness, Z, and layer

density, p, with time, t. The second and third terms represent the mass flow

to and from the layer, respectively. A transient energy equation for the same layer is given by the expression

WLZ - WLZpCp = 0 (4-2)

where the first term accounts for energy used to change the total pressure

within the layer, the second term accounts for energy used to change the

temperature of the layer, E is the rate of energy release within the layer,

R • • E^ is the rate of energy loss by heat transfer, and Ej^ is the rate of energy

loss by mass transfer.

9
In applying equation 4-2 to residential room fires of the type shown in

figures 1-1 and 2-1, it appears reasonable to assume that the total pressure,

p, remains nearly constant [15]. Therefore, the dp/dt term can be ignored.

If time derivatives are expressed as simple forward finite differences, the

resulting set of conservation equations for the upper and lower gas layers is

a. Mass (upper plus lower layers)

^1 = (r 1. . 1 1

T (t)

gu J

Z^(t + At)

At

Z^(t) Z^(t)

T^,(t + At) - T,(t) H - Z,(t) (t + At) - (t) gl gl d gu gu

At

(t)

gu

At

(4-3)

+ m (t + At) - m. (t + At) - m (t + At) = 0

o i V

b. Mass (lower layer);

*2

= ]v(t)

it) - Zj(t)

it

Zd<t) T^^(t + it) - T^^(t)

+ mpie (t + At) - m. (t + At) - m (t + At) = 0

c. Energy (upper layer)

f- = -WLp T c —_

3 'oo 00 p T (t)

gu

H - Z,(t) (t + At) - (t)

d gu gu

At

+ E^^(t + it) - Ej_^(t + it) - E„^(t + it) = 0

d. Energy (lower layer):

(4-4)

(4-5)

f , = WLp T c

4 <» “ p T - (t) gl

Z^(t) T^^(t + At) - T^^(t)

At

+ Ej^(t + At) + E^j^(t + At) = 0

(4-6)

10
• • * where etc. are obtained from the subsidiary equations in

Appendix A using Z^(t + At), Z^(t + At), T ^(t + At), etc. If Z^, Tg^, and

Tgj^ are independent of time, the above equations reduce to quasi-steady equations 2-2 to 2-5.

4.2 Transient Wall Conduction Equations

Heat conduction through the walls of the enclosure will be treated as a one-dimensional process. A solution to the transient heat conduction equation

L. k (T) = D c (T) 3x w^^^ ax ^w pw^^^ at

(4-7)

is needed throughout the thickness of the wall. In this equation

represents the thermal conductivity of the wall material, is its density,

and c is its specific heat. A very good approximate solution can be

pw

obtained by using a finite-difference numerical technique. This technique can

be viewed as a process in which 1) the solid is divided into a group of slabs or layers (figure 4-2) and 2) an energy balance approximating equation 4-7 is

written for each layer in terms of finite temperature, time, and coordinate

differences. The energy equations can be rearranged to express the current

slab temperatures, T(t + At), in terms of the previous temperatures, T(t), and

the previous surface fluxes, ^(b) resulting equations are

1

"

T (t + it) = Tj(t) (1 - a^it) + a^it

f

w

(4-8)

+ T.yo] (4-9)

yt + it) = yt) (1 - a^it) + ajjit y^(t)

q" (t + it) k „ ^wo

wN

(4-10)

11
where a 2k . — (A-11)

j pc AX

2

pwj

‘‘wj ' MTj>

At ^ minimum of

N _< 20

^1 “ ^wu ^wl upper wall slab temperatures when Tj =

^2 lower wall slab temperatures when = T^l

The energy balance at the upper layer gas-wall Interface can then be

expressed as

is the net flux radiated to the inner (exposed) surface of the

upper wall,

q" is the flux convected to the inner wall surface, and

^cu

T

2

is obtained from equation 4-9.

Similarly, the energy balance at the lower gas-wall Interface can be

expressed as

fc = q^ (t + At) - q" (t + At) - q" (t + At) » 0

5 ^ku ru cu

(4-12)

where

T (t + At) - T (t) wu wu

At

(4-13)

12
(4-14)

where

At

(4-15)

q", is the net flux radiated to the lower inner wall surface

rl

Vi is the flux convected to the lower inner wall surface, and

cl

T2 is obtained from equation 4-9.

5. TRANSIENT MODEL FOR CRIBS BURNING IN AN ENCLOSURE

The conservation equations presented in section 4 must be augmented with

a transient sub-model for pyrolyzing cribs in order to complete a transient

model for cribs burning in an enclosure. The following pyrolysis model

In this expression A^ and A^ are the internal and external surface areas

of the crib, respectively. Details of the calculation of these areas and q^^

are presented in reference [12].

In summary, the basic equations of the transient model for crib fires in an enclosure are

relates the time-dependent fuel pyrolysis rate, m^, to the free-burn pyrolysis

rate, the effective heat of vaporization, L^^p; the oxygen concentration

in the lower layer, and the radiant flux incident upon the cribs,

f^ = m (t + At) - m" (t + At) A. (Y , (t + At)/0.23)

7 V VO i 0x1

(5-1)

(q" (t + At) A /L ) = 0 ^rc c vap

13
a.

b.

c.

d.

f.

Mass balance for the enclosure:

(equation 4-3)

Mass balance for the lower layer:

(equation 4-4)

Energy balance for upper layer:

(equation 4-5)

Energy balance for lower layer:

(equation 4-6)

Energy balance at the upper gas-wall interface:

(equation 4-12)

Energy balance at the lower gas-wall interface:

(equation 4-14)

Fuel pyrolysis rate

(equation 5-1)

6. RESULTS OF CALCULATIONS FOR WOOD CRIBS BURNING IN AN ENCLOSURE

The following calculation simulates the burning of four closely spaced

wood cribs located near the center of the room shown in figure 6-1. The room surfaces are calcium silicate board. The experimental time-dependent freeburn (no enclosure) mass loss rate of the four cribs, m - m" (t) A., is

* vo VO i

shown in figure 6-2. For the calculation, the actual mass loss rate was approximated by the two linear segments shown in the figure. Unless otherwise

specified, the remaining thermophysical properties and model parameters used

in the calculation are the same used in reference [12]

.

Initially, the effect of the gas layer transient terms was evaluated.

Two calculations were made; one with and another without the time derivative

terms in equations 4-3 to 4-6. Transient wall effects (equations 4-13 and

4-15) were included in both calculations. The heat of combustion, AH, was set

at 15,000 kj/kg and the effective heat of vaporization, L^^p, was fixed at

1900 kJ/kg. Results depicting the first 150 seconds of simulated time are presented in figure 6-3. The effect of the gas layer transients is shown to

be small after approximately 60 seconds.

14
The calculation which retained the gas layer transient terms was repeated

using a slightly lower heat of combustion, AH = 13,000 kj/kg. Figures 6-4 to

6-6 compare these results with experimental results for the same configuration. Two of the solid-line plots in figure 6-4 are identified as experimental upper wall and ceiling temperatures. The remaining solid lines in this

figure represent experimental floor-to-ceillng gas temperatures measured at

equally spaced locations along a vertical line. These solid lines tend to

cluster in the upper and lower regions of the figure, thereby indicating the

two-layer nature of the gas widely-spaced traces between these clusters

correspond to the transition region between layers. The comparisons in

figures 6-4 to 6-6 demonstrate the ability of the transient model to predict

the major features of the experimental results.

7. DEVELOPMENT OF AN ENCLOSURE CORNER WALL FIRE MODEL

The transient model for a crib or cribs burning in an enclosure was

treated as a uniform fixed-area fire which changed in intensity depending on

its free-burn behavior, the incident radiant flux, and the available oxygen.

The corner wall fire situation is more complicated because 1) the burning or pyrolyzing occurs over an extended area which is subject to non-uniform radiation and oxygen exposures and 2) the boundary of the involved area changes

locally with time, because of the non-uniform exposure conditions.

In the following sections a wall fire spread zone model will be developed

which features local pyrolysis rates and flame spread rates as functions of

incident radiation and "local" oxygen concentration. In the context of the

two-layer zone model, "local" oxygen concentration is the concentration in the

gas layer (either the upper or lower layer) adjacent to the pyrolyzing

surface.

7.1 Burning and Flame Spread Rates

Tewarson [1,2] and others have experimentally shown that, for relatively

small horizontal upward-facing specimens under non-flaming and generally

steady conditions, the mass loss or pyrolysis rate, m^, is a linear function

of applied radiant heat flux, q^. They have also shown that under flaming or

15
burning conditions is both a linear function of q" and local oxygen concentration, Such functional dependencies will be assumed in this analysis

for wall fire spread. Thus, for non-flaming conditions m^ per unit area will

be expressed as [1,2,15]

m

0 ; **

‘’l -’i ‘'i i "i

(7-1)

where is the effective heat of vaporization and q” is the minimum heat

flux to cause mass loss. Similarly, under flaming conditions m" will be taken

as

m" = m" + ?(Y - Y ) + (qV - qV )/L

V V ox ox ^i 1 vap

(7-2)

where 5 is a constant and the "starred” terms depend on the reference state.

For the example to be considered, the reference state will be at normal air

. * * . * conditions so m^ represents the burn rate in air, Y^^ = 0.233, and q^ is the

minimum heat flux to sustain burning. Over a range of oxygen and flux levels,

it has been found [1,2] that a critical fuel supply rate is needed to sustain

burning; i.e..

m"(Y

V ox

q") > m" , ^i — v,crit

(7-3)

For a number of materials this burning limit appears to be approximately

5 g/m-s. Also it is obvious that burning will not be possible at Y^^ = 0 (or

a small, but finite value) regardless of the heat flux. Hence, the specific

mass loss rate is given by Eq. (7-2) as long as Eq. (7-3) holds and Y^^^ is in

excess of some specified critical value, Y^^ crlt* Otherwise m^ is given by

Eq. (7-1). Consequently, the overall mass loss rate is given by

m = // m" dA (7-A)

inv

where A^^^ is the involved area (burning plus pyrolyzing) . The overall

burning rate is

16
= smaller of

/ / m" dA

A

orm /r (ventilation limit)

a

(7-5)

where A^ is the burning area, m is the air flow rate through the doorway, and

r is the stoichiometric air-to-fuel mass ratio.

Quint iere [1] has also shown that the results obtained from a radiant

panel lateral flame spread apparatus can be expressed in the form

''£ ' - qp]

-2

(7-6)

where is the maximum flame spread rate for an incident flux q” which is not

"too close" to the minimum flux for piloted ignition, and C is an

empirical material constant.

Both theoretical and experimental work [3-6] show flame spread rate to

,m vary as Y , where m is in range 0.9 to 3.6, depending upon the material, its ox

thickness, and the value of relative to Y^^ crit* Incorporating this

oxygen dependence into Eq. (7-6) yields

v^ = [C(qj,ig-qp]'(7-7)

For q" approaching and exceeding q" , modification to Eq. (7) is

i 1 » ig

necessary. As a first approximation an upper bound, v , is specified for

. * ”

v^. There is also a lower flux limit, q^ , below which no opposed-flow flame

spread occurs. Therefore,

0; when q" < IXq" ^ smaller of v and Eq. (7-7); when q" ^ q" < qV . " 11 1)1S

l_ • •• N. • •• V , when q. > q. . «>* ^i — i,ig

(7-8)

In this formulation the spread will cease when Y^^ is zero, although, in

reality, the pyrolysis front would continue to advance if the incident flux

17
were sufficient. In the implementation of the wall flame spread examples only

lateral opposed flow flame spread will be considered up to the time when

in the adjacent layer becomes zero. It is recognized that flame spread in the

ceiling jet and downward spread are also present and will have to be Included

in a more complete analysis.

7.2 Wall Grid

Since the radiation incident upon a wall area element and the oxygen

concentration to which it is exposed are position dependent, it is necessary

to establish a grid for locating the involved wall elements. The grid for

tracking the area of Involvement on one wall is shown in figure 7-1 . The grid

for the adjacent wall is merely the mirror image of the first with respect to

the left hand boundary (room corner). Each wall is represented by k equalheight horizontal strips. Initially, burning is assumed to exist on both

walls from floor to ceiling over a horizontal distance, 6^, measured from the

corner. Subsequent flame spread is restricted to the horizontal direction.

At the end of each time step. At, new flame-front positions are established

for each horizontal strip. Differences between flame-front positions at the

beginning and end of each time step establish the added area of involvement.

The center coordinates of each area element are used to compute the local

radiant flux and oxygen concentration which are then used to compute m^ or mj^

for the element. The coordinates at the centers of leading edges of the

newest elements are used to compute the flame spread rates at these edges.

These coordinates are included in figure 7-1.

7.3 Local Radiant Flux to Wall

Incident radiant flux is needed as a function of position on the wall to

evaluate the local pyrolysis rates and flame spread rates of the developing

wall fire. This flux is made up of contributions from the hot upper gas

layer, the hot wall surfaces in both layers, and the flames on the Involved

portions of the walls. Details of the flux calculations are presented in the

following sections.

18
7.3.1 Radiation from Enclosure Surfaces and Upper Gas Layer

For this calculation, all enclosure surfaces are assumed to be nonburning and at a temperature or Twl (figure 7-2). Radiation from these

surfaces and the upper gas layer to any point on a wall is computed using

standard geometrical exchange factors [16] and the mean-beam-length approximation [17,18].

Consider a target wall element located within the lower gas layer

(figure 7-3). The element receives radiation from the four quadrants designated i = 1 to 4. Sub-regions i = 5 in the third quadrant and i = 6 in the

fourth quadrant identify the portions of these quadrants which lie in the

upper layer. The parameter ^ is the geometrical factor between the target

element dA and the enclosure surfaces which bound the ith volume.

The effective emissivity of the upper-layer gas within a quadrant is

obtained from the mean-beam-length approximation

Eg

i = 1 - e ^g^m,i (7-9)

where the mean beam length

[

volume of upper layer in

^ ^ ^ quadrant containing 1th reglo

m,i rsurface area of upper-layer

volume in quadrant containing

Lith region.

n.

]

(7-10)

If the emissivlties of the solid surfaces are assumed to be unity, then

it can be shown (Appendix B) that the radiation from the upper gas layer and

enclosure surfaces to a target element located in the lower layer can be

expressed as

*q" = I OF-. .T , + Z OF,. ,T„ + (1 - e ^

.) gw dA,i wl dA,!!- g,i gu g,i)t'^ - t"^ J

wu wl-* (7-11)

When the element is located in the upper layer the expression becomes

19
(7-12)

4

I i=l

r A oF [e ,T +

dA.i*- g,l gu

(1

6

I i=5

aFdA,i (1 - - ) wu

where the geometrical factors apply to the inverted forms of the geometries

shown in figure 7-3; i.e., quadrants i = 1 and 2 contain portions of the upper

layer and i = 3 and 4 contain portions of the upper and lower layers.

7.3.2 Radiation from Burning Wall Area

7. 3. 2.1 Radiation from Burning Wall to Adjacent Burning Wall

Each burning area element contributes to the radiant flux incident upon

the adjacent burning wall (figure 7-4) according to the expression

-(k il )

‘‘'•f = % ^s.r ^ 8 8 (7-13)

where dq^ is the incident flux at the "receiver".

k is the upper-layer attenuation coefficient,

o

I is the path length through the upper layer (Appendix C) , and

S

F„ _ is the geometrical factor between the receiver at surface

s , r element r and source at surface element s.

The flux emitted from the source element, q^, is expressed as

q = aT^(l-e ff) + e oT e ff (7-14)

e t s s

where the first term on the right represents flux from the flame and the

second term accounts for flux from the wall surface behind the flame. The

parameters k^ and are flame attenuation coefficient and flame thickness,

respectively. It appears from equation 7-14 that q^ will vary from material

to material. However, since data of this type are scarce, q^ was approximated

20
In the current calculations as a constant of 20 kW/m^. This value falls

between the "thin" and "thick" flame values estimated by Modak and Orloff [19]

for vertical polyurethane surfaces. The total flux incident upon a burning

wall element from the burning areas, A^, of an adjacent wall is obtained by

summing equation 7-13 over all burning elements;

7 .3.2.2

q" F

e s ,r -k

e

I g g

Radiation from Flames on Burning Wall to Same Wall

(7-15)

It is assumed that radiation transferred from flames on a wall to a spot

on the same wall has been accounted for in the test-method-based analytical

expressions for burning and flame spread rates; that is, these fire properties

are independent of burning area.

7.3.3 Total Incident Radiation

For the purposes of computing pyrolysis and flame spread rates, the

radiant flux, q^, to a point on a wall is

• it , • •• gw f - Z F

K

s.r

a T

wj (7-16)

where q" is defined by either equation 7-11 or 7-12, q" is given by

gw r equation 7-15, and the remaining term on the right compensates for the fact

that the wall area behind the burning region is included in both q^^ and q^.

The parameter T^j is either T^^ or T^j^.

7.4 Wall Fire Entrainment

A complete theory for wall fire entrainment does not exist. Nevertheless, an approximate entrainment model will suffice for the present task.

Indeed, a simple point source buoyant plume model will be adequate in view of

the approximations made in previous sections of this report and the preliminary nature of this project.

21
The mass flux, m^, a distance above an isolated point buoyancy source

of strength Q is given [20] by

ra = 0.0625 Q (7-17)

For the wall fire, Q will be set equal to the total heat release rate of the

burning wall area located in the lower gas layer and the point source will be

located at the floor. Since the mass flux to the upper layer is needed, Z^

will be equated to the gas-interface height, Z^.

7.5 Summary of Wall Fire Model Equations

The basic equations for the corner wall fire spread model are gas layer

equations 4-3 to 4-6, wall equations 4-12 to 4-15, fire plume entrainment

equation 7-17, and pyrolysis equation

f_ = m - m, - E m V- At AZ = 0

7 V D ^ V f

A

py

where

m = V

0;

q'i < qj

** **

(q'i -

qj «l > qj

(7-18)

(7-19)

m, = E m"v^ At AZ

b . b r ^b

(7-20)

Vc =

0; q;: < q"

± ± * smaller of v^ and equation 7-7 ; q’^ q^ ^ q^

’I ^ H.iS

(7-21)

m. -

*i; ^

'(’'ox - \x> ^ vap

(7-22)

subject to the constraints

. ft V . M m, > m, .

b — b,crit

(7-23)

22
Y > Y ox — ox,crit (7-24)

< m^/r

i

(7-25)

where Y^^ is either Y OX Ioxu

is given by equation 7-16

Subsidiary equations A-1 , A-2, A-4 to A-13, and A-16 to A-19 listed in

Appendix A complete the model.

8. RESULTS OF WALL FIRE SPREAD CALCULATIONS FOR A HYPOTHETICAL WALL MATERIAL

The wall material is termed hypothetical because it was arbitrarily

assigned artificial flame spread, burning rate, and stoichiometric properties

which represent the properties of wood, polymethyl methacrylate (PMMA), or calcium silicate board. The values used in the calculation are presented in

table 8-1. This arbitrary choice of properties was convenient and consistent

with the limited objective of demonstrating the qualitative features of the

model.

The allotted simulated time span for each calculation was 0 to 590

seconds. The time increment. At, was initially set at 10 seconds. In the

event that mathematical convergence was not achieved at a given simulated

time. At was halved, conditions from the previous time reinstated, and the

calculation repeated. If this process reduced At to less than 0.3125 seconds,

or if Y^^„ became equal to 0, the calculation was terminated. Finally, it

should be noted that the last term in equation 7-16 — the wall area correction term — was not included in these calculations. Therefore, q" is exaggerated during the later portion of the simulation when T^^ becomes large.

The first configuration considered is shown in figure 7-5. At t = 0,

uniform burning exists from floor to celling over a width 6^ = 0.5 m on each

of two walls meeting at a corner. Calculated results at subsequent times are

shown in figures 7-6 to 7-9. Since the area of involvement is symmetrical

with respect to the room corner, only the pattern on one wall is shown. The

23
burning rates listed in these figures represent the totals for both walls. As

previously stated, on account of the forementioned approximations and assumptions, results are intended to be viewed in a qualitative rather than a quantitative perspective.

Figure 7-6 shows the predicted conditions at t = 100 seconds. Only a small increase in area of involvement occurs up to this time because the

incident radiant flux is still relatively low. Note that the spread is

slightly less in the upper layer relative to that in the lower layer despite a higher incident flux in the former. The significantly lower oxygen concentration in the upper layer = 0.135 versus = 0.209) more than

offsets the higher radiation level, thereby producing less spread. Little

change occurs in the area-of-involvement pattern until some time in excess of

200 seconds. The situation at t = 240 seconds is shown in figure 7-7. Spread

in the upper layer begins to outdistance that in the lower layer because

radiation has become the controlling factor. Within the next 5 seconds

(figure 7-8) the spread increases rapidly in the upper layer as the incident

radiation increases. Rapid growth occurs in both layers until t = 250 seconds

(figure 7-9) when Y^^^^ goes to zero and the calculation terminates.

The above results are summarized in figure 7-10. The properties of PMMA

lead to a relatively low flame spread rate with a fire nearly fixed in area

(Ainv) for an extended period. However, the high energy release rate per unit area of PMMA which was used in the calculation, and the subsequent feedback of

a portion of this energy by the enclosure to the burning area produces a steadily increasing burning rate, m^, which appears to drive the system to an

unstable or flashover condition.

The effect of the initial width of the burning area on each wall, 6^, was

then investigated. Figure 7-11 compares the upper layer gas temperature

results, T , for calculations of corner wall fires with 6 equal to 0.50,

0.35, and 0.25 m. In each case, the initial burning area extended from floor

to ceiling. Reducing 6^ from 0.50 to 0.35 m delays flashover by approximately

300 seconds, whereas the 0.25 m Initial width falls to produce flashover

within the simulated time span.

24
Figure 7-12 presents the results of a calculation representing a corner

wall fire in a room with combustible lining on only the lower half of the

walls. In this case, with 6^ = 0.5 m, flashover does not occur within 590

seconds. Increasing 6^ to 0.8 m (figure 7-13) produces results only to 300

seconds, at which time mathematical convergence ceases. Nevertheless, the

rates of Increase of Tg^, m^, and just prior to 300 seconds suggest

imminent flashover. Indeed, the failure to converge could be a consequence of

rapid changes in the variables at the onset of flashover.

9. CONCLUSIONS

On the basis of the work carried out during this project the following

conclusions are drawn.

a. A wall spread model has been produced which takes into account the

oxygen and external flux dependencies of the flame spread and burning

rate properties of lining materials. These dependencies in conjunction with a simple horizontal flame spread model were shown to

produce aspects of the area-of-involvement pattern and flashover

condition which are qualitatively reasonable. Many improvements are

possible such as adding a ceiling-jet model and allowing for vertical

as well as horizontal flame spread.

b. The constantly expanding wall-grid scheme used in the model was inefficient with respect to computation time. Alternative schemes,

such as a predefined number of area elements which "turn on" as the

fire spreads, should be explored.

c. A transient version of the quasi-steady enclosure fire model from

reference [12] was a consequence of this project. Fair to good

agreement between calculated and experimental results was demonstrated for a configuration of four wood cribs in a room. This model

should be useful in future analyses of isolated fires within rooms.

25
d. Advancing the wall spread model from a qualitative to a quantitative

tool will require experimental verification of the flame spread,

burning rate, and entrainment sub-models for vertical surfaces.

Efforts in these areas are underway at NBS.

1 0 . ACKNOWL EDGMENTS

This work was sponsored in part by Armstrong World Industries, through

the National Bureau of Standards Research Associate Program, and the

Department of Health and Human Services. Appreciation is expressed to

Mr. Hank Roux for the support and encouragement he provided throughout the

planning and execution of this project, to Dr. James Quintiere for the technical guidance he gave, and to Messrs. David Messersmith and Charles Nauman for

the computer support they provided.

1 1 . REFERENCES

[1] Tewarson, A. and Pion, R. F. , Flammability of Plastics - I. Burning

Intensity, Combustion and Flame, 26, 85-103 (1978).

[2] Tewarson, A., Experimental Evaluation of Flammability Parameters of

Polymeric Materials, FMRC J.I., 1A6R1 , Factory Mutual Research,

Norwood, MA (February 1979).

[3] Magee, R. S. and McAlevy, III, The Mechanism of Flame Spread, J. Fire

and Flammability, Vol. 2, 271-297 (October 1971).

[4] Fernandez-Pello, A. C. and Classman, I., The Effect of Oxygen

Concentration on Flame Spread in Opposed Forced Flow, 1979 Spring

(Western States Section) Combustion Meeting, Brigham Young University,

Salt Lake City, UT (April 23-24, 1979).

[5] Frey, A. E. , Jr. and Tien, J. S. , A. Theory of Flame Spread Over a Solid

Fuel Including Finite Chemical Kinetics, Combustion and Flame, 36,

263-289 (1979).

[6] Maradey, J. F., Tien, J. S. and Prahl, J. M. , The Upward and Downward

Flame Propagation Limits of Rigid Polyurethane Foams, FTAS/TR-77-13 1

,

Case Western Reserve University, Cleveland, OH (September 1977).

[7] Quintiere, J. G. , Growths of Fire in Building Compartments, Fire

Standards and Safety, ASTM STP 614, American Society for Testing and

Materials, Philadelphia, PA, 131-167 (1977).

[8] Emmons, H. W. , Mitler, H. E. and Trefethen, L. N. , Computer Fire

Code III, Home Project Tech. Rept. No. 25, Harvard University,

Cambridge, MA (1978).

26
[ 9 ]

[ 10 ]

[ 11 ]

[ 12 ]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[ 20 ]

Pape, R. and Waterman, T. , Modifications to the RFIRES Preflashover Room

Fire Computer Model, IITRI Project J6400, IIT Research Institute,

Chicago, IL (March 1977).

MacArthur, C. D. and Meyer, J. F., Dayton Aircraft Cabin Fire Model

Validation, Phase 1, Rept. No. FAA-RD-78-57 , University of Dayton

Research Institute, Dayton, OH (March 1978).

Smith, E. E. and Heibel, J. T. , Evaluating Performance of Cellular

Plastics in Fire Systems, PRC Proj . No. RP-75-1-36, Ohio State

University, Columbus, OH (1977).

Quintiere, J. G. and McCaffrey, B. J. , The Burning of Wood and Plastic

Cribs in an Enclosure; Volume I, NBSIR 80-2054, National Bureau of

Standards, Washington, D. C. (1980).

Rockett, J. A., Modeling of NBS Mattress Tests with the Harvard Mark V

Fire Simulation, NBSIR 81-2440, National Bureau of Standards,

Washington, D. C. (1981).

Tanaka, T. , A Mathematical Model of a Compartment Fire, B.R.I. Report

No. 70, Building Research Institute, Japan (February 1977).

Quintiere, J. G. , An Approach to Modeling Wall Fire Spread in a Room,

Fire Safety Journal, Vol, 3, Nos. 2-4, 201-214 (January-March 1981).

Sparrow, E. M. and Cess, R. D. , Radiation Heat Transfer,

Washington, D. C. , Hemisphere Publishing Corp. (1978).

Modak, A. T. and Matthews, M. K. , Radiation Augmented Fires within

Enclosures, Tech. Rept. 22361-8, Factory Mutual Research Corp.,

Norwood, MA (February 1977).

Siegel, R. and Howell, J. R. , Thermal

NASA SP-164, National Aeronautics and

Washington, D. C. (1971).

Modak, A. T. and Orloff, L. , Fires of

Tech. Rept. J.I. 0A0R9.BU RC79-BT-11,

Norwood, MA (October 1979).

Radiation Heat Transfer, Vol. Ill,

Space Administration,

Insulations on Tank Exteriors,

Factory Mutual Research Corp.

,

Zukoski, E. E., Kubota, T. and Centegen, B., Entrainment in Fire Plumes,

Fire Safety Journal, Vol. 3, Nos. 2-4, 107-122 (January-March 1981).

[21] Tewarson, A., Physics-Chemical and Combustion/Pyrolysis Properties of

Polymeric Material, NBS-GCR-80-295 , National Bureau of Standards,

Washington, D. C. (November 1980).

[22] Quintiere, J. G. , A Simplified Theory for Generalizing Results from a Radiant Panel Rate of Flame Spread Apparatus, Fire and Materials,

Vol. 5, No. 2 (1981).

27
WALL

FIRE

SPREAD

FROM A

CORNER

1

2R

Figure

1-1.

All

spread

phenomena
APPROACH

Figure 2-1. Approach to solving wall fire spread problem

29

31

Figure

4-1.

Equation

of

continuity

applied

tog

or

layer
I

\

%

32

Figure

4-2,

Finite-difference

slabs
i'i

Figure

6-1.

Room

geometry

used

in

calculations
40

30

20

10

0

T

FOUR WOOD CRIBS - FREE BURN

\

\

\

-j I I I 300 600 900

TIME (s)

Figure 6-2. Fire-burn mass loss rate for four wood cribs
Transient

terms

absent

Transient

terms

included

in

s

i

s

(3) 3tiniVll3dN3i

Figure

6-3.

Effect

of

layer

transient

terms

on

wood-crib

calculation
400

350

300

250

200

150

100

50

0

rt- (S-4

*

II 12

PRC7-W-4-1

TEMPERPTURE

ROOM

150 300 450 600 750 MO

TIME (s)

Comparison of experimental and theoretical enclosure

temperatures - - four wood cribs
40

30

20

10

0

FOUR WOOD CRIBS —measured —predicted

free bum

300 600

TIME is)

900

Figure 6-5. Comparison of experimental and theoretical mass loss

rate - - four wood cribs

37
32

FOUR WOOD CRIBS —measured —predicted

8

0 I I I

I \ 1 1 0 300 600 900

TIME (s)

Figure 6-6. Comparison of experimental and theoretical oxygen

concentrations - - four wood cribs

38
WALL 1

|w=1|

39

Element's

horizontal

centerline

Z„

u

^(center)

-

(k

-

1/2)AZ
4(1

Figure

7-2,

Radiation

from

non-burning

enclosure

surfaces

and

upper

gas

layer

to

dA
ELEVATION

SIDE VIEW END VIEW

T

H ~r

V

i_.

1

^ L ^

1

2

Figure 7-3. Geometrical factor for radiation from non-burning enclosure

surfaces and upper gas layer

'4 I
Flaming wall elements

element s

Wall 1 (w=1)

element r

q" = ol* +€saTs^e

q;' = 2q;'eV«F,_,

*b

kf^f

Kigure 7-4. Radiation from burning elements on adjacent wall

42
=0

sec 4 j

Figure

7-5.

Initial

area

of

involvement

for

wall

spread

calculations
ihb

=

30.0

g/s

44
liib

=

47.5

g/s

5i H t = 240 sec

A 5
rfib

=

68.9

g/s

A6
riib

=

91.6

g/s
80

70

60

50

40

30

20

10

0

Tgu (10°C)

mb(g/s)

^inv (^0

q j (kw/m^)

t = 250

0.5 m

H h

t = 0

I I I I I I 100 200 300 400 500 600

TIME, t |sj

Figure 7-10. Summary of theoretical corner wall snread results:

initial burning area on each wall = 0.5m X H

J

)
Figure 7-11, Theoretical upper. gas temperatures From corner wall

spread calculatioVis : initial burning area on each

wall = 6 i X H

49
80

70

60

50

40

30

20

10

0

t = 590

Tgu (10°C)

mb (g/s)

Ajnv (10

qf, (kW/m2)

•d

0.5m

^ hW,w

w

100

l'i;.'ure 7-12.

200 300 400 500 600

TIME, t Is)

Summary of theoretical corner wall spread results:

initial burning, area on each wall = 0-5m X H/2

5lJ
Figure 7-13. Summary of theoretical corner wall spread results:

initial burning area on each wall = 0.8m X H/2
Table 8-1. Thennophyslcal and Fire Properties of Hypothetical

Wall Material

Parameter

m" * 0.011 kg/m^-s

D Y* “ 0.233

ox

q" = o = 0.4 kW/m^

X 3

C = 0.108 kg/m^-s

q"** = 11 kW/m^

Lvap = 1620 kj/kg

TMb,crit

= 5 g/m^-s

Equatlon(s) in

which used

(7-1. 7-2)

'^ox.crlt ® ^ arbitrarily selected)

C = 5.06 m^/2gl/2/j^y

2 q" =22.4 kW/m (estimated)

1 9 Ifi m = 2

Real Material

basis

PMMA [15, 21]

PMMA [22]

= 0.1 m/s (arbitrarily selected) (7-9)

4H = 15000 U/kg (A-3, A-5, A-llc, A-17) Wood [12]

r = 4.92 (A-3a, A-8d, A-llb, A-19) Wood [12]

k = (0.125 X lO"^ kW/m-K) • w

[1 + 6.7 X 10"^K"kT„ - T^)] w a

c = (1.12 kJ/kg-K) • pw

[1 + 6.7 X 10”^K“kT^ - Tg)]

(4-8 to 4-11, 4-13, 4-15)

Calcium

Silicate

Board [12]

p = 696 kg/m^ w

6 = 1.59 X 10"^m w

52
Appendix A - Subsidiary Equations

All subsidiary equations are functions of input parameters and/or one or

more of the seven independent variables - Z^, Tgu» "^gl* "^wu* ^wl » TMv*

”o 3 ”o

gu ('-H

(H - Z ) o n 3/2

m. = 4 W p C. /2g • 1 a^^^{Z ~ Z.) +

1 3 o ai n d

(a + b Z - .3/2 1

d

(A- 1 )

(A-2 )

where a = f 1 - -—j

(Z^ - (A-2 a)

b = 1 - gl

(A- 2 b)

8 ^

+ 0c055 m^ AH (Z^ - H^)

m =

P

8 m.

when Zj >

a — c when Z^

(A-3 )

where m, =

b

m : r < m./ra

V — 1 V

m— ; r > m./m

r ’ IV

(A-3 a)

. , "^a ^n ^d /W \ 0.025 . ra = k -=— ( -rr- I m,

e m T Z I W / i

gu n \ o /

(A-A)

\u = (' - I) “b “ (A- 5 )

E, = fm. + m + m ) c (t -T , 1 - m T - T - m c [t -T J (A- 6 ) Mu V e'^p^gugl^ vs gl vpp^sgl' V pp'' s g:

53
where the last term which accounts for pyrolysis gases emerging from the

surface at Tg will be neglected.

E = V + q" 1 A + q" A + q" , A_

Lu ru cu' wu ro ou rd F

i" = e o + Y (l - e 1 o ru g gu u g' wl

- [i - (1 -

-r ) (1 - € )] L V. o / 'u-* '> g-'J wu

Y.. =

u Awu

1 -k L

e = 1 - e g m

g

k = g

K (l + X r) m

a ' V

(m'' + m ) i V-'

L = m

2WL (h-Z^)

[h-Z J(VH-L) + WL

... 4 r ^ 4 4

q = e 0 T + [l-e J o T - o T

ro g gu '' g-* wu a

... r„4 . c , „4 _4 q,=e oT +(l-eJoT -oT- rd g gu ' g'' wu wl

V = h [t - T 1 CU CU ^ gu WU"^

h = p c /gti CU a p ° ^ \H/

~)* = (1-f) AH

p c T /in H*" a p a

(A-7)

(A-8)

(A-8a)

(A-8b)

(A-8c)

(A-8d)

(A-9)

(A-10)

(A-11)

(A-lla)

(A-llb)

54
= c m * Ml p i gl

PI, (t T - T ) + c m (t , - T ) i gl a' p e ' gl gu-^

(A-12)

\x -

*'=1 ^,l (Tgl - ^wl) (A-13)

• M _ 2 wall

(t - T ) ^ wu a

*' (A-14)

... ^ 2 ^

/( k p

*kl

/tt *

(t - T ) ^ wl a-' (A-15)

^cl ’"cl^^gl ~

'^wl^

(A-16)

^rl = Yi e (3 1^ + (l-e 1 a T

'1 g gu r' 1 _4 ^

f \ iwjL •aT,+‘;r —

g' wu wl 2 A

AH

wl

(A-17)

^1

= (A-17a)

0X1

0 .23 + fra /m . ] Y ^ e 1 - oxu

1 + fm '/m,

> e 1

(A-18)

0.23 ® [l -

oxu

1 + (A-19)

55
Appendix B - Radiation from Upper Gas Layer and

Non-Burning Enclosure Surfaces

A. Target Element in Lower Layer

Using Figure B-1 and following Siegel and Howell [18] , the incident

radiative flux on a target element in the lower layer due to radiation from

the upper gas layer and enclosure surfaces is given by the expression

dA,l

a T , + F

wl dA,2

a T

wl

radiation from quadrants

i“l and i»2

+ F,. - [as . + (l-e .) a 1

dA,5

|_ g,5 gu g,5^ wuj

+ , oe , + fl-e , 1 a dA,6

[_ g,6 gu '' g,6-' wuJ

radiation from gas

and upper surfaces in

quadrants i=3 and 1*4

t^dA,3

" ^dA,5^

^

"^wl

'* f^dA,4

" ^dA,6^ ^

"^wl

radiation from lower

surfaces in quadrants

1=3 and i=4

(B-1)

or

4

Z i=l

a FdA,i

6

Z i=5

o FdA,l

(B-2)

where

^dA,l

^dA,5

^dA,6

FdA,2

^dA,3

^dA,4

^dA,3

^dA,I

^dA,I

^dA,4 0.25

W-x

L

9 L

FdA.II

(B-3)

(B-4)

(B-5)

56
and [16]

^dA,I

_1_

2tt X ^ -1 Y . Y . -1

tan + tan

^1+X^ ^1+X^ ^1+Y^ ^

=

(tan”^ Y

”

1+Y

where A = 1/ Wx^ + Y^

(B-6)

(B-7)

(B-7a)

B. Target Element In Upper Layer

Using Figure B-2 and following Siegel and Howell [18]

a + (l-e J a ] g.l gu ^ g,l^ wu-*

4 a T + (l-e o) a ] S >2 gu ^ g,2'' WU*^

^dA,5

®

"^wl

^

"dA,6 " 4

^dA,3 ^g,3 ®*^gu ^^dA,3

” ^dA,5^

• (l-e ) a

g,3-' wu \

*

^dA,4 ^g,4 '^'^gu t^dA,4

" ^dA,6^

|

• (l-"g,4) ’4

radiation from quadrants

1=1 and 1=2

radiation from lower

surfaces of quadrants

1=3 and 1=4

radiation from gas and

upper surfaces In

quadrants 1=3 and 1=4

(B-8)

or
(B-9)

q" = Z a F., , [e ,T^ + (l-e ,1 gw dA,i L g,i gu g,!-* wu

+ Z 0 F.^ . (l-e ,) (T^. - dA,l g,l V wl wu

where the geometrical factors are defined by equations B-3 to B-7.

58
Figure B-1- Radiation target element in lower layer
Figure B-2. Radiation target element in upper layer

60
Appendix C - Radiation Path Length Through Upper Layer

D = + (Z^ - Z^]‘

when

(C-l)

z, = z. s > \

= “

^2 < ‘‘g = ° (C-2)

h > 2d

Z ^7 h “

^2 ^ ^d= '‘g

22 < ^d= *g

"2^2,;

^2 < ^d’ %

- Z

z, - z.

(C-3)

Zi < Z^

^2 < = 0

61
NBS*114A (REV. 0-76)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR-83-2765

«D)

4. TITLE AND SUBTITLE

A Calculation of Wall Fire Spread in an Enclosure

5. Publication Date Nov 1983

1. PerfonningOrganiution Code o

7. AUTHOR(S)

Kenneth D. Steckler

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Wort( Unit lb,

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsrre.r, City, State, ZIP)

Armstrong World Industries U.S. Department of Health

Lancaster, PA 17604 and and Human Services

Washington, D.C. 20201

13. Type of Report 8. Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas lactuat aummary ol moat aighillcant Information, If document Ineludea a aigni ficant bibliography or

literature survey, merttion it here.) Mathematical models of fire growth in enclosures offer a potential for assessing the

risk of materials with respect to a hazard such as flashover. The development of a model for fire spread over wall lining materials is presented. The work covers the

development of a transient two-layer zone model, initially formulated for crib fires

in a room, then adapted to simulate a spreading corner wall fire. The local wall flame

spread rate and burning rate per unit area are expressed in the model as functions of

the external radiation incident upon the burning surface and the oxygen concentration

in the adjacent gas layer (zone). Flame spread is limited to the horizontal direction.

Results for a fictitious wall lining material are presented. Major elements of the

results are shown to be in qualitative agreement with experience. Finally, areas for

improvement and the direction of future efforts are noted.

17. KEY WORDS ('s/x to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

burning rate; fire models; flame spread; flashover; mathematical models; room

fires; walls

18. AVAILABILITY (^Unlimited

I I For Official Distribution. Do Not Release to NTIS

i Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

20402, SD Stock No. SN003-003-

[K1 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

19. SECURITY CLASS 21. NO. OF

(THIS REPORT) PRINTED PAGES

UNCLASSIFIED 74

20. SECURITY CLASS 22. Price (THIS PAGE)

UNCLASSIFIED

USCOMK^-DC