NBSIR 82-2622

The Development of Hazardous

Conditions in Enclosures With

Growing Fires

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

December 1982

100

,U56

82-2622

1982

C.2

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU

OP STANDARDS

LIBRARY jTP 2 1983

noV OXC '

NBSIR 82-2622

THE DEVELOPMENT OF HAZARDOUS

CONDITIONS IN ENCLOSURES WITH

GROWING FIRES

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

December 1982

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
1 k ' . . »< UA3M 9 .. \m TAJ
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

NOMENCLATURE v

Abstract 1

INTRODUCTION 1

A DESCRIPTION OF THE MODEL AND ITS GOVERNING EQUATIONS 3

The Model 3

The Governing Equations 4

Discussion of the Equations 6

Solutions From Ignition to t q

6

Solutions From Time t q

to the Time When £ = - 6 9

Solution for <j> When £ = -6 10

USING THE SOLUTIONS TO PREDICT FIRE ENVIRONMENTS 10

A Problem in Smoldering Combustion 10

Hazard Development in Enclosures Containing Some

Larger Scale Fires 13

SUMMARY AND CONCLUSIONS 15

ACKNOWLEDGMENTS 16

REFERENCES 17

iii
LIST OF FIGURES

Page

Figure 1. Fire-in-enclosure flow dynamics consistent with

the model 19

Figure 2. Plots of ?(x) for different values of n and e, 0 < x < x 20 — o

o _ * 2(n+3)/[3(3n+5)] . . - Figure 3. Plots of e x and ij) as functions

of 1 /e for different values of n 21

Figure 4. Plots of £(o) and <J>(a) , 0 _< £ < 1, for different

values of e, and for n = 0 , n = 1 , and n = 2 22

Figure 5. Plots of 0

Q

(e) and <t> Q(e) ^ or different va lues of n 23

Figure 6 . Plots of c(o) and <j>(o), -6 < 5 _< 0, for different

values of o 24

o Figure 7. Plots of £(o) and <p(a) ,

z, 0, for different values

of a , and for 6 = 0, 6 = 0.2, and 6 = 0.4 25

o’ Figure 8 . Plots of ?(a) and <j>(o) for n = 1 and e = 0.00016,

and plots of p data from cotton and polyurethane

smoldering experiments 26

iv
NOMENCLATURE

A enclosure floor area

ASET available safe egress time

B a constant

b a constant

C combustion product generation rate

C CO generation rate

LU

Cp specific heat at constant pressure

f a function

g acceleration of gravity

H fire-to-ceiling distance

H

c heat of combustion

M combustion product concentration

m mass loss rate

n an integer

•Q energy release rate

•Qq a characteristic energy release rate

• ic • Q dimensionless Q

o o R ratio of leading terms in a £ expansion

RSET required safe egress time

T upper layer temperature

Tq ambient temperature

t time

t Q value of t when Z^ = 0

t DET time at detection

t HAZ time at hazard

v
a

6

Y

A

6

e

C

C (n)

X c

X r

w

p a

a

a

o

T

T O

dimensional unit of a combustion product

elevation of interface above fire

a constant of proportionality

a constant of proportionality

ratio of mass of CO generated to mass of material

elevation of fire above floor

A/H

a dimensionless parameter

Z

t /H

nth term in a £ expansion

fraction of Q transfered to enclosure surfaces

fraction of Q radiated from combustion zone

dimensionless M

ambient density

dimensionless t

dimensionless tQ

dimensionless t

dimensionless t Q

T/T

a value of <J> at t = t Q

vi
THE DEVELOPMENT OF HAZARDOUS CONDITIONS

IN ENCLOSURES WITH GROWING FIRES

Leonard Y. Cooper

Abstract

A mathematical model for simulating the environment in enclosures during the

growth stage of hazardous fires was developed previously. To use the model one must specify the energy release rate of the fire, certain heat transfer

parameters, the area and height of the enclosure and the elevation of the fire

above the floor. Solution to the model's equations would yield the time-varying

thickness, temperature, and product of combustion concentrations of an upper

smoke layer which starts to drop from the enclosure ceiling at the time of

ignition. In this paper the model equations are solved for the general class of

fires whose energy release rate, Q, and product of combustion generation

rates, C, are approximately proportional to tn(t is time and n_> 0). For such

fires, general results for the complete solution history of the enclosure environment are obtained and presented in the form of graphs, and, where possible, by

closed form analytic expressions. Use of the results is illustrated in two

example problems. The first of these involves a problem in smoldering combustion

where, according to experimental data, the combustion zone can be simulated by an

n = 1 fire. The second involves a prediction of the environment produced in an

enclosure which contains an n = 2 fire, which simulates a specific, large-scale,

flaming fire hazard.

INTRODUCTION

An excess of available safe egress time (ASET) over required safe egress

time (RSET) as a criterion for life safety in buildings under fire conditions is

a concept which has been introduced in references 1-4. ASET is the interval

between the time of fire detection (and successful occupant alarm), t^j^* and the

time of onset of hazardous conditions, tj^g* i»e», ASET = - t^-p. RSET is

the actual time required for occupants to evacuate from threatened spaces. If

the life safety criterion is to be used in an evaluation of the safety of an

existing building or of a tentative building design, then estimates of ASET and

RSET under characteristic potential fire scenarios must be obtained.

1
An estimate of ASET requires estimates of and t^^. t^j^ depends on

the characteristics of available detection and alarm devices, and, in general, on

the interactions of these with fire generated environments which develop around

them. Besides depending on the physiological characteristics of occupants,

also depends on the fire environment. It is therefore evident that estimates of

t DET an<* t HAZ re£l ui re predictions of the dynamic environment which develops in

buildings during fire conditions. The capability of providing such predictions

under a wide variety of fire scenarios is a general goal of enclosure fire

modeling.

A relatively simple model which predicts the dynamic fire conditions in

compartments of fire origin has been proposed in references 1 and 2. Besides the

height and floor area of the compartment and the elevation of the fire above the

floor, the free burn characteristics of the fuel assembly in which the potential

fire originates is a required input to the model. (Free burn is defined as a burn of the fuel assembly in a large ventilated space which contains a relatively

quiescent atmosphere.) The model has been shown to provide useful analytic

simulations of fire conditions in single room enclosures and in one-level, freely

connected, multiroom enclosures where the overall external boundaries of the

2 5 enclosures were sealed except for leakage paths near the floor ». The model is

capable of giving excellent results up to and beyond the onset of hazardous

conditions.

While the model involves governing equations that are easily solved with the use of a computer, it does not lend itself to generalized solutions which can be

displayed "once and for all" by charts, graphs or tables. Nevertheless, for any

specific fire [i.e., for a fire of specified free burn energy release, Q(t), and

product of combustion generation rate, C(t), where t is time] solutions to the

model's equations for arbitrary enclosure height and area and for arbitrary fire

elevation can indeed be obtained and displayed. It is the purpose of the present

work to solve the governing equations for the broad class of fires whose Q can be

reasonably approximated by growth proportional to t n (for arbitrary n^ 0), and

• • whose C s of interest can be reasonably approximated by growth proportional to Q.

2
A DESCRIPTION OF THE MODEL AND ITS GOVERNING EQUATIONS

The Model

A rational basis for the following idealized description of real enclosure

fires as well as a development of the subsequent equations are presented at

length in references 1 and 2. Figure 1 depicts the developing fire conditions.

A fire located a distance H below the ceiling is initiated in an enclosure

of area A. The ambient conditions in the enclosure are described by the density, p , the specific heat at constant pressure, C , and the absolute tempera3. r ture, T_. The fire is located at an elevation. A, above the floor (i.e., the

total height of the enclosure is H + A). An estimate of the total energy release

rate of the fire, Q, is assumed to be available, as are the fire's generation

rates, the C's, of any products of combustion of interest. In practice it is

recommended that free burn heat release rates and product of combustion genera-

• • tion rates be used as surrogates for Q and the C's, respectively. This

recommendation is consistent with the fact that onset of hazardous conditions

within the enclosure will occur at temperature and combustion product

concentration levels which are low compared to those levels at which variations

from free burn will begin to be significant.

As the fire develops from ignition at t = 0, buoyancy forces drive the high

temperature products of combustion upward toward the ceiling. In this way a plume of upward moving elevated temperature gases is formed above the fire. All

along the axis of the plume, relatively quiescent and cool ambient air is

laterally entrained and mixed with the plume gases as they continue their ascent

to the ceiling. As a result of this entrainment the total mass flow rate in the

plume continuously increases and the average temperature and average concentration of products of combustion in the plume continuously decreases with

increasing height.

When the plume gases impinge on the ceiling they spread across it forming a relatively thin, stably stratified upper layer. As the upward filling process

continues, the upper gas layer grows in depth, and the relatively sharp interface

between it and the cool ambient air layer below continuously drops.

3
• • With regard to the disposition of Q, it is assumed that A^Q is the net

amount which is actually radiated from the fire's combustion zone while the

• • remainder, (l-A

r )Q, is convected upward. It is assumed, further, that A^Q is the

net rate at which energy is transfered to the internal bounding surfaces of the

enclosure. A Q includes all heat transfer to surfaces by both convection and

c radiation. Reference 2 recommends choosing A^ = 0.35 for typical hazardous

flaming fires while A

c typically varies in a range of 0.6-0. 9, depending on enclosure geometry, the fire's proximity to walls, and on the value of A^.

At every instant, it is assumed that the gases in the upper layer are fully

mixed. Accordingly, the model provides a time-dependent description of the

environment in the enclosure by predicting the elevation, Z^(t), of the interface

above the fire, and the absolute temperature, T(t), and combustion product

concentration, M(t), of the upper layer. If the dimensions of C are taken to be

u

Q per unit time, where u

c is a dimensional unit appropriate for the particular

combustion product, then the dimension of M is the average amount of product

(dimension u

c

) per unit mass of upper layer mixture.

There are basically two mechanisms for the increase of the upper layer

volume (decrease of Z^). The first is a result of significant rates of mass injection from the plume into the upper layer on account of entrainment (mass

addition from the fuel itself is assumed to be negligible). The second mechanism

is a result of gas expansion due to energy transfer from the fire to the gas.

This comes about because relatively low density (high temperature) upper layer

gases replace relatively high density (ambient temperature) air, with a net

efflux of ambient air from assumed floor level leakage points. As the interface

drops from the ceiling its downward velocity is continuously reduced. The

entrainment mechanism, which is totally dominant early in the fire, eventually

becomes small compared to the expansion mechanism. By the time Z^ = 0 and the

interface drops below the fire (if A * 0), only the expansion mechanism remains

viable for further reductions in Z^.

The Governing Equations

For arbitrarily specified Q and C, the initial value problem for Z^, T, and

M is presented in reference 2. Here, fire scenarios which develop from a

4
specific class of Q and C, namely, Q ~ t and C ~ Q are being investigated. To

• • be definite, it is assumed that Q and C of free burn fires of interest can be

approximated by

Q(t) = Q

o [tH3/2

g

1/2

/A] n

; C(t) = 3Q(t) (1)

where Q

q

represents a characteristic energy release rate, n is any non-negative

number, 3 is a constant of appropriate dimension, and g is the acceleration due

to gravity.

It turns out to be convenient to introduce the following dimensionless

variables and parameters

C = Z^/H (interface elevation)

<j> = T/T (upper layer temperature)

SL p = ( 1—X )M/ (3C T ) (upper layer product concentration)

t = 3[(1-X )Q*] 1/3 (tH3/2

g

1/2

/A) (n+3)/3/(n+3) (time)

r ° ( o') £ - (l-i

c )[(n+3)/3] 2n/(n+3) Q*2/(n+3) /(l-Xr

) (n+1)/(n+3) (fire strength)

.* . 1/2 5/2

Q

q = Q0 /(Pa

c

p

T

a

8 H ) (characteristic energy release rate)

6 = A/H (fire elevation)

Using the above definitions in the model equations of reference 2 eventually

leads to the following equations for £, <J>» and p:

- eT

2n/< n+3) . 0.210? 5/3

; 0 < 5 < 1

- eT

2 "/(n+3

); - s < t < 0

dt 1 — (3)

d<|)

0; c - -S

(n+3)ex

2n/ (n+3)

<*> = {!- (n+3)eT 3(n+1)/(n+3) /[3(n+l)(l-?)]} 1

; -6 < c < 1

/(1+6); c = -6

(4)

p = <|>-1; -6 < z < 1 (5)

where Eq. (3) must be solved subject to

C(t-O) = 1 (6)

and where early time estimates for <{> and p are

n = 0:

lim(£-l)/(l+e/0.210) = -0.210t + higher order terms in t

t 0 (7)

lim <}>/(l+e/0.210) = lim(p+l)/(l+e/0.210) = 1 + 5er/6 + higher order terms in x

x •> 0 x + 0

5
n > 0:

lim(c-l) = -0.210t + higher order terms in x x > 0 (8)

lim(<J>-l) = lim p = (n+3)ex^

n^n+^V[3(n+l)(0.210) ] + higher order terms in x

x 0 x 0

Eq. (3) describes the rate of descent of the interface as it passes through the

regions above the fire (0 < £ < 1), below the fire (-6 < z, < 0) , and at the

floor (5 = -6). Eqs. (4) and (5) describe the corresponding upper layer

temperature and product concentration. Eqs. (7) and (8) are useful in starting a numerical solution to Eqs. (3) and (4).

Discussion of the Equations

The last section presented the equations which govern the dynamics of £,

(J> and p. From Eq. (5) the solution for p would follow directly from the solution

for <f>. From the time of ignition to the time that the interface drops to the

floor of the enclosure, a solution for <J> could be obtained from Eqs. (4), (7) and

(8) provided a solution for z, was available. Beyond that time the solution

for <J) could be continued by a direct integration of the second of Eq. (4).

With the above observations, attention is drawn to the solution for £. From

ignition until the time, xq = x(^=0) , when the interface drops to z, = 0, C is

governed by Eq. (6) and the first of Eq. (3). No general closed form solution is

possible, and a numerical solution for £(x;e,n) is in order. Once; this has been

obtained, the solution can be extended beyond x^ by direct integration of the

second and third of Eq. (3).

Solutions From Ignition to xq

In general there is no particular problem in obtaining a numerical

integration for £. However, in terms of generating a display of working graphical solutions which include times when Z, is small and positive, a problem does

arise in the limit as e + 0. Applying such a limit to the first of Eq. (3)

leads, in a first approximation, to the total neglect of the earlier referenced

(left hand) expansion term in comparison to the (right hand) entrainment term.

This corresponds, physically, to the situation of an interface that approaches

6
the elevation of the fire asymptotically in time. This is the incompressible

limit of a heat (buoyancy) source in an enclosure studied for the n = 0 case by

both Baines and Turner^ and Zukoski^. In the present nomenclature, and for a source whose strength grows as t n

, this incompressible solution is simply found

to be

£(T;e=0,n) = £ (0 ) (t) = [1 + 0.210(2/3) t]" 3/2

(9)

This result is plotted in Figure 2 along with other numerically obtained,

nonzero e solutions for £.

The above result for e = 0 leads to the result that £ - 0 as t 00

. But,

for a fixed n and an arbitrarily small but nonzero e, a £ = 0 position of the

interface will in fact be attained at some finite, large t. Thus, there will be

no uniformly valid solution for £(x;e,n) in a neighborhood of £ = 0 which is

independent of e in the limit e + 0. Away from the £ = 0 point, however, such a uniformly valid solution does exist, i.e., for £ in the range 0 < B <_ £ _< 1, and

for some fixed B,

lim £ (t ; e, n) = £^(t)[ 1 + 0(e)]

e > 0 (10) n fixed _>_ 0

This small e behavior of £ can be observed in Figure 2.

The small e limit is very important in problems of physical interest. As an

example, consider a constant (n = 0) smolder source of 0.5 kW located a distance

of 2 m below a ceiling and with X =0.1 and X =0.75. This leads to

-4 r c

e = 5.3(10 ). As an example of a relatively strong fire, consider a constant

flaming fire of 5,000 kW located 5 m below a ceiling, and with X =0.35

-2 r and X^ = 0.75. This leads to e = 6.0(10 ). In terms of a "small e" criterion,

the latter fire is still relatively weak.

From the above discussion it is reasonable to seek a regular perturbation

solution for £ of the form

00 lim £ = Z e P

£ (p) (t; n) (11)

e 0 p=0

0 <_t fixed

for 1 _> £ > 0, where £^^ has been obtained above. Furthermore, it is reasonable

to expect that the useful range of the above expansion can be extended to

large t values which satisfy

7
|eC (p+1) (x;n)/c (p) (x;n) <b<l (12)

for all p greater than some fixed value P. Provided e was small enough, but not

identically zero, such large x values would include the particularly interesting

value x , which corresponds to 4 = 0.

The suggested perturbation analysis was carried out for the first few terms

of the Eq. (11) expansion. This leads to

li" ^

(1)

/C

<0) -

- TU&& (0.410/3)

X

-»• 00

3/2 £x3(3n+5)/[2(n+3)]

5 _R

lim 5(n+3)

12(5n+9)

R

X

-> 00 and the resulting estimate for 5

lim

T m f" lim 5

"!

•+ 00 [_£ Oj

- r+ t!S- r2 +

Applying this estimate at £ = 0 leads to

0 = 1- R(x

q

;e ,n) + y^^ *

2

( V£ » n) + . . .

(13)

(14)

(15)

(16)

which, together with the Eq. (13) definition of R, suggests the general result

2(n+3)/[3(3n+5)] lim e

e 0

XQ(e;n) = f(n) = 0(1) (17)

The f(n) = 0(1) estimate is a result of the fact that the quotients

(n+3)/ ( 1 ln+2 1) and (3n+5)/(n+3) of Eq. (13) and 5(n+3)/ [12(5n+9) ] of Eq. (14) are

relatively insensitive to n in the entire range 0 n <

00

, the latter of these

always having a value of approximately 0.1.

Guided by the tentative result of Eq. (17), numerically computed x , e pairs were obtained and plotted in Figure 3 for a variety of different n values. The

corresponding values for <J)(x Q;e,n) were also obtained and plotted. The plots of

the numerical data are seen to be consistent with the asymptotic result of

Eq. (17). Indeed, over a very broad e range of practical interest the value of

the ordinate for a given n is seen to be relatively uniform in e. For example,

in the constant fire case,

e 2/5x (e ;n=0) = [

(

l-X) ( l-\ ) 2gQ

2

/ (p C T ) 3

] 1/5

t /A = 4.2(1 ± 0.15) (18)

o rcoapa o -4 -1

in the range 0.5(10 )< e < 0.5(10 ), where t Q is the value of t when = 0.

8
The above observation for n = 0 leads to a remarkable practical result;

namely, in the case of constant fires, and for fire elevations and room heights

of practical interest, the time for a smoke layer to drop from the ceiling to the

elevation of the fire is relatively independent of H.

Solutions From Time t to the Time When z, = -6

o

This section will be devoted to an analysis of the £, <J> solution beyond the

time t q

. It is convenient to present results in terms of a new dimensionless

time variable, a, which is defined by

a - ( n+3 ) 3(n+l)/(n+3)

° n<n

3(n+l) £T

Accordingly, working plots for ?(a) and 4>(a) have been numerically generated and

plotted in Figure 4 for selected values of n. Also, plots of (e;n) which

correspond to the results of Figure 3 are plotted in Figure 5, where o

q

is

defined as the value of a when x = x . o

After the upper layer interface drops to the fire elevation, Z, = 0, its

further drop until z, = -6 is determined by the second of Eq. (3) which can be

written as

« -1, -6 < C < 0 (20)

where

?(a

Q

) = 0 (21)

where a and o q are defined in Eq. (19), and where a

Q

(e;n) is plotted in

Figure 5. It is noteworthy that a o

q = 1 estimate of the time for a C = 1 to

£ = 0 interface drop corresponds to a result which can be derived from a neglect

of the entrainment term of Eq. (3). In this range of £, the real rate-of-drop of

the interface will be enhanced by entrainment, and, therefore, < 1

.

During the times of present interest it is still possible to compute <j> from

the first of Eq. (4), which can be written as

* = [l-o/a-or1; -<$ < e < o (22)

The solutions for £(a;a

Q

) and 4> ( a ;

)

are

z, = o

Q

-a, <}> = l+a/(l-a

Q

); -6 < z, <_ 0 (23)

9
and they are plotted parametrically in Figure 6 for different values of

0 _< o < 1. In order to use these results to obtain histories for c and <f> for a particular fire scenario one computes e, finds the value for a from Figure 5,

and then uses the appropriate C and <}> plots of Figure 6 for times in excess of

0

q

, and up until C = -6 .

Solution for <{> When £ = -6

The parameter 6 does not enter into the solutions for t or 4> until

? = -6 at which time the layer has just filled the entire enclosure. The floor

level leakage gas changes from ambient air to upper layer gas. Subsequent to

this time, <f> is governed by the second of Eq. (4) which can be rewritten as -^= <t»/(l+6 ), a > o

q

+6 (24)

For the solution to Eq. (24) to be continuous with the <J> solution of

Eq. (23) it is necessary that

4>(o = a *5) = l+(a +6)/(l-a ) (25)

o o o The solution to Eqs. (24) and (25) is

4

> = [(l+6 )/(l-a

o)]exp[(a-0

o-6 )/(l+6 )], a>.oQ

+6 (26)

Using the results of Eqs. (23) and (26), the solutions for 5 and <j> as functions of a with as a parameter, and for the 6 values 0, 0.2, and 0.4 are

plotted in Figure 7.

USING THE SOLUTIONS TO PREDICT FIRE ENVIRONMENTS

The solutions of the last section will now be applied to two example

problems. The first example will involve a problem of smoldering combustion

where results of the present theory will be compared to some full scale

experimental results. The second example will illustrate the use of the theory

in predicting the environment produced in an enclosure which contains a specific

large-scale flaming fire hazard.

A Problem in Smoldering Combustion

8 Smoldering experiments reported by Quintiere et al . were carried out in an

o enclosure of height 2.44 m and area 8.83 m . The opening to the enclosure was

10
formed by a closed undercut door where the undercut formed a 0.76 m x 0.025 m open horizontal slit at floor level. A smoldering source was placed in the

enclosure where the top surface of the source was at an elevation of 0.33 m. Gas

analysis was carried out at four equidistant elevations from the ends of sampling

tubes extending horizontally approximately 0.5 m from the walls.

The tests evaluated two different smolder sources; a loosely packed bed of

cotton, and blocks of flexible polyurethane foam. Mass loss rates, m, were found

to be approximately linear in time throughout the first hour of the two tests,

i.e.

,

m = at 0 < t < 60 min (27)

where

j °« “ \ 0.

2

21 g/min^ for polyurethane

33 g/min for cotton

The heats of combustion, H

c

, of the materials as well as the ratios, y, of mass of CO produced to mass of material lost were obtained in a separate small

scale apparatus. These were found to be

=

{ t H H k^gcotton (28)

Y

= (29)

(15 ± 8) kj/gp0 ^yuret.Jlane

0.11 8c0^cotton

+ 0.0l\ . - 0.04/ ®C0/ ^polyurethane

The results of the previous section will be used to predict the environment

which developed in the enclosure during the course of the two different material

evaluations.

Comparing Eqs. (27), (28) and (29) to Eq. (1) leads to

Q

* <>H„t « QjtH3/2

g

1/2 /A]"

c.CO

c # o

1 = ayt = 8Q

(30)

where C^

0 is measured in g^,Q per unit time. From the above it is concluded that

(31)

Also

n - 1; Q

q « aAHc /(H3/2

g

1/2

); 6 = y/H.

H = 2.11 m, A = 0.33 m, g

= 9.8 m/s

2

, A = 8.83 m

2

T

a = 294°K, p a = 1.18 kg/m3

, C

p = 240 cal/(kg°K) (32)

Radiant losses from the combustion zone are neglected, i.e., \ =0.

o Considerations in the Appendix of Cooper , together with experimental results of

Mulholland et al .^, and Veldman et al .^ indicate that for a = 0 combustion

11
zone in an enclosure with proportions similar to the present one, X * 0.6.

This X value will be used here,

c

Using Eqs. (28), (29), (31) and (32) along with the above X values in

Eqs. (2) and (19) leads to

<J> = 135 M + 1 = 204 M . . +1

cotton polyurethane ( 33 )

where, e.g., Mcotton is the upper layer concentration of CO during smoldering of

the cotton source. Using the above data in the e, t, and o definitions of

Eqs. (2) and (19) leads to the following representations for these variables

which happen to be valid for both the cotton and polyurethane tests

e = 1.6(10 ^

) , t = 4.1(10 3 )t^3

, a = 2.8(10 3

)t

2

; (t in seconds) (34)

Either Figure 3 or 5 provides both an estimate for the time, t Q

, when the

interface drops to the elevation of the combustion zone, and an estimate for the

temperature, <t> Thus,

Q

, of the upper layer at this time.

e = 1.6(10

n = 1

-4

1

e 1/3t = 3.8

o

<J> = 1.066

o

or a = 0.062

o (35)

Using these results in Eqs. (33) and (34) leads to the estimates

t Q = 1 . 5( 10 3 )s

Mcotton (t=t

o> = 4-9O02

) PPm CO

^polyurethane < t=to> = 3 * 2 < 1

0

^ ^

C0

Also, the time when the interface would reach the floor elevation,

£ = -6 = -0.156, can be estimated from Eqs. (23) and (34), and from the

a = 0.062 result of (35).

° 3

o(c=-6) = a +6 = 0.218 -*• tU=-6) = 2.8(10 )s o Thus, the theory indicates that the interface reached the floor somewhat

prior to the 60 minute duration of the tests.

With the use of the transformations of Eqs. (33) and (34), M

cotton and

^polyurethane data (sampling port 1.68 m above the floor) are plotted in

Figure 8 up to the time a = 0.37 [t = 3.6(10

3

)s] . Also included in the figure

-4 are figure 4 type y(a) and 4(a) plots for e = 1.6(10 ) and n = 1. As can be

noted, there is excellent agreement between computed and measured values for y.

Relative to this agreement, however, it should be stressed that the measured

12
values for y = 4>-l = T/T -1 are from CO concentration measurements rather than

a temperature measurements. Values for T/T -1 which would correspond to actual

1

2

measured temperatures are less than an order of magnitude smaller than the

indicated theoretical values. This is consistent with H

c estimates of

1

3

Ohlemiller , which are an order of magnitude smaller than the reference 8, H

c estimates of Eq. (28). Thus, for example, if the above calculation is carried

through for polyurethane using H

c = 1.5 kJ/g instead of 15 kJ/g, then at 3600 s the predicted pairs of values for M and T/T

& are 1.9(1 0^ ) ppm CO and 1.04

(H

c = lo5 kJ/g) compared to 2.0(10^) ppm CO and 1.4 (H

c = 15 kJ/g).

Hazard Development in Enclosures Containing Some Larger Scale Fires

Table 4-2 of reference. 14 provides a catalogue of experimentally determinec

energy release rates obtained by Heskestad^ for the growth stages of flaming

fires in practical fuel assemblies. The Q of all items in this listing is

2 • proportional to t . For example, the Q of many items can be estimated by

(t = 1000 s in the nomenclature of reference 14)

O

Q = 0.10 t

2 kW/s2 (t in seconds) (37)

These latter items include wood pallets stacked 3. 0-4. 6 m high, many

different types of polyethylene, polypropylene, polystyrene and PVC commodities

in cartons stacked 4.6 m high, and a horizontal polyurethane mattress.

The results of the present analysis will be used to characterize the hazard

development in enclosures which contain Eq. (37)-type fires. Toward this end,

Eqs. (1) and (37) lead to the result

n = 2; Q

o = 0.010(A /H)kW/m (38)

Using Eq. (38) in Eq. (2) together with the values of p^, C

p

, T

a and g of

Eq. (32) leads to

Q* = 9.4(10" 6

)A

2H~11/2

m

3/2

; e = 0.015(l-X

c )(l-Ar )“3/5A

4/5H"11/5

m

?/5

x°= 0.085(1-* ) 1/3H

2/V1 t 5/V/3

s- 5/3

" 1 (39>

r With Eq. (39), the results of the present analysis can now be used to answer

a wide variety of hazard related questions. For illustrative purposes two such

questions will be addressed here.

13
Question 1:

Flaming ignition is initiated in stacked commodities of the "Eq. (37)

2 variety" which are contained in a warehouse of height 6 m and area 1500 m . At

what time does the upper layer attain the potentially untenable temperature* (due

to downward radiation) of 183° C, and what is the elevation of the layer interface

at this time?

Answer (Solution):

Consistent with recommendations in reference 2, assume X = 0.35, and, for

the purpose of a hazard analysis of this type, conservatively assumed X^ = 0.6.

Take H to be the floor-to-ceiling dimension, 6 m, and A to be zero. Then for

A = 1500 m^, Eq. (39) leads to

-4 5/3

e = 0.052; t = 1.6(10 ) t (t in seconds) (40)

From the above, T , = 183°C = 456°K and T = 294°K. Then for e = 0.05 hazard a and n = 2, Figure 4 and the x, t transformation of Eq. (39) provides the result:

when

<j> = <J>, = T, ,/T = 1.55 hazard hazard a

x = x, , = 3.0 + t, , = 37 0s hazard hazard

C = ^hazard

= 0,44 * Z

i = Zhazard

= 2,6 m

(41)

Thus, the hazardous condition will occur (no earlier than) 370 s following

ignition, and when the layer interface is (no lower than) 2.6 m from the warehouse floor. Note from Eq. (37), that the strength of the fire at this time is

estimated to be 14 MW.

Question 2:

Flaming ignition is initiated in a polyurethane mattress 0.6 m above the

floor of a hospital ward with floor-to-ceiling dimension of 3 m and area 10 m . At what time does the upper layer interface reach the potentially hazardous 1.5 m

elevation, and what is the upper layer temperature at this time?

Answer (Solution):

As in the previous solution, take X^ = 0.6, X^ = 0.35 and T^ = 294°K. Also,

H = 2.4 m, A = 0.6 m, hazard = m » anc* ^ = *^0 m ^* '^ien » from Eq. (39)

e = 0.045; x = 1.3(10

2

) t^2 (t in seconds) (42)

14
As before, the e = 0.05, n = 2 plots of Figure 4 once again provide a basis

for obtaining the desired solutions. Using these plots and the new T, t transformation of Eq. (42) leads to the result:

^hazard Zhazard

/H °’ 38

when

t = t =3.3

hazard

* t. . = 110 s hazard

4» = 4>. , = 1.64 -» T « 482°K = 2.09°C

hazard hazard

Thus, the layer interface will drop to the 1.5 m elevation (no earlier rhau)

110 s following ignition, and its average (untenable) temperature will bt (no

higher than) 209°C. Note from Eq. (37) that the strength of the fire at this

time is estimated to be 1.2 MW.

SUMMARY AND CONCLUSIONS

A set of general model equations which simulate the dynamic environment in

enclosures containing fires has been previously formulated and reported in

references 1 and 2. To use the model one must specify the energy release rate of

the fire, certain heat transfer parameters, the area and height of the enclosure,

and the elevation of the fire above the floor. If predictions of the

concentration of a product of combustion is desired then the fire's rate of

generation of that product must also be specified.

The present work has solved the model equations for the class of growing

fires which can be characterized by energy release rates, Q, which are proportional to t n

, where t Is time and n _> 0 is any fixed number. This class includes

O all constant energy release rate fires (n = 0), "ramp fires" (n = 1) and t fires

(n = 2). These descriptions of energy release rate have been used extensively in

the literature to characterize the growth of fires in practical fuel assemblies

which are common to a wide range of real building occupancies.

The model equations and the present solutions predict the time-varying

temperature and thickness of a combustion product laden upper layer which, at the

time of Ignition, grows from zero thickness at the ceiling of the enclosure.

General time-varying solutions for the variables, in the form of dimensionless

working plots, have been presented in Figures 2-7.

15
For those fires which generate any specific product of combustion at a rate

• n proportional to Q ~ t , solutions have also been obtained for the upper layer

concentration of that product. Plots of these are presented in Figures 3-7.

Specific results for the solution variables at the time, t Q

, when the upper

layer interface reaches the fire elevation are presented in Figures 3 and 5. For

the important special case of constant fires (n = 0), these latter results led to

the remarkable conclusion, presented in Eq. (18), that t Q is relatively independent of the fire-to-ceiling distance, H.

The results presented in this work can be used to solve a wide range of fire

safety problems. In order to provide a hint of the utility of these results two

types of example applications, one relating to smoldering combustion hazards and

one relating to flaming combustion hazards, were presented. In the case of

smoldering combustion, present results from the theory were compared to data

which were previously acquired during two full scale experiments. In the case of

flaming combustion the theory was used to predict the onset of hazardous conditions in both a hospital and a warehouse fire scenario.

The fire environment model equations solved here represent a compromise

between accuracy in real fire environment simulation and practicality of implementation. In this regard it is reasonable to envisage future implementation of

somewhat more modeling detail which would not seriously detract from the overall

model practicality. For example, the use of a somewhat more detailed description

of heat transfer phenomena (i.e., X^), as reported in references 15 and 16, would

lead to more reliable upper layer temperature estimates. An accounting of

significant upper layer-lower layer mixing due to wall effects would also lead to an important model improvement. Model improvements of this type are the focus of

present investigations.

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Department of Health and Human

Services, and the Bureau of Mines and National Park Service of the U.S.

Department of Interior.

16
REFERENCES

1. Cooper, L. Y. , A Concept for Estimating Safe Available Egress Time in Fires,

to appear in Fire Safety Journal.

2. Cooper, L. Y. , A Mathematical Model for Estimating Safe Available Egress

Time in Fires, to appear in Fire and Materials.

3. Cooper, L. Y. , Measuring the Leakage of Door Assemblies During Standard Fire

Exposures, Fire and Materials, Vol. 5, No. 4, 1981.

4. Cooper, L. Y. , Calculating Escape Time from Fires, Proceedings of

1980 SFPE/NBS Workshop on Applications of Fire Technology, Soc. Fire Prot.

Eng. publication.

5. Cooper, L. Y. , Harkleroad, M. , Quintiere, J. , Rinkinen, W., An Experimental

Study of Upper Hot Layer Stratification in Full-Scale Multiroom Fire

Scenarios, 20th Nat'l. Heat Transfer Conf., Milwaukee, 1981, to appear in

J. Heat Transfer.

6. Baines, W. D. and Turner, J. S., Turbulent Buoyant Convection from a Source

in a Confined Region, Journal Fluid Mech., Vol. 37, Part I, 1969.

7. Zukoski, E. E. , Development of a Stratified Ceiling Layer in the Early

Stages of a Closed-Room Fire, Fire and Materials, Vol. 2, No. 2, 1978.

8. Quintiere, J. , Birky, M., Smith, G. , An Analysis of Smoldering Fires in

Closed Compartments and Their Hazard Due to Carbon Monoxide, to appear in

Fire and Materials.

9. Cooper, L. Y. , Estimating Safe Available Egress Time from Fires - Appendix,

NBSIR 80-2172, Nat'l. Bur. Stand., Feb. 1981.

10. Mulholland, G. , Handa, T. , Sugawa, 0., Yamamoto, H. , Smoke Filling in an

Enclosure, 20th Nat'l. Heat Transfer Conf., Milwaukee, 1981.

11. Veldman, C. C., Kubota, T. , and Zukoski, E. E. , An Experimental

Investigation of the Heat Transfer from a Buoyant Gas Plume to a Horizontal

Ceiling - Part 1. Unobstructed Ceiling, Calif. Inst. Tech. Rept.

NBS-GCR-77-97 , prepared for U.S. Dept, of Commerce, Nat'l. Bur. Stand.,

1975.

12. Smith, G. , Research Associate (1981), National Bureau of Standards, private

communication.

13. Ohlemiller, T. , National Bureau of Standards, private communication.

14. NFPA 204M Guide for Smoke and Heat Venting, National Fire Protection Assoc.

,

1982.

15. Heskestad, G. , Factory Mutual Research Corp., private communication.

17
16. Cooper, L. Y. , Convective Heat Transfer from a Buoyant Plume to an Unconfined Ceiling, J. Heat Transfer, Vol. 104, No. 3, 446-451, 1982.

17. Cooper, L. Y. , Convective Heat Transfer to Ceilings Above Enclosure Fires,

19th Inter. Symp. on Combustion, Haifa, 1982.

18
19

Figure

1.

Flre-ln-enclosure

flow

dynamics

consistent

with

the

model

.
20

Figure

2.

Plots

of

c(t)

for

different

values

of

n

and
l+(u‘3! °x) rl= (u ‘9 ! °-i ) (p

(ul9) 0

-i (S+UElC 9

(E+u)Z

21

O

.

P

2(n+3)/[3(3n+5)]

.

.

*

Figure

3.

Plots

of

e

r

and 4

a*

function*

of

1/e

for

different

values

of

n

.
3.

ii

ii -eo

3.

ii

n•©.

2.0

1.8

1.6 3.

II

II

1.4 £

1.2

1.0

o

Figure 4. Plots of c(o) and <j>(o), 0 _< C < 1> for different

values of e, and for n = 0, n « 1, and n 2 . 22
I

I

1+ (u‘9! °o) ri= (u‘9 ;°£> ) (p

23

Figure

5.

Plots

of

o

(e)

and

<j>

(e)

for

different
l

+

rl

Figure 6. Plots of c(o) and 4>(o), -6 < c _< 0, for different

values of o . o 24
Figure 7. Plots of £(o) and <p(a),

of o q, and for 6 0, 6

25 ^ S °» f°r different values " °«2, and 6 « 0.4 .
CM

OO

l + rl = B L/L

=

(f)

b

H/!Z= 5

26

Figure

8.

Plots

of

c(o)

and

$(o)

for

n

1

and

e

0.00016,

and

plots

of

data

from

cotton

and

polyurethane

smoldering

experiments

.
NBS-114A (rev. 2-ieo) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See, nstructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 82-2622

2. Performing Organ. Report No. 3. Publication Date January 1983

4.

TITLE AND SUBTITLE

THE DEVELOPMENT OF HAZARDOUS CONDITIONS

IN ENCLOSURES WITH GROWING FIRES

5.

AUTHOR(S)

Leonard Y. Cooper

6.

PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

FINAL

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

It. SUPPLEMENTARY NOTES

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

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bi iography or literature survey, mention it here) A mathematical model for simulating the environment in enclosures during the growth

stage of hazardous fires was developed previously. To use the model one must specify

the energy release rate of the fire, certain heat transfer parameters, the area and

height of the enclosure and the elevation of the fire above the floor. Solution to

the model’s equations would yield the time-varying thickness, temperature, and

product of combustion concentrations of an upper smoke layer which starts to drop

from the enclosure ceiling at the time of ignition. In this paper the model equations

are solved for the general class of fires whose energy release rate, Q, and product

of combustion generation rates, C, are approximately proportional to t n

(t is time

and n > 0). For such fires, general results for the complete solution history of

the enclosure environment are obtained and presented in the form of graphs, and where

possible, by closed form analytic expressions. Use of the results is illustrated in

two example problems. The first of these involves a problem in smoldering combustion

where, according to experimental data, the combustion zone can be simulated by an n=l

fire. The second involves a prediction of the environment produced in an enclosure

which contains an n=2 fire, which simulates a specific, large-scale, flaming fire

hazard

.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Combustion products; compartment fires; egress; enclosure fires; fire detection;

fire growth; hazard analysis; mathematical models; room fires; smoke movement;

tenability limits.

13. AVAILABILITY

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14. NO. OF

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