A Review of Compartment Fire

Models

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

April 1983

4** °' %

/ \

33-2634

1983

C.2

S DEPARTMENT OF COMMERCE

aiONAL BUREAU OF STANDARDS

NBSIR 83-2684

A REVIEW OF COMPARTMENT FIRE

MODELS

NATIONAL bureau of STANDARDS

libra r.y MAY 19 1983

QC foy

f lA —' ^

O'-2 " c?

Walter W. Jones

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

April 1983

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Dkmctor
.

•' •• 10 i . • .
TABLE OF CONTENTS

Page

1 . INTRODUCTION 1

2. THE GOVERNING EQUATIONS 2

2.1 Form and Special Assumptions • A

2.2 Equations 5

3. RADIATIVE AND CONVECTIVE LOSS AND GAIN 8

3.1 Radiative Loss and Gain 9

3.2 Implementation of Radiation Formulae 13

3.3 Convection 1A

A. ENTHALPY AND MASS FLUX 15

A.l Plumes 16

A .2 Door Jets and Mixing •••••• 17

A.

3

SpecLal Cases and Selection Tables 20

5. SUBSIDIARY EQUATIONS 21

5.1 Fire Source 21

5.2 Radiation Between Objects 25

5.3 Conduction •••••••••••••• 25

6. CONCLUSIONS 27

7. REFERENCES 27

Appendix A - Nomenclature

Append Lx B - Summary of the Models

iii
FIGURE CAPTIONS

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Geometry used in calculating the radiative transfer. The box

is a special case of the general geometry used in ref. [17],

Schematic of the dichotomy used in all of the zone models.

Selection rules for determining the direction of flow driven

solely by buoyancy.

The four flow regimes considered in the single compartment

flow models. The external region is assumed to be ambient.

One possible configuration of the neutral plane height (N)

and the lower layer/upper layer discontinuity (Z^). Also

indicated in the figure is the notation used for vent

specification (B, H^.

The nine possible flow conditions. The "SAjj" notation is

taken from Tanaka [18]. Compartment two Is assumed to be

ambient. The selection rules determine which flow formula is

to be used to calculate the vent flow.

iv
1 . INTRODUCTION

The purpose of this review Ls to bring together the various fire

models which have been constructed over the years, in the hope of

defining a general framework for further research in the field of fire

modeling. In this context we will attempt to formulate each model (or

numerical implementation) in terms of similar variables. No attempt

will be made to justify or rederive each facet of each model. Rather we will elucidate the assumptions and enumerate the equations behind each

model. References (1-16, 18-19) describe the models in detail, and

Appendix B summarizes them.

The specific topic is the one-room model. All of the physics and

chemistry which has been developed for a one-room model will be

discussed. The model is assumed to be embedded in a world of uniform

temperature T

fl and reference pressure Pfl . (The outside wall is at T

g

which may not be the same as T

fl .) A more general treatment is possible,

but in discussing doorway phenomena in particular, the problem of multiroom connections would simply cloud the issue of what physics is

actually present in each of the models.

The discussion is broken down into the basic conservation equations, the source and sink terms for these equations, and then the

subs Ldiary equations which deal with interaction of various objects in a single compartment fire. The governing equations are divided in this

manner to facilitate the development of a model (and numerical code)

which will be modular in scope as well as general in application. With

such a model we would hope that we can test various fire, fire spread,

and entrainment models.

The notation used Ls detailed in Appendix A. In addition to being

consistent within the monograph, the symbols chosen were those most

commonly used by the various authors. In other cases, for example where

several symbols were used, it is hoped that a judicious choice will make

the text clearer. For symbols which are used only within one sect Lon,

1
or where the usage differs slightly, these symbols are defined Ln the

text. Appendix B contains a blblLography of the models discussed in

this report. Within the text they are referred to as (NBS-I, NBS-II,

BRI, Harvard, Cal Tech and Dayton). This indicates the general

geographic origin of the models. There are several references for each

one, usually generated as one aspect or another is studied in detail.

Rather than attempting to reference each model completely at each occurrence Ln the text, they will be referred to by the above generic names,

and for the interested reader a more detailed explanation is provided in

Appendix B.

2. THE GOVERNING EQUATIONS

The basic equations describe mass, momentum and energy transfer

from point to point Ln a fire related environment. These equations will

be formulated Ln a way which describes the flow from one volume to

another. Thus they are couched in the integral formulation of "control

volumes". These control volumes will be of sufficient size that we wLll

require only a few to describe any system of interest. The choice is

based on the premise that the details which occur within such a volume

do not concern us (at present), but their mutual interaction does.

In considering dynamic systems, it is necessary to solve a problem

self-consistently in terms of conservation of mass, energy and momentum. If such is not done, then some of the dynamics may be obscured or even lost. In particular, discussion of movement of the zone interfaces

should be self-consistent.

By appropriate manipulation the conservation equations for mass and

energy can be transformed into the form

dm

It

c m

P

dT

dt

A Z "

V dt

= Q + h

l“i

( 1 )

( 2 )

2
and

P = pRT (3)

where Q is the net energy input to the volume due to radiation (from all

sources), convection and conduction, and h^ (the enthalpy of the L t ^

object in the control volume) is relative to the initial temperature of

the volume from which the mass parcel m^6t came. The equation of state

for an ideal gas is usually used for closure of the system. More

correctly it should be written

P = P(p,T) (4)

especially for applications to fire problems which are not ideal gas

problems. However, for the case of an ideal gas, the derivations and

discussions are simplified, and generalizations can be discussed later.

The sign convection is that positive fluxes on the right hand side of an

equation will increase the quantity being calculated on the left hand

side, that is, transfer into a volume is indicated by a positive flux on

the right-hand side.

The basic differences which arise among the models can be discussed

using these equations as a basis. In an alternative form of the energy

equatLon, written in terms of the internal energy (not enthalpy) an explicit work term appears

P6V

in place of the time derivative of pressure in equation (2). If the

energy equation is to be solved in this form, then this work term should

be carried explicitly.

Table I shows which of the zone models include, at least in

principle, either the pressure derivative or the volume work term in

their derivations but none of these models retain this term in the

energy equation.

3
Table I Model Includi

NBS-I Yes

BRI No

Ha rvard No

Cal Tech Yes

Dayton No

The most important contribution to the pressure term

\ V6t 4^- > is due to exothermic reactions and forced ventilation.

I dt ext ) This term arises, for example, when chemical energy release results in an increase in the volume occupied by the gas as well as the internal

energy of the gas. If the gas were not allowed to expand, then the

pressure in the compartment would increase. With the usual leaks

present In rooms, the internal pressure change is essentially negligible

and in general this term is small. However, it may be appropriate to

retain this term since not all problems will necessarily allow such an approximation.

2.1 Form and Special Assumptions

The general form of the zone (control volume) model is to divide

each compartment into two zones: an upper zone which contains a hot

layer, and a lower layer which is often, but not always, considered to

be at ambient conditions. There may exist one or more fires and plumes

in a room and these can usually be considered to be part of the upper

zone. Mass and energy transfer between the zones is provided by the

plumes and mixing at vents as well as radiation between layers. This is

true for all of the models and is possibly a deficiency since there Ls experimental evidence that shows mass flow can occur along walls and

other boundaries. In general the plume, once created, simply transfers

mass and energy from one zone to another. Another set of equations

could be written for the plume, but as long as it is in quasi-steady

state, considering It to be part of the upper zone is quite sufficient.

4
Another way of looking at the plume Is to consider it so small in mass,

energy content and volume that it can be ignored except as a transfer

mechanism. For some problems, however, the plume must be considered a separate zone along with the concomitant conservation equations.

Mass and convective enthalpy transfer to and from a layer can exist

at an arbitrary number of ports, although only the BRI model allows

direct vert Leal transfer for multi-level buildings. Finally, radiation

and convection are allowed to and from walls, floors and ceilings. The

Harvard model includes detailed heat transfer to the fire source and

other internal objects as well.

The specific formulation for a single compartment will be

discussed. It will be in a form which allows general connectivity to

other compartments. Then a discussion of the physical phenomena is in

order. These are the pieces which then become part of the general flow

equations.

2.2 Equations

For a given compartment there exist three equations and four

unknowns for each zone. There are several variations of these equations

which have been used.

The general form of eqns. (1-3) together with the subsidiary

equations

H - Zu + Z

*

*

< 5 > and

at the zone boundary (6)

provide closure for the model. An implicit assumption is that the cross section (A) is known, and is usually assumed to be constant, A £ A (z),

where AZ = V. The subscripts "u" and "i" represent "upper" and "lower",

respectively, whereas "i" refers to openings to other zones, other

compartments or external volumes.

5
With the assumption of two zones per compartment, there exist two

sets of equations (1-3), one for each layer. In some cases, of course,

one zone may be infinitesimally small, such as the upper layer at t = 0.

Specifically there are four differential (1-2) and two algebraic equations (3), together with the two subsidiary equations (5-6) for each

compartment. Substitutions can be made using the subsidiary equations

which still leaves unknown the manner in which the other variables will

be found. For numerical purposes, as well as having common grounds for

comparison of the models, we will write the equations in terms of

m

u

= W = z

iApi

y V V r V T r

p

u and P

Jl

This yields six equations in eight unknowns,

dm

u „ . (7)

dm

* _ . dt

± m

i t i

( 8 )

dT dP

c m -r— = V —r—— + Q + E h m

p u dt u dt u

^ i,u i,u (9)

dT

l

dP

* • c m. —— = V„ — ( Q n + E h. . m. . p l dt l dt

^ i,A i ,

£

( 10 )

(ID

together with the subsidiary assumptions = P^ at the interface

between the zones and V = + V^. The set of equations can, and

usually is, simplified somewhat. The pressure consists of a reference

pressure at the floor and a hydrostatic term

P = p RT

u u

P„ =

6
z P(Z) = P(floor) - / pgdZ

.

o

The hydrostatic term is considerably smaller than the reference

pressure, P(floor). Thus, change in pressure as a function of height is

important only for calculating the height of the neutral plane and the

relative pressure between two vents, the latter being important for vent

flow calculations. This simplifies the equations in the following

manner

P

u

p RT

u u

p

* RTi

p,

which reduces by one the number of independent variables. It should be

noted that the reference pressure P = P

r = P (floor) is not independent

of time, so that the tern dP/dt ^ 0. This yields, finally, five equations in five independent unknowns. The specific set differs for each

of the models, but all of them can be recast into this form.

The exact form in which these equations are solved varies from

model to model. Nevertheless, they all have the same source terms; that

is, the right-hand side of equations (7-11) will be similar for each

model, with differences showing up in the number of terms and the exact

formulation for each term.

If we ignore, for the moment, the problem of enthalpy of formation,

then the enthalpy transfer terms can be written as

h.m

3 C

P

(I

3 - V

where T^ is the temperature of the layer which is being changed and Tj

is the temperature of the layer from which the gas is coming. With

these assumptions, approximations and insertions, the four differential

equations can be written as

d

TT m

dt u m + I m.

P

L i->u (12a)

7
d

dt l = " fi (12b)

e +

[

*i,A

Cp% 57 T

u

' V

u 57

+ %+

*f C

p

(TR~T

u

) + Vp( VV

+

I "iWV

(12c)

c m.

P * — T

dt l \ 57 + \

- VP

< W+

\

VP

< W ( 1 2d )

• • where Q and Q are the heating, or cooling, terms for the upper and

U X lower layers, respectively. In the present discussion

Qu + (>c + V %

and

= 0 (assumed in all the models under discussion)

.

The term includes the heat of pyrolysis.

In the following sections we will discuss the source term for eqn.

(12) and discuss the differences which occur in each of the models.

3. RADIATIVE AND CONVECTIVE LOSS AND GAIN

The radiative and convective heat loss and gain terms are contained

in eqns. (12c, 12d) and are represented by the term Q. The formalism

fr

m used here is somewhat more complex than is needed strictly for loss and

gain to a layer; however, the same terms will show up again in discussion of the subsidiary equations that describe heating of the walls,

floor, celling, and other objects. Sources of heat which increase

• • • • • internal energy are positive. Once again Q = QR + Q

c + + Q .

8
3.1 Radiative Loss and Gain

The terms which contribute heat to an absorbing layer are the same

(in form) for all layers so we will discuss these in terms of a general

layer contribution. Radiation can leave a layer by going to another

layer, to the walls, exiting through a vent, or heating up an object.

Similarly, a layer can be heated by absorption of radiation from these

surfaces and objects. The formalism which we will use for the geometry

is that used by Siegel and Howell [17] and is shown in figure (1). The

radiative transfer can be done with a great deal of generality; however,

most models at least make the assumption that zones and surfaces either

radiate and absorb like a black body, or a grey body with some constant

emissivity (e<l).

A further assumption consonant with the stratified zone assumption

is that emission and absorption are constant throughout a gas layer. In

application to a two layer model, the assumption is made in all implementations that the lower layer is diathermous. This is not a necessary

assumption but greatly simplifies the radiation transfer calculations.

In future work, the case of a lower layer which can emit and absorb

should probably be included, especially if interlayer mixing is

included.

The radiation transfer problem is always broken into two parts:

first is transfer to and from gases, walls and surfaces; second is

transfer to and from objects and fires. There are two reasons for such

a division. The first is that, although complicated, radiation to and

if- from walls, gases and surfaces can be treated more or less completely.

The primary problem which arises here is that not a sufficient amount of

information is known about wall emissivity. For objects and fires,

however, not only are the absorption and emission coefficients not well

defined, the geometrical factors are not well defined and may change in

time as well. Usually the models assume some average shape for the

flame and plume and make reasonable estimates for the various view

factors. For the most part, the discrepancy is not important. However,

9
when flashover Is about to occur, this is probably not a valid assumption. Furthermore, with current fire sources, positive feedback, may

increase the pyrolysis rate and may lead to unstable numerical solutions. For these reasons, we will describe the two sources of radiation

transfer separately.

The notation used for the wall, gas, surface interaction is shown

in figure (1). All of the zone models can be cast into this form. The

primary differences are the sophistication with which the geometrical

factors are calculated, and the approximations which are used in finding

the eraissivity and transmission factors for the layers. With the

assumption of blackbody emission, and using eq. (17.20) of Siegal and

Howell [17] we obtain

N

j=l

I, [

(4kj - F

kj WT

j - w.<]

L J

<5, . /1-e.

.iL. F x —i

e, kj kj \e

1

' Q . =

for radiation to surface (k) from surface (j).

Of interest here is the net influx of radiation to the upper gas

layer, which can be written as

- - ?

k W

The notation used is:

Fkj

T kj

\i

= geometrical view factor of surface (k) by surface (j)

= transmission coefficient from surface (k) to surface (j)

= absorption coefficient

2 = net radiation flux to surface (j) - watts/m

= net radiation to the upper gas layer - watts

10
• o Q , Q

= net radiation to the upper and lower layer (boundary)

U Xr surfaces

o o A a = Stephan-Boltzman constant = 5.67 x 10 watts/m K

L = mean beam length - (m) a = absorption coefficient of the upper gas layer (m

-

*)

We assume that the grey, upper zone, can be treated as an equivalent radLating .sphere with a mean beam length given by

L = 4V/A

where V Ls the volume of the gas and A is its surface area. This yields

an effective emissivity for the upper layer of

e = 1 - exp (-a L)

.

O

For a two zone model, with the assumption that the lower layer is

diathermous, we have the two terms, as shown in detail by Tanaka [10]

Q

= A e R /D, Q= A e 0

II /D and Q

= - Q + Q

u u u u

7 l l l SL g u l

plus additional loss terms for a vent. The factors Au and A^ are the

upper and lower boundary surface areas minus the vents and other

openings. The terms are given by

D - « - a-suKi-VF

uU

} {1 - (1 -e P

F «>

- ((l-

£u )(l-e),)(l-e

g

) 2 F^} (14)

n u=

[(I - <lg

) Fuu

} {1 ~ (1 "£ £

)F U}

(15)

" [1 + ( 1 - e £ ){( 1 - £

g

)Fu£F

£u

11
"* 1 {1 - (1 -£

u

)(1'£

g

)Fuu

1(1-F U) - O-^Xl-e/Fu^JoT^

- (1-e ) F e 0T

h

g Hu u uw

(16)

- [ { l-( 1-e ) ( 1-e )F F + (1-e )(l-e )F_ ]e aT

u g uu Jtu u g Zu g g

Equations (14-16) represent the most general case. These equations

can be simplified somewhat if we do not consider the effect of grey

walls. Then we have, as an example,

e= e = 1. Z u

With this approximation, the factors become

6 = A P /D, u u ^

= WD *

D = 1,

n = { l-( 1-e )F }aT

g uu uw

(1-e )F 00 oT

4 - e aT4 g ZZ Zu g g

n. = {l-F 00 }aT4 - (1-e )F. 0 aT

4 - F„ e aT

4

Z ZZ Zu g ZZ uw Zu g g

uhere

u u Z Z

Finally, the net radiation to the upper gas layer is given by

Q

g - - s Vk nk /D (17)

12
3.2 Implementation of Radiation Formulae

The various models differ in the number of these terms whLch are

included, and when included, the form taken for the view factors and

calculation of emissivity. Table II shows the level of completeness of

each model.

Table II

Model Comments

NBS-I Complete

NBS-II No radiative transfer - effective radiative

proportional to H

c

loss

BRI Complete - no radiation through vents, does

include radiation from fire source

not

Harvard Complete - ambient lower wall, vent not treated selfconsistently.

Cal Tech No radiation transfer - effective radiative

proportional to H

c

loss

Dayton Considers only total radiation (F^j = 1),

Tuw = T^ , no radiation through vents

NOTE: "complete" refers to the formalism discussed above.

Two terms which have not yet been discussed are first, the radiation due to a fire source, and second, the radiation interchange through

a vent. The former is a very difficult problem in that*ra fire sourqe is

generally very irregular in shape, and usually changing in time. The

latter is done very much in an ad-hoc manner, that Ls an overlay on the

usual radiation treatment, primarily due to the difficulty of calculating the actual configuration factors for vents and other arbitrary

‘openings. For these reasons, the effects of these two sources of radiation will simply be presented for the models where appropriate. The source terms for radiation exchange with vents, walls and a zone are

13
shown in Table III for the three models which have a complete radiative

transfer scheme. This example is presented for comparison purposes and

is for e = e„ = 1. The notation is that used in figures (1) and (2), u 1

and given earlier. All three use e (lower) = 0.0; therefore we use

8

the notation refers to the upper gas layer only, whereas and

refer to the upper and lower wall emisslvities, respectively.

TABLE III

Note: A=A +A.+A , A =A + A. ; e u d uv v uv Jcv u

1.0

E

f = radiation from a fire source and fiZ^Z^) is its view factor to the upper layer

3.3 Convection

Convection is the mechanism by which the gas layers lose (or gain)

energy to the walls. The other physical process, conduction, which is

intimately connected with convection, will be discussed in the section

on subsidiary equations - heat loss by the wall and heating of interior

objects

.

The basic formulation is the same for all models which incorporate

convective heat loss. The primary differences lie in the method used to

change the wall temperature, redistribute energy and the calculation of

the heat loss coefficient - h

c

. Table IV summarizes the results.

14
These formulae are applied in the models in a general way although

very little data exists for detailed validation. The formula used in

NBS-I is specific to crib fires Ln a single room and that used in NBS-II

was an effort to make such a detailed comparison between a model and a particular experiment.

TABLE IV

Model Form (q

c ”) u/i Note Constant

NBS-I h (T -T )A

c g w w

both 1

[ h^ - 10 W/m2-K

h - j or «h - 0.88 Q

1/3/A

3/8

(1-X )Q t . • • NBS-II

p iC

p

T

aL

c

q upper q - Q(t)/Q

0

A c

» parameter

t , L are characteristic

scales for the room

BRI a

f0. 21 (Gr.Pr) 1/3

. T

g

> T

u

j W^(0.012(Gr-Pr) 1/3

, T

g

< T

u |

upper 2 8*

3 lYT—J

v T

8 Pr - 0.7

Harvard h <YVAw

+ upper 3 h-min{50 ,5+4s(-^Q^IW/m2 -K

Cal Tech CiQ upper

Dayton hc<YTvall)Awupper h - 11.21 W/m2-K

c

1. Applies to both upper and lower walls; as a slice of the layer reaches a new section of the wall, the wall is assumed to equilibrate instantaneously.

2. v = 7.18 x 10

3-8 T2 ^, g

= 9.8 m/sec, i - length over which convective heat transfer takes place.

Applies to the upper layer only.

3. Applies to upper surface only; "£q" is an approximation for the amount of energy required to bring the new slice of heated wall to the same conditions as the hot upper wall. As the layer retreats 6q is set to zero, that is, no energy is put into the lower layer by cooling of this new slice of lower wall.

6q = [/*" q dt][c It ] [A ], where A is the area of ceiling, and

. c u u w w ^(thermal energy in the upper layer) >0

4. ENTHALPY AND MASS FLUX

The primary mechanisms which provide for mixing and transfer of

mass and energy between the two layers are flow provided via the plume

and mixing at a door and other vents. These phenomena are complicated

If one considers the general multiroom case. In the following section

we will consider the situation as it exists in current models with one

15
room vented to an ambient atmosphere and make no attempt to descrLbe the

general interaction. In the case of the plume, this implies that the

plume starts at a fire source and no combustion occurs during the ascent

of a smoke parcel. As the gas rises, it entrains air from the lower

layer. This mixture is then deposited in the upper layer. For the

problem of vent flow, the exterior will be a single "layer" whose

pressure is calculated from hydrostatic equilibrium at temperature T_

.

This is the external air temperature and is generally somewhat (usually)

below T , the external wall temperature.

4.1 Plumes

A fire generates a plume which transports mass and energy from the

fire into the upper layer. In addition, the plume entrains mass from

the lower layer and transports this mass and energy into the upper

layer. The energy and mass (Q^ and m^) generated by the fire are common

to all models. The form of entrainment used by each of the models is

shown in Table V.

There are several possible problems in the implementation of plume

theory in numerical applications. Specifically, the enthalpy which is

generated by a heat source due to combustion is relative to some reference temperature. It is important, to be a self-consistent, to use

this same temperature as a reference in calculating the energy necessary

for pyrolysis and the net energy added to the fuel to bring it up to the

pyrolysis temperature. In particular, in order to go from the fuel

temperature to the pyrolysis temperature, a certain amount of energy

if must be subtracted from the lower layer; of course there is also some radiative heating from the fire and from both layers. Finally, in using

Table V, one must be careful that the formula is applicable to the

problem. For example, the form given for NBS-I is specific to a crib

fire.

16
m [m m + m£ ] e p e f

TABLE V

Comments

NBS-I 0.055(m

bAH) 1/2 (D-0.53)n[D-0.53] n(x) Heaviside step function

( 0 x < 0 |1 X > 0

NBS-II 0.21 p L(gZ) 1/2

Z

2 (Q*) 1/3

BRI

/

2

' \ 1/3 o.522 (irr~j z5/3 (1 + 0,44 ?

)_2/3 • . 4.35 /_L^V/3 z-5/3 TL WphJ

2 2 Harvard TTp^(b u - R u^,)

Cal Tech 1.2 x 0.21 P<>) (gZ) 1/2Z

2 (Q*) 1/3

Uy - <C

o

/X) 1/3

; x - R/l. 2a;

u - (Co/H

p

) 1/3

; b - 1.2aH

p

;

C

Q

- 25gE

f /48wa2C

pT

L

P

L H

p - hR

-h

f-h^+X - plume length

Q* - Q/P^c T Z

2 (gZ) 1/2

Dayton +i

)

- 1]}

z

8

“ - YC

pT

L

/(YC

pT

L + XL

AH

c

)

c - 0.80 ^|7

j<l-(u) 1/5

(gyo

/U

2

) 1/5

U)

C - 1.09E4/5

(gy /u

2

) 1/5 (l-p /p,) 1/5

Z p 8 8 8 L

1.49E‘4/5

[(1./(l-U)] 1/5

(a.p L

P o+Y/XL02 ) 2/5

(po

u

o

/po /i^)

2/5

L “ layer

4.2 Door Jets and Mixing

Flow at vents is governed by the pressure difference across a vent

which occurs at each zone level. In the control volume approximation

the general momentum equation for the zones is not solved (a possible

source of error). Instead, the momentum transfer at the boundaries is

Included by using Bernoulli's solution for forced flow. This is

augmented for restricted openings by using "flow coefficients." These

latter mod if Lcat Lons deal with the problem of constriction of velocity

streamlines at an orifice. As shown in figure (3), the general case for

multilayer flow can be quite complicated.

17
There are actually two cases which apply to this type of flow. The

first, and usually thought of in fire problems, is the case where air,

smoke or some other fluid is pulled from a compartment by buoyancy. The

second type of flow is the piston effect, and is particularly important

in the initial stages of a fire. Rather than depending on a difference

in density between two gases, the flow is forced by a pressure difference across an orifice generated, for example, by combustion in a compartment.

Figure (4) shows the four configurations which will be considered.

The sLtuation is specific for one room venting to an unconfined, ambient

atmosphere. Cases (b and c) shown in figure (4) can be considered as

two manifestations of the same geometry. An important distinction must

be made in the physics of the two cases, however. In case (b) both

buoyancy and the piston effect are present whereas in case (c), the flow

Ls driven only by buoyancy.

A further complication which arises is that of determining the

position of the neutral plane relative to the layer discontinuity

height. The somewhat complex interaction leads to a large set of flow

conditions even in the case of a single compartment, at least for cases

(c) and (d). In general, all models proceed on the basis that flow is

governed by Bernoulli's equation at an opening and the driving force is a combination of the ground (reference) pressure and the hydrostatic

change in pressure as a function of height. The differences in the

implementation of the models lie in the formalism actually used to

calculate the flow, which terms are included (i.e., mining between the

upper and lower layers), and which regimes are included.

Table VI shows which of the four regimes (filling, buoyant, flow

and choked flow) are included in each model. The terms are explained in

figure (4).

18
Table VI

Flow regimes included in each model

Model Regime

NBS-I c NBS-II a BRI a-d

Harvard a-c

Cal Tech a-c Dayton a-c

The most systematic approach is done by Tanaka [ 18 ] • Therefore we will use this notation, since all other models can be cast into this

form. The one piece which is missing from the BRI model is the intraroom layer mixing which will be taken from NBS-I.

The notation is:

S = smoke

A = air

ij = flow from room i to room j

P = reference pressure (at the floor)

The order of the letters indicate which type of layer is flowing

into which other types. For example, SA^ indicates that fluid from a smoke layer in room (j) is flowing into an air layer in room (i). Of

course, in the present case, only one room exists with the remaining

f?' space being an infinite, ambient atmosphere. Therefore, only the term

AA^> SA^2> AA21, and AS21 can be non-zero. Table VII shows the various

terms where the notation is shown in figure ( 5 ).

Mixing between the upper and lower zones can be important at doorways, and possibly at the walls. Little energy will be transported by

this mechanism, but it does provide a means by which smoke can be

injected into the lower layer, thereby providing a means to change the

19
opt Leal properties of that layer. At present, the only work which

incorporates this phenomena Ls that of Quintiere et al. [13], and is for

vents only. The contribution to the mass input to the lower layer is

given by

N-Z

m

21/sa

12 ‘ °- 5 ( VV “FT

1

• (18)

Neutral Plane Flow

N < z — £

TABLE VII

Term

SA P

12 '

\ 0

2/3CB(2gp

u

|p u

-pa

|) 1/2 {(min(Hh

, Z

£ )-N) ‘-(maxCH^, Z^-u)’"'

1'}

0

C'B(2p

a

|p-P

a

|) 1/2

{ min(H

n

, Z

£ )-Ht

>

.3/2 »3/2,

AAi2 i:

Condition W«£<Z

a vz -

SA.12 i:

2/3CB(2gpJpu

-pa

|) 1/2 {(H

h

-N) 3/2

- (maxCHj,, n) - N)

3/2

> “h i X ’ «£ < “

“h

< X °r H

t

*

2/3CB(2gp

a

|pa

-Pu

|) 1/2 {fo-max^, H^))

(K-mln(H

h> X))

J "‘} \ 1

Z^, < X

0 ’Si " Z

£ or H

£ ” X

.3/2 3/2. AS

21

AA.21

C’B(2p

a

|Pa-P|) 1/2 {mlnO^, Z^) - H^} H < Z„

£ — £ H > Z„

£ £

C = C’ = 0.68

4.3 Special Cases and Selection Tables

Figure 6 shows the possible combinations of the neutral plane and

vent configurations. The neutral plane is given from the hydrostatic

equation by knowing the pressures at the floor and realizing that at the

neutral plane the pressure differential is zero. For the single room

case, there is only one possible choice for the neutral plane. If the

pressures are equal at the floor and everywhere (within a vent) the

densities are the same, then the neutral plane is well above the opening

height (Hy). If, however, the situation is as shown in figure (5), then

we have

20
P

i

(Z) = P - P^gZ^ - Pu

g (z - Z^) (internal)

and (19)

P (Z) = P - p gZ. (external)

e a a

At the neutral plane we have P^(N) = P

e (N) and therefore

defines the position of the neutral plane. Normally the flow coefficients will be approximately equal

C ~ C' ~ 0.6 - 0.7

for both air and smoke.

5. SUBSIDIARY EQUATIONS

The term "subsidiary equations" covers all of the interactions

which do not directly influence mass or energy balance of the upper or lower layer, or terms which are not yet on a firm mathematical basis.

The former includes effects such as thermal conductivity through walls

and an example of the latter would be fire sources and the effect on

them of radiation for the upper gas layer.

5.1 Fire Source ^

The most difficult aspects of modeling a fire (source) are deciding

where the fire actually burns and elucidating the effect of radiation on

the fuel source. The latter problem includes both self-radiation, that

is radiation from the flame back to the fuel source as well as radiation

from walls, layers and other objects to the fuel. Many of the models

avoid the problem by specifying the burning rate as a function of time.

For some purposes this is adequate. However, for studying flashover,

multiple ignition and similar problems, a fire model must be specified.

21
The heat release rate per unit mass being h

c (generally not a strong function of temperature), the enthalpy flux into the upper layer

is

Q c = h m

f c v

(T - T ) m - Q U V V p

Essentially, the energy introduced into the upper layer is a product of the heat release rate at the pyrolysis temperature with a reduction for the energy required to gasify the fuel and subsequent

pyrolysis.

For a spec Lf Led burning rate, this is sufficient. In order to

treat a self-consistent fire, however, one must Include the interaction

of a fire source with its environment. A self-consistent fire source

must include the effects of oxygen vitiation, radiation to the fuel

surface, radiation by the fuel surface, and heating by conduction. The

NBS-I, Harvard and Dayton models have fire source algorithms whLch

include one or more of these effects, as shown in Table VIII.

Table VIII

NBS-I

Harvard

Dayton

Oxygen, radiation

Radiation, re-radiation, conduction

Oxygen, radiation

All three models attempt to include the effects of radiation from both

the upper layer, walls and the flame itself. Clearly, prior to ignition, the radiation from "other" sources is most important. However,

once ignition occurs, radiation from the flame to walls, the gas layer

and other objects should generally dominate. In addition, the Harvard

model includes re-radLation from the fuel source itself. This latter

effect should not be Important for gas burners or pool fires but may be

effective in reducing the burning rate of charring materials.

22
An important difference in the models is the method of including

the geometrical (view) factors. As before, the intent is to include

only the radiation which an object "sees." A fire changes shape in

time, and rather quickly at that. This leads to the use of approximate

flame shapes, the most common being cones (Harvard) or cylinders (NBS-I,

Dayton). When including incident radiation due to walls and the gas

layers, a complete treatment of the problem allows for absorption by the

plume and flame. Finally allowance has to be made for effects of oxygen

vitiation. In principle, a more complete kinetics scheme should be

included, especially as the detailed species which are released in

combust Lon become accessible experimentally. Nevertheless, the effect

of oxygen (or lack) must be taken into account. The (NBS-I) model does

these in an ad hoc but reasonably effective way, namely

*f

= fifree

+ + V (19)

where m^ = free burning rate of a fire

m = enhancement due to radiation effects

K

m = decrement due to vitiation

o

For NBS-I, which considered the burning of cribs, the m^ term is

given by

_. _ ^top ^top +

^side ^side

m

R

“ L

vap

where L

ya p

is the gasification energy and is a measured quantity. For

this particular experiment, Lva pignored the contribution (decrement)

due to heating of the fuel. Thus to do this problem more generally a loss term, ^

targ et > should be included in eqn. (19). The term "Top" and

"Side" refer to the parts of the cribs which were being modeled. The

Harvard model proceeds in a similar fashion except that the view factors

which determine the q term are calculated numerically based on the

object position and geometry relative to the ceiling and walls. In the case of NBS-I, the view factors are calculated analytically, once again

based on the geometry of the room and object (fire source). One aspect

23
of the Harvard model is the inclusion of absorption of the flame radiation by the fire source.

Oxygen deprivation has two parts. The first is the oxygen concentration near the fuel source. The second is the amount of burning which

occurs in the plume (a flame) above the fuel source. This latter is

important since, as the upper layer descends, the functional form of

combustion changes. The NBS-I model does this by

. . ) , ox

ra o m

free ( 0.23

and the Harvard model accomplishes this same effect by changing the size

of the cone of the flame. As the layer moves down, the cone is truncated, reducing the pyrolysis rate, thus reducing the mass flow. The

Dayton model accomplishes this using a method similar to NBS-I. The

Dayton model uses the Steward [20] and Fang [21] plume models (see Table

V), however.

The last piece in this analysis concerns the pyrolysis itself. In

addition to the problem of heating the fuel from its reference state T^,

there is also the problem of a phase change, for example, from solid to

gas, where the combustion finally occurs. From the point of view of

modeling a fire, the only reason to be concerned with this process is

that a phase change is sensitive to temperature and pressure; thus the

fire history affects the burning rate m^. An important part of this

process is the production of toxic substances, such as carbon-monoxide

and cyanide. Since this is in large part the motivation for studying

the fire problem, a great deal more effort must be spent on this aspect

of modeling a fire source.

As is obvious, our understanding of the fire source itself is

minimal. Further, the source to be used is specific to the problem to

be studied. At this point there is no agreed upon way to include a general, self-consistent fire source in a model.

24
5.2 Radiation Between Objects

The calculation of radiation effects between objects suffers from

problems similar to those of the fire source itself. Interobject radiation is an important phenomenon only if multiple targets are present and

one is interested in flashover. The only two models which address this

issue are the Dayton [2] and Harvard models [19]. The primary difficulty arises in calculating view factors between the objects. This has an analogy in several other fields. One must calculate the projection

onto a surface (the target) of the surface subtended by the solid angle

(view factor) of the emitter. Except for simple objects such as spheres

and planar surfaces, this calculation is difficult. Current research in

modeling of solid objects, which deals with piecewise continuous approximations to a surface, will aid in automating and generalizing these

calculations. At the present time, however, the methods used in the

above mentioned models will have to suffice.

5.3 Conduction

Conduction of heat through solids occurs in two places: the

compartment walls and interior objects. The techniques used are similar

for both cases. Generally a slab is cut into N intermediate slices (N+l

nodes). Then the heat conduction equation

dT

dt

V (aVT) * aV T = a

is solved for each element of this finite grid. On either side of a

if- finite slab, boundary conditions are imposed for radiation (net) and

convectLve heating and cooling. The number of nodes is chosen to reduce

the error to some reasonable value, say 5%. Use of N > 20 will improve

precision without a concomitant increase in accuracy. For very thin

walls only the surface nodes are necessary.

For the first time step, an initial temperature profile must be

assumed. The usual assumption is uniform ambient temperature throughout

25
the solid. After the fLrst step, the previous distribution can be

assumed. As can be seen, this implementation differs markedly from the

usual "control volume" approach.

The actual calculation usually employs a finite central difference,

forward (explicit) t Lrae step scheme.

T (t + fit) = (1 - afit) T

± (t) + + T

j-i + 1}

with fit = time step interval

2 a = differential coefficient = 2 a/ (fix)

At the surface, the boundary condition

b fit q

with b = (pc fix) ^

,

c = heat capacity per unit mass.

The flux q may arise from convection or radiation and may be positive or negative.

When this formulation is applied, particularly to walls, ceilings

and floors, a decision must be made as to the division of surfaces which

are In the upper and lower layers, and what Is done to the energy

balance as the wall surface immersed in each layer changes. This method

of heat conduction is done by NBS-I, BRI, Harvard and Dayton, although

the Dayton implementation is not as complete as the others.

An alternative is to use an average conduction coefficient and assume that the energy loss is proportional to the temperature difference times this heat loss factor. Such a formulation is applicable to

heat loss in early stages of a fire when only the ceiling is affected

and principally by convective heating. At later stages, however, this

model is clearly not appropriate. The models which use this method are NBS-II and Cal Tech.

26
6. CONCLUSIONS

We have examined the models which use the control volume concept in

analyzing fire growth and spread. While none of the models is complete,

all of them have features which can be well utilized in the formation of

a general fire model. These features reflect the specialties of the

respective authors.

7. REFERENCES

[1]

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

Plastic Cribs in an Enclosure: Volume I, Nat. Bur. Stand. (U.S.),

NBSIR 80-2054 (1980).

[2]

MacArthur, C.D. , Dayton Aircraft Cabin Fire Model Version 3,

Volume I and II, University of Dayton Research Institute (1981).

[3] Emmons, H.W. , The Prediction of Fires in Buildings, The

Seventeenth Symposium (International) on Combustion, The

Combustion Institute, Pittsburg, PA (1978).

[4] Mitler, H.E., The Physical Basis for the Harvard Computer Fire

Code, Home Fire Project Tech. Rept. #34, Harvard University

(1978).

[5] Cooper, L.Y., Estimating Safe Available Egress Time from Fires,

Nat. Bur. Stand. (U.S.), NBSIR 80-2172 (1981).

[6] Zukoski, E.E. and Kubota, T. , Two-Layer Modeling of Smoke Movement

in Building Fires, Fire and Materials _4^, 17 (1980).

[7] Zukoski, E.E . , Kubota, T. and Cetegen, B., Entrainment in Fire

Plumes, NBS-GCR 80-294 (1980).

[8] Emmons, H.W., The Growth of Fire Science, Fire Safety J. _3, 95

(1980). :«r-

[9] Emmons, H.W. , The Calculation of a Fire in a Large Building, ASME

81-HT-2 (1981).

[10] Tanaka, T. , A Mathematical Model of a Compartment Fire, Building

Research Institute (Japan), Research Paper #70 (1977).

[11] Tanaka, T. , A Model of Fire Spread in Small Scale Buildings,

Building Research Institute (Japan), Research Paper #79 (1978) and

#84 (1980).

27
[12] Quintiere, J.G., McCaffrey, B.J. and Kashiwagi, T. , A Scaling

Study of a Corridor Subject to a Room Fire, Comb. Scl. and Tech.

18 , 1 (1978).

[13] Quintiere, J.G. , Steckler, K. and McCaffrey, B., A Model to

PredLct the Conditions in a Room Subject to Crib Fires, First

Specialists Meeting (International) of the Combustion Inst.,

Talence, France (1981).

[14] Quintiere, J.G., Growth of Fire in Building Compartments, ASTM

Special Technical Publication 61A (1977); The Spread of Fire from

a Compartment - A Review, ASTM STP 685 (1979).

[15] Rockett, J.A., Fire Induced Gas Flow in an Enclosure, Comb. Sci.

Tech. I2j 165 (1976).

[16] Quintiere, J.G., An Approach to Modeling Wall Fire Spread in a Room, Fire Safety J. _3, 201 (1981).

[17] Siegel, R. and Howell, J.R., Thermal Radiation Heat Transfer,

McGraw Hill Book Co., New York (1981).

[18] Tanaka, T. , A Model of Multiroom Fire Spread, to be published as a Nat. Bur. Stand. Internal Report (1983).

[19] Mitler, H.E. and Emmons, H.W., Documentation for CFC V, The Fifth

Harvard Computer Fire Code, Home Fire Project Tech. Rep. #45,

Harvard University (1981).

[20] Steward, F.R., Comb. Sci. and Tech. _2, 203 (1970).

[21] Fang, J.B., Analysis of the Behavior of a Freely Burning Fire in a Quiescent Atmosphere, NBSIR 73-115 (1973).

28
Appendix A - Notation

area (m z

); A

u

, A^, A^ are the upper and lower compartment surface

areas in contact with the upper and lower gas layer. Fig. (1),

respectively. A^ is the interface area between the upper and

lower layers. In section 4.1, "A" is used as a variable :Ln the

flow equations to indicate air.

width of a vent (m)

flow coefficient - 0.6-0. 7 for both smoke and air

specific heat - Cp, cv (J/kg/k)

energy release rate (J/s)

view factor - relative area of "i" as seen by "j" (dimensionless)

o acceleration of gravity (9.8 m/s )

height (m) , H

u

, H^, are the upper and lower limits of a vent -

Fig. (2)

enthalpy (J/kg/k)

heat of combustion - theoretical (J/kg)

compartment indices

mean beam length (m) equivalent opaque sphere

mass (kg)

mass flow (kg/s): m - rate of release of volatiles

vm

e - entrained into a plume

m^ - fuel release

m - mass entering room "i" from room "j"

m - flow rate in plume (m = in, + m ) P p f e height of the neutral plane (m)
P pressure (pa): P -* P - floor reference pressure

P

e - Eqn. (19)

P^ - outside ambient pressure

Q rate heat is added or lost (J/s):

• • Q , Q - upper, lower zones,

U X respectively

Q,. - fire (h m ) f c v

•Q

q - objects

Q

- radiation

K

•Q - convection by walls

c

• - radiation added to upper

gas layer

Q. - radiation from surface "k"

k

Qp

- combustion energy lost by

formation of volatiles

R gas constant for specific mixture

S smoke - section 4.1

t time (s)

T temperture (k) : T

a - ambient

T

c - external wall

T

u

- upper wall *-

- lower wall

- reference temperature for enthalphy flow

T - upper zone temperature

oT

v - volatile temperature

O V volume (m ) 30
z

a

Y

e

P

<

layer thickness (m)

absorption coefficient of upper gas layer (m *), thermal

diffusivity (m^/s)

ratio of specific heat (c

p

/cv

)

emissivity (dimensionless): - surface "i"

e - upper gas layer

8 - upper compartment surface

- lower compartment surface

O mass density (kg/m )

thermal conductivity (j/msk)

Kronecker delta =0 i £ j

- 1 i - J

Subscripts - In general "u" and "Z" indicate upper and lower gas layer,

respectively. For area and emissivity variables, reference

is to the compartment itself.

31
Appendix B - Summary of the Models

The references contain reasonably complete descriptions of the

models. In addition, copies of the computer codes are available from

the authors. The following li6t is a suggested starting point for each

of the models, and is the primary summary of the contents and capabilities of each:

Model

NBS-I

NBS-II

BRI

Harvard

Cal Tech

Dayton

References

13

5 18

19

6

2

To summarize, the most complete transport phenomena are contained

in the BRI model, and the most consistent radiation transport in

NBS-I. Both the Dayton and Harvard models attempt to deal with the

problem of a self-consistent fire source model. As indicated in section

5.1, this is a difficult problem given our understanding of flames

shapes, turbulent mixing and the chemical kinetics of fire oriented

combustion.

None of the models handles the numerics well. Part of the problem

is mixing differential and algebraic equations, both of which are

stiff. (Stiff in this context refers to two or more processes with

widely (greater than a factor of ten) varying time constants.) Another

part of the problem is in the discreteness of the source functions and

their derivatives. Finally, one has to look carefully for possible

instabilities in the numerics themselves when the approximation,

discussed in section 1, are made.

32
RADIATION

PATH

BETWEEN

TWO

SURFACES

WITH

AN

ISOTHERMAL

GAS

BETWEEN

THEM

33

Figure 1.

Geometry

used

in

calculating

the

radiative

transfer.

The

box

isa

special

case

of

the

general

geometry

used

in

ref. [

18

].
Tu,

£u

34

Figure

2.

Schematic

of

the

dichotomy

used

in

all

of

the

zone

models.
SIGN

[5P1

SIGN

(<5p)

dp

6P/Sp

^ f CO f CM f X CM »—

rvi tMi

35

Figure

3.

Selection

rules

for

determining

the

direction

of

flow

driven

solely

by

buoyancy.
36

Figure

4.

The

four

flow

regimes

considered

in

the

single

compartment

flow

models.

The

external

region

is

assumed

to

be

ambient.
H

37

Figure

5.

One

possible

configuration

of

the

neutral

plane

height

(N)

and

the

lower

layer

/upper

layer

discontinuity

(Z^)

.

Also

indicated

in

the

figure

is

the

notation

used

for

vent

specification
NEUTRAL

PLANE

BELOW lC

38

Figure

6.

The

nine

possible

flow

conditions.

The

"SA^j"

notation

is

taken

from

Tanaka

[18],

Compartment

two

is

assumed

to

be

ambient.

The

selection

rules

determine

which

flow

formula

is

to

be

used

to

calculate

the

vent

flow.
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

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

SHEET (See instructions) NBSIR 83-2684 April 1983

4.

TITLE AND SUBTITLE

A REVIEW OF COMPARTMENT FIRE MODELS

5. AUTHOR(S)

Walter W. Jones

6. PERFORMING ORGANIZATION (If joint or other than N BS, see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

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

10.

SUPPLEMENTARY NOTES

U 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

bibliography or literature survey, mention it here)

We have examined the concept of zone ^modeling as it has been applied to fire

problems. The existing models which embody the zone concept are compared in

order to ascertain the state-of-the-art in understanding of fire growth and

fire spread processes. This review is intended to be a starting point for

future modeling efforts and thus the discussion centers around the use of the

various submodels.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) compartment fires; fire models; room fires; smoke movement; zone models

13. AVAILABILITY 14. NO. OF

Uni i mi ted PRINTED PAGES

Q] For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

41

15. Price

TXT] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

U S COMM* D C 6043-P80
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