NBSIR 83-2718

A Model of Multiroom Fire Spread

US. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Labora ory

Center for Fire Research

Washington, DC 20234

August 1983

U S. DEPARTMENT OF COMMERCE

#</*c*u o»

o

.U56

83-2713

1983

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

LIBRARY AUG 19 1983

NBSIR 83-2718

A MODEL OF MULTIROOM FIRE SPREAD

Takeyoshi Tanaka

Guest Worker

Building Research Institute

Japan

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

August 1983

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF SYMBOLS

I . INTRODUCTION

II. THEORY

1. Conceptual Model 4

2. Zone Model Equations 7

2.1 Zone Conservation Equations 7 2.2 Practical Equations for Layer Properties 11

3. Component Process Models 21

3.1 Combustion Model 21

3.2 Entrainment 35

(i) Fire plume 35

(ii) Door jet 37

3.3 Heat Transfer 39

(i) Radiative heat transfer 39

(ii) Convective heat transfer 45

(iii) Thermal conduction 46

3.4 Flow through openings 48

(i) Rate of flow through openings 48

(ii) Solution of pressure equation 54

Ill . PROGRAM 59

4. Program Structure 59

Page

v

vii

viii

1

4

iii
Page

5. Description of the Subprograms 61

5.1 Main Program 62

5.2 PARAM 66

5.3 TPROP 72

5.4 DFIRE 75

5.5 WPRS 80

5.6 RKG 82

5.7 DFNC 85

5.8 INTPOL 95

5.9 FPLUM 97

5.10 DRJET 99

5.11 ABSORB 102

5.12 HTRAD4 104

5.13 HTCNL 107

5.14 CNDCT 110

5.15 VENTL 113

5.16 RFLOW 118

5.17 PSET 122

6. Data Structure 124

IV SAMPLE CALCULATION 127

7.1 Sample 1: Comparison with FMRC Test 6. 132

7.2 Sample 2: One story, fuel rich fire 132

7.3 Sample 3: Two story, fuel rich fire 141

7.4 Sample 4: Two story, fuel lean fire 141

7.5 Sample 5: Two story, fuel rich fire with outdoor wind 146

7.6 Sample 6: Two story, fuel rich fire with narrow 146

external openings

7.7 Sample 7 and 8: Fire in tall buildings 160

V. CONCLUDING REMARKS 166

ACKNOWLEDGEMENT , 167

REFERENCES 168
LIST OF FIGURES

Page

Figure 2.1 Schematic Diagram of the Model 5

Figure 3.1 A Simplified Steady State Combustion Model 22

Figure 3.2 An Alternative Diagram of the Combustion Model 24

Figure 3.3 Relation Between the Heat of Combustion of Gasified 26

Fuel at Different Temperatures

Figure 3.4 Combustion Model in Calorimetry 28

Figure 3.5 Doorjet Model 36

Figure 3.6 Conceivable Doorjet Configurations 38

Figure 3.7 Radiative Heat Transfer Model 40

Figure 3.8 Convective Heat Transfer Model 40

Figure 3.9 Pressure Difference Profile Across an Opening 49

Figure 3.10 Conceivable Pressure Difference Profiles Across an Opening 53

Figure 3.11 Pressure Calculation Chart for Two Rooms 56

Figure 3.12 Pressure Calculation Chart for a General Case 56

Figure 3.13 An Explanatory Building Model and Its Pressure 58

Calculation Chart

Figure 4.1 Program Structure 60

Figure 7.1(a) Predicted and Tested Temperature Elevation 130

(Sample 1: Comparison with FMRC Test 6 )

Figure 7.1(b) Comparison Between Predicted and Tested Gas Concentration 131

Figure 7.2(a) Upper Layer Temperatures 133

(Sample 2: One Story, fuel rich fire)

Figure 7.2(b) 0

2

and Fuel Concentrations of the Upper Layers 134

Figure 7.2(c) Fuel Burning Rate in the Rooms 135

Figure 7.2(d) Upper Layer and Flow Through Openings 136

Number under a ceiling with no arrow: temperature elevation

Number at a doorway with arrow: flow rate (kg/s)

Figure 7.3(a) Upper Layer Temperatures 138

(Sample 3: Two Story, fuel rich case)

v
Page

Figure 7.3(b) 0^ and Fuel Concentration of the Upper Layers 139

Figure 7.3(c) Upper Layer and Flow Through Openings 140

Number under a ceiling with no arrow: temperature elevation

Number at a doorway with arrow: flow rate (kg/s)

Figure 7.4 Upper Layer Temperatures 142

(Sample 4: Two Story, fuel lean fire)

Figure 7.5(a) Upper Layer Temperatures 144

(Sample 5: Two Story, fuel rich fire without door wind)

Figure 7.5(b) Upper Layer and Flow Through Openings 145

Number under ceiling with no arrow: temperature elevation

Number at an opening with an arrow: flow rate (kg/s)

Figure 7.6(a) Upper Layer Temperatures 147

(Sample 6: Two Story, fuel rich fire with narrow external

openings)

Figure 7.6(b) 0^ and Fuel Concentrations of Upper Layers 148

Figure 7.6(c) Upper Layer and Flow Through Openings 149

Number under a ceiling with no arrow: temperature elevation

Number at an opening with an arrow: flow rate (kg/s)

Figure 7.6(d) Shaft Pressure (at ground level) 150

Figure 7.7(a) Upper Layer Temperatures 152

(Sample 7: Five Story, fuel rich fire with narrow external

openings; Note: Rni denotes i-th room on n-th floor)

Figure 7.7(b) Upper Layer Gas Concentrations 153

Figure 7.7(c) Upper Layer and Flow Through Openings 154

Number under a ceiling with no arrow: temperature elevation

Number at an opening with an arrow: flow rate (kg/s)

Figure 7.8 Upper Layer and Flow Through Openings 162

(Sample 8: Ten Story, fuel rich fire with narrow

external openings)

vi
LIST OF TABLES

Page

Table 3.1(a) Flow Rate of Hot Gas and Air 51

Table 3.1(b) Flow Rate of Hot Gas and Air 52

Table 7.0 Typical Conditions of Sample Calculations 128

Table 7.1 Input Data for Sample 1 129

Table 7.2 Input Data for Sample 2 129

Table 7.3 Input Data for Sample 3 137

Table 7.4 Input Data for Sample 4 137

Table 7.5 Input Data for Sample 5 143

Table 7.6 Input Data for Sample 6 143

Table 7.7 Input Data for Sample 7 151

Table 7.8 Input Data for Sample 8 161

vii
LIST OF SYMBOLS

A Area

a Thermal diffusivity

AA Air flow rate through an opening (air to air layer)

AS Air flow rate through an opening (air to smoke layer)

B Width

C Specific heat

Cp Specific heat at constant pressure

D Depth of room

F Configuration factor

f Volume fraction of soot

v

g Gravitational acceleration

H Height

AH Enthalpy difference, Heat of combustion

h Specific enthalphy

k Thermal conductivity. Absorption coefficient

L Heat of gasification, Length

£ Length, Thickness

M Molecular weight m Water content

m Mass rate

P Total pressure. Relative pressure. Partial pressure

Q Heat per unit mass

Q Heat Rate

q" Heat Flux

R Gas constant

r Specific heat ratio (r=C /Cv) s Mass fraction of original fuel that turns into soot

SS Hot gas flow rate through an opening (smoke to smoke layer)

SA Hot gas flow rate through an opening (smoke to air layer)

SA* Flow rate of gas that enters upper layer through doorjet

SE Air entrainment rate into a doorjet

T Temperature

t Time

u Specific internal energy

V Volume

W Mass fraction of species w Mass fraction of original fuel that turns into residual char

X Neutral zone height

x Distance from the surface of a wall

Y Mass fraction of a species

Z Layer thickness, Vertical distance

z Vertical distance

a Flow coefficient

y Mass generation rate of a species

c Emissivity

n Fraction of carbon that turns into CO

v Kinematic viscosity, Stoichiometric coefficient

viii
p Density

a Stefan-Boltzmann constant

A Fraction of air flow AS that mixes into upper layer

Superscripts

* complete combustion

(*) per unit time

" per unit area

(~) vector

n iteration number

a air, ambient

p pyrolyzed fuel

Subscripts

a lower layer, air, ambient, actual

as between lower and upper layer

C convection, carbon

d discontinuity, decomposition

e entrainment, effective

ex experiment

f fuel

G hot gas, hot gas layer

g gas, gasification

H hydrogen

i room number, in

j room number, time step

ij from i to j , between i and j

k opening number

1 sill, species

m mean

0 oxygen

o original, reference state, out

p pyrolysis, pyrolyzed fuel, constant pressure

pr products

R room

r reference state

S upper layer, soot

SS between the two upper layers v vaporization, constant volume

W wall
«

<>' !

‘

• < .

• :

. ~ i

*
A MODEL OF MULTIROOM FIRE SPREAD

T. Tanaka

ABSTRACT

Some refinements have been made on a multi-room fire

spread model. The primary improvements are: (a) that a model on excess fuel burning in arbitrary room has been

introduced; (b) that a model for the prediction of gas

concentrations has been added; (c) that subroutine ABSORB,

which was created by Modak, has been introduced to predict

the upper layer emissivity; (d) that a new fire plume

model developed by McCaffrey has been included to remove

the inaccuracy and inplausibility of a vertical point

heat source plume model; and (e) that the code has been

revised so that it can deal with tall buildings with

somewhat less computer memory size.

Also, some sample calculation results with the new code

and a documentation of the code have been included.

I . INTRODUCTION

When the previous model for hot gas flow in multi-room structure

was devised [10,11], an intention was to use the model to predict or analyze inter-room fire spread in small buildings, particularly residential structures. The previous model had assumed that the combustion

of input fuel is completed within the room of origin. But it was soon realized that in order to raise the temperature of an adjacent room

high enough to ignite the combustible material in there, quite a big

fire is needed in the room of origin. In such cases, it is no longer

plausible to assume complete consumption of input fuel within the room

of origin and the burning should be sensitive to the air supply into

the room. In fact, it is a well known fact that the pyrolysis rate of

1
an intense fire is almost controlled by the air inflow rate alone. As can be seen when one observes window flames brought forth by a big fire, excess fuel burns outside the room of origin. If this occurs in a room

of the structure, it will affect even the flow in the structure, not to

speak of the temperature elevation. Therefore, emphasis was placed on

the pursuit of modeling the transport of excess fuel in a structure in

refining the model. This appears to be indispensable for the purpose of

predicting fire spread.

Additional refinements have also been made. First, species conser- vation equations were introduced to predict the concentration of soot

and primary gases in any room in a structure under the condition that

their generation rate due to the burning of unit mass of fuel can be

specified. The importance of the prediction of the species concentrations

in fire growth lies in that it relates the prediction of the burning rate

of gaseous fuel and the estimation of the upper layer emissivity. With

these predictions, subroutine ABSORB, which is an excellent outcome of

Modak’s work [7], conveniently provides the layer emissivity. With

regard to element process modeling, a major refinement is the replacement

of the point heat source fire plume in the former model with McCaffrey’s

most recent model [6]. Generally, a point heat source model does not

provide a good prediction for a fire plume in a room in which our primary

concern is the region significantly close to fire base, even if some manipulations such as a virtual point heat source are introduced.

McCaffrey’s model has an advantage in that it is based on accurate

measurements in this region and it accounts for the size effect of the

source

It might seem to some readers somewhat redundant that this paper has

duplicated, to some extent, the same discussion already stated in the

previous papers [10,11]. This was done partly because it was suspected

that some readers may feel it is more convenient to be able to read this

article without frequent reference back to the previous papers, and

partly because the layer equations were derived in a somewhat different

way as based on the derivations of Baum [3], Quintiere [5] and Zukoski [5].

2
Finally, a documentation of the code, which was revised so that it

can deal with tall buildings with less computer memory capacity, has been

attached in this paper.

Before proceeding into the details of this report several points

should be emphasized about its general applicability and accuracy.

Primarily the model and computer code should serve as a framework for

multi-room fire growth predictions. But some elements of the model are

not accurately and completely formulated because of our current lack

of understanding. Therefore these elements or their omission will contribute to uncertainties in the overall results. The weakest element

is the transport of gases in vertical shafts. The treatment of this

phenomenon as an effective plume in a tall room is not valid once the

plume interacts with the walls or with structural elements in the shaft.

Pressure drops due to walls and obstacles in vertical shafts and

corridors are not taken into account. These phenomena must be accurately

modeled before the code is sound. Also the effect of vertical mixing

so as to heat the lower layer must be better understood, and the

assumption of a cold lower layer must be quantitatively examined.

Finally the fuel supply rate by the fire must be specified in this

model. For building design purposes that may be sufficient; however,

a model for combustion efficiency will be needed to predict multi-room

fire growth. Despite these limitations, thoughtful use of this code

should give insight into a variety of fire growth problems.

i
II . THEORY

1. Conceptual Model

The problem considered here is the motion and state of the hot gas,

which is generated by a fire and considered to be responsible both for

human casualties and property damage.

A fire usually starts from the ignition of a combustible item in a room. The ignited item emits pyrolyzed fuel into the air, which, by

reacting with the oxygen, releases heat and combustion products. The

heat and combustion products are pumped up through the buoyant plume

induced above the fire, entraining more air as they rise to form a hot

gas layer of mixture of the combustion products and fresh air under the

ceiling. This layer may contain even unburnt fuel as well when the fuel

input rate is so large that it cannot be completely burnt with the

oxygen available in the room.

At the very early stage immediately after the start of a fire, the

hot gas layer will descend simply pushing only air out of the room openings because the gas at high temperature expands to displace the cold

air that has occupied the upper part of the room. But before long, the

hot gas layer will drop below the soffits of the openings, which of course allows the hot gas to flow out of the room of origin into the

adjacent rooms to form other hot gas layers there. Those upper layers

will behave basically in the same way as that of the room of origin,

thereby reproducing the similar filling process and spreading the hot

gas throughout the structure. Thus, a fire in an arbitrary structure

at an arbitrary elapsed time after ignition may be illustrated as in

Fig. 2.1. In such situations, the behavior of the fire depends on the

conditions of the structure and the fire source. The mathematical zone modeling approach will be a useful approach for predicting the fire

behavior in such complex structures and can be a viable alternative to

expensive and time consuming full scale fire experiments. To formulate

4
5

Fig.

2.1

Schematic

Diagram

of

the

Model
a model for this purpose, however, we will be obliged to significantly

simplify the problem since the true phenomena involved in fire are quite complicated while our current knowledge is still limited. Among

many simplifications and approximations which need to be made in zone modeling, the following assumptions are considered to be the most

essential and commonly used in current zone models.

(a) In any room which hot gas enters, two well defined stratified layers,

i.e., a hot gas upper layer and a cool layer, are formed.

(b) Each layer is homogeneous in any physical or chemical property.

Besides these basic assumptions common in any zone model, the following

are also considered to be essential in this model.

(c) The upper and the lower layer are divided by a flat discontinuity

such that there is no mass and heat transport due to interface

mixing unless through a fire plume or door jet.

(d) Both radiative and convective heat transfer between any lower layer

and the surfaces that are in contact with the layer, namely floor

and lower part of walls, are ignored.

These two assumptions (c) and (d) are introduced to keep the lower layer

temperature constant so that many processes, particularly the formulation

of flow rate through openings, can be simplified. These assumptions are

not as essential as the first two given in (a) and (b), which are generally used in zone modeling approach. In fact, some models for single

room fires have already relaxed the latter two restrictions and allowed

interface mixing between the two layers and thermally non-transparent

lower layer [1,2]. Some more elaborate formulation of door flow rate

and advances in our understanding of layer interface mixing may make it

6
possible to handle the transitional lower layer even for multiroom structure cases. On the contrary, we will hardly be able to remove the former

assumption without complicating the zone modeling approach. Indeed,

various advantages given by zone model approach would disappear at the same time when the assumptions (a) and (b) are abandoned.

In Fig. 2.1, p , T and Y tf , respectively, stand for the density, the

S S X* absolute temperature and the mass fraction of species £ in the upper

3. layer of room i, and p , T and Y stand for those of the lower layer

3 3 36 where subscript i is omitted from every symbol for brevity. Z and Z

S 3

are the thicknesses of the upper and the lower layers respectively.

Both SS. . and SA. . denote the rate of hot gas flow from room i to i ij 13

through k-th opening between room i and j , while AS^ and AA_ likewise

denote the rate of air flow where k is omitted again for brevity. It

might be confusing that each of the hot gas and the air flow has been

given two symbols, but each symbol corresponds to each of the characteristic opening pressure difference profiles as may be suggested in Fig. 1

and will be discussed later. A temporarily convenient interpretation of

these symbols may be that SS„ and SA_^ . are the hot gas flow above and

below the discontinuity height of room j, i.e., Za ^, respectively, and

so forth with the air flow AS. . and AA. .. It will be obvious in Fig. 1

ij iJ

that when seen as a doorjet, SA can entrain the air from the lower layer

as it travels to the upper layer while SS can only entrain the hot gas

from the upper layer thereby contributing no additional mass to the

layer. The sum of SA and the rate of air entrained into it is termed

SA* . The mass loss rate of fuel and the air entrainment rate into the

fire plume are termed m and m respectively and m (= m + m ) denotes

p e s p e the total rate of gas that enters the upper layer through the fire plume.

2. Zone Model Equations

2.1 Zone Conservation Equations

In formulating the physical basis of the mathematical model, we

have to consider first the conservation equations for mass, species and

7

i
energy of the layers. To do this, however, it is necessary to introduce

some additional assumptions regarding the flows through openings, which

have been termed as SS, SA, AS and AA in Fig. 2.1. The assumptions introduced here are as follows:

(a) Hot gas flows SS and SA enter into the upper layer.

(b) A part of air flow AS enters into lower layer and the rest of

AS flows into the upper layer.

(c) Air flow AA enters into the lower layer.

The assumption (a) may not always be self-evident in some conceivable

situations in fire: the hot gas may not be able to enter the hot upper

layer when the temperature of the latter is much higher than that of

the former. But it is very difficult to abandon the assumption (a)

because if we introduce another layer or let the hot gas go into the

lower layer to cope with this problem, we are forced to sacrifice the

advantages given by the set of assumptions in section 1. The assumption

(b) is introduced to take account of tall shafts filled with a thick hot

gas layer. The air flow AS may no longer be able to penetrate the upper

layer when it has to travel a long distance as it is heated by mixing on its way to the lower layer.

Then, under these assumptions, the layer conservation equations can be written down for any room i on any floor of a multiroom structure as follows

:

Overall mass conservation of upper layer

(SS. . + SA.

.

ij ij

+ m + m

P e

( 2 . 1 )

8
Overall mass conservation of lower layer

(P ) = £ i (1 - A) AS. . + AA. . - (AS. . + AA. .) dt a R a ( 31 Ji ij ij

- (SA! . - SA. .) > - m

Ji Ji /

( 2 . 2 )

where A^ denotes the area of room i, and A(0 _< X 1) is the fraction of

air flow AS that mixes into upper layer; so X = 1 indicates that the air

flow AS wholly mixes into the layer and no portion of it reaches the

lower layer, while X = 0 indicates the opposite. The summation Z is

taken with respect to j and k, i.e., with respect to every opening

between the room i and the adjacent spaces, although subscript k has

been omitted from the flow terms for brevity. Needless to say, Eqs.

(2.1) and (2.2) hold for any room in the structure. The fire plume flows

Ap and m

e> can simply be given zero values for rooms other than the room

of origin.

Mass conservation of species in upper layer

fr <VsVs> Y

* %

+ Y

? *. + + z Kj(SS. . + SA. .) 31 3i

- Y (SS. . + SA. . ) + Y*(SA' . . - SA.,) + Y* . XAS

.

£ 13 x3 l 31 3 i £.3

(2.3)

where Y^ and Y^ stand for the mass fractions of species l in the gasified

fuel and air respectively, i represents either fuel gas, C^, CC^, ^0,

1^0 or N

2 and m^ is the burning rate of gasified fuel, which is controlled

either by fuel or oxygen supply rate, and y^ is the mass generation rate

of species Z due to the burning of unit mass of gasified fuel, i.e.

,

Yj = <v" - vptyvf

M

f

where v" and are stoichiometric coefficients of species 1 in product

and reactant system respectively.

9
Energy conservation of upper layer

dt

|^

u

£

Y

«. P

s^ AK

Z sJ + P

d(A

R

Z

s> dF~ * %et + KY

£ ”p + Ktf

+ E

fl Eh

£ > j

Y

£,j)

(SSji + SV - (E W (SS

ij + SAij>

+ (SAji

- SAji> + AASji

Energy conservation of lower layer

31

j(J

“Kk) Vaj +P <Va>

dt A

e

+ £ (Ehtj Yl: ) {

(1 - A)AS

ji + - WhK)(ASij + Mij

- («^)(sa:. - sv]

where u, h and p respectively stand for specific internal energy and

enthalpy and absolute pressure; q is the net heat added to the hot

net

layer by radiative and convective heat transfer. These Eqs. (2.4) and

(2.5) are an expression of the first law of thermodynamics applied to

this particular case.

The pressure in Eqs. (2.4) and (2.5) may vary with height due to

hydrostatic effect as

m

e

(2.4)

)

(2.5)

10
where subscripts o and z represent a reference level and height from the

level. However, hydrostatic term of this equation is so small compared with

P

Q

that in many practical conditions it can be reasonably considered as

negligible. So the pressure P in the above equations can be regarded

constant with height.

Finally, with the same consideration regarding the pressure, the

equation of state may be written as

Equation of State

P =

p T R =

s s p T R

a a

( 2 . 6 )

where gas constant R is assumed the same regardless the difference of

composition among the upper and lower layers.

2.2 Practical Equations for Layer Properties

It is useful to transform the above equations so that the

layer properties with which we are more familiar are given explicitly.

Let's restrict the consideration here to multiroom structures such that

the area of any room does not vary with height, unlike such a structure

as an aircraft fuselage, then can be regarded as independent with time

too.

11
First, noting that left hand side of Eq. (2.3) can be expanded as

d

dY o d

dt (

'

1 a

P

sAR

Z

s^ ” Ws dtT~ + Yl dt ^ P s¥s^

subtracting Eq . (2.1) multiplied by Y. . from Eq. (2.3) yields:

)c , 1

,Va 5? - (

Y

I

- h) \

+

(

Y

?

- Y

*) K

+ Y

* S,

+ E hTt>j - Y

t

(S8 + S* ) +(Y

t - Yt

(SA- - SA )

+ |Y

1.J - Y

1

US

Ji

(2.7)

Next, noting the following relations

Ws Ws ( 2 . 8 )

where

p n= Ptet) RT

s

12
substituting Eq . (2.8) into the left side of Eq. (2.4) yields

if{(;W3 ) Vs} +p

~-

" dF {(

Eh*Vs) Vs } - IF {( EPl) Vs }

+ P

dtT"

2-

= E h if <V.V.> } + E { <V,Vs> ar } - 3f

(pVs>

where use was made of EP = P. Then calculation of Eq. (2.4) -

l 1

E |h^ x Eq. (2.3)

|

gives

E (VsVs if

1

)

- Vs If Vt -( E Vi) *b

+

!

E

(

h?- h

«) j5}v {

E K- h*K H

+

[

E

{

E(hi.j - Wj} (s V + SV> +

{

E (

h

?

- h

*)

Y*

}

( SAh- SA

Ji )+

{

E K-J

The term -Eh Y, , which has shown up in the right hand side, is

£

£ 1 the heat of reaction of the gasified fuel AH, as shown below.

• E

X/ v* X/

EViWf - E

XV?Wf

^reactant ^product

= AH (2.9)

( 2 . 12 *)

13
When no phase change is involved, the enthalpy state equation can be

written as

h

t - hr ,

i

+ C

p.»

< T - V (2 ' 10)

where subscript r represents a reference state. Since is always considered

to be the most dominant composition of both hot gas and air, it will probably

be reasonable to assume the specific heat of any gas as constant regardless

the change of mass fractions of species, i.e..

IC „Y n = C (constant)

£ p,£ fi. p

( 2 . 11 )

Substituting Eqs. (2.9) - (2.10) into Eq. (2.12') yields

dT

dP

C p A Z ~rz~ ~ AnZ — = q „ + AH-m, + C (T - T )m

p s R s dt R s dt net b p p s t

+ Cpa(T - T )m + C l < (T . - T )(SS. . + SA. .) s e p ( s,j s jx ji

+ (T - T )(SA! . - SA..) + (T . - T )AAS. . I a s ]i jx a, j s jx f

( 2 . 12 )

Following the similar procedure as above, for the lower layer, we

obtain

dT

dP

C

p

P aA

R

Za dt \Za dt

= C E (T . - T ) P a, j a

(1 - X )AS . . + AA

Ji .4 (2.13)

14
' The equation for the layer thickness can be obtained as follows:

From the equation of state.

dP 0 , p dT

s 1 dP _ _s s dt " RT dt T dt

s s

Substituting this relation into Eq . (2.1) yields

, dZ

dF < P sVs>

= Ps^ dT + A

R

Z

S df

dZ

1 dP . „

P

s dTs

P

sAR dt + Vs RT * dt Vs T

* dt

Noting this and adding Eq. (2.1) multiplied by C T to Eq. (2.12) gives

P s

dZ / C \ , C p T L -T1- + A Z V - 1 It- = q + AH-ITL + C T ro p s s Tl dt R s \ R

J

dt Mnet b P P P

+ C T m + C Z | T .(SS.. + SA.,) - T (SS.. + SA..)

P a e p \

s,j v ji ji' s

v xj ij

+ T (SA! . - SA. .) + T . AAS

a Ji Ji a, j Jji

}

(2.14)

Finally, let’s obtain the equation for pressure. Multiplying

Eq. (2.1) by C T and adding the result to Eq. (2.12) yields

P s

dT ' HP d

CAVs dT - Vs dT + Vs dt (P sVs> - Vt + AH

-*b

+ CTm + C T m + CI i T . (SS. . + SA. .)

p p p p a e P \ s,j 31 ji

- T (SS.. + SA..) + T (SA’ - SA..) + T . XkS . A s iJ iJ a ji ji' a, j ji>

15
The left hand side of the above equations turns out as

dT dP d

C p Ad Z - A Z ~ + C T — (p AZ ) p s R s dt R s dt p s dt s R s

dT dp

s . m s (arV dP AZ C p —- + T

-

7— + C p T — A^Z — Rsplsdt s dt / pss dt R s dt

V

C CP d (AZ ) _E dP + __p _ a z dP

s R dt \ s dt +

R dt

Therefore, it follows that

V

C

P dP

, C

pP d <Vs)

s R dt R dt

dP AZ -7Z = q _ + AH-ih, + C T

R s dt net b p

m

P P P

+ C T in +

p a e C Z (t .(SS.. + SA..) - T (SS^ . + SA..)

P \ S,J 31 31 s ij 13

+ T (SA! . - SA. .) + T . XAS

a 31 Ji a, 3 3>4

( 2 .

Likewise, a similar equation can be obtained for the lower layer by

adding Eq . (2.2) multiplied by C T to Eq. (2.13) as follows

pa

C CP d(AZ ) _p dP p R a dP

V — — 1 r - A_.Z — = -C T ma R dt R dt R a dt p a e

+ C E

P T . \ (1 - X) AS . . + AA .

a, 3 \ Ji Jif

- T (AS . . + AA . .

)

a

v 13 13

- T (SA! . - SA. .) a 31 Ji ( 2 .

15a)

15b)

16
Then, noting that Z + Z = H„ (i-th room height : constant) and adding

s a R

Eqs. (2.15a) and (2.15b) yields

a

r

hr r - 1 dP

dt q + AH-iil + C T m + CZ (T .(SS.. + SA..) net d PPP P l s,j 31 3 1

T (SS.

.

s 1 J SV T . (AS . . a > J Ji + AA. .) 31

T (AS .

.

a ij AA^)

\

(2.15)

where use was made of the relations C - C = R and r = C /C . p v p v

Eq. (2.15) could be substituted into Eqs. (2.12) through (2.14) to

eliminate the pressure term dP/dt in those equations, and a careful

numerical computation with small time step might be executed for the

set of ordinary differential equations (2.7) and (2.12) through (2.15)

to predict the fire under any opening condition. This kind of calculation

method may have an advantage when we deal with fires in buildings with

very many rooms or very narrow openings. But unfortunately we have not

yet established this method, therefore let us consider whether the pressure

term can be neglected in practical fire conditions.

It is implied by Eq. (2.15) that the pressure in a room will rise

proportionately to the net rate of the heat and fuel added if there is

no opening, in other words no outflow; however, usual rooms of a building

are not sealed so tightly that an extremely high pressure can be attained

[3], [4]. As the pressure rises, the larger outflow will be induced through

the openings to reduce the pressure elevation. It may be useful to think

17
of j single compartment with only one opening for simplicity. Then,

Eq . (2.15) may be written approximately as follows

V^r-hr) ar - Vt + - c

P

TAe^2pp

*

* A where fuel input is neglected. P represents P = P^ - P , i.e.,

pressure difference between indoor and outdoor, and is the effective

opening area. This equation implies that the larger heat input, the

smaller room volume and the smaller opening area, the larger the rate

of pressure elevation, and

* i / yt + flH,lhb \

2

F 2o \ C TA I

is the maximum pressure attainable. So the larger opening area causes

a smaller maximum pressure. In addition to the leaky condition of

usual rooms, there is another reason which may make the pressure unable

to rise very high. In the case of usual fires, in which heat is generated

by the burning of gaseous fuel emitted into rooms at a moderate rate,

vitiation of oxygen and heat transfer to the walls will soon decrease the

net rate of heat addition to the rooms if the openings are narrow.

The above discussion has not yet justified ignoring the pressure

terms in general but at least has implied that in most of usual cases

pressure elevation is not significant due to the presence of sizable

openings or oxygen vitiation. So hereafter the consideration is

restricted to the condition where the following assumptions hold:

18
(a) dP/dt terra in every equation can be ignored.

(b) Pressure elevation is so small that the thermodynamic pressure

in the equation of state can be replaced with a constant

reference pressure.

With these assumptions, the temperature and the density of any lower

layer are kept constant all the time if they are the same as those of

ambience at the beginning, in other words Eq. (2.13) is no longer

needed. The former assumption (a) implies that the pressures are determined such that the right hand side of Eq. (2.15), which are actually

simultaneous algebraic equations for the pressures of the rooms, is

balanced [5].

As a result of the neglecting of the pressure terms in the

preceding equations, the final equations can be summarized as follows:

Temperature

dT

s dt

(2.16)

19
Species

dY. T

a _ s dt p T A_.Z a a R s-

{

Y A +

(

Y i- Y0"p +

(

Y

"-

Y0

ws- v}

+ 4t£ {"t.j- V<ss,i + s V +

(

Y?- Y0

(SA! . - SA. . + XAS. .) i Ji Ji Ji J (2.17)

Thickness

dZ

dt C p T

p a a7— / q + AH*iil + C T m + C T (m - m ) 1 A

r \

net b ppp pas p )

+ —J—— E i T . (SS. . + SA. .) - T (SS. . + SA. .) P a

T

aA

R I S ’ J J1 J1 s

T (SA! . - SA. . + XAS. .) 1 a Ji ]i Ji f (2.18)

Pressure

q + AH-itl T

net b . P . h =— m

C T

P a T p a a + E <f T . (SS. . + SA. .) - T (SS. . + SA. .) \

T \ s,j ji ji s ij iJ J

+ E J (AS. . + AA. . ) - (AS. . + AA. .) I = 0

\

J 1 J 1 13 iJ f (2.19)

The equations for the flow terms (SS . . SA. . etc.) are taken from

J 1 > J 1 > Ref. [10] and presented in Table 3.1.

20
3. Component Process Modeling

The ordinary differential equations for the layer properties, i.e.,

Eqs. (2.16) - (2.19), have to be complemented with the equations for the

component processes involved in fire, such as heat and species generation due to the burning of fuel, radiative and convective heat transfer,

fire plume and doorjet entrainment, flows through openings. This section

will be devoted to the modeling of these processes.

3.1 Combustion Model

Generally, it is difficult to model the combustion of actual fire

with high accuracy. One of the difficulties may arise from the fact

that the fuels and the rooms involved in fire are not ideal from the

viewpoint of combustion. We have to estimate the heat release of real

fire based on calorimetry of the fuel, however, unlike calorimeters or

well-designed combusters, fuels in real fire may contain water, leave

residual char and discharge smoke and carbon monoxide. Also attention

has to be paid to fuel rich combustion, where all the fuel is not burnt

within the room of origin because of insufficient inflow air. In such

cases, particularly in case the fuel is from a pyrolyzed solid combustible,

it is not easy to estimate the heat release due to the burning of the fuel,

since we do not have enough calorimetry data for such fuels.

To deal with this problem, let's introduce a simple steady state

combustion model as shown in Fig. 3.1, where original fuel that contains

21
INCOMPLETE

BURNING

PRODUCTS

22
water is being steadily heated to its decomposition temperature, gasified

leaving some portion of it as residual char, and incompletely burnt producing

soot and CO as well as CO^ and l^O. In this diagram, it is the fuel at the

gasified state that may be transported outside the room of origin and burnt.

The reference state of the fuel which is inserted in the diagram is not a

state which real processes pass through, but the virtual state where the heat

of combustion can be theoretically calculated using the heat of formation of

species

.

In the following, the various quantities which need be inputed into the

layer equations will be derived based on the combustion model in Fig. 3.1 or

the thermodynamically equivalent model in Fig. 3.2.

(i) Heat required for unit mass loss

Let’s consider of the pyrolysis of the fuel composed of 1 kg of dry

fuel and m kg of water, namely 1 + m kg in total. In this case, 1 - w kg

of gasified portion of the fuel and m kg of water vapor are counted as the

mass loss. Hence

where w stands for charring ratio. On the other hand, the heat required

to give this amount of mass loss is, as indicated by Fig. 3.2

mass loss = 1 - w + m (3.1)

o o o

+ Lgo(T ) + mLVO(T )

23
0

24

Fig.

3.2

An

Alternative

Diagram

of

the

Combustion

Model
where Ch „ , C and C , respectively stand for the specific

fuel vapor char

heat of original fuel, water vapor and residual char; T , T and T

° ops

are respectively the temperature of a reference, pyrolysis and residual

char, and L (T ) and L (T ) are the latent heat of gasification of the

g o v o fuel and water at a reference temperature respectively.

The heat required per unit mass loss of the fuel Q is given as

/ (1 - w) C . + mC ( dT + w // C , dT + L (T ) + mL (T )

) fuel vapor r char go v o

o ’ o

1 - w + m

(3.2)

Although the specific heat of the mixture of the fuel and vapor may change

with the value of w and m, it will be assumed for consistency with the layer

energy equations that this is constant and the same as the C

p

in the layer

equations, i.e.

,

(1 - w)Cjr . + mC fuel vapor

1 - w + m

(3.3)

then, Q becomes

P

w / C , dT + L (T ) + mL (T ) char go v o

Q

= C (T - T ) +

P P P o 1 - w + m

(3.4)

25
26

Fig.

3.3

Relation

Between

the

Heat

of

Combustion

of

Gasified

Fuel

at

Different

Temperatures
Eq . (3.4) may be interpreted as the heat transferred to the fuel from the

surroundings, e.g., the flame, the upper layer, which is obtained from the

viewpoint of fuel pyrolysis.

(ii) Heat release due to incomplete burning of gasified fuel

What we need to obtain as the heat of combustion should be the enthalpy

difference between the combustion products and the fuel of the gasified state

at as was indicated in Eq. (2.9). Let’s consider here of the combustion

of unit mass of the gasified fuel. Looking into the diagram in Fig. 3.3

reveals that the following thermodynamic relations hold.

AH (T ) +

a p

C dT = AH (T ) +

g a ° (3.5)

AH (T ) = AH (T ) + L, (T ) a o r o do (3.6)

where and are specific heats of gasified fuel and product. From

Eq. (3.5)

-AH (T ) = -AH (T ) +

a p' a o

jr

c

p

/ (C - C )dT

4

g Pr

(3.7)

The left side of Eq . (3.7) is the heat of combustion at temperature T^,

which we are seeking, and Eq . (3.7) implies that

-AH (T ) = -AH (T ) a p a o (3.8)

27
Incomplete

burning

(at

T^at T0) I 0

<o>u Eoo

28

Fig.

3.A

Combustion

Model

in

Calorimetry
if the second term in the right hand side of Eq. (3.7), which is trivial

compared with the first term, is neglected.

From Eq. (3.6) and (3.8) it is obvious that

-AH (T ) = -AH (T ) - L, (T ) a p r o d o (3.9)

In this equation, -AH^OT^) can be calculated using data for heat of formation

of the species involved, but L

d

(T

Q

) cannot easily be obtained. So let’s

consider the complete burning process as in calorimetry. Looking into the

diagram in Fig. 3.4, we see the following relation

L (T ) do = -AHro(T )-{-AHexo(T )}-L~(T ) g o (3.10)

where -AH^ (TQ

) is the so-called heat of combustion, which has been measured

for many existing materials (oxygen bomb) calorimetry, and -AH (T ) is the

r

v o'

enthalpy difference between the reference state fuel and the complete combustion

products, which is again calculable as is -AH^(T

o

). It is not always easy to

obtain the gasification energy L^(T

o

) unless the fuel is gas or liquid. Many

* attempts have been made to obtain Lg(T

Q

) for solid fuels, but the measured values

have ranged widely, probably because of the difficulty in the measurement and

the dependence of volatile composition on heating condition. Theoretically,

* the range of L (T ) should be

g °

.JU 0 < L (T ) — go—< -AH (T ) - {-AH (T )} r o ex o'

jS* Although the attempts to provide the data of Lgo(T ) have not yet 7 been so successful in case of solid materials, let’s dare to assume that the data is

available.

29
The next bold assumption is to regard that the L^(T

q

) in Eq . (3.10) is

the same as L,(T ) in (3.9), despite the likelihood that the composition of

d o the gasified fuel is inevitably different between the real and ideal burning

because of the residual char in the real case. This assumption and the combination of Eq . (3.9) and (3.10) yields

-AH (T ) ap = -AHro(T )-{(-AH''(T ro))-(-AH*exogo (T ))-L*(T )} (3.11)

Accordingly, it follows that the heat of reaction of the gasified fuel can be

estimated, in principle, by using experimentally or theoretically available

data.

In calculating -AH^(T

q

) for a specific fuel, it is necessary that the

chemical composition of the fuel is known. Usually, this is given in terms of

chemical formula or weight fraction of chemical species, but the latter is more

widely available. So, let’s start from the weight fraction. If v^, and

are the molecular number of the species C, H, 0 and N in unit mass of the fuel

respectively, they are given from their weight fractions as follows:

W, w. V

1 = 12 x 10

-3 V

2 = H

W

1 x 10

-3 V

3 =

0

w.

16 x 10

-3 v

4 = N

14 x 10

-3 (3.12)

where W , W , W and W represents the weight fraction of C, H, 0 and N

L H 0 N

respectively

The chemical equation for the complete combustion of this fuel may be

written as:

30
v

2 v3 * * C H 0 N + (v- + -7 — )0_ 4- (L + L,)

v v v v 1 4 2 2 g d

^lC+ -2^2 + -l °2 2

2

V V

V

3 n +

V

4 N„ + (v + r 4) 0,

1 4

2' 2

v

2 v

4 * ^V

1

C0

2 + ~2 H2° + ~T N2 + (“ AHr

^ (3.13)

and -AH (T ) is given as

r o

-AH*(T ) r o 97.6 + 68.3 —

Y

97.6 ~ x 10

3

+ 68.3 -| x 10

3 (kcal/kg) (3.14)

Next, let’s calculate -AH (T

q

). Various species may be discharged in case

of incomplete burning, however, it is assumed here that only soot, CO^, CO, ^0

and ^are involved in the product. The gasified fuel no longer has the same

chemical composition as the original fuel since a part of carbon has been left

as solid char through the gasification process. The new weight fractions of

C, H, 0 and N should be (W - w)/(l-w), W / (1-w) , W_/ (1-w) and W. / (1-w)

C rl (J IN respectively. Let s/(l-w) be the fraction of the carbon that turns into soot

and n be the fraction among the rest of the carbon, i.e., (W^ - w-s)/(l-w),

that turns into CO. Then, the chemical equation for the incomplete burning

can be expressed as

31
C , H , 0 , N , + (v!

v

1

v

2 v

3 v

4

1

V

2 V — 3 + -T--f ) °2 + (L

g + L

d

)

V

2 V

3

V

4

V

1 ->v|'[soot] + v^''C H 2 ^2 2 ^2 2 ^2 +

1

*

' _

~2

v ' vf

> °2

ii 2 4 -v [soot] + (v” ’-v” ' ')C0

2 + \>" "CO + —- H2 0 + -y N

2 + (-AH.) (3.15)

where the stoichiometric coefficients are:

, w

c

- w

V

1 1 - w

12 x 10

W„ - w-s

1 , , s -3 ’ V

1 1 - w

12 x 10

-3

it? _

1 - w

12 x 10

-3 ’ v

i

' nv

i

(3.16)

W, w„ w. v ' =

. < = N

'2 1-w in-3 ’ 3 1-w in-3 * v 4 1-w

1 x 10 16 x 10 14 x 10

-3

Then, -AH^_ can be given as follows:

-AH = 97.6 (v' ' ' - v' ' ’ ') + 27.6 vV ” +57.6-4

r 11 1 2

= {97.6(l-n)+27.6n}

W -w-s

V^< 1 W

12 x 10

3

W

1 +57.6 I H

2 1_w lx 1<T

3

(3

Substituting thus obtained Eqs. (3.14) and (3.17) together with experimentally

& & obtained -AH (T ) and L (T ) into Eq . (3.11) yields the value of -AH (T ). ex o g o a p

(iii) Mass generation rate of species

The mass generation rate of each species due to the burning described in

the chemical Eq. (3.15) is given by applying the following general formula:

.17)

32
- "PVv

f

M

f

Y = (v' (3.18) ' Y a

K i

where and are stoichiometric coefficient of species £ in product and

reactant system, and = 1 because in this case y is considered for unit

mass loss. So each y^ turns out as follows

"fuel

- (0-DM

f

/lxM

f = -1

y *. - (v '

' - 0 ) x 12 x 10

3 = -~-

soot 1 1-w

V**** V* V*

{0 - (v”' - + -j - )} x 32 x 10

" 3

W -w-s W„ , . -{ (i _ —) -1— . _£ + — . — . -L- . _JL_ U } x 32

2

; 12 1-w 4 1-w 2 16 1—

w

Y

_ q W -w-s , , Y C0

2 = {(v^”-v| ,,, )-0} x 44 x 10 = (1-n) j_w - • ^ (3.19)

W -w-s

Y co

= <4’”-0) x 28 x 1° -'-hr- * if

, V

2 ia in-3 1

W

H 18

YH

2 0

" C-2 - 0) x 18 x 10 = y

* 1^7

* ~

v* W W ,4 9ft , n-3 1 W

N 28

W

N

y. T = (—r- - 0 ) x 28 x 10 = -y • 3

• 77- = 3 N

2

2 2 1-w 14 1-w

(iv) Burning rate of gasified fuel

Generally, burning rate is a function of the local concentration of fuel

and oxygen and temperature, however, since the flame temperature is usually

so high, it is practically quite reasonable to assume that fuel and oxygen cannot

i 33
coexist at the same spatial point and the burning rate is governed by the

supply rate of fuel or oxygen into the flame region. In the case of combustion

in the upper layer the supply rates will be controlled by local turbulent mixing

of the fire plume, ceiling jet, door jet and so forth, however, it is now

difficult to reasonably model every mixing process involved. So, let’s assume

that the mixing rate in an upper layer is so large that fuel and oxygen cannot

coexist in the layer and by this reason the supply rates of either fuel and

oxygen into the layer becomes the controlling mechanism of the burning process.

With this assumption, the burning rate of the gasified fuel can be given

as follows:

^

Y'Iy. .(SS.. + SA. ,)-Y.(SS. ,+SA. .)+Y*(SA! .-SA. .+AAS. .) \ Ji f 13 ij f 31 31 31 '

+ Y^ m + Y^(m -m ) f p f s p

(Y > 0 :fuel control)

°2

“b V|y„ ^ (SS . .+SA. . )-Yrt (SS. ,+SA. .)+Y® (SA'. .-SA. .+AAS . . ) | o

2>j 31 31 o

2 13 13 02

v 31 31 31 »

+ Y^ m + Y^ (m - m ) 0

2 p 02

s p

(Y = 0 toxygen control) (' °2

Note that since the mixing rate in the layer is so large that fuel and oxygen

cannot coexist in the layer, the criterion to determine mass burning rate

must be the oxygen concentration Y^ .

. 20 )

34
3.2 Entrainment

Entrainment by fire plumes and door jets play an important role in hot gas

transport in a structure. The nature of fire plume, which rises up vertically

from a horizontal fire source, has been comparatively so well investigated,

both theoretically and experimentally, that a fairly reliable model is available

here. Regarding door jet entrainment, however, no reliable data are available, so a somewhat crude model will be presented by just stretching the

result on fire plume entrainment.

(i) Fire plume

The following equations proposed by McCaffrey are used to calculate the

rate of flow that enters the hot upper layer of the room of origin m

s [6].

• 2/5

(a) Continuous flame region (0<Z/Q <0.08)

2/5

(b) Intermittent region (0.08<Z/Q <0.20)

(3.21)

0.909

2/5

(c) Plume region (0.20^Z/Q )

1.895

35
r

36

Doorjet

configuration

Modeled

doorjet
where m, Z and Q respectively stand for mass flow rate (kg/s), height above

heat source (m) and nominal heat release rate (kW) . Since the flow rate m

should not be smaller than the mass loss rate m an adjustment is made as

P

follows

m

s max {m , P

m by Eq. (3.21)}

(ii) Door jet

(3.21)

Since little is known about door jet entrainment at least for the purpose

of quantitative estimation, it is assumed that the door jet behaves as if it

were vertical thermal plume rising up from a source horizontally placed on a

plane so that the fire plume equation given by Eq. (3.21) can be utilized. A

fictitious source is assumed at the distance Z below door jet center level so

that the flow rate given by Eq. (3.21) becomes equal to the door jet flow rate

at the level. The distance Z

q can be obtained by substituting m = SA^ and

Q

= C SA..(T . - T ) into Eq. (3.21), and the following relations are substituted

P 1 1 s , j a again into Eq. (3.21) to yield the rate of door jet flow that enters the upper

layer of i-th room (Fig. 3.5).

m 53 SA. . P Ji

Q = C SA. . (T . - T ) P Ji s,j a (3.22)

z- = Hr - Z

s - (min. (Hr. Hr - Zs

) + H,,, - Z

s>j )>/2«0

where SA^ and T

g are respectively stand for the flow rate and the temperature

of door jet; and Z

g are room height and upper layer thickness; is the

soffit height of the opening; X is the neutral zone height; and subscripts

i and j denote room numbers (Fig. 3.6).

37
38

Fig.

3.6

Conceivable

Doorjet

Configurations
3.3 Heat Transfer

Heat transfer is the precursor of fire spread since every ordinary material

has to be heated before it ignites. Also heat transfer may have an effect on

the gas flows in structures because the flows are driven by buoyancy. Every

mode of heat transfer is involved in fire. This section will be devoted to

the practical modeling of these heat transfer process.

(i) Radiative heat transfer

The procedure to calculate the radiative heat transfer may be classified

into two, i.e., how to calculate the heat transfer under given configuration,

temperature and emissivity, and how to estimate the emissivity of the hot layer

which contain soot, ^0 and CC^. The former question is solved by using netradiation method, which was first introduced by Hottel and developed by many

others, and the latter is. conveniently obtained by using ABSORB [7].

a) Radiative heat transfer in two layer configuration

It may be said that a room which contains a hot gas layer under its ceiling

is one of the most characteristic heat transfer configurations in building fire

(see Figure 3.7). With the highly advanced outcome in radiation heat transfer

science, it may be possible to consider furthermore complicated situations, but

it still may be useful to simplify the problem as follows to fit the current

purpose.

So, let's assume that the interior wall of a room can be divided by

discontinuity surface into two parts, i.e., one that is in contact with the

upper layer and the other that is in contact with the lower layer, and each

39
QJ X)oS

OJ

4-1COc

(QMH

(0a) PC

QJ>

•H

4-1 CJ

QJ>

(CouM

•H IIiI I

40

Fig.

3.7

Radiative

Heat

Transfer

Model
part is uniform in temperature respectively. It is also assumed that the

surfaces of the walls and the gas are gray. For this heat transfer system,

net -radiation method gives the following relation

Q

1 A

i i-£

1

(ctT

i ^0,1 )

Q

1 = A

1

{ «0,1 - (1 " e G

) *11*0,1 ' (1 - £ G

)F12^0,2 } - A

l*0,,

Qo = A

2 2 1-e, (0I

2 - ^, 2

> (3.23)

Q

2 ’ A2

l %,2 - - F22’0,2 ) ' Vo.G

4o,g

£ goT

g

Qg

- -(Q, + Q

2

)

where subscripts 1, 2, G and d respectively stand for ceiling, floor, upper gas

and discontinuity; q” ^is respectively radiative heat flux leaving from

surface k; is the net radiative heat loss of surface k; a, e and T are

respectively Stefan-Boltzmann constant, emissivity and temperature, and

F, . is the view factor between surfaces k and j , which is given as follows

kj

in this specific case.

F

11

1

A

1

* F

12

F

ld A

x

* F

21

F

2d A

2

’ F

22

1 A

2 (3.24)

where A denotes surface area.

The solution of Eq. (3.23) gives Q^, Q

2 and Q

G as follows

41
(3.25)

. p p

Q

1 = A

1 G1 d” ’ Q2 A

2e

2 [T ’ QG

" (Q

1 +

where

D = {l-(l-e

1 )(l-eG

)F n

}{l-(l-e

2 )F

22 }-(l-e

1 )(l-e

2 )(l-eG

) 2 F

12

F

21

P

1 = [{l-(l-e

G

)F

11 Hl-(l-e

2 )F22 }-(l-e

2 )(l-e

G

) 2 F

l 2 F

21 ]oT^

(' 1 “ e G^

F l2e 2° T

2 _ t 1 + " c

2') ^1-£

G^

F l2

F

2d

“ F22^ £ G°

T

G (3.26)

P

2 = [{1 - (l-e

1 )(l-e

G

)F

11 }(l-F

22 )-(l-e

1

)(l- eG ) 2 F

l2

F

21 ]aT

2

- (1 -C G)F

21

e l0T r

[a- (1 -e l ,(1-e G

)FU

)F2d + (1 -£

l)

(1 “E G

)F21 l£G

OT

2

In spite of the significant simplification on the configuration, the solution

given by Eqs. (3 . 25)- (3 . 26) may seem still lengthy, but it is one of the

simplest solutions in the sense that every view factor can be obtained only

from surface areas.

If the upper gas is black, i.e., £ = 1, it follows from Eqs. (3.25)-(3.26)

that:

Q

1

e l

AlO (T

4

1

e 2A

d°

(T

2 - T

G

) l-(l-e

2 )(l-Ad

/A

2

)

(3.27)

42
b) Emissivity of upper layer

The calculation code ABSORB [7J, gives the absorptivity of isothermal

and homogeneous mixture of soot, CO^ and 1^0 whose total pressure is 1 atm.

Modak's calculation may functionally be expressed as

a = ABSORB (T , s T, k

s

, L ) m

(3.28)

where T

g

and T represent the temperature of blackbody source and mixture; k

g is

the absorption coefficient of soot at a wavelength of 0.94 ym; P and P

CO^ i^O

are the partial pressure of C0~ and H„0; and L is the mean path length of the

L L m

layer.

The emissivity of the layer may differ from its absorptivity when the

temperature of radiation source is not the same as that of the layer but the

difference is not significant so practically the layer temperature can be

substituted into T of Eq. (3.28) to obtain the emissivity.

s

The absorption coefficient of soot at a wavelength of 0.94 ym is given

as

k

s

7 0.94 x

10' fv

(3.29)

where fv

is the soot volume fraction and given as

43
f V

Y o / p soot s soot(3.30)

where Y , p and p are the soot mass fraction of the layer, the layer

soot s soot

gas density and the density of soot.

The partial pressure P

2 of gas species of the layer as

and Ph

2 o are given in terms of the mass fraction

(3.31)

and

( 3 . 22 )

where Y and M are mass fraction of species in the layer and the molecular weight

of the species respectively.

The path length L is given as m

L = m

0.9

4V

A (3.33)

where V and A are the volume and total surface are of the upper layer.

44
(ii) Convective heat transfer

In the room of origin, whose temperature is considerably high, the role

of convective heat transfer may be relatively small as compared with radiation,

but in the rooms other than the room of origin it may still be relatively

s ignif leant

A conventional method is used to estimate the convective heat transfer

to ceilings although it may not be adequate for the room of origin, in which

special patters of flow due to fire plume, ceiling jet and so forth, are

involved

For a horizontal surface surrounded by thermally insulated wall, the

mean Nusselt number is given as (see Figure 3.8).

1

0.054

(Gr • Pr)

0 * 38

(Tg > T

w )

(3.34)

0.003 (Gr • Pr)

0 ' 38

(Tg < T

y

)

where T and T are the temperature of the upper layer and the surface of the

G W

wall that is in contact with the layer. Grashof number Gr and Prandtl number

Pr are given as

Gr =

g* < t

g

- V

v

2

T„

(3.35)

45
where g denotes gravitational acceleration, v and a are respectively kinematic

viscosity and thermal diffusivity of the gas, and i is the reference heat transfer

length

Using Nu given in Eq . (3.34), we can obtain the convective heat flux q^' as

where k is the thermal conductivity of the gas. The physical properties of the

gas may be substituted by those of air and approximated as follows:

(iii) Thermal conduction

The importance of the wall thermal conduction lies in the fact that both

the radiative and the convective heat transfer to the walls are related to the

wall surface temperatures. The calculation of the thermal conduction will

turn out to be very complicated if the effects of water content, thermal

property variation with temperature, decomposition and so forth, which are

commonplace in real fire, are taken into account. However, since little has

been considered in dealing with those problems, a one dimensional thermal

conduction equation for uniform thermal properties as follows is solved in

this model to obtain the wall temperature.

(3.36)

Pr = 0.72

(3.37)

46
Thermal conduction equation

1

'!

j

![

|

i

( 3 . 38 )

Boundary condition

x=0

— A M x=l qout

( 3 . 39 )

where k, c, p and Z are respectively the thermal conductivity, the specific

heat, the density and the thickness of the wall; qV and q" are the net in out

incident heat flux to the wall surface and the net heat loss flux from the back

of the wall.

47
3.4 Flow Through Opening

The primary tasks that have to be considered in the calculation of the

flows through openings of a structure are to describe the rate of the flows

as a function of the pressures, and to solve the pressures so that Eq . (2.19)

can be satisfied.

(i) Rate of flow through opening

The rates of the flows of hot gas and air are affected by the temperature

and the thickness of the upper layer and the pressure of each side of the

opening. Let's consider the case where each side of the opening has a hot

layer of arbitrary temperature and thickness as the most general case, which

is shown in Figure 3.9 .

The hydrostatic pressure drop with height of each room is also shown in

Figure 3.9 and it is obvious that there are three regions where the increase

rate of pressure difference between the rooms with height differ from one to

another; i.e., below Z . the pressure difference is kept constant, while it

a , l

varies proportionally to n - o . between Z . and Z . and to p . - P a

K s,i a,

a,

j

s,j s,i

above Z . . The pressures of the rooms at the height Z are described as

a »

follows

48
s

cu iH

•H

4-1OU

P-i

<uoa

<v

<u

•HP

<uu3 co

CO CO CO o

a)

u

u 2 PM <3

er\ co

60

•H

Pn

49

an

Opening
P

i

(z) =

P

i - p agZ (Z—Za,i >

P

i - P agZa,i

- p s,i

g(Z- Za,i ) < Z>Z.,1>

P (Z)

P

J - P agZ

P.

3 - P agZ

a, i

P s, j

g( z-z

d *

(z^j>

(Z>Za,j>

(3.40)

Before formulating the rate of the flows, it is useful to obtain the neutral

zone heights associated with the above pressure profiles. The first one is

obtained by equating the second and the third equation in Eq. (3.40) as follows

X

nas

P. - P.

_j (. g

( VP s,i)g a>i (3.41)

Likewise equating the second and the forth of the equation yields

X

nss

P ~ P

i

(p s,j-p s.i)g

S,1

p ,-p

S,J S,1

Z . + -

a, i p jlI

. - P s ,3 s,i a,

(3.42)

It should be noted that the densities and the pressures varies as the fire goes

on, so the pressure difference at an opening may take various profiles. All

the possible pressure difference profiles are covered in Figure 3.10 and the

rates of flow of hot gas and air for each case are shown in Table 3.1. It may

be clear that the symbols for the hot gas flow rates SS and SA and for the air

flow rates AS and AA are assigned to each different pressure difference profile,

and are convenient to formulate the rate of the flows. Also, it should be noted

that although the formulas in Table 3.1 are described using the neutral zone

heights, they are still functions of the pressures because the neutral zone

50
Table 3.1(a) Flow Rate of Hot Gas and Air

(PK .< P )

s,i s,j y

S3&;»

sAij = %-

M; =

AAj.i ~

jo/B»&&I Ssfcl {(HrXnxf2- MHtl Z..})-X*sfil

0. (HtlZ..,)

" 3°^^ (Hft, 2i^} (ma* fHs, Za, J

0. if H^i;J

0.

ASj.i — 0.

• 0.

0.

Za.i.^Xnas-^'i 5% » SSy

SAii

5

S/fyi.

AScj.

AS^

AAcy

Mp

i^hm,rh\ f(H W«/f

I

^MJf?r £ I

LA

a

0. 0. 0.

3 °^UJ^-% fa | fa” ?s, i | (OiwT^^fc/Ht)/"

a

f 1 yz/S’iTj-Pci

a

(Ht.>Z.,i)

(Ht<^)

(H<>XW 1 H,'

(Ht<Xw«rH,>2^)

(Hjt< 2xi)

Za./X,MAS

1 m

j

il

5S«;

55ji

SAj

%

A$i

AA;j

AAt

>

°.

0. 5/4ji = 0.

(Ht>Xnss)

(Ht^Xntt)

(Ht>Z». xHt<Xfl5i

er Hj2/'rl s)

= j BrS.; I fa,

df] (H«.>z,, 1 h,<^)

0. (Hi^Z* , 5Trii>Zij)

a

0<%(m;i.(H4/ ?4,;)-Hi i /zP»IVT:| (H»<2.,.)

0. 51 (H.iZ.„)
Tabic 3 1(b) Flow Rate of Hot Gas and Air

(P .> P •) s ,

s,j

Xr\*s^. 2a, u *5*.. 3 [(Hf/na)^ (i»X(Hj,Xm) Xwj) }

la

i

v — I %i &,j

}

5/11, Jfo( KfffiEPU {m(H^;)-x4

l o. SAji ~ 0-

ASij = ASji = <7.

A4iy

o^^n^Z^-Hi }/2 filft-Pj

A^jc= 0_

(Ht>X*„)

(Ht<x.„)

(H^Z* tH,<X»)

(HcA* jsH^Z.,,)

(Ht&, « tel:)

(Ht <Z.,.)

(H.2Z..J

2a,*XXrtfl3<iia,i ;-& t |

3 vj23)s,i § rSjlf (X,5i_

i« > n (HtyOjj

'(Hfi,?*,}) "Xius/” ltu^(^ttXMs)~XiaJ'

j

SAji= a

ASij ~ °- fOU-*«0VA!)A(X*** (Ht,Xn4

Mj = <?• M.. » j°<K f }j2&\?j -ft |

L o.

(Ht^Xno)

(H-fcSXnji)

& Kjt^Xub)

(H«<Z*)

(tUZ*;)

2 Xw

5S

i‘

ij — a.

. = &,r fti I SHcxJ*- Hw,i)-x»«r}

5Sii= „

._

(H(.>Z«,i)

SAj = Mji= 0.

ASij ~

t 0.

0 (Ht<Z.,»l4,2r,i) ASji=i

MJW= Mj = /2fJ?rp;

|

o. 52
53

Fig.

3.10

Conceivable

Pressure

Difference

Profiles

Across

an

Opening
heights are functions of the pressures as known from Eqs. (3.41) and (3.42).

(ii) Solution of pressure equation

The equation that the pressures in the flow rate formula in Table 3.1 must

satisfy is given by Eq . (2.19). Eq . (2.19) is actually simultaneous algebraic

equations each of which holds for a room in the structure and may involve any

of the room pressures. Since the real form of Eq. (2.19) is too complex and

lengthy to discuss the solution procedure, let's rewrite the equation functionally

as follows

where n denotes the total number of the rooms in the structure. In the real

f

x (P r

p

2

, ... , Pn

) = 0

(3.43)

configuration of a structure, only some of the pressures P^, ..., P^ are

actually involved in each equation of Eq. (3.43), however, the equation for

k-th room f (P) = 0 always includes the pressure of k-th room P^.

A primitive method to solve Eq . (3.43) is explained as follows. Let’s

consider first the case where n=l, then Eq . (3.43) is reduced to

(3.44)

54
Although still nonlinear, there is no substantial difficulty in solving this

equation numerically. In a numerical procedure, P^ may be changed iteratively

until A^, which is defined as follows, reaches a convergence criterion.

A

1 = f

l

(P

1

) (3.45)

The feature that is a monotonous decreasing function of P^ may be utilized

in devising some efficient numerical method. Noting that numerical solution of

the pressure equation for single variable pressure is never difficult, let's go

on to the case where n=2, i.e.,

f

l

(P

l»

P

2

} = 0 (3.46a)

f

2 (P r

P

2

) = 0 (3.46b)

Let's assume that P^ is solved by any means from Eq. (3.46a) as

P

x = f* (P

2

) (3.47)

then, this can be substituted into Eq. (3.46 b) to yield

f

2

(f*(P

2

), P

2

) = 0 (3.48)

This equation has only one variable pressure and can be solved as stated above.

Eq. (3.47) can not be obtained analytically. A numerical method can be applied

to Eq. (3.46a) to obtain P^ only when P

2 is fixed, however, it can be applied

whatever fixed value P

2 takes, which is practically equivalent to the existence

of analytical solution as given by Eq. (3.48). What is only needed is that P^

55
Fig.

3.12

Pressure

Calculation

Chart

for
should be renewed by solving Eq . (3.46a) whenever is changed on the process

of numerical solution of Eq. (3.48). This procedure may be made clearer by

the chart in Figure 3.11. Likewise, in case of n rooms, the solution procedure

is given by the chart in Figure 3.12. In a real structure, however, it is quite

rare that one room is connected with every other room in the structure. Looking

into the way the rooms are connected in a specific structure will provide more

efficient charts of the procedure. For instance, in case of the building shown

in Figure 3.13, Eq . (3.43) is reduced to

f

2

<P

2

, P

4

> - 0

f

3

(P

3

. P

4

) =0 (3.49)

f

4

(P 1’ P2> P

3* V = 0

Eq. (3.49) implies that any of P^, ¥ and P^ is the function of P^ alone, i.e..

P

1 “

*1 (P4>’ P

2 * f2 <V- P

3 = f3

(P4> (3.50)

so these pressure need to be renewed only when P^ is changed. The chart for

this procedure is also given in Figure 3.13. This kind of simplification can

be made for almost every real building and will help save the computation

time.

57
cn CM

58

Fig.

3.13

An

Explanatory

Building

Model

and

Its

Pressure

Calculation

Chart
III. PROGRAM

The theories of the model to predict the state and motion of the fire

induced hot gas were outlined in the previous sections. In this section,

the skeleton of the program to compute numerically the model is briefly

described and also the features of the subroutines included in the program

are summarized. A full listing of the program has not been included due

to its length, but may be acquired from the author at the Building Research

Institute, Japan.

4. Program Structure

The structure of the program is, as can be seen in Figure 4.1,

comparatively simple. Four of the five subroutines called from the main

program, namely PARAM, TRbP, DFIRE and WPRS, primarily deal with input and

pre-treatment of the data to prepare the necessary information for subsequent

computation

The essence in computation of this model is to solve the ordinary

differential equations given by Eqs. (2.16) - (2.18) and the algebraic equations

(2.19). RKG is a numeric subroutine to solve the differential equations

by Runge-Kutta Gill method together with DFNC, which yields the value of the

differential coefficients of the equations. Most subroutines referenced from

DFNC are related to physics and devoted to yield the value of the terms in the

59
s

I
right hand side of Eq. (2.16) - (2.18). Subroutine VENTL, together with

RFLOW, PSET and DRJET, solves the algebraic equations (2.19) and yields

the flow rates through openings.

5. Description of the Subprograms

In the following, brief descriptions on each subroutine of the present

code are given. These descriptions are organized into several components.

The task of the subroutines is stated, followed by some remarks which elaborate

on the nature of the subroutine. The symbols corresponding to the computer

coding and mathematical variables addressed in the subroutine are described.

The input and output variables are listed along with their relationship to

other subroutines. Lastly, the calculations made in the subroutine are listed.

In the input /output list in the descriptions, the following symbols respectively

are defined:

Data: Data Source

/AAAA/: Blocked common of which the name is AAAA

BBBB: Subroutine of which the name is BBBB respectively, when they

are listed under the heading From of To .

The readers may feel that the description about the calculation in each

subroutine is not sufficiently given, but this is so partly because the model

itself is truely so simple that redundant explanation would make its point

rather confusing.

Further details of the routine ABSORB can be found in ref. [7].

61
5.1 MAIN PROGRAM

(i) Task

To prepare the data and the initial condition for the model computation.

(ii) Remarks

The Main program primarily deals with the data input and initiation of

the computation. Universal constants and common data are directly given here

while the specific data for a calculation are given through subroutines

PARAM, TPROP DFIRE and WPRS , which are called from the Main program.

(iii) Symbols

Code Paper Description

TITLE Comments about the specific calculation

G cro

2 Gravitational acceleration (= 9.8 m/s )

SIGM 0

-11 2 Stefan-Boltzmann constant (= 1.356 x 10 kcal/m s

CP C

P

Specific heat of gas (= 0.25 kcal/kgK)

TA Ta Temperature (K) , 1 of ambience and

RA P a

_ . /, / 3. ( lower layer Density (kg/m ) j

RSOOT p soot

Soot density (= 2,000 kg/m^)

ALP a Flow coefficient

ALD a Flow coefficient

CR \ Fraction of air flow AS that mixes into upper layer

of a room

CF A Fraction of air flow AS that mixes into upper layer

of a shaft

62
YP(L) Mass fraction of species £ in volatile where £ is assigned as in 5.6 RKG

YAM(L) Mass fraction of species Z in air

QF (N, I) AH-n^ Heat release rate of gasified fuel of i-th room

on n-th floor (kcal/s)

QR(N,I)

QC(N,I)

q

r’ qg

Radiative

Convective

heat gain of the upper layer of

i-th room on n-th floor (kcal/s)

EPG (N, I) £ G Layer emissivity of i-th room on n-th floor

YO(L,N, I) Initial property of upper layer of i-th room on n-th floor where Z is assigned as in 5.6 RKG

PP(L,N,I) Initial partial pressure of species Z of i-th room

on n-th floor where £ is assigned as 5.6 RKG

P (N,NRMX2+I) Relative pressure of i-th outdoor space due to

wind (Pa)

H At Time interval for the computation (s)

NX Number of time steps of the computation

XO Initial time

w w Charring ratio of fuel

PM m Water content per unit fuel

NFLR, NFLR1

NRMX, NRMX2, NRMX4 }

See 5.2 PARAM

DDT, NTMAX See 5.4 DFIRE

PWIND1, PWIND2 , PWIND3 ,

\

) See 5.5 WPRS

PWIND4 J

SS(N,I,J,K), SA(N,I,J,K) \

AS (N, I, J ,K) , AA(N,I,J,K) l See 5.7 DFNC

SAD(N, I, J,K) ) EMP(N,I),EMS(N,I),EME(N,I) See 5.7 DFNC

63
(iv) Input

Input From Input From

TA Data W /FUEL/

DDT /FIRE/ PM /FUEL/

NTMAX /FIRE/ PWIND1 (1) /WIND/

NRMX /BULG/ PWIND2 (1) /WIND/

NRMX2 /BULG/ PWIND3 (1) /WIND/

NRMX4 /BULG/ PWIND4 (1) /WIND/

NFLR /BULG/

NFLR1 /BULG/

(v) Output

Output To Output To

G /UNIV/ YO(L,N,I) RKG

SIGM /UNIV/ H RKG

CP /UNIV/ NX RKG

TA /UNIV/ xo RKG

RA /UNIV/ NFLR RKG

RSOOT /UNIV/ NFLR1 RKG

ALP /UNIV/ NRMX RKG

ALD /UNIV/ NRMX2 RKG

YP(L) /CONS/ NRMX4 RKG

YAM(L) /CONS/ P(N,NRMX2+I) /PRES/

PP(L,N,I) /CONS/

64
(vi) Calculation

(a) Ambient density

a

(b) Mass fraction of fuel and 1^0

v

P . 1-w

f 1-w+m

Y

P

= m

1^0 1-w+m

65
5 . 2 PARAM

(i) Task

To input and pre-treat the data needed to set up the image of the

objective structure.

(ii) Remarks

Generally, four conditions may be mentioned as the major physical factors

that control fire behavior, namely, configuration and thermal condition of the

structure, outdoor weather, and nature of fire source. This subroutine deals

with the first matter, i.e., how to provide configuration data needed for the

computation.

(iii) Symbols

Code

N

I

J

K

NFLR

NEFLR

NSHFT

NROM

HFLR

Paper

n

i

j

k

Description

Floor number in the structure

Room number on each floor *1)

Room number on each floor

Opening number between each two rooms

Total number of floors in the structure

Number of specially designed floors

Number of shafts in the structure

Number of rooms on a commonly designed floor

Height of a commonly designed floor (m)

66
NEF(L) Floor number where 2.-th specially designed floor

is located

NEROM(L) Number of rooms on 2.-th specially designed floor

HEFLR(L) Height of 2,-th specially designed floor (m)

BRN(I) Vi Width

DRN(I) Vi Depth of i-th room on a commonly

designed floor (m)

HRN(I) Vi Height

BRE(L, I) BR,i Width

DRE(L, I) Vi Depth of i-th room on 2,-th specially

designed floor (m)

HRE(L,I) Vi Height

BRS(I) B

R 1 Width ) K, 1

of i-th shaft (m)

DRS(I) Vi Depth !

HRST(I) Top height i 0 f i-th shaft from a HRSB(I) Bottom height S reference (ground) level (m)

BWO Width of opening (m)

HHO Soffit height )> of opening from floor level

HLO Sill height )

IPR(I) The room number to which pressure calculation

proceeds after Eq. (2.19) is cleared for i-th

room (common floor)

IRE(I) The room number to which pressure calculation

returns when Eq. (2.19) is not cleared for

i-th room (common floor)

IPRE(I) Equivalent to IPR(I) (but for specially

designed floor)

IREE(I) Equivalent to IRE(I) (but for specially

designed floor)

NRMX The maximum number of the rooms on any floor

NRMX1 = NRMX + 1

NRMX2 E NRMX -+ NSHFT (number of the shafts)

NEX Floor number of a specially designed floor

67
NRMX3 = NRMX2 + 1

NRMX4 = NRMX2 + 4

NFLR1 = NFLR + 1

BR(N, I) Vi Width (m) \

DR(N, I) Vi Depth (m) I[of i-th room on n-th floor *2)

HR(N, I) Vi Height (m) 1

AR(N, I) Vi

2'

Floor area (m )

HRP (N, I) Ceiling height (m) ) . . , , f of i-th room on n-th floor

HRL(N, I)

, . , _ , N ( from a reference level Floor height (m) J

BW(N, I, J,K) BwWidth (m) of k-th opening between room i and j on n-th floor

HH(N, I, J,K) “h Soffit height (m) )- . . . , ( of k-th opening between room

HL(N, I, J,K) H

*

_ . i , , . , /x ( i and i from n-th floor level Sill height (m) J J

HHP (N, I , J, K) Soffit height (m) ) of k-th opening between room

> i and j from a reference (ground)

HLP (N, I, J,K) Sill height (m) j level

NW(N, I, J) Number of openings between room i and j on n-th floor

IPRG (N, I) The room number to which pressure calculation

proceeds after Eq. (2.19) is cleared for i-th

room on n-th floor

IRET (N, I) The room number to which pressure calculation

NRM(N)

NDMX(N)

returns when Eq. (2.19) is not cleared for

i-th room on n-th floor

Number of the rooms on n-th floor

Maximum number of the openings between every

combination of two rooms on n-th floor

*1) A shaft space must be numbered from NRMX+1; i.e., the smallest number that

is assigned to a shaft must be one larger than the maximum number of rooms

on any floor.

*2) As a concept, a shaft space is considered to be NFLR+1 times lapped over,

in other words, AR(N,I), HRP(N,I), HRL(N,I) etc. are given the same value

for any N (N=l, NFLR+1).

68
(iv) Input

Input From Input From

NFLR Data BRS(I) Data

NEILR Data HRST(I) Data

NSHFT Data HRSB(I) Data

NROM Data BWO Data

HFLR Data HHO Data

NEF(L) Data HLO Data

NEROM(L) Data IPR(I) Data

HEFLR(L) Data IRE(I) Data

BRN(I) Data IPRE(I) Data

DRN(I) Data IREE(I) Data

HRN(I) Data

BRE (L , I

)

Data

DRE(L, I) Data

HRE (L , I

)

Data

69
(v) Output

*

Output To Output To

NFLR /BULG/ HR(N, I) /ROOM/

NEFLR /BULG/ HRP (N,I) /ROOM/

NSHFT /BULG/ HLP (N,I) /ROOM/

NROM /BULG/ AR(N, I) /ROOM/

HFLR /BULG/ BW(N,I,J,K) /OPEN/

NFLR1 /BULG/ HH(N,I,J,K) /OPEN

/

NRMX /BULG/ HL(N,I,J,K) /OPEN/

NRMX1 /BULG/ HHP(N,I,J,K) /OPEN/

NRMX2 /BULG/ HLP (N,I, J,K) /OPEN/

NRMX3 /BULG/ NW(N,I, J) /OPEN/

NRMX4 /BULG/ NRM(N) /BULG/

NEF (L) /BULG

/

NDMX(N) /BULG/

NEROM(L) /BULG/ IPRG(N, I) /PRES/

HEFLR(L) /BULG/ IRET (N, I) /PRES/

BR(N, I) /ROOM/

DR(N, I) /ROOM/

70
(vi) Calculation

(a) Floor area of i-th room on n-th floor

AR(N,I) = BR(N, I) * DR(N, I)

(b) Ceiling height of i-th room on n-th floor from the ground level

n-1

HR(N,I) +

^

(n-th floor height) (room)

n=l

HRP(N,I) = / HRST(I)

999 .

(shaft)

(outdoor)

(c) Floor height of i-th room on n-th floor from the ground level

n-1

0 +

^ (n-th floor height) (room)

HRL (N , I ) = n=l

HRSB(I)

0 .

(shaft)

(outdoor)

(d) Height of openings from the ground level

n-1

HHP(N,I,J,K) = HH(N,I,J,K) + (n-th floor height) (soffit)

n=l

n-1

HLP (N , I , J , K) = HL(N,I,J,K) + (n-th floor height) (sill)

n=l

71
5 . 3 TPROP

(i) Task

To input the information on thermal conditions of the rooms in the

objective structure.

(ii) Remarks

Generally, the thermal properties of ceiling, floor and walls of a

room in a structure may be different from one to another, moreover, each of

the walls may consist of various materials. However, here the thermal

properties only for homogeneous ceiling and floor are input because the

current radiation model has assumed the system composed of only two surfaces,

and thermal conduction model has assumed homogeneous slabs.

(iii) Symbols

Code Paper Description

NDIV N Number of wall element slices fictitiously

assumed for numerical temperature computation

NDIV1 = NDIV + 1: number of points where internal

wall temperatures are calculated

FKWO (I , IW) k Thermal conductivity (kcal/msK)

CWO(I,IW) c Specific heat (kcal/kgK) of the wall of

i-th room on RWO (I , IW) P Density (kg/m ) commonly designed

floor

FLWO(I, IW) l Thickness (m) *2) /lW=l; ceiling \ *1

\IW=2 ; floor )

EPWO (I , IW) e Emissivity
FKW(N,I,IW) k

CW(N, I , IW) c

RW(N, I, IW) p

FLW(N,I,IW) £

EPW(N, I, IW) e

NFLR, NFLR1 , NRMX1, NRMX2

NROM, NEROM(L), NEF(L)

Thermal conductivity (kcal/msk)

Specific heat (kcal/kgK)

3 Density (kg/m )

Thickness (m) *2)

of the wall of

i-th room on n-th

floor

/ IW=1 ; ceiling

\

\ IW=2 ; floor )

Emissivity

See 5.2 PARAM

(iv) Input

Input From Input From

FKWO(I, IW) Data NDIV Data

CWO(I,IW) Data NFLR /BULG/

RWO(I,IW) Data NFLR1 /BULG/

FLWO(I.IW) Data NRMX /BULG/

EPWO(I, IW) Data NRMX1 /BULG/

FKW (NEF (L) , I , IW) Data NRMX2 /BULG/

CW (NEF (L) , I , IW) Data NEF(L) /BULG/

RW (NEF (L) , I , IW) Data NEROM(L) /BULG/

FLW(NEF (L) , I, IW) Data

EPW (NEF (L) , I , IW) Data

73
(v) Output

Output To

FKW(N,I,IW) /WALL/

CW(N, I , IW) /WALL/

RW(N, I, IW) /WALL/

Output To

NDIV /WALL/

FLW(N,I,IW) /WALL/

EPW(N,I,IW) /WALL/

*1) In this model, internal surface of a room, which consists of the ceiling,

the floor and the vertical walls, is divided into two parts, i.e., one

that is in contact with the upper layer and the other that is in contact

with the lower layer for the purpose of simplification of both radiative

and convective heat transfer calculation, and it is assumed that each

surface is all the time homogeneous, and represented by either the ceiling

or the floor. Consequently, when the thermal properties of the walls are significantly different, there is a doubt whether the heat transfer model

provides accurate enough predictions.

*2) So far as a fire of relatively short duration is considered, the wall

thickness here does not have to be actual thickness. Since it is somewhat

tedious to take account of heat flux to the back surface of the wall, this

model has assumed the wall thermally insulated on the back whose thickness

is a half of that of the actual wall.

74
5.4 DFIRE

(i) Task

To input the data on fire source and derive the necessary information

on heat and species generation due to the combustion of the fuel.

(ii) Remarks

In this subroutine, various information on fire source, e.g., location

of fire, chemical composition of fuel, fuel mass loss rate and some other

parameters are input. And using these input data, heat of combustion of

gasified fuel and rates of species generation are derived based on the combustion

model described previously in section 3.1.

(iii) Symbols

Code Paper Description

NFF Floor number where fire is located

IFR Room number of fire is located

LFP Parameter regarding fire location in the room

of origin (LFP =1, 2, and 4 represent center

wall and corner respectively

DDT At Time interval of data input on mass loss rate

area and height of fuel (s)

NTMAX Number of input data on mass loss rate, area

and height of fuel

WC w

c Mass fraction of C

WH W

H

Mass fraction of H /

\ in the dry fuel

WO w

o Mass fraction of 0 l

WN w

N Mass fraction of N

75
QEXP -AH Higher calorific value (enthalpy) of the fuel

exp

due to complete combustion (kcal/kg)

TE T

P

Mean temperature of pyrolysis (K)

QGFY

k

L

g

Heat of gasification (kcal/kg)

WMFL M

f Molecular weight of fuel (g/mol)

r (defaulted value: M^ = 50)

W w

Fraction of dry fuel that turns into char

S s Fraction of dry fuel that turns into soot

PM m Water content to unit dry fuel

ETA n Fraction of the carbon in gasified fuel that

results into CO

DHCO

* -AH Enthalpy difference between a reference state

r fuel and products of complete combustion (kcal/kg)

DHC1 -AH Enthalpy difference between a reference state

r and products of actual burning (kcal/kg)

DHA -AH , AH Heat of combustion (enthalpy) of gasified

a fuel (kcal/kg)

QP Q

P

Heat needed for unit mass loss (kcal/kg)

GAMA(3) Y

f

Fuel

GAMA (4) Y

s Soot

GAMA (5) V

°2 ^2 ( generation rate due to the combustion

GAMA (6) Y C0

2

( of unit mass of gasified fuel (kg/kg)

| GAMA(7) Y co CO

GAMA(8) YH 0H

2 0

GAMA(9)

>

N

2 BFIRED(I) m Mass loss rate (kg/s) \ P

2

J of fuel where i is

AFIRED(I) Area (m ) (not referenced) \ a data number numbered

l from time = 0

HFIRED(I) Height (m)

CP C

P

Specific heat of gas (kcal/kgK)

TA T Ambient temperature (k) a

76
(iv) Input

Input From Input From

NFF Data QEXP Data

IFR Data TE Data

LFP Data QGFY Data

DDT Data WMFL Data

NTMAX Data W Data

WC Data S Data

WH Data +

WO Data PM Data

WN Data ETA Data

CP /UNIV/

TA /UNIV/

(v) Output

Output To Output To

NFF /FIRE/ WMFL /FUEL/

IFR /FIRE/ W /FUEL/

LFP /FIRE/ S /FUEL/

DDT /FIRE/ PM /FUEL/

NTMAX /FIRE/ ETA /FUEL/

WC /FUEL/ DHA /FUEL/

WH /FUEL/ QP /FUEL/

WO /FUEL/ GAMA(L) /UNIV/

WN /FUEL/ BFIRED(I) /FIRE/

QEXP /FUEL/ AFIRED(I) /FIRE/

TE /FUEL/ HFIRED(I) /FIRE/

77
(vi) Calculation

(a) Enthalpy difference between the fuel at a reference state and the

products due to complete combustion

* W

c WH 3 -AH = (97.6 + 68.3 4r ) x 10

r 12 2

(b) Enthalpy difference between the fuel at a reference state and the

products due to incomplete combustion

W -w- S W

-AH

r - [{97.6 (1— n) + 27.6a) + 57.6 ] x 10

3

(c) Actual heat of combustion of gasified fuel

* *. -AH = -AIL -{-AH - (-AH ) - L } a ^ r exp g

(d) The heat needed for unit mass loss

0. 32W(773-T ) + L + 580m

Q

= C (T - T ) + S

p p p a 1 - w + m

(e) Mass generation rate of species

Y

f

* 1-w

= -{(1 - W -w-s W - W,_ 00

n_v C H _ 1_ _0 , 32

2 12 4 2 16 1 1-w

78
W -w-s

Y C0

2

= (1"n) 12

W

c-w-S 28

YC0 n

12 1-w

_ 1 TT 18

YH

2

0 2

W

H 1-w

r 5l

N„ 1-w

44

1-w

79
5 . 5 WPRS

(i) Task

To yield the data on outdoor pressure due to wind.

(ii) Remarks

This subroutine inputs outdoor wind velocity and wind pressure

coefficient for each of the four external walls of a structure and

calculates the pressures of outdoor spaces.

(iii) Symbols

Code Paper Description

NWMAX Number of input data

DWT At Data input interval (s)

WV(J) V j-th input value of wind velocity (m/s)

COEW(I) C Wind pressure coefficient for i-th

outdoor space

PWINDl(J) p

i 1-st 1

PWIND2 ( J) P

2

2 -nd

( outdoor space pressure due

1 to wind (P ) *)

PWIND3 ( J

)

P

3

3-rd i a

PWIND4 ( J) P

4

4-th

RA P,

3 Ambient density (kg/m )

80
(iv) Input

Input From Input From

NWMAX Data WV(J) Data

DWT Data COEW(I) Data

RA /UNIV/

(v) Output

Output To Output To

PWINDl(J) /WIND/ NWMAX /WIND/

PWIND2 (J) /WIND/ DWT /WIND/

PWIND3(J) /WIND/

PWIND4 ( J) /WIND/

(vi) Calculation

P <=C. 1 p v

2 (1=1,4)

i 1

2 a

*) Elsewhere the four outdoor spaces are numbered following maximum number

of indoor spaces on any floor, including shaft spaces, in other words,

their number begins with NRMX2+1. But in this subroutine, renumber them

from one to four in numerical order of the spaces.

81
5 . 6 RKG

(i) Task

To solve the ordinary differential equation for temperature, thickness

and species concentration of every upper layer given by Eqs. (2.16) - (2.18)

with Runge-Kutta-Gill method.

(ii) Remarks

One of the two main strategies of numerical calculation of this model

is in short to integrate the ordinary differential equations for the temperature, the thickness and the species concentrations of the upper layers. In

this program, Runge-Kutta-Gill method is picked up among many available methods.

(iii) Symbols

Code Paper

XO

YO(L)

H

NX

IX

X

Description

Initial time

Initial value of upper layer properties

Time interval (s)

Number of the time steps

Time step number

Time (s)

82
Upper layer properties of i-th room

on n-th floor.

YW(L,N,I)

Y1(L,N,I)

Y2 (L,N, I)

Y3(L,N, I)

FRKG(L,N,I)

L

dT / dt

s dZ /dt

s dVdt

l

NFLR, NFLR1

NRMX, NRMX2, NRMX4

The value of differential coefficient

Number assigned to each upper layer

property as follows

1: Temperature of upper layer

2 : Thickness of upper layer

3 Fuel

4 Soot

mass fraction of

upper layer

H2°

See 5.2 PARAM

W(L,N,I)

j R1 (L,N, I) , R2 (L,N, I) , R3(L,N,I), R4(L,N,I) j Work variables

Q1((L,N,I), Q2(L,N,I), Q3(L,N,I), Q4(L,N,I) )

83
(iv) Input

Input From Input From

XO MAIN NX MAIN

YO MAIN IX MAIN

H MAIN FRKG(L,N,I) DFNC

NX MAIN

(v) Output

Output To Output To

YW(L,N, I) DFNC H DFNC

Y1 (L,N, I) DFNC X DFNC

Y2 (L,N, I) DFNC

Y3 (L ,N, I) DFNC

(v) Calculation

Since the scheme used here, Runge-Kutta Gill method, is one of the most

popular ones for solving ordinary differential equations, the description

is omitted. The readers who are interested in this scheme are advised to

reference some appropriate documents, or coded subroutine RKG itself.

84
5 . 7 DFNC

(i) Task

To calculate the value of the differential coefficients given by the

right hand side of Eqs. (2.16) - (2.18).

(ii) Remarks

The value of the differential coefficients at a given time can be

calculated by substituting into each term in the right side of Eqs. (2.16) -

(2.18) its value at the time. But the way each term relates to the layer

properties is so complex that some subroutines are needed to yield its value.

Consequently, physics subroutines are called before the value of the differential

coefficients are calculated.

(iii) Symbols

Code Paper Description

Y(L,N,I) T , Z , T^ Upper layer property of i-th room on n-th floor

that is assigned as follows according to L

Y(L,N,I) =

L = 1

L = 2

L = 3

L = 4

L = 5

L = 6

L = 7

L = 8

L = 9

Temperature (K)

Thickness (m)

Fuel

Soot

0 r2

CO,

CO

h

2 o

mass fraction

N„

85
TIME

DT

EMP (NFF, IFR)

EMS(NFF.IFR)

m

m

EME(NFF.IFR) m

HFEL h

HPLM Z

QPLM Q

P (N, I) P.

a

AW (1) A

TW (1) T

] EP(1) e

3 QRAD(l) Q1

QSRAD(l)

AW(2 )

TW(2 )

EP (2)

QRAD(2)

QSRAD(2)

QCV q

c QSCV

QR(N,I) -Q

( QC(N,I) Q

( TWJ (N, I ,IW,K) T

Q

l

/A

l

Q

2 /A

2

Elapsed time from the start of fire (sec)

Time interval (s)

Mass loss rate of fire source in the room

of origin (kg/s)

Rate of gas flow that enters the upper layer

through fire plume (kg/s)

Air entrainment rate into the fire plume

(m = m - m ) (kg/s)

e s p

Fire source height from the floor level (m)

Distance between fire source and discontinuity (m)

Heat release rate of fire source (kg/s)

Relative pressure of i-th room on n-th floor

at reference level (P ) a

the wall

that contact

with upper

layer

Total area (m ) of

Surface temperature (K) of

Surface emissivity of

Total radiative heat (kcal/s) to

2 Radiative heat flux (kcal/m s) to

2 Total area (m ) of

Surface temperature (K) of

Surface emissivity of \ the wall that

contact with

Total radiative heat (kcal/s) to | lower layer

2 Radiative heat flux (kcal/m s)

Total convective heat transfer to wall (kcal/s)

2 Convective heat flux to wall (kcal/m s)

Radiative heat gain

Convective heat gain !

of up

of in-th

upper layer (kcal/s)

th room on floor

Temperature of the wall (IW = 1: ceiling,

IW = 2 : floor) of i-th room on n-th floor

at k-th interior point

86
QRADW(N, I, IW)

Q SRADW (N , I , IW)

QCNV(N,I, IW)

QSCNV(N,I,IW)

Total radiative heat (kcal/s) -

2 Radiative heat flux (kcal/m s

Total convective heat (kcal/s) 1 O Convective heat flux (kcal/m^s)

1 to the wall

I of i-th room

r on n-th floor

^ where IW=1,

i 2 respectively

| denote ceiling

and surface

VGAS V

3

Upper layer volume (m ) AGAS A

2 Entire boundary area of upper layer (m )

PATH L

m

Upper layer mean path length (m)

TEGAS Temperature defined as

300. (Y(1,N,I)<_ 300.)

TEGAS = j Y (1 ,N, I) (300<Y (1 ,N, I) < 2000.)

2000. (2000. £ Y(1,N,I)

RSOOT p soot

Soot density (kg/m )

FV f

V

Volume fraction of soot

SOOTK k

soot Absorption coefficient of

of 0.94 y® (m”l)

soot at a wavelength

EPG (N, I) £

G Emissivity of upper layer

n-th floor

of i-th room on

PP(4,N,I) = SOOTK of the upper layer of i-th room on n-th floor

PP(L,N,I) Partial pressure of the gases in the upper

layer of i-th room on n-th floor where L is

assigned as in Y(L,N,I)

SS(N,I,J,K) ss 4 . ij

Hot gas flow rate above ii the discontinuity

of j-th room through

SA(N,I,J,K) SA. . ij

Hot gas flow rate below f k-th opening between

} i-th and j-th room

AS(N,I, J,K) AS. . ij

Air flow rate above l on n-th floor

(kg/s)

AA(N,I,J,K) “ij Air flow rate below

SAD(N,I,J,K) SAh

Flow rate of air that enters the upper layer of

j-th room through a door jet SA^ (kg/s)

YAM(L) Mass fraction of species i in the ambient air

YP(L) Y? Mass fraction of species i in the fuel volatile

87
BURF(N.I) of gasified fuel in the

upper layer of i-th room

on n-th floor (kg/s)

Burning rate

QF (N, I) AH*m^ Heat release

CP,TA,RA,CR,CF See 5.1 MAIN

NFLR,NFLR1 ,NRM(N) ,NW(N, I, J)

NRMX , NRMX1 , NRMX2

,

NRMX3 , NRMX4 See 5.2 PARAM

HR(N, I) , HRP (N, I)

,

HRL(N, I)

BR(N, I) ,DR(N, I) , AR(N, I)

BFIRED,DDT,NTMAX

DHA,QP,TE,GAMA(L) ,WMFEL ] See 5.4 DFIRE

PWIND1 , PWIND2 , PWIND3 , PWIND4

] DWT , NWMAX J

See 5.5 WPRS

FRKG (L,N, I) See 5.6 RKG

(iv) Input

Input From Input

Y(L,N,I) RKG NFLR

TIME RKG NFLR1

DT RKG NRMX

BFIRED /FIRE/ NRMX1

HFIRED /FIRE/ NRMX2

DDT /FIRE/ NRMX4

NTMAX /FIRE/ NRM(N)

NFF /FIRE/ HR(N,I)

IFR /FIRE/ HRP (N , I)

LFP /FIRE/ HRL(N,I)

PWIND1 /WIND/ BR(N, I)

PWIND2 /WIND/

From

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

/BULG/

88
PWIND3 /WIND/ DR(N, I) /BULG/

PWIND4 /WIND/ AR(N, I) /BULG/

DWT /WIND/ NW (N, I , J) /OPEN/

NWMAX /WIND/ CP /UNIV/

SS (N, I, J,K) /FLOW/ TA /UNIV/

SA(N, I, J,K) /FLOW/ RA /UNIV/

AS(N, I, J,K) /FLOW/ RSOOT /UNIV/

AA(N,I,J,K) /FLOW/ DHA /FIRE/

SAD(N,I, J,K) /FLOW/ QP /FIRE/

YAM(L) /CONS/ TE /FIRE/

YP(L) /CONS/ GAMA(L) /FIRE/

EMP (NFF,IFR) INTPOL WMFL /FIRE/

EMS (NFF,IFR) FPLUM TWJ (N,I , IW,K) /WALL/

EME(NFF, IFR) FPLUM EPW(N,I,IW) /WALL/

EPG (N,I) ABSORB BURF (N, I) /QGAS/

QR(N,I) HTRAD4 QF(N,I) /QGAS/

QC(N,I) HTCNV1 P(1,NRMX2+I) INTPOL

CR /UNIV/ QRAD(IW) HTRAD4

CF /UNIV/ QSAD(IW) HTRAD4

QCV(IW) HTCNV1

QSCV(IW) HTCNV2

89
(v) Output

Output

FRKG (L,N, I)

TIME

BFIRED

HFIRED

DDT

NTMAX

PWIND1

PWIND2

PWIND3

PWIND4

DWT

NWMAX

QSRADW (N, I , IW)

QSCNV (N, I , IW)

DT

EMP (N, I)

EMS (N, I)

EME(N, I)

To

RKG

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

INTPOL

CNDCT

CNDCT

CNDCT

FPLUM, /PLUM/

/PLUM/

/PLUM/

Output

EMP (NFI , IFR)

QPLM

HPLM

TEGAS

PATH

SOOTK

Y(6,N, I)

Y (8,N,I)

PF(L,N,I)

TW(IW)

AW(IW)

EP (IW)

Y(1,N,I)

QR(N,I)

QC(N,I)

Y(L,N,I)

To

FPLUM

FPLUM

FPLUM

ABSORB

ABSORB

ABSORB

ABSORB

ABSORB

/CONS/

HTRAD4, HTCHV1

HTRAD4, HTCNV1

HTRAD4

HTRAD4, HTCNV1

/QGAS/

/QGAS/

VENTL

90
(vi) Calculation

4

(a) Mass loss rate

m

p = INTPOL(t)

(b) Outdoor space pressure

PNRMX2+i * INTP0L(C)

(c) Fire plume entrainment

Q = (AH -

a Q + C (T - P P

T )} a m Heat release rate

Distance between discontinuity and source

m

s

m

e

= FPLUM(Q, Z, m

p

) Rate of plume flow and air entrainment

(d) Emissivity of upper layer

A

1 - *R

+ 2(BR

+ V Z

S

Total wall area that contact with upper layer

A - + A

1 v '¥.

Entire boundary area of upper layer

Upper layer volume

L = m

Mean path length

91
I

P p T Y

s a a soot

f = y = • v p soot T p soot s soot

Volume fraction of soot

Ifv SOOt 0.94 x 10

6

Soot absorption coefficient

Y Y Y Y

. °2

, C°2, Y

C0

, H

2°

, N

2 , Y

f

b

32 44 28 18 28 M„

C0

2 \ 44

/S CO^ partial pressure

. .LV

H

2 0 \

18

7s H

2 0 partial pressure

e G

- ABSORB (T

s

, L

ffi , k^, , P^) Hot gas layer emissivity

(e) Radiation heat transfer

A

1

- *R + 2(B

R + VZ

S

A

2 = A

R

+ 2(BR + D

r

)(HRP - HRL - Z

g

) Total wall area that contact with

lower layer

•1 VA

i

q

2

/a

2

HTRAD4(T

1

, T2> T,, A

2

, A

R>

Eg)

92
(f) Convective heat transfer

q c = HTCNV1 (T , Tr

A

x

)

(g) Thermal conduction

TWJ = CNDCT (q^n , ^out^Wall temperature

(h) Rate of opening flow

SS. . ij

SA

ij

AS

ii

AA

ij > = RFLOW(Y)

SA

ij

“b

93
I

(i) Differential coefficient

dT T

s s dt VaTaV~ <Qp +

Qr + Qc " 9

P V

+

p a

Ta\Z

S [£ {(Ts,r

T ) (SS . ,+SA. . ) + (T -T ) (SA! .-SA. .+AAS. .)}

s ji 31 a s 31 31 31

+ (T -T )m + (T -T )(m -m )1

p s p ass p J

dZ

dt C p T A

(QF + QR + Q

C Q

p

m

p

) p a a R

P aT

aL

Tl

ElT . (SS..+SA..) - T_, (SS . .+SA. . ) + T (SA! . -SA . .+AAS . . )

}

s,3 31 31 S 13 13 a 31 31 31 »

+ T m + T (m - m ) 1 P P as p J

dt P T

a a [E{CYlij

-Y

t )(SS. i+SA..) + CYa-Y

£

)(SA’

1

-SA..«AS.

i)f

+ y A + (YrV*P + (YrV (VV]

in which

summation j and k (when i represents a room)

taken with respect to n, 3 and k (when i represents a shaft)

94
5.8 INTPOL

(i) Task

To interpolate the input data for a given time.

(ii) Remarks

This subroutine is used to obtain mass loss rate, fire source height

and outdoor pressure for a given time by interpolating their data which have

been input at every constant time interval.

(iii) Symbols

Code Paper Description

TIME t Elapsed time after the start of fire (s)

DLTT At Time interval (constant) of data input (s)

DSRCE(I) d

i

Value of i-th input datum (i=l corresponds

to t=0)

Til t i+l

Time at which (i+l)-th datum is located

(= At x i)

TI c

i

Time at which i-th datum is located

(= At x (i-1)

)

NMAX nmax

Number of input data

TMAX t

max

Time at which the last input datum is

located

QATT d

t Interpolated value of data at a given

time t

95
(iv) Input

Input From Input From

TIME DFNC NMAX DFNC

DLTT DFNC DSRCE DFNC

(v) Output

Output To

DATT DFNC

(vi) Calculation

(a) When t < tmax

d

t

(t i+i

- t) d

i

~ (t

i - r > d i+i

At

(b) When t < t max

d

t nmax

96
5.9 FPLUM

(i) Task

To calculate the mass flow rate of plume gas that enters the hot upper

layer of the room of origin.

(ii) Remarks

This subroutine calculates the mass flow rate of plume gas that

enters the upper layer. The equations here were presented by McCaffrey

(NBS) based on his experimental analysis of burner fire.

(iii) Symbols

Code Paper

z Z

QFPLUM Q

R m

P

FMZ m

s

EMZ

•m

e LFP

(iv) Input

Description

Distance between discontinuity and fire

source (see 5.7 DFNC)

Heat release rate (kcal/s)

Mass loss rate (kg/s)

Flow rate of plume gas that enters upper

layer (kg/s)

Air entrainment into the plume (kg/s)

Fire location parameter

Input From Input From

QFPLM DFNC Z DFNC

LFP DFNC R DFNC

97
(v) Output

Ou tput To Output

FMZ DFNC EMZ

(vi) Calculation

Q

= 4.187 Q

• LFP

0.566

0.011 (^ 275 ) /LFP

0.909

0.026 (t|ts

) /LFP

Q

'

7 1.895

0.124

(^f/ 5

) /LFP

(0 <

(0.08

( 0.2

m = m - m

e s p

To

DFNC

.08)

< 0 . 2 )

98
5.10 DRJET

(i) Task

To calculate mass flow rate of doorjet gas that enters the hot upper

layer.

(ii) Remarks

The nature of buoyant doorjet entrainment has not yet been fully

clarified, nevertheless we need some means to estimate the entrainment rate.

So this model is devised based on the crude assumption that doorjet behaves

as if it were a vertical thermal plume rising up from the doorjet level, which

means the same equations as for fire plume entrainment are used only by replacing

the heat release rate of fire source with the enthalpy of the doorjet. It

may be more desirable to take account of heat release of combustion when the

doorjet flow contains fuel. but this is ignored for its complexity.

(iii) Symbols

Code Paper Description

Z Z Distance between discontinuity and doorjet

TJ T

s,j Doorjet temperature, which is the same as upper

layer temperature of j-th room (K)

SA SV Opening flow rate of hot gas below the discontinuity

of i-th room

FMZ sA

'i

Flow rate of doorjet gas that enters the upper

layer of i-th room (kg/s)

CP c

p

Specific heat of gas at constant pressure (kcal/kgK)

TA T

a Ambient temperature (K)

QJ Q Door jet enthalpy (kW)

99
(iv) Input

Input From Input From

T.I VENTL Z VENTL

SA VENTL CP /UNIV/

TA /UNIV/

(v) Output

Output To

FMZ VENTL

(vi) Calculation

Q

= 4. 187 C SA. . (T

P J 1 s,j

T ) a

i y

/0 - 1.0011

J

566

JL_)

1.026 /

1/0.909

± \ 1/1.895

0.124

J

Z (0 < ~~ < 0.08)

Q‘

z (0.08 < < 0.20)

( 0.20

100
0.011 ( < 0.08)

Z+Z 0.566

2. ) Q

2/5

Z+Z

(0 < -

2/5

SA^__ Z+Z 0.909

0.026 ( ) Q

'

Z+Z 1.895

0.124 ( -s-t-7?- ) ’2/5

Z+Z

(0.08 < -r^ < 0.2)

Q

'

Z+Z

( 0.2 < Q

;

101
5.11 ABSORB (Code by Ashok T. Modak)

(i) Task

To compute the emissivity of hot upper layer.

(ii) Remarks

This subroutine, which is a Modak* s creation, together with subroutines

ASYMP, CHEBY, DLECK, EGAS, PENTA, SOOT and SCRTCH, computes the absorptivity

of isothermal and homogeneous mixture of soot, CO^ and H^O at a total pressure

of 1 atm and a temperature 300 ^ 2,000 K [7].

The gas emissivity generally differs from its absorptivity when

temperature of radiation source is not the same as that of gas, however,

the difference between the two is not so significant in many practical fire

situations. Accordingly, the gas temperature is simply input in place of

external radiation source temperature to obtain the gas emissivity.

102
(iii) Symbols (arguments only)

Code Paper Description

TS Blackbody source temperature (K)

T T

s Mixture temperature (K)

PATH L

m

Mixture path length (m)

S00TK k

soot Absorption coefficient of soot at a wave

length of 0.94 Jim (m-1

)

PC02 p „ C0„ . , „ co_ 2 | partial pressure whose

PH20 P„ ~

[ total pressure is 1 atm

h

2 o Ho 0 ’ 2

ALPHA CG Absorptivity (emissivity) of the mixture

(iv) Input

Input From Input From

T DFNC PC02 DFNC

PATH DFNC PH20 DFNC

SOOTK DFNC

(v) Output

Output To

ALPHA £

G

(vi) Calculation

See ref. [7] , [8]

103
5.12 HTRAD4

(i) Task

To calculate radiation heat transfer to the walls in a room that

contains a hot upper layer.

(ii) Remarks

This subroutine computes the radiation heat transfer to the internal

room surface assuming that the surface is divided into two uniform surfaces,

i.e., one that is in contact with upper layer and the other that is in

contact with lower layer. The two internal surfaces and hot gas layers

are assumed gray.

(iii) Symbols

Code Paper Description

I i Wall surface number (i=l,2 respectively

represents the wall that is in contact

with upper and lower layer)

T (I) T.X

Surface temperature (K)

AW (I) A.

1

2 Surface area (m ) of wall i

EP (I) ei Emissivity

AD A

d

Area of discontinuity (the same as the

floor area) (m^)

EG £G Upper layer emissivity

TG V T

s Upper layer temperature (K)

QRAD(I) Total radiative heat transfer to wall i

(kcal/s)

104
QSRAD(I) -VA

i

QRADG -v «

Fll F

11

FI 2 v

12

F21 F

21

F22 F

22

F2G F

2G

SIGM o

Radiative heat flux to wall i (kcal/s)

Radiative heat gain of upper layer (kcal/s)

View factor between the surfaces

Stefan-Boltzmann constant

(iv) Input

Input From Input From

T (I) DFNC EG DFNC

AW(I) DFNC TG DFNC

EP (I) DFNC SIGM /UNIV/

AD DFNC

(v) Output

Output To Output To

QRAD(I) DFNC QRADG DFNC

ASRAD(I) DFNC
(vi) Calculation

View factors

ii

i-H i

—

(Jh 1-AjA.

d

F

12

* VA

i

F

21 A

d

/A

2

F

22

1-A

d

/A,

F

2G

= A

d

/ A

2

(b) Radiative heat transfer

Q

l

/A

l = " £ 1

P

1

/D

Q

2

/ a

2 = “ e

2 P

2

/,D

qg -«1 + v

where

D = {l-(l-

£l

)(l-eG )F11 }{l-(l-e

2 )F22 }-(l-ei )(l-e 2 )(l-eG) 2 Fl 2 F

21

P

x = [{l-(l-e

G

)F11 Hl-(l-£

2 )F22 }-(l- £

2 )(l-£

2 )(l-£G ) 2F

12

F

21 ]aT4

-(l-e

G )Fi2 e 2 aT2“*- 1+(- 1_£:

2') ^ (- 1_£

G')F12

F

2G_F

22 ^ £ G

aT

G

P

2 = r^-(1 " £

1 )(1_£G

) F

11

Hl-F22 )-(l-e

1

)(l-e

G

) F

12

F21^ aT

2 - (1-VF2lVTr[{1- (1-e

i )(1 -e G

)Fll }F2G+(1-e i )(1-£

G

)F21

]e

G

aT

G

106
5.13 HTCNV1

(i) Task

To calculate convective heat transfer to the walls that are in contact

with upper layer.

(ii) Remarks

This subroutine includes the formulas for convective heat transfer for

vertical wall, ceiling and floor. However, only the heat transfer to the

wall that is in contact with hot upper layer is computed using the formula

for ceiling because the heat transfer to lower layer has been ignored in the

current model.

The gas properties associated with the heat transfer are substituted

by those of air. The formulas for Nusselt number were quoted from the

results for natural convection, which may not be fairly adequate in the

presence of turbulence caused by fire plume and/or doorjet.

(iii) Symbols

Code Paper Description

TG T , T Gas temperature (K)

Or 5

AW

TW T

w

L l Reference length of heat transfer (m)

107
of the gas

V V

A a

LMD k

G 8

PR Pr

GR Gr

NU Nu

ACC h

QSCNV

QCNV %

(iv) Input

Input From

TG DFNC

TW DFNC

AW DFNC

(v) Output

Output To

QSCNV DFNC

2 Kinematic viscosity (m /s)

2 l Thermal diffusivity (m /s) /

Thermal conductivity (kcal/msK)j

2 Gravitational acceleration (m/s )

Prandtl number (Ev/a)

Gashof number (Eg£^(TGW-T )/v^T ) G

Mean nusselt number (Eh£/k)

2 Mean heat transfer coefficient (kcal/m sK)

2 Convective heat flux to the wall (kcal/m s)

Rate of convective heat transfer to the

wall (kcal/s)

Input

G

From

/UNIV/

Output To

QCNV DFNC
(vi) Calculation

(a) Reference length of heat transfer

* = ^

(b) Gas properties

v = 7.18 x 10

-10

k = 6.50 x 10

-8

0^)'

P = 0.72

G = s* CVV

v

2

T„

(c) Nusselt number for a ceiling and heat transfer

0.054 (G • P ) 0,38

(T > T)

r r GW

Nu =

0.0029 (G • P ) 0,38

(T < Tri ) r r GW

h -tNn u

<c

= h(T

G

- V

q = q" A

c c w

109
5.14 CNDCT

(i) Task

To obtain surface temperature of a wall by solving thermal conduction

equation for the wall.

(ii) Remarks

This subroutine solves one dimensional (inward direction) thermal

conduction equation by Gauss-Seidel iteration scheme. Temperature dependence

of thermal properties and moisture content of a wall are neglected for simplicity,

and only a wall of uniform material is considered.

(iii) Symbols

Code Paper Description

NL, NDIV N Number of wall element slice fictitiously

assumed for numerical temperature computation

NL1 Number of points where internal wall temperature

are calculated (=N+1)

FKW (N, I , L)

FLW(N,I,L)

CW (N, I ,L)

RW(N, I ,L)

k

a

c

p

Thermal conductivity (kcal/msK)

Thickness (m)

Specific heat (kcal/kgK)

3 Density (kg/m )

of the wall of

i-th room on n-th floor

(L=l: ceiling

L=2 : floor)

TWJ (N, I,L,K) T Temperature of £-th wall of i-th room on n-th floor at k-th internal point

110
TJ (K)

TJ1(K)

QDIN

QDOUT

ACR

DT

DX

K

(iv) Input

k.j

k,j+l

l in

out

At

Ax

n

k

j

Temperature of k-th internal point at

preceding time step (K)

Temperature of k-th internal point at

current time step (K)

Heat flux to the front surface of the

wall (kcal/m s)

Heat loss flux from the back surface of

the wall (kcal/m^s) *1)

Conversion criteria

Time interval (s)

Distance between internal points where

temperature is calculated (m)

Iteration step number

Internal point number where temperature

is calculated

Number to distinguish the current time step

from the preceding step

Input From

FKW(N,I,L) /WALL/

flw(n,i,l) /WALL/

cw(n,i,l) /WALL/

RW(N,I,L) /WALL/

twj(n,i,l,k) /WALL/

(v) Output

Output To

twj(n,i»l,k) /WALL/

Input From

NDIV /WALL/

QDIN DFNC

QDOUT DFNC

DT DFNC

111
(vi) Calculation

(a) Constants

Ax = N

r = At

(4r>

(Ax)

2 cp

(b) Temperature of internal points

n+1

o,j+l 1+r

Z- (Tn + ^q" )

+r ^l,j+l ^ k

q in)

b

1+r

rn+l _ k,j+l 2 (1+r

)

n+1

N, j+1 1+r

— (

n + T

n+^ ^

1+r) I k+l,j+l k-1, j+1 I

£_ (T

n+1 . \ + bid

+r

y

N-l,j+l k

4out / 1+r

b, . +

k*J

1+r

where superscript n denote iteration step number and

b . = T

o,3

b. . - T. . +

k, j k ,

2

. + r ( T . - T . + — qV \

0,3

y

1.1 o »3 k mJ

( T. , , . - 2T. . + T, . J y

k+1 ,3 k, 3 k-1,

3

y

= T + r IT -T - — q" i f,j N, j y

N-l,j N,j k

qout

y

(front surface k=l)

(internal point)

(back surface k=N+l)

c l) In this program, q" has been taken as 0 assuming t is so short that i?aoes not raise the temperature othat a fire duration

temperature of the back surface

of the wall.

112
5.15 VENTL

(i) Task

To solve the pressure equation given by Eq. (2.19) to yield the flows

through openings.

(ii) Remarks

This subroutine primarily deals with the solution of the algebraic

equation given by (2.19) together with subroutines RFLOW, PSET, and DRJET.

The calculation is executed iteratively changing the pressures until a

convergence criterion is cleared for every room in the objective structure.

And once Eq. (2.19) is solved for every pressure, the solution of the opening

flows throughout the structure and the burning rate of gasified fuel are

automatically obtained.

(iii)

Code Paper Description

ACR Convergence criterion

ACR2 Convergence criterion *1)

KA(N, I) Work variable

M(N,I) Work variable to count the iteration step

ZD Z Distance between discontinuity and

door jet (m)

XNAS Xnas Neutral zone height at the opening

*1) A convergence criterion to ease the criterion that is given by ACR when

convergence for a shaft has not been obtained within certain number of

iteration steps

113
LCNTL

YLCNT (N, I)

BURF (N, I)

P(N,I) P.

SGQ (N, I) A

±

Y (L,N, I) , YAM(L) , YP (L)

SS(N,I,J,K) ,SA(N,I,J,K)

AS(N,I,J,K) ,AA(N,I, J,K),SAD(N,I,J,K)

QF(N, I) ,QR(N, I) ,QC (N, I)EMP (N, I) ,EMS(N,

NFLR , NFLR1 , NSHFT , NW (N , I , J) ,NRM(N)

NRMX2 , NRMX4

HHP (N, I, J,K) , HLP (N , I , J , K) , HRP (N , I

)

IPRG (N, I )

, IRET (N, I)

CP, TA, RA

DHA, GAMA(L) , TE

The number of species that control

burning rate

(

3: fuel

5 : oxygen

Mass fraction of burning control species

of the upper layer of i-th room on n-th

floor *2)

Burning rate of gasified fuel in the

upper layer of i-th room on n-th floor

(kg/s)

Relative pressure of i-th room on n-th

floor at a reference level (Pa)

The value of pressure equation for guess

pressures

See 5.7 DFNC

See 5.2 PARAM

See 5.1 MAIN

See 5.4 DFIRE

*2) This variable equals Y (LCNTL, N, I) for Y (LCNTL, N, I) >_ 0 and equals 0

for Y(LCNTL,N,I) < 0. This is introduced to avoid the concentration

o-f species that controls burning rate from taking negative value for

numerical reason.

114
(iv) Input

Input From

Y(L,N,I) DFNC

YAM(L) /CONS/

YP(L) /CONS/

SS(N,I,J,K) /FLOW/

sa(n,i,j,k) /FLOW/

AS (N, I , J ,K) /FLOW/

AA(N,I,J,K) /FLOW/

sad(n,i,j,k) DRJET

QR(N,I) /QGAS/

QC (N,I) /QGAS/

EMP (N, I) /PLUM/

EMS (N, I) /PLUM/

CP /UNIV/

TA /UNIV/

XNAS RFLOW

(v) Output

Output To

KA(N, I) PSET

M(N, I) PSET

SGQ(N, I) PSET

P(N,I) PSET, RFLOW

Y(L,N,I) RFLOW

Y (1 ,N, J) DRJET

Input From

HHP(N,I,J,K) /OPEN/

HRP (N, I) /ROOM/

HRL(N,I) /ROOM/

NRM(N) /BLDG/

NW(N,I , J) /OPEN/

IPRG(N,I) /PRES/

IRET(N,I) /PRES/

NFLR /BULG/

NFLR1 /BULG/

NSHFT /BULG/

NRMX2 /BULG/

NRMX4 /BULG/

TE /FUEL/

QP /FUEL/

GAMA(L) /FUEL/

DHA /FUEL/

Output To

ZD DRJET

SA(N,I,J,K) DRJET, /FLOW/

SAD(N, I , J ,K) /FLOW/

BURF(N,I) /QGAS/

QF (N,I) /QGAS/

115
(vi) Calculation

(a) Rates of opening flow

SS. . ij

SA. . ij

AS. . ij

AA.

RFLOW (P.,P.,T,T . , Z , Z .) i 3 s’ s,j’ s’ s,3

ij

(b) Doorjet entrainment

2 - H

R

- Z

s - {mln(H

hHR

- V + max(X

nas> Hr Yj '

SA’ . . = DRJET (T . , SA. . , Z) 3i s » 3 31

(c) Burning rate

E{Y £ . (SS. ,+SA. . )-Y (SS . ,+SA. .)+Y*(SA| ,-SA. .+AAS. .)}

f >3 31 3 i f 13 13 f 31 3 i 3 i

m = ( + Y^ m + Y^ (m — in ) P ) f p f s p

(Y

0

2

> «

-— [Z{Y . (SS . ,+SA )-Y_ (SS. ,+SA. . )+Y* (SA! . -SA. .+AAS . . )

}

Y

0

0

2

,3 ji 31 0

2

v 13 13" 31 3 i 3 i

+ YP m + Y* (m - m )] (Y =0)

0

2 p 02 s p 0

2

in which summation is taken with respect to j and k in case of a room and n, j

and k in case of a shaft.

116
(d) Heat release rate of gasified fuel

(e) Pressure equation

4 - Q

r + Q* + Qc

- Q

p %

i C T

P a + 7- E{T

s,j

(SSjl+SAji) ‘T

s (SSi3

+SA

iJ )}

cL

T

+ E{(AS

j± + AA

j± ) - (AS±j + AA

±j )} + m

P

in which E is taken as in (c) Burning rate

(f) New pressure guess

I

IPRG(i) (lA^ £ ACR)

IRET(i) (1A 1 > ACR)

and

P

± = PSET (Ai

>

117
5.16 RFLOW

(i) Task

To calculate the rate of flows through an opening when the conditions

of each side of the opening are given.

(ii) Remarks

This subroutine computes the rates of hot gas and air flows through a

given opening when the relative pressure at a reference level, and the

temperature and the thickness of the upper layer of each side of the opening

are known.

(iii) Symbols

Code Paper Description

ZA (N, I) Z . a, i

Distance between the discontinuity in i-th room

on n-th floor and a reference (ground) level (m)

RS (N, I) p s,i Density of the upper layer of i-th room on n-th

floor (kg/m^)

XNAS X

nas Neutral zone height between the upper layer

and the lower layer in the adjacent room (m)

XNSS X

nss Neutral zone height between the two adjacent

upper layers (m)

Y(1,N,I) T . s,i Temperature (K) /of ) , the upper layer m

Y(2,N,l) Z . s,i

ml . , , N [ i-th room on n-th floor Thickness (m) )

P(N,I) P.

i

Relative pressure of i-th room on n-th floor

at a reference (ground) level (Pa)

118
See 5.7 DFNC

SS(N,I,J,K) , SA(N,I,J,K)

AS(N, I, J,K) , AA(N,I,J,K)

HHP (N,I,J,K) , HLP(N,I,J,K), HRP (N, I) See 5.2 PARAM

BWI (N,I,J,K) - BW(N,I,J,K) See 5.2 PARAM

RA, TA, ALP, ALD, G See 5.1 MAIN

(iv) Input

Input From Input From

Y(1,N,I) VENTL RA /UNIV/

Y(2,N,I) VENTL TA /UNIV/

P (N, I) VENTL ALP /UNIV/

HHP(N,I,J,K) /OPEN/ ALD /UNIV/

HLP (N, I, J,K) /OPEN/ G /UNIV/

BWO (N,I, J,K) /OPEN/

HRP (N, I) /ROOM/

(v) Output

Output To Output To

SS (N,I, J,K) /FLOW/ AS (N, I , J ,K) /FLOW/

SA(N, I, J,K) /FLOW/ AA(N,I,J,K) /FLOW/
(vi) Calculation

(a) Height of discontinuity from a reference level

, i ^RPji Z s,i

> i HrP, j Zs,j

This subroutine provides the formulas of opening flow rates only when Z

is smaller that Z . ; so subscripts i and j are exchanged when Z . is

^ > 3 & , i

larger than Z

a, i

(b) Hot layer density

P = P T /T

S a a s

however, the following manipulation is made to avoid zero divide in the

calculation of neutral zone height when the concerning layer temperatures are

the same.

p = p 1.0 x 10

s a

-4

(p - p < 1.0 x 10

a s —

and

p .=p . - 1.0 x 10 s,i s,j

-4

(0 < p . - p . < 1.0 x 10

s * 3 s , l —

p .=p .-1.0x10

s,j s,i

-4

(0 < p . - p . < 1.0 x 10 - s ,

s,j -

120
(c) Neutral zone height

P - P

= J 1L

naS (p a

- p s.i)g

P. - P,

X = -4 ±

nSS (p s,j

- P s,i)g

(d) Rate of opening flows

See Table 3.1
5.17 PSET

(i) Task

To give the pressure of a room new guess value.

(ii) Remarks

This subroutine to guess the pressure solution, which is basically a

Regula-Falsi method, is devised considering the fact that the pressure function

A. is a monotonously decreasing function with respect to .

(iii) Symbols

Code Paper

DP AP

P (N , I) P

PI (N , I) P

x P2 (N , I) P

2 SQ(N,I) A

SQ1 (N , I) A

x SQ2 (N , I) A

2 KA(N , I) , M(N , I)

Description

Pressure increment

The latest

A preceding

A preceding

The latest

A preceding

A preceding

See 5.8 VENTL

pressure guess of

i-th room on n-th floor

value of pressure

function of i-th room on n-th floor

(iv) Input

Input

P(N,I)

From

VENTL

Input

SQ(N,I)

From

VENTL

122
(v) Output

Output

P(N,I)

To

VENTL

(vi) Calculation

(a) Before marching first changes the sign of A

?! <- P, A

x ^ A

and

P =

+ AP (A > 0)

?!

- AP (A < 0)

(b) After marching first changed the sign of A

?! «- P (Ax

A

1

> 0)

P

2

<- P (Ax

A < 0)

and

P = P

2

A

1 - P

1

A

2

A

1 - A2

123

J
6. Data Structure

1=1, NEROM(L)

Symbols

TITLE

TA

* 1 ) *2 ) NFLR , NEFLR, NSHFT '

* 2 ) NROM HFLR

* 2 } NEF(L) , NEROM(L) , HEFLR(L)

BRN(I)

DRN(I) 1=1, NROM

HRN(I)

BRE(L, I)

DRE(L, I)

HRE(L, I)

BRS(I)

DRS(I)

HRST(I)

HRSB(I)

I, J, K, BWO, HHO, HLO

999

NEX, I, J, K, BWO, HHO, HLO

999

IPR(I) )

IRE (I)

IPRE(I)

1=1, NSHFT

1, NRMX2

1=1, NRMX2

IREE(I)

Format Input

20A4 MAIN

F5.1 MAIN

315 PARAM

15, F5.1 PARAM

215, F5.1 PARAM

5X, 15F5.1

i

PARAM

5X, 15F5.1 PARAM

5X, 7F10.1 - PARAM

5X, 315, 3F5.1 PARAM

5X, 315, 3F5.1 PARAM

415, 3F5.1 PARAM

415, 3F5.1 PARAM

5X, 1515

j PARAM

5X, 1515

| PARAM

124
N

FKWO(I.l)

CWO(I.l)

RWO(I.l)

FLWO(I,l)

EPWO(I,l)

FKWO(I, 2)

CWO(I,2)

RWO (1,2)

FLWO(I,2)

EPWO(I, 2)

FKW(NEF (L) , 1,1)

CW(NEF(L), 1,1)

RW(NEF(L) , 1,1)

FLW(NEF(L) , 1,1)

EPW (NEF (L) , 1,1)

FKW(NEF (L) , 1,2)

CW(NEF (L) , 1,2)

RW(NEF(L) , 1,2)

FLW(NEF(L), 1,2)

EPW (NEF (L) , 1,2)

FKW(1,I, 1)

CW(1,I,1)

RW (1,1,1)

FLW (1,1,1)

EPW(1,I,1)

1=1, NROM

1=1, NROM

1=1, NEROM(L)

1=1, NEROM(L)

I = NRMX1 , NRMX2

8E10.3

8E10.3

8E10.3

8E10.3

8E10.3

TPROP

TPROP

TPROP

TPROP

TPROP

125
I

FKW (1,1,2)

CW(1 , 1, 2) J ]

RW (1,1,2) > I = NRMX2 , NRMX2 > 8E10.3 > TPROP

FLW(1 ,1,2)

) EPW(1 ,1,2)

NFF, IFR, LFP, DDT, NTMAX 315, F5.0, 15 DFIRE

WC, WH, WO, WN 4F10.3 DFIRE

WEXP, TE, QGFT, WMFL 4F10.2 DFIRE

W, S, PM, ETA 4F10.3 DFIRE

BFIRED(I) , 1=1, NTMAX 5E10.3 DFIRE

AFIRED(I) , 1=1, NTMAX 5F10.3 DFIRE

HFIRED(I) , 1=1, NTMAX 5F10.3 DFIRE

NWMAX, DWT 15, F5.0 WPRS

WV(J), J=l, NWMAX 10F5.1 WPRS

COEW(I) , 1=1,4 4F5.2 WPRS

Note

*1) Total number of floor NFLR should be less than five unless corresponding

arrays are enlarged.

*2) Max (NROM, NEROM(L)) + NSHFT should be less than seven unless corresponding

arrays are enlarged.

*3) Regarding the symbols, reference the descriptions of the program in

which the data are input.

126
IV. SAMPLE CALCULATIONS

Some results of several sample calculations are depicted in this

section. Since the conditions given for each sample are somewhat artificial

especially in that the fuel input rates are given independently of the

thermal conditions in the room of origin, one cannot be too eager to draw

from these sample results conclusive insight about how a real fire behaves.

One should keep in mind that what is shown here is nothing more than how the

current model, particularly when the model of excess fuel transport is introduced,

behaves under various conditions. As can be seen in the following, some

examples have yielded predictions that may be somewhat surprising to us. In

this regard, the newly introduced excess fuel transport model has to be

subjected to experimental validation.

Another purpose of the sample calculations is, though somewhat

incidentally and unsystematically, to show the kind of features that have

been provided in the current program. This program can give additional

information, other than those variables shown in the following examples,

such as mean radiative and convective heat flux to the room surfaces,

temperature of the walls, room pressures, and the concentration of smoke and CO.

Some of the typical conditions for each sample calculation are shown

in Table 7. Only for the first test case do we have experimental results to

assess verifications. The input data required for these sample calculations

are listed in Tables 7.1 through 7.8. These tables would serve a user of the

computer program. The documentation in Section III together with the program

listing are needed to fully understand the tables.

127
Table

7.0

Typical

conditions

of

sample

calculations

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Fig.

7.1(b)

Comparison

Between

Predicted

and

Tested

Gas

Concentration
7.1 Sample 1: Comparison with FMRC Test 6 (Figure 7.1)

'

This sample is an attempt to compare the prediction with the FMRC

Test 6 [9]

The experimental configuration is a spreading fire on a

horizontal surface within a room with an open window. The predicted

temperature is closer to the window temperature of the test than to the

layer average temperature. The predicted gas concentrations are not in

excellent agreement with the test data. Differences may stem from the fact

that the predicted concentrations are the values under the presence of H^O

whereas the measured values are based on dry gas analysis. It is difficult

to fully identify the causes of the discrepancies between the predictions and

the measurements. Some of them may have to be attributed to the model itself,

but some must be attributed to measurements and the inconsistencies of comparing

local results with average predictions.

7.2 Sample 2: One story fuel rich (Figure 7.2)

This is an example of a fire in 1 story structure. The fuel input to

the 2nd room linearly increases up to 1 min. after the start of the fire,

remains at a constant value of 200 g/s until 4 min., and then linearly

decreases so that it becomes zero at five minutes. The results are

presented in Figures 7.2 a-d. The arrangement of the rooms are displayed

schematically in the figures with the rooms labeled as Rl, R2, etc. In

Figure 7.2 (d)

,

the flow rates are given in kg/s at the arrows and the

upper layer gas temperatures are given in °C. This notation will be

repeated in subsequent examples. A striking feature of the upper layer

temperature is that each of them has a sharp peak in their histories, and

132
1200 T T T T

0 1 2 3 4 5

TIME (min)

Fig. 7.2(a) Upper Layer Temperatures

(Sample 2: One Story, fuel

rich fire)

133
20

15

10

5

0

2 3

TIME (min)

Fig. 7.2(b) 0

? and Fuel Concentrations of

the Upper Layers

134
I

1

I

I

I

I

I

200

150

M

to*

ca 100

50

2 3

TIME (min)

Fig. 7.2(c) Fuel Burning Rate in the Rooms

i 135
I

0.06 0.06

0.49 061

0.40 0.49

0.66 0.67

Fig. 7.2(d) Upper Layer and Flow Through

Openings (Number under a ceiling with no arrow: temp,

elevation; Number at a doorway

with arrow: flow rate (kg/s))

136
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137
Fig. 7.3(a) Upper Layer Temperatures

(Sample 3: Two Story, fuel

rich case)

138
20

TIME (min)

Fig. 7.3(b) 0 and Fuel Concentration of

tne Upper Layers

139
Openings (Number under a ceiling with no arrow: temp,

elevation; Number at a doorway

with arrow: flow rate (kg/s))

140
the peak time coincides with the time when the oxygen concentration of

that room becomes zero. And after the time at the temperature peak in

the room of origin, the burning rate of fuel becomes far less than the

fuel input rate. This means that some of the input fuel has flowed out

through external openings and the rest is accumulated in the rooms. Theoretically, these phenomena are understandable, but it should be noted that this fuel

input has been arbitrarily specified. A real fire, in which the fuel input

and the thermal conditions are closely coupled, might yield somewhat different

results

.

7.3 Sample 3: Two story, fuel rich fire (Figure 7.3)

This is a case where the room of fire origin is located in 2nd room on

the 1st floor of 2 story structure. The fuel input rate is the same as in

sample 2. The point addressed in this example is that the same fuel input

may cause very different results for a different structural conditions. As

can be seen in Fig. 7.3(a), the temperature of the room of origin remains

at considerably high values unlike the preceding case. The reason is that

in this 2-story case stronger flows are induced than in the preceding 1-story

case because of the presence of a shaft and external openings on the 2nd floor.

7.4 Sample 4: Two story, fuel lean fire (Figure 7.4)

In this example, all the conditions are the same as in sample 3 except

that one half of the fuel input is specified. There is a striking different

between the shaft temperatures for the two cases, whereas the temperature

differences are not very substantial in the room of origin. This example may

help to recognize the importance of excess fuel transport on fire spread.

141
TIME (min.)

Fig. 7.4 Upper Layer Temperatures

(Sample 4: Two Story, fuel

lean fire)

142
Table

7.5

Table

7.6

SAMPLE

5

:2

STORY,

FUEL

RICH,

OUTDOOR

WIND

CASE

SAMPLE

6

:2

STORY,

FUEL

RICH,

CLOSED

WINDOW

300.

300.

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Fig. 7.5(a) Upper Layer Temperatures

(Sample 5: Two Story, fuel

rich fire without door wind)

144
0J93

2.14 2.28

2.41

Number at an opening with an arrow: flow rate (kg/s))

145
7.5 Sample 5: Two story, fuel rich fire with outdoor wind (Figure 7.5)

In this example, the outside wind with constant velocity of 3 m/s blows

against the right hand side of the structure, and exerts a positive pressure

on the right of the building and a negative pressure on the opposite side.

The fuel input is specified as in example 7.3. As can be seen by referring

to Figs. 7.3(a) and 7.5(a), the wind results in raising the fire room temperature and lowering the shaft temperature compared with sample 3. The wind causes

more air to be blown into the fire room, by which more fuel is burned in the

room and less fuel is transported into the shaft than the case of sample 3.

7.6 Sample 6: Two story, fuel rich fire with narrow external openings

(Figure 7.6)

This is an example in which the width of every external opening is

narrowed to 0.1 m. The temperature of each room again behaves in a complex

manner as in sample 2. The flows through external openings are, as

expected, generally weak except at early stages. Even under the very narrow

external opening condition, the temperatures behave nearly the same as under

wide opening condition at the very beginning, where the burning in the room

of origin is not obstructed by oxygen vitiation. But this results at the cost

of an extreme pressure elevation in the structure. In Fig. 7.6(d), in which

the shaft pressures are compared between samples 3 and 6, a very different

behavior of the pressures can be observed.

146
0 1 2 3 4 5

TIME (min.)

Fig. 7.6(a) Upper Layer Temperatures

(Sample 6: Two Story, fuel

rich fire with narrow external

openings)

147
Fig. 7.6(b) 0 and Fuel Concentrations

or Upper Layers

148
0.23

0.14

Fig. 7.6(c) Upper Layer and Flow Through

Openings (Number under a ceiling with no arrow: temp,

elevation; Number at an opening

with an arrow: flow rate (kg/s)

149
280

Fig. 7.6(d) Shaft Pressure (at ground

level)

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Fig. 7.7(a) Upper Layer Temperatures

(Sample 7: Five Story, fuel

rich fire with narrow external

openings; Note: Rni denotes

i-th room on n-th floor)

152
20

15

10

5

0

2 3

TIME (min.)

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153
154

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159

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7.7 Sample 7 and 8: Fire in tall buildings (Figures 7.7 and 7.8)

These are the cases to which the present model is applied to very tall

buildings. It has been almost commonly accepted that the hot gas, which is

contaminated with smoke and toxic gases, is the major cause of human fatality

in buidling fires, especially in the case of highrise buildings. Therefore,

some reasonable method to predict the smoke movement will greatly help to

develop the countermeasures to alleviate the hazard of smoke. Of course,

several efforts are already underway for this purpose, for example by

Wakamatsu [12], Tamura [13], and Klote [14], however, I believe

this is the first attempt in which a two layer zone modeling approach is

applied to tall buildings. By now there is no adequate experimental data

by which the results shown in Fig. 7.7 and 7.8 can be validated, but it

seems that the computed results are plausible.

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Fig. 7.8 Upper Layer and Flow Through

Openings (Sample 8: Ten Story,

fuel rich fire with narrow

external openings) 0.5 - 4.5 min.

023

162
Fig. 7.8 cont’d

0.78

163
3.78

n 3.77— 3.77—*• 3.78 — -*—4.08 -*— 4.07 n

1

1

Fig. 7.8 cont'd

TIME: 3.0 min.

164

4.08
Fig. 7.8 cont'd

533

165
V. CONCLUDING REMARKS

The preceding model, which had been devised in an attempt to predict

multiroom fire spread has advanced a step toward that goal. The refinements

have been made so that excess fuel transport and combustion can be predicted.

As a result, some sample calculations with large fuel inputs dramatically

display predictions with characteristic temperature peaks for each room as

excess fuel departs. These results may have been caused by the specification

of an arbitrary fuel input rate, or may in fact be truely characteristic of

actual fire growth in buildings. In any case, some properly designed experiments will be needed for the validation or further refinement of the model.

Some trial calculations were also made of the prediction smoke movement

in tall buildings. The results seem to be fairly encouraging for utilizing

the two layer zone model for smoke movement in highrise buildings. But

substantial refinements must be made in the calculation schemes for this purpose.

Eventually, ventilation effects need to be added, and experimental studies

conducted for validation.

Finally, for any complete fire spread model, the fuel input must be

predicted in the context of the local environmental conditions. The

current model has not yet addressed this, and is a logical next step which

can be developed consistent with other current fire growth models.
ACKNOWLEDGEMENT

I would like to express my gratitude to many people of the Center for

Fire Research, at the National Bureau of Standards, who helped me work

comfortably.

Also special thanks are addressed to the people of the Fire Modeling

Group, headed by Quintiere, and some other people such as McCaffrey

for much valuable advice, discussions and encouragement.

167
REFERENCES

[1] Quintiere, J.G. and McCafferey, B.J., "The Burning of Wood and Plastic

Cribs in an Enclosure", Vol. 1, Nat. Bur. of Stand. (U.S.), NBSIR 80-2054

(1980).

[2] Rockett, J.A., "Modeling of NBS Mattress Tests with the Harvard Mark V

Fire Simulation", Nat. Bur. of Stand. (U.S.), NBSIR 81-2440 (1982).

[3] Rehm, R.G. and Baum, H.R., "The Equation of Motion for Thermally Driven,

Buoyant Flows", J. Res. Nat. Bur. of Stand., Vol. 83 (3), May-June (1978).

[4] Quintiere, J.G., "An Approach to Modeling Wall Fire Spread in a Room",

Fire Safety Journal (3), (1981).

[5] Zukoski, E.E. and Kubota, T. , "A Computer Model for Fluid Dynamic

Aspects of a Transient Fire in a Two Room Structure", Daniel and Florence

Gruggenheim Jet Propulsion Center, California Inst, of Tech., Jan. (1978).

[6] McCaffrey, B.J., "Momentum Implication for Buoyant Diffusion Flame",

Vol. 52, Combustion and Flame Journal, 1983.

[7] Modak, A.T., "Radiation from Products of Combustion", Fire Research, 1

(1978/79).

[8] Mitler, H.E. and Emmons, H.W. , "Documentation for CFC, the Fifth Harvard

Computer Fire Code", NBS-GCR-81-344 , (1981), National Bureau of Standards.

[9] Alpert, A.L., J. de Ris, et al. , "Influence of Enclosure on Fire Growth

Volume I: TEST DATA, Test 6: Window and No Doorway", FMRC Tech. Rep.

No. OAOR2.BU-6, (1977).

[10] Tanaka, T. , "A Model on Fire Spread in Small Scale Buildings", BRI

Res. Paper No. 79 (1978), Building Research Institute (Japan).

[11] Tanaka, T. , "A Model on Fire Spread in Small Scale Buildings (2nd Report)",

BRI Res. Paper No. 84 (1980), Building Research Institute (Japan).

[12] Wakamatsu, T. , "Calculation Methods for Predicting Smoke Movement in

Building Fires and Designing Smoke Control System", Fire Standard and

Safety, ASTM STP 614 (1977), Amer. Soc. of Testing and Materials.

[13] Yoshida, H. , Shaw, C.Y. and Tamura, G.T., "A FORTRAN Program to Calculate

Smoke Concentration in Multi-story Buildings", NRCC, DBR Computer Program

No. 45 (1979), National Research Council (Canada)

[14] Klote , J. H., "A Computer Program for Analysis of Pressurized Stairwells

and Pressurized Elevator Shafts", NBSIR 80-2157 (1980), Nat. Bur. of Standards.

168
NBS-114A (rev. 2 -ec) 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- 2718 July 1983

4. TITLE AND SUBTITLE

A Model of Multiroom Fire Spread

5. AUTHOR(S)

T . Tanaka

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

QU 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) Some refinements have been made on a multi-room

fire spread model. The primary improvements are: (a) that a model on excess fuel

burning in arbitrary room has been introduced; (b) that a model for the prediction of

gas concentrations has been added; (c) that subroutine ABSORB, which was created by

Modak, has been introduced to predict the upper layer emissivity; (d) that a new fire

plume model developed by McCaffrey has been included to remove the inaccuracy and

inplausibility of a vertical point heat source plume model; and (e) that the code has

been revised so that it can deal with tall buildings with somewhat less computer

memory size.

Also, some sample calculation results with the new code and a documentation of the

code have been inducted.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

compartment fires; computer programs; fire models; fire plumes; fire spread;

high rise buildings

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

Uni imited

| J For Official Distribution. Do Not Release to NTIS

f

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

20402. 15. Price

'23 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

USCOMM-DC 6043-P80
,

" ~ —- -