L. .». .0

•, <-2,•:-,...;-•-,,c:.'• .,.• • -.- .,,-;.• . •• .•• . ;; -.•-,• . ,,'•,, • :• :-*., :• ••-. ..

'•'•:: !:Probabilistic, M c•dels :f

Behavior Of Compartr

Manuscript Completed: November 1980

Date Published: August 1981

. Prepared by

N.O. Siu

School of.Engineering and Applied Science

* University of California:

Los Angeles, CA 90024

Preparedfor

q..; Division of Risk- Analysis.

Office of Nuclear. Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, D.C. 20555

NRC FIN B6284

...........

* 0

:b I:;z• ::•:NU REG/jC R-2•

oretheFie

ne0nt Fire.'s:
Q,

L .ACKNOWLEDGMENTS:

The author gratefully acknowledges the aid and- guidance provided by

Professor,.G. Apostolakis in this worký. -A special acknowledgment goes to

Dr. M. Kazarians. for. his helpful comments and constructive criticism.

Finally, the author thanks Pickard, Lowe, and Garrick, Inc-. for the useful information provided for the nuclear •power plant: compartment fire

analyses presented in" Chapter 5 of-this-report.

.. ,,-
I

-~>-.2 j 

S"

Physical models 'which predict the thermal hazards (including tempera-

-tures and heatfluxes).during aacompartment fire as functions of space

.,and.,time are developed. Since -large uncertainties are inherent to the

analysis the models areprobabilistic.

General. models are constructed for the periods of fire growth and

f.ullydeveloped burning. These models are used in sample-analyses to

-_:estimate:.the fire hazard in particular compartments.

The' overall methodology requires the synthesis of a detern.inistic

physical model from information available in the literature. Uncertaint1es in, the' input parameters required by the deterministic model are assessed and are incorporated with. the uncertainties in the model itself, to form state of knowledge distributions for the thermal hazards

'ABSTRACT

- iii -

-..- ~--.-~t . ~ - .-.-- --.

-LTABLE' OF-.-CONTENTS

ACKNOWLEDGMENTS .1

AfSTRACT. ....... ............... . . .-. ".. .. ... . J.............

7:'i-:"i• •TABLE OF: CONTENTS'-......... .............................................- . iv

LIST OF TABLES . . . . . . . . . . . .

LIST OF-zFIGURES

LIST OF VARIABLES...............

Chapter

1. INTRODUCTION AND SUMMARY..................

2.. OVERVIEW.-OF THERMAL HAZARD MODELS ...... . ..

2.1 Static Models. ....................

2.2 .Dynamic Models . . .. .... . . . . . . . . . .

2.2.1 Compartment.Fire.Development

- 2.2.2 Fully-Developed Burning Period Models

2.2.3 Fire Growth Models... ...........

3. GENERAL`METHODOLOGY . . . .. .. . . . . .. . ..

3.1 Identification of-Uncertainties ... ...........

3.1.1 Statistical Uncertainties ....... .

3.1.2 State of Knowledge Uncertainties ........

3.2 Frequency-Magnitude Relation For Fire Severity

3.2 1 A Statistical Approach .... ...........

" 3-2.2 The ReferenceModel Approach........

3.2.2.1 SimplificationsDue to Diffuse

:State-of-Knowledge ..... .

. . .. . . . viii

xiii4

. - 1

............................... 6

7

7

10

. . . . . . 12

•.' . .. -. . '20. 2

. . . . . . 21

24

24

27

. . 27

.. .. . . . 29

•i

i

3.2.2.2 The Deterministic Reference Model . . . .

3.2.2.3 Incorporating Parameter Uncertainties..

3.2.2.4 Incorporating Uncertainties in the DRM

29

32

33

37

- iv -

----- ~-
TABLE :OF CONTNT3 cntne

P~age.

4. %:THE GROW-TH PERIOD MODEL- 41

.4 .. 1 -,The_.Refere nce -Model 4

4.1.1-,HeatRelease Rate*. . - .. . ...... .............. . 43

4.1.2 -The Ceiling Gas Layer ....... . . ........ . ..48

4.1.3 Heat Transfer to Fuel Elements ....... ........ ......... 1...

4..,3;. Radiative Heat.Transfe.r.From.Flame . 5 1.

4.1.3.2 Radiative Heat Transfer From Hot Gas Layer 52

4.1.3.3 H.eat. Trans f _r. Within the Flm..... ....

4.1.3.4-ConvectiveHeat Trasfer .............. 533............. 

4,1:3.5 Heat Reflected and Emitted from thie Walls . 4

4.1.4 Fuel Bed Ignition and Fire Spr-ad ....... . .. .... 55.....5

4.2 Parameter Uncertainties . . :........... . ...... ..... .. . 4.. 56..

4'.2.1 Burning Rat.e Paraeters .. ... . . . . . . ... 56

. :"4.2.2 Hea Release Parameters........... ..................... 59

4.2.3 Ignition Temperatoures . .•.... ...... ........ 59

4.3 Modeling Uncertainties. ............... . ...... . ... ..... 61

{~

,t

4.3.1 Uncertainties in Basic Modeling

4.3.1.1 Heat Release Rate Modeling.Uncertainties

4.3.1.2 Ceiling Gas.Layer Modeling Uncertainties

4.3.1.3 Heat Transfer Modeling Uncertainties ....

4.3.1.4 Fuel Element Ignition t'cdeling Uncertainties

4.3.2 Uncertainties in Numerical Implementation of DRM .

4.3.2.1 Fuel Bed Discretization. ...... ............

4.3.2.2 Radiative Heat Flux Attenuation ... ........

4.3.2.3 Ignition of Fuel Elements .... ...........

4.4 Reference Model Verification . . . . . . . . . . .. . . . .

4.4.1 Wood Crib Fire ...........................

4.4.2 Vertical PMMA Slab Fire . . . . . . . . . . . . .. . .

.4.4.3 Vertical. Cable Tray Fire........ .........

• 61

61

63

64

64

66

66

67

68

69

69

75

75

4.4.4 Compartment Fire Simulation 82
_JJ

j• -. • .. " j9 • ••. • 6 : . : . •. " • • 7 -.-

TABLE OFCONTENTS- (continued) -

.5. APPLICATION OF GROWTH PERIOD .MODEL ... .... .. . . .. ........... 87

:5.1 Electrical. Cabinet Fire . .. ......... 8 7.. •:1 , .1 , E x . .i . ..... .,; .:. : i8 8

.5.1.1.Exposure Fire Mode . ................ 88

5.1.2 Thermal-Response Model .". ........................ ... 90ý

5.2 Cable Tray Fire .............................. 94

. 5.2.1 The .Det'-rministic Reference Model ..................... ".94

5.2.2 Parameter Uncertainties . ......................... . 98

5.2.3 Modeling Uncertainties ..... .................... .. 104

5.2.4 Combined Probability Distribution for Spread Time 107

- 5.3 Fire Spread Frequency Including Suppression ....... . . .. 108

.. 5.4 Comments on Analysis .................. ..................... 110

6. FULLY-DEVELOPED BURNING PERIOD MODEL DESCRIPTION....... .... . 113

6.1 The Deterministic Reference Model . . ............ 114

6.2 Parameter Uncertainties ........... .................... 116

6.3 Modeling Uncertainties ........ .................... 118 4 .. 6.4 Reference Model Verification ............................... 119

r 6.5 A Simple Application . .................................... 122

7. REFERENCES ................... ............................ 128

- vi -

------- ---- ..... ..
- .: . : :. .y _- ,: : .- ? :,. 4, • '.4::•

SAppendiýce s

A. SHAPE-FA&TORS

B.. LATIN HYPERCLJIE SAMPLING:'

C. CONVECTIVE.HEAT TRANSPER DURING FIRE GROWTH'

C.1 Hot Gas'Temperature . .. ........... .. . . .'... C.2 kHeat Trans~fer C6efficients -.

D. HE AT FLUX'"FROM; CEILING GAS LAYER...................

E. VERIFICATION 7 OPEF' QUATrON(4.3)....... .......

Page

1:34 -

" 1 2,6. 

. . 146

* - 147-

•" . 149

154

f

- vii -
<

LIST._OF TABLES

I.

Number 'Title -Page

4.1 Best Estimates for Cs . . . . . . . . . . . . .

4.2' Best Estimates for H

f

4.3 Best Estimates for Y . ... . . . ... . . . . . . .

4.4 ignition Temperatures ( 0 C)

'4.5 Properties of Wood Crib in [3 7 ] .. ... ...

4.6 Parameters Used'in Cable Tray.Fire. Simulation

4.7 . Upward Flame Spread Velocity of Cable Tray Test

.4.8 Properties of Wood.Crib in [53]

5.1 Exposure Heat Flux in W/cm2 ...........

5.2 Cabinet Air Temperature (OF) ...........

5.3 Parameters for Cabinet Air Simulation ......

5.4 Important Parameters for Cable Tray Simulation

5.5 LHS Bounds for. , f , . . . . . . . . . . . . 0' f' s

5.6 Burning Parameters for Pilot Fuels ........

5.7 Response Surface Coefficients (Eq. 5.9) .....

5.8 Histogram for Vertical Spread Time .......

5.9 Histogram for the Conditional Frequency of. Fire

Spread to an Adjacent Tray, Given a Fire

6.1 Fixed Parametei.s Used in Fully-Developed Fire

Simulation . . . . . . . . . . . . . . . . . . . .

E.1 Parameters Used in Cyclohexane Simulation ....

E.2 Parameters Used in PMMA Simulation ........

45 

45

45

60 

60 

. . . . . 78

.79 

. . .. . . .. 79

91 

. . . . . . 91

. . . . . . 93

. . . . . . 97

. . . . . . 100

. . . . . . 100

. . . . . . 102

. . . . . .102

.111 

. . . . . 123

155 

.155 

-viii-,

LIST OF FIGURES

Number: Title Page

2.1 Compartment Fire B.ehavior .......................... ............... ....

4.p1~~.. ( 2.2 ~Experimentafj Ver'if icatinoMde [1] . . .. 3

2.3 Time-Temperature Curves for Two "Equivalent" Fires . ..... 14

2.4 Fire GrowthEvent Tree 1ode [181 .............. 1

3.1 Flammability Ratings of Wallboard [21] . 3

3.2 Data fdr'Cohipartm~ent- Fire Severity [23] . . 8

4 3.3 Uncertainty. in Frequency-Distribution of X.... .... ... . ...-.. .1

3.4 .Uncertainty in Frequency Distribution of X (small

statistical, uncertainty): . ... .... "..... . ... ..-. .. .31

3.5 Comparison of Thomas' and Steward's Flame Height

Correlations [28]1 .-. .- ."-............... .. .. . . ". ... ... 34 "... 

3.6 Schematic for PropagationofUetntie Thug th

Reference Model . . .. ......... . ...... 36

4..1 Flow Chart for Growth Model. Calculations .. ....... .. .. .. -43

4.2 Burn..ng Rate vs. External Heat Flux [35] . . ...... ... 46.. 

4.3 SurfaceRegression Rate .

4.4 Burning Rate Correlation for Wood Cribs [37]........ ...... .. 46 .

4.5 Correlation for Burning Rate [38 ... .... ... .... ......... .47

4.6 Fire Scenario Asumed for Ceiling Gas Model [~]J l]. ... 5

4. 4.7 ~~Histogram and Fitted Lognormal for CV (data from [3) .... 5

4.8 Thermal Decomposition Curve for Polystyrene [49] ....... 65

4.9 Spontaneous Ignition Data Versus Theoretical

Predictions [38]..... .. . -. . .. . . ..... ... ... . . . .. . ....... 70

4.10 Variations of Experimental Spontaneous Ignition .... Times [38 .. . ................................. . 71

4.11 Diagram of Wood Crib Used in [37ty .2] .... . ....... ... 281

-ix-
4v0."

W.. .

JLI S.ft OF;FIGURES (continued) .

Number Title Page

4.12 Referenr,-e Model Predictions for Burning Rate of Wood

Cribs .Versus Experimental Data,[37].... ........................... 73 -

4.13 -Reference M.del:-Prediction of Burning Rate of :PMMA--, Vertical Slab Versus,.Experimental Data [511 ........ 76

4.14 Cable Tray Experimental Set-up [52]. ....... ............. 80

4.15 Comparison of Experimental and Predicted Burner Heat

Fluxes Along Cable Tray . . ........ . . .. . . . . . . . .. .. . . . 81

4.16 'Experimental and Predicted Burning Rates.[53] . . .... 83

4.17 Experimental and Predicted Heat Fluxes [53]) ................. 84

4.18 Experimental and Predicted Gas Temperatures [53]. .... ....... 85

5.1 Exposure Fire for Cabinet . . .. ............. ............... 89

5.2 Heat Fluxes for Heat Balances at Each Node ... ....... ....... 89

5.3 Cable Tray Fire Set-up ....... .......... ............... 95

5.4 Distribution of Propagation Time, Due to Parameter

•Uncertainties .. .... ...... ..... .......................... 105

6.1 Comparison of DRM Prediction with Experimental Data from

Ref. [43] ............ .......................... .......... 120

6.2 Distribution of the-Ventilation Factor for Two Nuclear

.Power Plants [58,59].. .... 1 ...... . . . . ..... . . . . . . 121

6.3 Distribution of the Averageý Compartment Temperature

Uncertainties Due to Input. Parameter Urtcrtainties ... ...... 125

A.1 Transfer Between Two Differential Surfaces ....... .......... 136

A.2 Transfer Between a Cylinder and a Parallel Surface .. ...... ..136

A.3 Transfer Between a Cylinder and a Perpendicular

Surface ........................ ........................... 136

A.4 Transfci- Between a Rectangle and a Parallel Surface ......... 139

4

.............

I

- X -
LI ST"OF. fIGU RES, (continued)"

Number Title

p

i

~

Page

A. Transf..er Between , a e.tang eand a.Pr.pen.dicular,..

,Surface

A.6 ''Transfer Between Combinationsofý Surfaces

B.I Illustration of Example Latin HypercubeSampling

Scheme . .....................

D.l Model for Ceiling Hot Gas Layer

D.2 -Model of Ceiling . . ....... -....... .

E.1 Model of Experiment [62] . . . . . . . . .... . . .

E.2 Comparison of Predicted Results with:Experimental

Data [62]

E.3 Model of Experiment [63]

E.4 -Transfer Between a:Differential Volume and a

Differential Surface ....... . .

E.5 Comparison of Predicted'Results with Experimental

Data [63] . . . . . . .. . . . . . . . . . . . . .

... . .. .139

*. . .. 143

- . .... 150

" .150.

. . . . 150

. . . . 156

S. . . 158

. . . . 158

. . . . 158

- xi -

•. K

*i•

LIST 'OF.VARIABLES.

Variable

A

B'

CO0

CS

CV

D

E

F1 2

Hf

M

p

T

V-*

V

W

Definition

Area •i),)

Height'of. doorway- -(m)

Orifice coefficient for opening

Surface controlled, burning rate constant .(kg/J)

Ventilation controlled burning rate constant

Diameter (m)

Error 'factor for quantifying modeling uncertainties

Shape: factor from-object I to object 2

Heating value, of. fuel, (J/kg)

Mass (kg)

Heat content of pilot fuel (J)

Rate of heat production (W)

Temperature (.OK)

Ignition temperature (OK),

Volumetric flow rate (m'/s)

Rate cf air flow (kg/s)

Height (m)

i.

c p

f c

Absorption coefficient (m-1)

Wood crib stick width (m)

Specific heat (J/kg0 K)

Conditional frequency of fire involving two

cable trays

Porosity factor

Gravitational constant (9.8 m/s 2 )

f

p

- xiii - Preceding page blank

k2A2
L

10 ft

;T -,OF'-VAR IABLES...(Continued)'

h Heat transfer, coefficient (W/m2 OK)

k Thermal conductivity (W/m.K)

mi Burning rite, -of fuel (kg/s)

Specific burning rate "(kgm/m s)

m' Specific-burning rate constant (kg,

Heat tflux .(W/m 2 )

q- Radiative heat flux from flame to fi

fl,r

s Wood crib stick spacing (m)

t Time (s)

.:t* ...- 'Time. to ignition_ -_(s)

V Flame front average velocity .(mls)

- - z

£.-

/M 2 s).

uel bed. (W/m )

Ei

J1

.T0

P

p

0

w

Thermal diffusivity (m•/s)

Plume entrainment coefficient (Eq. 4.9)

Fraction ofQ radiated away from fire

Emissivity

Combustion efficiency

Characteristic dimensionless parameter.

for fire growth

Length (m).

Kinematic viscosity C(m2 Is)

Density (kg/mi)

Stefan-Boltzmann constant (5.6697 x 10 W/m mK 4 )

Characteristic time (s)

Dimensionless parameter for fire growth

Plume entrainment coefficient (Eq. 4.9)

- xiv -
'5 .,; - 55~<* ~

d,

'C - -

LIST OF V .ARIABLES', -continued)

Subsc ripts Definition

A Ambient

d Neu (see n" ..below)

-DRM" . .'::JResuilltof "deterministic referencemodel calculation

e Environment

ext External to fuel bed -

fl .Flame

f . Fuel, fuel bed

FV ':Forced ventilation

G ý"Gas

n _-'Neutral-plane,(divides hot gas layer from.

cold, 'air. beneath)

0 . Opening (e.g. window)

o Object, or property evaluated at level of fuel bed

p Pilot

W Wall

- xv -

-- , c°.,t. . ..~ ,. v .. , " W .#-,. -. . -.

'Chapter 1

INTRODUCTION AND SUMMARY

Fire can represent a threat to the safe operation of nuclear power

plants,: not only because it can damage machinery in the immediate vicinity of the flames, but also because it can damage control and ýpower- cables passing nearby. Since power plants rely heavily upon these: cables

to link-their various systems, a fire can cause the failure of many com-

. "ponents well;.removed from the blaze.

'U

~iL~

In order to proceed with a quantitative analysis of the.-fire risk to

nuclear power plants, the frequency that a given component will fail due

to fire-is needed. The determination of this frequency in turn requires

models -for the frequency of fire occurrence within a power plant, the

distribution of the magnitude of fire severity given that a fire occur.;,

and the probability of component failure upon exposure to fire of a given severity.

In this work, we concentrate on developing the second model of these

three,- i.e. the probabilistic model for the time-dependent thermal hazard a component is exposed to during a nuclear power plant compartment

fire. Our approach requires the construction of a physical model for

the fire scenario of interest. We call this model our deterministic reference model (DRM). The uncertainties in the input parameters used by

the DRM are propagated through the DRM, and are then combined with a

probability distribution quantifying our uncertainty in the accuracy of

-1 -

4 ,-. ~ w,
the, RM to form a probability distribtition forthe frequency that a

component is exposed-to d aspecified thermal hazard (eg. high temperature). RelevanIdata to aid ourt constrution of the"parameter and modelin'g uncertainty distributions are often unavailable, and so subjective judgment is :an important :input in this Drocess.

To assist the-fire, hazard 4analysis, we.develop general models for two

different phases of a Icompartment fire: the fire grbwth period and the

period of fully-developed.burning.

In Chapter 2,: we -outline the time history of an enclosure fire, and

review some of the models available in the literature. We observe that

the majority of work on-.compartment fire models has, been devoted to the

period of fully-developed burning,. where the fire essentially involves

the -entire compartment, and that there are few models available to describe the period of-fire growth preceding fully-developed burning. We

further note that the methods of approach used are generally deterministic.

In Chapter 3, we argue that any analysis of a freely burning fire

murc deal with large uncertainties, and that a pr6babilistic approach is

appropriate for this problem. We identify the various classes of uncertainties which arise in our models, and formulate the, methodology

(briefly outlined above) employed to handle them.

In Chapter 4, we apply our general approach to the period of fire

growth. The reference model is synthesized from a number of separate

correlations and models "for individual fire phenomena. The uncertainties in the reference model's predictions are discussed, and the model

is used to simulate a number of experiments documented in the literature, in order to test ics validity.

-2-

-~1~
I•

Unless the fuel geometry is, particularly simple, a large amount of

bookeeping is required during the fire growth computations. Tht reference model, therefore, is in the form of a computer program, called

COMPBRN, and is described in more detail in Reference (1).

In Chapter 5, we consider fire hazards in two specific cases. The

first scenario concerns the exposure of an electrical component housed

in a metal cabinet to high temperatures induced by an external firs. We

construct a simple model to predict the air temperature within the cabinet, and show that moderately sized oil-fueled fires may be of some concern. In the second case, we utilize our complete probabilistic methodology in 3ssessing the frequency that two cable trays in the cable

spreading room, each in a different electrical division, are involved in

a fire.. We incorporate models not only for the growth of fire but also

for the elapsed time before suppression.

Although a great deal of emphasis has been placed on the modeling of

the fully-developed period in the compartment fire literature, we do not

consider it to be of as great importance to nuclear power plant fires,

not only due to the relatively low concentrations of highly flammable

material in the power plants and the large size of the rooms., but also

because many accident sequences may be initiated and well in progress

during the growth period. We treat the fully-developed burning period

model in Chapter 6, and conduct an analysis of a hypothetical fire in

the cable spreading room which has reached the fully-developed stage.

We remark that our models are primarily thermal in nature, that fluid

flow and chemical kinetics effects have been either neglected or treated

extremely roughly. Our approach appears to be reasonable during the

- 3 -
early portion of fire growth and during the fully-developed burning perj!-od, but is considerably weaker in between these:two regimes. A notable

deficiency is our lack of a model for the transition between..growth and

-~ .fully-developed burning, called flashover, other than a simple thermal

model involving a positive feedback term in the equations for fuel burn-

• 11

ii.I.

ii

I

I
'A'.

Chapt--r 2

OVERVIEW OF THERMAL,ý HAZARD; MODELS

During a fire .a given component may suffer damage by absoroing, a criti.-

cal amount of heat-. Our models for.the'thermal hazard -within a.burnihg

compartment must therefore predict the amount of- heat the component absorbs which in. turn requires knowledge of the amount of.heat the compo-

, nent is exposed. to. In this work we concentrate on the latter quantity;:...

components are modeled& rudely, although moredetailed modelscan be-developed for specific'.problems.

The two.primary-modes. of heat transfer 'considered are convection and

. radiation. The convective heat flux to the object is given by

h [ Te - To ] (2.1)

If the radiative heat flux impinging on the object is'qr ' the absorbed

heat flux is given by

. .. . = E " " ( 2 2 ) q o,r qr"

If the heat flux at the source is q(i, we can re-write Equation.(2.2) S

as

0F o-r qs

F represents the fraction~l solid angle of the object's surroundings

0-s

that the source occupies. We note that the object also re-radiates a

heat flux 41" given by rr

-5 -
4" =sT' (2.4) rr 0

In keeping with our simple modeling, we generally fix T at roem 0

temperature and c at unity. Thus, our models calculate the "cold wall

heat fluxes" to: black bodies; the resulting quantities ara conservative

estimates of the actual values.

The models in the literature which estimate o" , the sum of q' and 0 O01 c

q", can be divided into two categories: the static models and the dyor

namic models. Although we-describe the static models in the following

section, the remainder of this work shall concentrate on the dynamic models, since these form a more general set which includes the static models.

2.1 STATIC MODELS

The static models for predicting the thermal hazard from a fire are generally derived for fires in the open, or at least where enclosure effects are negligible. A representative example is presented in Reference (2), where a fire over a fixed diameter pool of fuel is assumed,

and the thermal hazard as a function of distance is calculated. Since

convection is only important in regions of elevated temperature (i.e.

in or above the, flames), only radiative heat fluxes are calculated.

Other similar ana'yses are presented in References (3) and (4).

In each case, the source heat flux q" and the shape factor F a 0-s

which contains the problem geometry, must be calculated. Typically, an

average flame temperature Tfl is assumed. If we further assume blackV 

•E"

body behavior, there results

qs O fl (2.5)

-6-
This equationshal behdiscussed .inmore detail'in Chapter:4, where an

alternate formulati n-of..q- i§s employed. As for F ,-numerous analyt-

:,ical- fo6frmuIae -(haveý ben,-,d-:rive'd-,-'for- a variet of- gebmetries" 15,6,]. If,

the flame is mod(".lIed asa cylinder, the cheight may -be cmue immedi-m;

l knowledge o1 .with burning ... rate) ,rmThm .s!:-rre

[7], and used in the':., shape, .factor'. computation. Thomas'. flame height

-. corre-lation.s~discussed :.briefly 1 - nChapter 3, and.shape factors for

somfe'repres entativel.geomfetr . -A,,---

':•.f ."-":!.::.~i••:~i'<"om::rpr~se~tvei:go~tie larepresented in Appendix.-A -. . -7 "' ' .:

2.2~ DYNAMIC MODELSý.

owe to grow, neither the-source-termqs' nor . !. :.i!. " A.: : ,sia - fa t .".. - S.. ...

the shape factor Fo is.const ant., any longer. Furthermore,. if, the fire'

-"-i""s locatedwithin aoomthe heating room boundaries and-the accumulat-

:in gZly, 1.a of hot, gases .near the ceiling become increasingly important

he 1 ,.r.s:ce. as*time progresses. .Thus we must review the available dynamic, ,models for compartment fires.

2.2.1"ý,ý -Comparttment" Fire Development

Conceptually, the time -history. of a compartment fire (a fire which is

contained within a. room) can be resolved into four stages or periods:

inifiation, growth, fully-developed burning, and decay (see Fig. 2. 1).

.Theinitiation phase "involves the ignition of a small fuel element in

the room .of interest..If the-primary fuel bed is cable insulation, "

Which- is generally difficult to -ignite, additional pilot fuels such as

-paper,.oil, cleaning flid, or:'plastic -foam may be important. The last -

fuel was a major ,contributor --co the initiation of the Browns Ferry Fire

-7-.

J
.1.

~.,:...j. .AA~, A'

- . . . *.

'A. A

A,

A'

A, -

I

A;,' A' ~ ~,

42

A'A"A"I' '~

's'- A

~

4"'. A4A. ."'"' *A.'~ )~A'

A, -, ". "JAW. .* A.

A A.. A'

~AA~

-'IA' ' A' II''"'

A IA ~

A A A

A" ~A.'AA 'A

'"A'AAA A.""-

0

Ime

Fig. 2.1 - Compartment Fire Behavior

A A A

A - - A

A A AAAII -

A A
' .- i[8.. :Ai c f f. r .. plant. compo.-

[8 t this stage ofie development. the only.:hazard t ln op

nents is in the immediate vicinity of the fire.

In the following- period 'of fire growth, the fire -spreads over the.

fuelbed•: by continuous flamefront movemnt.. by adiativeand-convective..,

: heatling of nýon-contiguous f"uel. e lements, and sometimes by. direct trans- .- .:

port of burning fuel, spread .by 'firebrands being a good example of the

- /:. }• " latter process. The growing fire also heats up the compartment walls.

and- ceiling, and- creates a layer of hot gases next to the ceiling.

These'bodies in *tu:rnact .as-heat sources which radiate and reflect heat-

... "back to both the burning and non-burning portions, of the fuel bed, in-

'i-creasing the volatiiization of the fuel, and thus increasing the rate of

fuel feed within the 'flames and enhancing the ease of ignition of the

•fuel outside the flames.. This positive feedback loop-leads to an accelerating fire giowth rate, which may become so large that the entire compartment becomes involved'- in the fire during an almost instantaneous

period called flashover.

..The period of fully-developed burning which follows flashover, if it

- occurs at all, is characterized-by burning rates which are limited only

by the available ventilztion or free fuel surface area. During this

period, the temperature and heat flux levels tend to be fairly uniform.

-- This can be contrasted with the behavior during the period of fire,

" growth, where the heat flux and temperature levels &re strongly peaked

U -- near the- fire.

The period of fire decay occurs when the majority of combustible fuel I " within the room has been- consumed. During this period, the intensity of

- . -.

I "the fire dies down and the fuel often smolders rather than burns in an

ý, • open flame.

1.i

AU

-- -- .- - - ~ --- <.9
It should be noted that the progress of a compartment fire can be

slowed down and halted at any moment by suppression efforts. Furthermore, even if such efforts are delayed or totally lacking, the achievement of fully-developed burning is not assured, since a fire over a

small fuel bed may not generate enough heat to ignite distant fuel elements. A crude suppression model is described in the application of our

model to a nuclear power plant cable tray fire in Chapter 5.

2.2.2 Fully-Developed Burnin9 Period Models

Of all the work done on compartment fires, the greatest amount has been

done on fires which have reached the fully-developed stage of burning.

This emphasis is due to the relative simplicity of the analysis, and the

fully-developed burning regime's great importance to general fire safety

engineering.

A fully-developed compartment fire is a fire which is burning all

available fuel in the room and is limited only by the total amount of

fuel surface area or by the total amount of ventilating air flowing into

the room. There is essentially no concern with the spatial growth of

the fire within the enclosure since at least most of the fuel is already

involved in the blaze. Furthermore, strong turbulent mixing of the hot

room gases and relatively uniform fuel distribution allow the analyst to

at least initially ignore spatial variations in the temperature and heat

flux fields. The room can thus be modeled as a point in space, and the

complex set of equations which govern the fire reduces to a simple heat

balance equating the heat generated to the %eat lost to the environment.

- 10 -
.2;

Although a detailed-formulation of this model is provided in Chapter

6, the gener l'scheme is as follows: -

* All of the fuel, in ýthe r6om is •assumed toý burn-at. a rate-:,(in kilo-

-grams per second) determined by the amount of available fresh air

or fuel surface area. The rate of heat liberation within the com-

...partment is directly proportional:.to this mass burning'rate.

* Heat is o~st from the compartment ..,by radiation and convection to

the room walls, by radiation out.the room windows and-.doors, and by

the. physical removal of hot gases 'by the room ventilation. The

heat loss to: the walls is a function of the wall temperature, which

is obtained by solving the transient heat conduction equation for

the wall.

;*"The.heat,-gained by the compartment air is equated to the heat lost,

and the air temperature is solved for. Time is incremented a!:d

Steps 1-3 are repeated until most of the fuel (typically 70%.of the

original mass is consumed.

The solution resulting from thisý procedure can be expressed as a plot of

the average room temperature as a function of time. The maximum temperature, achieved and the time it is reached are only functions of the fuel

loading, the fuel type, the fuel surface ares, and the ventilation-level of

the-compartment, the fuel loading being,-the -imount of fuel per unit area of

compartment.

This is the basic approach adopted by a. fair number of researchers, including Odeen-19], Xawagoe and Sekine [10), Harmathy [i11, and Tsuchiya and

Sumi [ 12], although each researcher incorporates varying degrees of sop- histication into his model. Some time-temperature curves from Reference

- I -
<I-, ~

.3

".

..: ,. .- . : . . ". A 'A ' .','A . -

(10.) are shown -inigure 2.2, as well as :actual data from the fires simuThe functional relationship'between time :and temperature derived in this

f~ashion :forms -the-basis for the mostý widely recognized design codes [13].

In fact, many nations employ some form of a standardized time-temperature

curve to determine the stress a material shouldbe exposed tboduring flammability tests. This curve is considered to characterize a standard exposure fire and. leads to descriptions of -fire severity, in terms ofthe equivalent duration of the standard fire. For instance, a fire of relatively

..low severity whi-ch burns for three hours may be,.described as being equivalent :t a fiir-e,-fol6ilbwing the standard time-temperature curve for. one hour. .

This equivalence can.be relate'd to the concept of.a critical amount of absorbed heat leading- to czomponunt- damage, .and to the integration of Equation

(2.1) over time. The Lime-temperature curves for these two fires are shown

in Figure 2.3.-

2.2.3 Fire Growth Models

During the period of fire growth, the heat fluxes oc and C,r are

highly non-uniform functions of space. qoc is essentially zero except

for objects in the immediate vicinity of the flames, the hct gas plume

above the flames, or. the hot gas layer near the ceiling. As for " - -"o , r

Fo0 s is a purely geometric factor and will vary strongly with location.

The prediction of fire. growth with time . is thus extremely important,

since this governs the behavior of q" (-,t) and q" (i•,t)-

-O c o ,r

A reasonable method for handling growth period fires is to use a quasistatic approach, an approach based on the static calculations dis-

- 12-
•w. ....... . - . . . _ _ _ " . . . / - 'i ! : : ; .I, , : -

... ,.....-..,..,..... , ..-

•: ~0 9

~q

506- 1f 00 m

L 1.16 W/nik

:1 000 .

.0 . -F. = 0.06 m

". AA,/A 0.06

- " k 1-.16 W/m0 K

0-

0 t f. ", 30

1000-

FO .= 0.09

. .. 00.0 7oo 

0 30 60 90

n,;n

1000

50,0 ./A. 0.03

k= T1.16 W/mOK

zt ~ 0-

0 30 60 90

Fig. 2.2 Experimental Verification of Model [10]

-"13 -
~: A-y

~-v- / -

A.

I -

2- ~ C'

I'., .. ,., *u~2 y4~ ?;tŽ~tCt~Th~Th5 ~ ~

,,.'-, 

.

.St:Ia~.nda Fd F, ,.o

S

0

0

0.

S

ITime'

Fig. 2.3 - Time-Tempe rature Curvtes fotTwo "Eqiivalent' Fires

- 14 -

-: ~ '' 2-CCC.sI'C>5 1. ~
cussed earlier b"u t, extended to account f or' fuel lement ignition- and

t u s fire g row t E ss n i l y at ea h Ro int .in time , : t ime -i ,nd ep nd nt

A-- r-_: _ _ _calculations, for% f lame ,size, burning -rates "heat re~lease -r tes:,..shape,::

t. os and._heat abso•ripto•..n are: done, fuelelementsare ignited if they

hav a- bsre enug ha, ieis incremented- -and the process, repeat-

." sed. 'Further models are also required. to account for the presence of.

compartment boundaries, since teabove static calculatiions are. con7_structed .either. to. analyz fire inthe -open or. fires wh ich are; considerdtoo-small to be affected by the surrounding room.

These..needed-refinements. are outlined by Quintiere [14], who not only

des----.,crb the.- calculations required to proceedithe a comparenrt fire

.. growth analysis, but also presents :a simp lemelfr fire spreading

continuously 'ovex the floor 'of a room.

:The essential differences in behavior betwe.n a fire in -an enclosure:

and a fire in' t he open ar casdb the restriction ot aipro: f plow abut

thedfire as wellas the physical pr esence Thi:

.)::-TL..:~~~~ compartoen sicThh.aov boundaries. of ttcclultosae'isn

W. :•restrictioný not- -only imiftis the a n ofp resh air which carn 7each the

f:ire butals inhibits the mcvement of hot gases away fron- the flames.

A layer ofthese gases accumulates in. the compartment and adds its radiatives heat fluxes lto the heat reflecting from the room boundaries back

to' the'-burning and unburnt poratsioesentscf the fuel Udr the inf

i•i.:,:, i""c~tiuosl-oe6th fo r o 1 room. n Une "h~nf

ence, of. th-is' feedback, the, fuel. bed 'releases volatile gases more rapidly; hence the burningl rates increhse and the unbuant fuel ignites morereadily.

The refinements suggested by Quintiere are simply relations needed to

-. determine the airflow into th e compartment, the size and temperature of

- 15 -
9,1

44

the gas layer, the wall temperature, and the radiative heat fluxes from

the gases and walls back to the fuel bed. Except for the fluid fl ow, ma.

de, which"s developed• by Rockett [1.15].-.,,most -of the relations are der

ived- from: heat balances over •the.fir p lume'.' the hot gas .ayer, and he

room walls and-ceiling. Unfortunately, Quintiere does. not advance his

model ýbeyond- the generalI formulation stage, :.save for. the simple spread -

model mentionedd and this model :isnot really directly applicable to. nu

,clear power plants: where fire spread :ýover .discontinuous fuel, bed'smust

be considered.

A more complete formulation'. o. f the, same- basic model is presented: by',

Pape and hi's, co-ogrkers!6,17] .( Theirmo~delis in:the form of a computer code, which- upohn ;input-of: the appropriate room -geometry and fuel

characteristic parameters, deterministically -yields a certain output

(e.g. the heat flux as a function of the distance from the fire). An

interestitig twist in- their model results 'from recognition that the fuel

burning rates wil.l .v'ary strongly ,'as a fution of .the configuration and -

composition.of the-ffuel bed. Sincc these-burning rates are difficult to

model for all but the most simplified of'fuel beds, subjective probability distributions•are employed.

A major .drawback of Pape's model is the neglect :of burning rate enhancement due to.heat feedback to the burning fuel .bed,(Quintiere references experimental results which show increases in the burning rate of

up to 60 per cent-.-for wood cribs and 300 per cent for plastic slabs due

to this effect). ,We also.note that their model is not completely probabilistic, since no uncertainties other than those in the burning rates

are explicitly addressed.

- 16 -

" 7::
Ol;:i~lne'of ' t i p ant problems arisng with the gane al.:app.oach ou-•,

lined .is .that of c om pcoa

synthesis of component models, which. d rbe ' buining erates, heat re

laerteflame ',~~ he 

s::i ,i heat reights; igntn times,':_ etc.;w it cannot deal with

phen no men inc ude in the component m de s For example,. Glassman, .. • - . I S.-h s,. comments:,- appen~ded.,.t!o:-. Ref•. 14.) sta .t es'. ýt h -a t- •; " " "

.- .- a t R mflashover might, not..:

,j ust l-be, simply thý rsult f anacceleratingý grow.th-`,proces aae

.: "Posit ive feedback 'to the f.ire, but may 'also he caused by the" sudden

aa. blt ofoa source'"of energy (such as a'_flgame) e near' the compartment ceiling.where': Trge.:quantities of:'unbur•t gas. !maye •reside. "The'energy- source ignitegs`ithe gasesandthe resulting EIa"-.ne preadsý almost in-

'stantaneoul noous y' trough" hr ,he.,entire.. h hot gas layer';: the -hu ge-increase ; in the '!-room heat flugxes caus'ed by such a proces' to the ignition of -lhe

remaining unburnt, portions of fuel in the compartment Clearly, aIt-.

'hough such a flash~over mechanism might be-very difficult to.'model, it-s

.neglect ould_-lead iLo.ove rly sim plistic predictions of ;the 'if'e hazard

for ,a given room.

An 'alternative approach.t6o the growth problem is presented by Fitzgerald 1.181. This 'probabilistic- analys~is ýviews.t 4le compartmentc.of in-

!teest as '"a group.of'.fuel packets' separated by, spatial barriers:.K - Fire"

propagation is:: trea.e8d. -with an'..event tree..type.-analysis, simiilar, to that

used in nuclear plant accident analyses, with the frequencies of "barrer failure" to be derived' from the heat:transfer cbnsiderations.

'For instance, suppose there eare three fuel ,pacets in the compartment

of interest, which -if burning; can only transmit -heat to each other by

convection or radiation (this defines a "fuel packet"). Assume in our

- 17 -
-•7CT72. -- , •:•

.... i A9 s.

scenario that packet I is burning and .that packet:2 .wi • 1 .ignite before

packet 3 does. Packet 1 will either self extinguish-befo e packet 2 ig-.

ntorit will :not, ':and -there, are:'frequencies -associated,"w th :each'of -

these -events which theoretically. can be-derived from-.heat transfer con--

siderations. Similarly, the frequency -that the fire propagates fromtpadket .2 to packet 3 can: be-.-found, and so thei frequency, of total room

involvement can be, computed. -,An event tree for this :-scehario is shown -

in Figure 2.4. - ...

An important advantage of this procedure is .that the exceedingly complicated problem of fire growth can be resolved into~smaller and simpler

calculations. However, sincethe growth rate of ý-a fire is strongly dependent on :the current' size anid shape of the fire, as will be discussed

in greater detail in Chapter 4., the barrier failure- frequencies are

functions of the fire configuration. If the room contains four fuel

packets, the frequency of: propagation front packet I to packet 2 will depend on whether packet 3, packet 4, neither, or both are already burning. Thus the number of barrier failure frequencies -which must be computed increases geometrically with the number of fuel packets. Adding

to these difficulties, the failure frequencies also depend on time,

since the temperature and extent of the hot gas layer as well as the

fuel burning rates increase as the fire continues to burn.

Thus, although the explicit separation of the probabilistic framework

from the heat transfer calculations in Fitzgerald's model shows promise,

its advantages over the synthesis approach seem small when analyzing

complex fire configurations.

-18 -
Self-.

Termination

of Fuel

Package I

Established 9Mon tofi igniionof Fuel

-.ion Self- Selflpiiti~i TernaionTermination Paka~ 

of Fuel of Fue"

Package 1Packeoe 2

Ignition lin

of Fuel Ignition

Packaoe 2 of Fuel

Non Self- SelfTermination Termination

of Fuel of Fuel

Packaoe 2 Packaqe 3

:.. Igniton ,•nelt.

of Fuel

Package 3

'ion Self-

.,. Termirnation

. Package 3

Fig. 2.4 - Fire Growth Event Tree Model [18]

11

• - 19 -

-- --- - - - - - - - -- - - -x'}..&~-
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1⁄2 -'

GENERAL METHODOLOGY

----'I-nw

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~

£

4-I-I.--.

A complete model of the thermal hazards caused by a compartment fire in

a nuclear power plant should incorporate sections to handle':each 'of ýýthe

four phases (irnitiation, growtfh, f ully-developed burning, and .extinction) .in- afire's history. We note that the first is important primarily because it determines the frequency at which components will be exposed to fires of: any magnitude. nI this report however, we shall -

neglect both the period of ignition and that of extinction. The period.

of ignition shall be ignored because the results of researchers [19,20]

indicate t hat the frequency of fires within nuclear power plants -can be

estimated to a good degree from the available data. In the remainder of

the report it shall be assumed that. a small.,fire already exists, in the

compartment of: interest.- As for the extiinction period,--it can be, seen

that-the heat-induced stresses on any component during fire decay will'

be at most equal to the stresses experienced during growth and fully-developed burning. Furthermore, a fire in a nuclear power plant compartment'probably will not be allowed- to burn to completion; suppression efforts are":expected to be finished be-fore all of the available fuel is

consumed.. 'Since there are a number of simple models in the literature

which will give reasonable order-of-magnitude estimates for stress level

decay as a function of -time [9,101, we. shall not emphasize analysis of

this phase either.

- 20-
Thei o ver all1 model proposed there ,foreý' has twoparts. The first deals

.::.-wt e%,growth periodl while the s econd part -concerns the fully-develpeburning phae' In both parts' th a pproacli ýsha-ll be pr~obabilistic

ie. our state of knowledge shall be exprss'edi terms of probability

distribu~tions), an approach which requires- that:-we .identify. and quantity

the ,uncertainties in our analysis. I n' th~e-remaintd e r. o f -th i-s chapter, w~e

halIl discuss the various types of'u nc e'r taýin tie s e countered in our

probleml and the the methods usedA to: inopoaa ths netinties into,

t he a n a l~'i~s.1

31 IDENTIFICATION OF UNCERTAINTIE

Ideally any pysical situlatiUri -can be- modeled .usingý th6 bsic' laws of

mass, -momentum, and energy conservation:.: The solution of the .resulting

equations, piu~s any~iecessary auxiiary equations modeling, for example,

the chemicaal. reactions involved, would in princip-l1e describe. fully the

situation. we wish to- exa Imine. However, a burningfire -frely is. an ex:

tremely complex phex-iomenon7 involving" tu rbulent' ýtran~sport-of dheat; masss;ýand momentum, radiative heat. transfer, and a multitude of chemical reactions. Its govIerning equations are- sufficientily. numerous and involved

that analytical 'or numerical solutions are presently impossible to find

for, all but' the most simplified configuraItions. Sinc'e the presence of,

room walls will 1Add f t. even more complexity to. the equations, an accurate

.ýp o s 

.f 

a< 

... .. description of the-progress of a compartment fire presently cannot be

derived.purelyl.from -first principles.

Owing t6o' the .lack of- a rigorous overall modl- for fire, much. of the

information incorporated in our fire model is empirical. This in-forma-

.1

4'

- 21 -
tion may be in the form of correlations of experimental data, using correlating variables obtained from dimensional analysis of the governing

equations, it may be ini the form of uncorrelated data obtained from

highly specific tests, or it may be in the form of engineering judgment

and experience. Such knowledge can tell us how the burning rate of a

given wood crib in a given experimental enclosure will vary with stick

spacing, or if a given electrical cable on a given tray and exposed to a

given heat flux will ignite, at least within certain degrees of accuracy. However, it is not immediately clear how accurate this information

is if the test conditions are changed slightly, and more importantly how

useful it is in the field where conditions are anything but standardized. An interesting picture of our empirical state of knowledge is

given by Emmons (21], who looks at the flammability ratings of twentyfour different wall materials. The results of standardized flammability

tests for six European countries are shown in Figure 3.1, and the correlation of the results is rather weak. As an extreme example, one material was rated the least flammable by one country's test, and the most

flammable in another's.

The uncertainties in our knowledge of fire which contribute to the

confusion mentioned above can be broadly classified into one of two

types: statistical uncertainties due to the random nature of fire, and

state-of-knowledge uncertainties in our modeling of fire.

- 22 -
A00

A A 0

0oA

.4

a,

I

I'

A

j

24

231

22-

21"

20.

19.

18'

17"

16'

. 15"

13'

12'

10'

9,

7-

6-

5-

44

3-

2-

1

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OA

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8 A

I, . . . 4 * I . 9 1 . 4 1 I A I 1 .5 .4 .5.

1 2 34A567 8 9 10 11 12 13 IA15 1617 IS19 20 21 2223 A

Varieties of Wallboard

Fig. 3.1 - Flammability Ratings cf Wallboard 121)

- 23-
,-:. '•:L- :,._,_ • : . . - . . % - _ .: -' • - L :, . : . - .-. • • • • ; i

3 1. 1 Statistical Uncertainties

Fire is a random phenomenon. The behavior of a flame at a given time is

strongly dependent ,or the surrounding' environment as well: as its own

current configuration. Random fluctuations. in the air flow field around

the fire and in the fuel bed beneath it act as randomly varying boundary

conditions for the flame. Since the burning rate of a fire is partially

dependent on the heat flux from the flame back to the fuel bed, and this

feedback depends on the flame's current shape, these fluctuations can

lead to different burning and growth patterns for nominally identical

rooms.

The behavior of the heat fluxes and temperatures in a compartment are

direct functions of the fire's be~havior; it therefore follows that these

fire-induced stresses are also random variables. This variation caused

by the randomness of fire is termed statistical uncertainty; if we perform a particular experiment a large number of times under "identical"

conditions and measure a specific variable, e.g., the temperature at a

particular point in space, we will obtain a frequency distribution for

that variable.

3.1.2 State of Knowledge Uncertainties

The statistical uncertainties described, above are inherent in the nature

of fire. Even if our knowledge of fire were to increase dramatically,

we would not be able to reduce these uncertainties significantly. Such

an increase in knowledge, however, would markedly reduce our modeling

uncertainties. These uncertainties arise when we attempt to predict the

frequency distribution for the parameter of interest, applying a "sin-

-24-
4 1

-- ~ I

~44~~44

*4

#~~~

I

P. moddi (sim le in the r.sen'se_ that-,.it is. solvable), to. a complex prob-

"22].

-.O urp un!certainties .:in them :odesling n f.ire, .agive ca. n.be further

categorized as being unce- rtainties in th'fundamental physical modeling,

in .the" additional' modeling .required, -:to aallowr !eas onnable`- tat ....o ... .i 

times, and. :in the.values of-,the parameters t6 be used when the model is

applied toward'-a real compartnent fire. To illustrate- these -different

,sources of uncertainty.ý we consider.. e growth period mode'l. presented in

-Chapter 4.

The- growth period model is a -synthesized. model; its components are

" the many. models.,available- in the .literature which.d-dscribe various ph; -g -_ -f in t . e . n rat 'as a. " nc I I -o..-•.-. 0:

sical phenomena,'. including models for-the burning-"rate as a -funci-n of .

ventilation, flame height as a func-tion4 of burning rate, -radiated :heat

flux as a: function of flame height, etc. In general, each. of. these models was developed independently. Our uncertainty in the modeling of

the basic physics,-of fire therefore is due not only to our uncertainty - .

•-in how good each-model's predictions are-under: the conditionss it.was developed'fdr,;but also to our uncertainty in- the ynthesis of these: independent -models. One of our primary concerns is whether or not the

synthesis contains enough component models (i.e. if all important phe- -.-

nomenahave been- modeled),.

The second type of modeling uncertainty arises because we usualily

cannot implement the synthesized growth period model directly..: Analytical formulae can, only be used -.for the simplest configurations; for more

general cases,-we must resort to the computer. The translation of our

model into a working computer code requires additional modeling of the

- 25 -
-.4 4

~ V'

givIen problem,: including' the"discretizati •i"'of all space, angle, 'and

t Oime:dependent variables. .This',additional. modeling leads.,to, a number 'of

..changes, in>:the :physi'caL: s ituat ion,: de~pic~tedi'by the mode L.' ýýSome o h

more' .important ones .iniclude the separation- of a single flame o6ver-a long.

fuel bed into a number of 'smaller separate flames, the realignment ofan

arbitrarily oriented object parallel to one of the room boundaries, and

the. modelingof :a.- three-dimensional fuel element as a'. two-dimensional

slab. The uncertainties due- to fuel bed discretization are discussed in

greater detail in Chapter 4. While this additional modeling is generally not- required from a theoretical standpoint, it is useful practically

speaking, if.a solution is to be obtained :within reasonable'time and expense limits.

Finally, we note that we cannot specify the dimensions, thermodynamic

characteristics,' and..geometry of a fuel bed within an arbitrary compartment to the degree of detail required by our models. A single slab of

wood may have strong non-uniformities in density; water content, and

.structure, all of which will lead-ito variations -in the burning .rate of

the slab. In the application of our model to a fire in a nuclear power

plant compartment therefore, we can only specify representative but not

exact fuel bed characteristics. If we are describing a fire in a class

of rooms (e.g. -cable spreading rooms), the fuel bed characteristics

will be even more uncertain, due to plant-to-plant variability. ''The

variations of the parameters specified from those which actually describe the room under consideration clearly lead to additional uncertainties in our modeling.

A""

- 26 -
- . r- - 1

•:-- -- YAA .--:I, .A-A g•-{ : 

'-..t; A . •+

•)! .};.]..... .,,,:3.2:". FREQUENCY-MAGNITUDE RELATION ýF6R FIRE SEVERITY'-.i: 1 .. '.-•

. " SA isticAAppeoach

3.2.1 Aproach. A' Sttistial _

In principle, a frequency. distribution for the fire severity, could be

constructed directly from.eix.amination of data from fires in experiments

and in the field. SSuch an:.approach is used by Flemding'et a L. [231, 'who

utilize data estimates from fires in both nuclear and non-nuclear ins

. - . tallations. An important feature of their analysisis theiruse of the

volume of the room directly involved in the burning as a measure of-fireseverity. Since :they assume that the burning volume may be roughly ',

-characterized as a cylinder whose height is the height of the compartA ment, their measureof soverity reduces to the effective diameter of

- _." , A•[ , . .A.oA 

ily-" "A

tha t-c'ylinder.. A plot of "the data and their fitted regression line is

shown in Figure 3.2; a frequency distribution for the actual fire severity and: the scopression time can be easily derived.

While this measure for fire. severity has some advantages, being certainly related to the maximum severity of the fire, and being easily

measurable after extinguishment, we recognize that a componer.t may' be - A

...damaged even if it is not directly involved in the flames. Although the

"volume :of involvement can be redefined to include areas damaged but

not necessari-U burnt, a more direct approach is to use additional meaEk.."; sures of severity more fundamental to the physics of the -fire, namely

. "the spatially dependent temeperature-and heat flux distributions. Unfortunately, there -is little data for the temperature and heat flux lev-

& els achieved during compartment •fires, either from experiments or from

•" " i ~real' fires. 

-•

....- 27-

- AAAAAAAAAAbAA - - -Aý,A.AAAA~IAAA A r AL ~ ~ A A-AAA~~~ A A -
toil AfUSorbat

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0

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E

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0' nSO.fl. Fa ,

C Ia blge red

0. 3/0

'S

ýO

0 5 15

20

25

Time To Pul OU, ýFure' hr

Fig. 3,2 - Data for Compartment Fire Severity [23]

- 28 -
3.2.2 The Reference Model Approach

Aside from- the fact that there is relatively little physical data avai-

.lSab f or r`fxi1la -le~ copatmnts-1. m fie another diffi culty. whicharises

if we attempt to, construct a statistical model for fire.severity is that

the.-data 'is note. usually direct~yapial to nuc-lear 'power -plantrooms., - As we .have emphasized earlier;,;a fire's environment plays a crucial role in thde.,d e yelopment.of the fire. Extrapolation of test results

'for a.wood:--fuee f i re in a wel.1Vvent. ated sma I roomto a large, e1ec-.

tr ica I insu lat idn- _fue-led, p•poor ly- ,venti lat"ed:: cable :spreading .room i.s a-,

.difficult procedure.

"-:In. orderto:ha-dle-fir insituatisfor.w. hich we-have little: or no

exper imenta l', data', "we utilize a deterministic model o0f-vfire which pro-

-vides.-us, .with predictiols of-fire severity .which incorporate the physica1 characteiifstics, of' the,-f ire'.. -These predictions serve as'reference

.points for our uncertaintv-analvsis;.

_A

3.2;.2.1 Simplificatios' Due.to Diffuse State-of-Knowldge.

Before we describe 'the refere. neemode Iapproach :in detail we firs assert that our. state-of-knowledge uncertainties in ,the. behavior of fire.

are much larger than the statistical uncertainty. Die result of this

assertion is that we 'need .ony prledict the average" -valueof any given

fire severity:-measure -,rather than -its! entire freque cy' distribution'..

As was discussed earlier in this' chapter, the -randomness of fire ensures that even if we are given a specific fire, any measure of thermal

hazardowe consider.:will'.hav'ea.frequenc distribution.' Thus, ii general

we-.must predict not .aý sinigle: value of fire stress-:. (at any particular

- -29 -

- 4 I

2" -"
point •in time) but rather a frequency distribution' for fire stress.

However, because our: modeling uncertainties-are. non-zero, our.predicted

.- .. disribtion- wi-il pro ably i f er from -the -.actua isru .Rather

than making ra-. sing le.,prlediction;, therefore, we produce a :sriesof esti-

-" " mating distributions. and -'subjectively assess the likelihood (i..e the

.probability) that each estimator is the true distribution., -In Figure

3.3, we;.illustrate the approach;. with a, simple example:, where, we believe

.:the left-most curve is the actuali. frequency distribution with probability pi' the central curve is the"actual -distribution with probability P2'

etc.

The general "probability -of frequencY" methodology -[24,25] outlined

above is simplified if our state of. knowledge is relatively weak (see

Figure 3.4). In this case, the statistical uncertainties (indicated by

the width of the three frequency distributions) are.overwhelmedby the

state of knowledge uncertainties (indicated by the distance between the

Scurves), even if the statistical uncertainties are not small in.an absolute sense. -We therefore-nmay treat the frequency distributions as high•

ly peaked curves loca'ted at -some representative parameter of central

tendency (i.e. the mean, median, or mode), and so our problem reduces

to the task of predicting this. parameter, rather than the entire. frequency distribution.

To make our following discussion of the reference model approach concrete, let us assume that we wish to predict T the temperature of the

air surrounding the component of interest. From the above discussion,

we decide to construct a probability- distribution for the median Te of

the frequency distribution for T The distribution of Te will *be com-

-"30 -

- .,Y.- .-PCP.C~CŽC. -p;----:cP~r-- p<-- --.--.- - - - - - - - - - - - - - - - --P
r

-A 1K

1'

.r.--~r

ii; I

P,

P

P3

P,

|1 I 

I

x

Fig. 3.3 - Uncertainty in Frequency

Distribution of X

Fig. 3.4 - Unce

Dist

stat

pp

P3

x

rtainty in Frequency

ribution of X (small

istical wicertainty)

A4.,

... ;, . ::: . ;. 7 -:
:~~~-<L

posed .of three elements:1. a- deterministic, reference model (DRiI) a set

of probabi.l ity :distributions: for: each -input parameter .,required by" theDR'.a nra prob y dstribution which meas es our.conf ience in the

accuracy -of the 'DRN.`

fri.

4

'it

I;

3.2.'2:2 The -Deterministic Reference Model

The deterministic reference model is a model which: explicitly incorporates the phrsics0 of .the.probhlem. Given a set of. input parameters, the

DRN will yield point estimates of each of the physical variables (e.g.

tetperatur.e, heat, fIux.,. fire size, etc;) used to measure f:ire severity.

We trealtthleURN's prediction as-an '.expert's opinion", whený we later

quantify the uncertainties in the DRM, we are expressing our belief in

the credibility-of the expert in a spirit similar to.-the approach used

by.Apostolakis and Nosleh [26] and Apostolakis et al. [27].

It should be noted that the DRN does not necessarily provide a

"best-estimate" output, although efforts.are usually made. to ensure that

it-does.; if there are competing modelsavailable, simplicity and usability are also important factors in the choice of the reference model. A

good example of this last point is illustrated by the choice of flame

height models.

The model for the flame height above a horizontal, pool used in this

work is that given by Thomas (7]. His semi-empirical correlation is

I:-" L

Z = 42 Df [ m/pAVgDf ] (3.1)

However, a better correlation for flame heights is- given by, Steward

[28]

log 1 0 (2Zfl/Df) 0.20 log, 0NCO± 1.21 (3.2)

- 32 - 7,

~ .14..>.

Ak
:W!_ 

~ L i:•

hwhere.

2 + 2

Q2

p r+ /2) )1

NA

V..,.+

V A '

W0

r: - -+°toicýiometric :,air-to fuel•jratio (kg air/kg f.ue

an h ubcitostand f or: the6 fuel proete at :the 'flame 'base:.

there the to rltionsare:shOwn in Pigure3.5, and Ste--

ward's p r edii age seon ýt e ttr f ir a i• de.r.. rangoffires. Howev

full~~~4 knw hsb .creain 

of, the ýburning

fe Suh detail is not easily -foundý. f or the various, fukels. Within' nu

clear power plants/ there.binga myriad of varieties of eetical in

.,,ultf~n~o~ Frthrtl -",Thoma' r quit dqate deut for the:-

7 type-and sizes of ,. fires Oexpected (solid or liquid fueled, and -moderately.

sized). -

3.2.2.3' Icortpoat~n9 POmeter ara 'Uhcertaiflties.

"-- -once theDR isconstructe I, the uncertainties in the input parameters

"ar- • propagated"hrough theDRII t obtain- distributions for. the output

var..iables.>.• : Suipposourrferc modl, which. may be a sihgle equationb: ~~ t.-11: *ý 

'i 

h m , a ,"

The these processes iparameters and..yields a single estimate TD-.

The relation bet,.een T and T'e;-shal+l be 66scribed later If'we input

:.two different sets of values,

33........ 7... ."...... .:-...:-

-,: " " .- -, --. '.. 

-- 

v '"'-...

J:,. .• . .

- '
2. 0'

IT :Thom a

 ̧!:0

0

low

0

SEOwa ~d

0.51

-1.0 0.'0 1.0

Fi.'.'S -tl -.- ' a s.. ei h

Fig".3.5 -Comparison.of Thomas' and Steward.s Flame Height SCo . rrelationsr [28],

- 34 -

-V - ...-..-..- ** 7~ ,t.*-V

................................- .*)-: -.
d

A

Z.. '

xx

thales for Tw i w ll resultw d if e e cosrc te oftu theeent eachT

LDRM RM

of m the input ectors eby nrand m saping fr"V~om h ros bl tyditi.'

buti-On for 'K (assum~n that the, p~arame~ters 1are Aindependent), th alues of -TD calculated will alsoý haeadsrbuto' seFgr .)

If our referene moelis -actually iml funto hspoaa

ti on of the'7 a'pram~erý uncertainties, nay beoaccomplished by, analytical

methods,-, e g. the methodý of moment ipropa'gato~[9' As the mode~ibe-

.- comes more complex;, we resort to numerical~mt~-

A s.imple :but brutal -numerical appro~ac iste Monte smltio 'aa 

.technique, where: the input vectors-are.,propagated through the 'actual ref erence model. major drawback of this: method is that. very large nunbers, ' of amples.- a6 required to.,obtain -goodresults, :and if the :model i

- in the form-of a comp~lex ýcode rather ha a s~ingle equaton th siua

t~ion Costs can be prohibitiv'e.--

"A similar but sghtly. more re fined approach utilizes :response sur- faces [130,31.1. .,The main -difference between eMonte

Carlo approach'is that in the-former, the reference model is roplaced-by

a response surface,. an approximating function for the model. Since this

reponse surface. slys, typically a sum of polynomials of each of the input

- 35 -

~ -.

3 -C;~,~A4 i 3- <3 C4.~3;3-- -.
I

C7

-J

Fig. 3.6 - Schematic for Propagation:

the Reference Model.,

of Uncertainties Through
."indepne: the tiespent ..per orming- ca. cu can :,'ios 

L]:" . . duceid (tremendously,. thi s. .bein~g of.. great import-ance, if: the number, of,-.ran - .: ./ .].:.:• :

/":• . ""dosinpt. vectors aghj.Frther,,, li~ if he response.,sur•face."s.o'.:"/):.:~ :i:•]i..•

:'•." ~ m t i•) e'n hmep 6f-i ls " Whih Fus Co tb d ce d. ace is~c cos'..[ n:t e.. -,i.$.(-::.!•~~ " 

L!- number nedefo aMon te: Ca rl16simul,!tion of comparable accurac.

8':',, .*2.2.4 1Incorp'orating Uncertainties in the DRM

Tetoremaining types of mobde ling uncertainty to be 'han'dled..are`the ..

S.,.L,.-•- uncertainIty .in the,-basic physical. modeling and the* uncer tvainty in "..the

•(!1-numerical1:implementation:o the physical modeling. Although they,"6ome

4:1-

•i#..from somewhat .different sources, we consider both-collectively .'

We recalp that we wristhe to construct our probability distributicn: for

cthe medianexternal temperature T and that sorfar we possessn abprobabilit3 . distribution for which incorporates our uncertainties in the

T[problem parameters. We now define an error function E such that

,.'-•b:"....Te T T' T RM .( 3.3 )

ET may be thought of as a factor which measures our confidence in the

mprediction of the DReM.

i f SZ snce even iffwe were absolutely certain of the..actuai e values of the

problem parameters, we would not know the exact factor by which thefo

SDRM's prediction varies from the actual value of f a we mustse construct

ba distribution for ET which incorporates our state of knowledge untertaintliesm A distribution weighted to the right of the point E equals

!i . .

TT

urile indipcates that we believe that our DRh tends to underpredict the

•'1 - 37-

.Am •i•. % ... " .. .... .. " ' ... ' "
actual 'value of,-Te ,--whil.e a: distr.ibution .weighted to the' le ft' wil II:in-7ý

dicate the opposite'. The distributicon -for ET may be' in the. form .ofý a. histogram; one possible, pro~cedure for conistruction of the histogr~am (ýinthe a3bsence of strong, data). ,is as-follows.

1.Choose the. upper and lower bounds for the histogram.

These are the absolute..values Above. and below .wh ic h the, asesr-osnot. realistically expect to see anyocr

rences given hspre$sent--s tate of knowledge. Any such

:occurrence would be classified as an extremely rare

event.

2. -Roughl quantify the degree to. which the model overpredicts. the actual stress.- This percentage will show. what

fraction' of. the histogram .area is to the left of the,

point ET 1.0.

3. Roughly quantify the accuracy of the reference model; decide'how often the, real value., will "be within a certain.ý

factor of theý predicted value. This will indicate the

.peakedness of 'the distribution about the point ET 1.0.

4. Select histogram divisions. These divisions should be

selected' in such a way that ýthey' are meaningful to the

assessor..- Thus the probability- that the error factor

will be between, 1/3 and 2/3, is easier to understand and

judge than the probability that it is between 0.42 and

0.71, for example. The number ofhistogram divisions

should be low enough that the assignment of probabilities

to each histogram block in step 5 need not be extremely

precise.

-38-

• . .; • O D b••?¥.L::-':•' " . % ' - •b. • - -. • •v , ,- ;.i•,L -• L -. •• • .• • •• % <
5. Asign pr-obab i ities to .-each. histogram block subject to

the' above con s t raints The assessed probabilities should

reflect•the analyst.s .state of.:know ledge. Too precise

values generallydo' not accurately-reflect the assessor s

uncertainties. ,Final-ly, the sum-of the probabilities

should clearly equal unity.

6. Check the resulting histogram and -adjust :if necessary for

consistency. The analyst should.. ensure. that the histogram not only complies withý the* ab6ve -restrictions but

also that he agrees with the statements made by the his-

.togram.- A good test of the histogram is .to--calculate its

-vari6dus percentiles. and Verify i f these values do agree

with the assessor's beliefs.

7. The analyst may use one of a number of techniques to fit

a continuous distribution to the histogram. In this

case, he should check and see if the percentiles of the

fitted distribution conform with his beliefs.-

In assessing the histogram block probabilities, a variety of different

forms of- evidence may be used. Thus, we may consider not only data from

experiments and .real fires but also predictions of their reference models -predictions- by experts in the field,. qualitative information from

the literature and, critical assessments of- the. limitations and assumptions in the DRM used.: Even., if data is available, it should be mentioned that a subjective weighting of the applicability of the data to

the fire situation considered must often be made.

- 39-

1--- - -

- - - .. - .-.-. ..-- 
iý 2

J4<.~.

~V<~~"

It is important to note that the distribution for the error' factor

strongly depends on the measure, of stress considered. -For example, we

expect that the error factor for -"t he:, DR,'s, predictionof, spatilly de-'

pendent, heat fluxes. will .exhibit diffeentbehavior from the error;-. factor for the predicted fire, plume. temperature .-. We further "note:that

-is dependent-on the problem parameters:-.our: assurance in"the DR!s -pred-.

ictions wi l': change ,-accordfngý, to the .,part icuiar fire .mode led As, an.

'.:: .-. iiexam ,,. w e w .l shfow,-...in. Chapt tr',4,,-that, our modeI does a f airly, good job

f orO,- ,sma l' wood-fueled fires..: on:.the other hand, we have .less confidence

Ain our simulations of larger fires, or fi res fueled by less-. flammable

materials suchl as"electrical-insul'ation

- :40 -

-J
Chapter 4

THE GROWTH PERIOD .MODEL

The emphasis of o'ur model fdr .the -ýfire, hazards during -the f'ile growthperiod is on the description o f fire, g rowth, since thle- temperature and

heat flux fields which control fire growt also" determine component damage• Ini other words-, the thermal :stressesvý. imposed on-a ;component, are

available as intermediate quantit~ies, in our., fire" growth model" and so no

additional- models are, needed.

The model forfire growth is coastructed as described in Chapter 3.

A reference model, which deterministically. predicts-the progressof a

compartment fire is derived in Section 4.1 from the physical considerations-of the problem. The .input parameter uncertaintihes to be propagated thro.ugh•• the...:ref.e.rence .. model are. described in Section 4.2,.and the mo-

,del ing uncertainties which contribute to the DRM's error ffactor fcr each

o °utput Li variable are treated in Section 4.3. In Section 4.4, we briefly.,

out:line some DRM simulations of actual experiments, and in Chapter 5, we analyze two different. nu•clear power plant compartment fire scenarios.

4.1 THE REFERENCE MODEL

As was seen earlier :in Chapter 2, there currently is no overall model

capable of dealing with-.the dynamic process of fi-re growth. While progress has been made towards detailed modeling of a growing, compartment

firestarting from first principles [32], the practical application of

such a model to real fire situations may not be possible for some time.

41
An alternative approach, used in this work, is to model the growing

f ire using a quasi-steady which -ra.ework incorporates the large number

of steady-state models available in the literature for the different aspectso ofa fire's behavior. Thus, the reference model. computations proceed as follows.

At each point :intime, the steady :state7,burning rate of an ignited

fuel element is computed with the aid of -an-.appropriate model (the

choice of model depends on the nature of the fuel whether or not-the

fire is ventilation controlled). The heat release rate is computed from

:.the burning rate and. the effective heating value for the fuel; this

":qantity inturn.is ,Usedto compute,, the' •heat transferred to the surroundings byradiation and t..o the accumulating layer of hot gases, near

. .the ceiling by convection. Upon determining the average temperature and

-.thickness of the ceiling gas layer, the heat fluxes to any non-burning

fuel element in the room are calculated, and the element is considered

ignited if various ignition criteria are met. The size.and-intenisity of

the fire are thereby ,updated and the process continues, marching forward

in time. A flow diagram for the basic computational scheme is shown iA

Figure 4.1.

In-the following reference model, it: will be-assumed that the primary

mode of heat transfer from the flame to an object will be thermal radiation. Radiation is important during the growth period for a number of

reasons, the most important being that the hot gases from the fire are

initially restricted to the buoyant plume immediately above the fire and

a thin layer next to the ceiling, leaving most of the room essentially

at ambient temperature where convective 'heat transfer is small. These

- 42-
ý t,

Cc Im pute Bur nin Ra teo~

Loyer Noro Ceil'ing

Rote e.nto Comprtmen

Determin IiNo bu rni

F~t lmenI gnt

--4

A.

•!•: •

Fig. 4.1 - Flow Chart for Growth Model Caiculations

- 43 -
-1-

effects wIl be. accentuatod in nuclear power plant rooms, since .these

rooi ms ted to be'- large (anid thus. possess large amounts',. of :thermal. iner-, . . "

ti-an thecombujstibleý fuels.'tend toi be. widely separated..

Of course, the quasi-steady and the'. radiative, .:dominance assumptions

may-lbecome-weaker as the f ir~e growth acclelerates- towards fully-developd,

burning.. Smaller .time steps and more detailed fluid flow models can be

used alleviat.theseý tto .respective ,'problems,' but o1r -k-owledge:of the

burning.conditions .during: the period immediately preceding'flashover 1y ".prec

so weak;,"that: these. improvementsomay not. be ofý much help.

'In- the remainde'r of this"-section, the. various component mo'dels: d'iag -

411

.rammed in Figure 4.1, which are neede.to synthesize the- reference mo ' -

del, are described broadly.- The computer program COMPBRN which actually, - -

implements the reference model calculations is described in Reference

k-...

4.1.1 Heat Release Rate

- The heat release: rate of a- fire'e, isthecentral variable forvany computaa

tion- of the fire's behavior, whether inside or outside an enclosure.

The -tot-al rate of heat released is given by

r", the burning rate, depends on.-the available ventilation, surface area

. of the -fuel, and: the nature of. the .fuel. A fire is termed "ventilation.

controlled" if 'the, rate 'of burning is limited by the amount of •oxygen

available, one -cubic meter of air being capable of combusting.approxi--

mately 113 Btu -of organic fuel [33]-. If the fire is not ventilation

- 43a -a
controlled, "ts 

bnstantaneous- burning& rateis 

b blimied thecurrent',.

fulsrace, area.

,!}.,- 

.. . . .

For ventiluation co6ntrolled faresa, 

re

For fuel suriace controlled fres the burning 

nit a

fuel surface is given by

-q~ 

4

0 4'.

Equation (43) is.a simplified version of a gross heat balance,,Itperfor 

med

- .the":. surface --of,,the fuel. As explained in 34], the ,raXte at, t,7hich.

•hea .be supplied to volatilize the.fueýl, ,bed equals, ýhe .,totalt I rate

i~~i.!it 

va•5lat:l.z. 

....

the," 

-

:thatý heat .s supplied, to' the fuel bed minus" the rate osf heat loss.from

-, th.ue I bd, e.

, - . "W 

1 f - f1,r + " qloss

.. Or 

.

ii:"""- . Clearly, qf"r"and q'4"oss can vary from fire to :f ire. ýWe introduce a

Constant m0 to model-this heat bal'ancFi,: but If experimentalev'idence.is:

-, - i,-"-Floss-.-

::<•:i•..not,,.IAvai~lahl6,, 

•the ,actual va~lue tOl. be•. .. .used. in., calculations- may." .n tb,.- be° : .:m 

SIeay, d. Table 4. 1 providey, fsome rIepresentative 

Wvaluesi, fioroC- e aor

a variety, of materials, Figure 4.2 provides empirical verification of

Equation (4-3), Figure 4.3 displays the.-behavior of m for liquid-pool

FFfires, and Figures 4.4 and 4.5 respectively provide correlations for in

0

for fires over wood-cribs (assemblies of wood sticks or. beams.stacked 

in

-evenly, spaced layers of alternating directions) and liquid pools.

- - - - 44 -

-. FFFFFF.F~F FFF FF1~'C4~F - -

- F'FF~F if-----------------u
Table 4-1- Best Estimates forý C

-C (kg/J) Fuel Refer'ence

553x10 Wood

1 9,_3 x10 Heptane.[4

[34]

Table 4.2 -Best Estimates for, Hf

i (J/kg)

18 6 x 107

43 0 x107

1.85., x 1 -7 2. 70 x 1o0

FuelI

Wood

Oil

Cable

Reference

Insulation

[44]

[45]

Tableiý.4.3 - Various Estimates for Y

Y , . ' : Fuel Reference Notes

At

41

4~>

-V.-

.35. - .36

.30 - .40

:20 27.

.16 17

17-

17 7

.215

.42 -

• Small-scale

Benzene [136]

Gasoline [36]

,-Butane , . [36 "

Methianol '[36]

Frir 1 [42]

"PTMA . [42]

Polyethylene [42]

PIMA [40]

7.6 cm <

D

30.5_cm <

71.6 cm <

D

D < 122 cm

= 122 cm

D < 76 cm

D < 122 cm

= 7.3 cm *

2 3 cm < k < 122cm

experiments at .highpressure

-45-

.7 ~...~..4444/ -> <.4~ ,> 4

4S,'A.,~'~-;~.. - ~.~47~~ ~A~~..7A~4.4j/.~ -~ .-- 4

-~.' 4

-. e,-~44 - ,-. 4.~~'~4444.,~
I0.4

Fig. 4.2 -Burning, Rate vs. External Heat Flux [35]

E.

CO

G60oi.%=n

I

40 100 400

Pool Diameter, cm

Fig. 4.3 - Surface Regression Rate for Liquid Fires [361

E

-o

LO

0

(A,/As)sib>. cm

Fig. 4.4 - Burning Rate Correlation foT Wood Cribs [37)

- 46 -

- .. - -. -- .,~%>r2•v.t1x- .z-xaAa4.~z~;,~?otc< s~x'. w~. r',r~z<vt-r <to~-,.
Burgess' Radiciion H y p o t h a z i s

214PO Y,

10- J.

Z:- cm

0

Spolding's 0

A

111 102 ConvoclivO 

C11 Theory 0.14 %P I 0 (SOa-0

0 3 A 57A d d An cm,

Og 9 = 3.215 10 .4 gmicm

1 10 10

10,

V/ . g d 3p6T /ugag

Fig. 4.5 - C orrelation for Burning Rate [38]
412 The 'Ceilinig Gas Layer.-

. The:laye.r e of hot' gases near the.. ceiling. which .accumulateý' during a cornpartmnent. fdire may play p significant part in the growth rate of-the

.fire. .Heatfluxes frm.'this..layer.-preheat-: non-burning fuelel.ements,

fire. .xe He fr.et _,ri

.. reducing.the time to ignition, of these elements

In. order to characterize this layer, we need to know both its depthý

and-.its average temperature. Although some work has been done on the

gas layer created by a s:ialI fire'under a high ceiling [39],,.we adopt a

simpler model, .employed~by Pape etsal [16], which allows for-more gener.-

al situations.

In this approach, we perform a gross heat balance on the gas layer in

a manner very similar to that used for the entire room in the fully-developed burning period analysis.described in Chapter 6.

The gas layer gains sensible heat by the transport of hot -gases from

the various: fire plumes in the room; it loses heat by the flow of hot

gases out of the: compartment (either by the forced ventilation system or

by flow through any doors or windows), and. by convection to the ceiling.

Although radiation exchange with ;the remainder of the. compartment also

-occurs, the net effect of this mechanism on the heat content of the gas

layer is fairly small during much of the growth period, and shall be neglected.

Equating the heat losses to the heat gains in a manner consistent

with our quasi-static approach to the growth problem, we obtain (assuming a single fire for simplicity)

.. (l-y)Q W c (T -T) + h A (T T ) (4.5)

out p G A ceil G ceil

48-
In the fully-developed burning:period model weassume that the: hot gas - ' >w :? a s ~ i m et he ., t hp e s s o fo a s r . ..

-2 ,,•('. ! es occupied the entire compartment volume, and s'so the expression for

Out 6u. t is very simple. When the hot ses gathernin. a layer, as-seen in

ga se _l ye ,

v igure 4.6 . the.computations a fesomewhat more"difficult.

Assuming that the stratification between :the ht epper layer and the .

cold lower layer is strong, the rate of gas flow out of the compartment

opening is given byj 15]

Wcut .67 C A0

0 A2gB(TA/T6)(I - TA/TG) -(1- Zn/B)

+ W IN (4 .6 ) .

and the rate fresh air flows into the compartment is, neglecting fric-

.tion losses within the room,

W•.: =.6 7 C A 0 p A / 2 g B ( 1. - T A / T ) ( Z n / B - Z d / B ) in0 CAA n d

x ,Z/B + .55Zd/B) + WFV (4.7)

If the fuel mass burning-rate is, much smaller than. either of these mass ... j

flow rates, as is usually the case, a mass balance on the entire con- -:

partment yields

W W.

out in

or

(TA/T) (B -Z) 1 (Zn Zd) (Zn + '.5Z )2 (4-8)

To find another relation between T Z and Z A-G n' an d,we ot tht hefr

pumps hot gases into the upper layer at a rate given by

'Z -Z + (1 w) (49)

w pi *: D /2

where

-49-
!: • :;' ,-', U• 

I

kB,

777

77,ra

r...

If •i•

I

Fig. 4.6ý -7 Fire Scenario Ns sdme d for;CeilIing -Gas Model' [15]

.5*5

K 55 Js .4. j

4

~s .5~?7 f 57 44~.' s p
~ i ! • ! •r E .* • • • • ! ! .. .i --., .... - .i j .'

Z = height of flame base above floor (i)

-0.5

The values of ý and w are somewhat dependent on the particular fuel

Iburnt.

Performing a mass balance on the hot gas layer,

- Wpl = Wout

and so

z -Z " "•d- + + m(l-o) =

.67CoA p f2gBT )(1-TA/T (IZn/B) 3A + W (4.10)

00 OA C n +AT 

Equations (4.5), (4.8), and (4.10) form a coupled system of non-linear

algebraic equations which can be solved iteratively for T , Z , and Zd S""G ' nd

I using a variety of approaches (e.g. a Newton-Raphson iteration scheme).

4.1.3 Heat Transfer to Fuel Elements

For the medium to large fires of interest to safety engineers, radiation

and convection are the dominant mechanisms for transferring the heat released by combustion from the flame to the surroundings.

... 4.1.3.1 Radiative Heat Transfer from Flame

I It has been shown experimentally that the total amount of heat radiated

away by a fire of moderate to large size (greater than 20 cm in diameter) is a constant fraction of the total heat released by the fire

j40,41,421, although there is some disagreement as to the actual value

of this fraction. There is presently no model derived from theory which

51
...accounts for this empirical observation but some

."Constancy of -the .radiait, output :fraction have been

Q . Q

As mentioned inm:hapter 2,most fruain o

r Tfl

justifications for the

.advanced [41]. hnus,

(4.11)

are of,; th6e form r

-. (4.12)

- since the black-body, radiative output'. of a -diffferential; hot gas .volume

element, is proportional toý the local TV product. However, determination of the average •flame temperature in practice %:is usually accom-

. plished' .bynmeas'uring 7sthe :radiative output- of the flame- and. then back

calculating for TJ fi . Therefore, there seems to.be little practical: advantage..for this latter approach over the one adopted.,

Once the amount of heat radiated away from a flame is known, the amount received by any object in the flame'svicinitycan be calculated

using the standard methods of radiation heat transfer analysis. Using

Equation (2.3), -:.and assuming all incoming radiation is absorbed

"( 1.), we have

o,r

F Qr/A o-f! f-I (4.13)

4. 1.3.2 Radiative Heat Transfer from-Hot Gas Layer

To calculate the radiative transfer from the hot gas layer to an object

outside the layer, we assume that the layer acts as a uniform source at

the ceiling height. In a manner analogous to our computation for radiative heat transfer from a flame, we then have

qo,ceil F o-elceil o n (4.14)

-52-

4,
-'r Sii ~ i ii: '"

The shape factor formulae necessary to compute Fceil are. given in

Appendix A..

If we model the 5

ceiling~ gas layer ass a. ~hmgnos 5 sterinal ifi .

niteislab of thickxess'D with wavelengih :independent properties , it can

be shown that with a few other assumpions.see Appendix .-D),

q aloT" iX4lI-)q'X2.A -T•(1X)[l+(l-E)X] . (4. 15)

where

X =exp(-l.,5aD)

q incoming h flux impinging ongas layer free surface

4.'..3 HetTransfer WitinthjFam

Since-.:"it is likely that any object. in. direct contact with :a flame will

be'.severely damaged in.:a short period~of time, an extremely detailed mo-.

del for the heat transferred is not necessary. We therefore conservatively assume that for objects.-touched by the flame,

* ,,2. '5..--,.. : : •

0 .. A-. (4.16)

'In other..words, the object is subjected to an average heat flux level

within the flame, where attenuation is ignored.

4;.1'.3.4 ý.Convective Heat Transfer.- A.

. The convettive heat transfer to an object is governed by Equation (2.1)'

= h [T TI (4.17)

o,c -e o

For the fire scenario postulated, where the hot gases released by the

fire stratify near the ceiling, we generally neglect convective heat

-. 53 -

7:;2T.•, .- v55
transfer ýoutsid(I of the ga s.- layer andý theboatpueabv h ie

since T isesnilyteabient ro tepature outside of these

zonwes. To compute convective transfer'.within these zones, we requ'ire

values h.nd forTh. ,for' A number of correlations and models for.both,

the object's rising heat cot• 'ent, e"cold and calculate the wall heat fluk-""". "

es by maintaining T at ambient levels.

'....Heat Reflected. and Eritted-from the•06 W I Iss 

The boundaries of an enclosur e reflect heat f rom a f iire. 'back. int .o t:,heenclosure, and also radiate increasingly larger amounts of heat as they

warn up. An accurate-des'ýcription of L/hese heat:if ues-would require de-

:tailed shape factor calculations of- the-type -outlinied:,jn., Appendix -A-.

However, sincthese tfuxesend:, to be smalI, only an order:6f-m gnitude..

assessment: is necessary

In order to -accountfor wall i•flec-tion, -the.enclosure.boundaries: can.

-be' assum'" i 7• to be uniform sources• of -hat-fluxes of-strength

(1~~J4 J - 418)

. ."-Thei heat: flux,`to an object from wall reflections can.then be found using

:the factor fshape roma .rectangular surface to -a differential surface,

as seen:-in 'the ce -il-ing. gas "layer computation.

AS forthe emissionn of heat" from the walls, the "'all temperatures are

:generally.- low .enough that this. effectcan be :neglected, a.t least until 1

the fi re becomes quite: large in size.

":-5.

. - -"->• 4--"
4. 1 4:. F u el Be d ignition and Fire Spread

Once the heat fluxes received by a fuel element are computed, the -ques-

. tion of wh•t•er .or not that element ignites.and contributo.the:fire

growth: must be answered.

As is g.erally. the case iz this model, we present a: simplistic descr:iption :of arathter, complicated phenomenon. In this -particular, instance-, :a fuel element is considered ignited-•simply if its -surface temperature. exceeds• a critical i-gnition temperature..

The concept of an ignit ion-s temperature ifs.z a somewhtat...imprecise one,

since the temperature at which the .volatile gases above the fuel surface

will ignite ýis .dependent upon.:a -number of :variablles.. These variables

include the concentration of.oxygen in the. vapr-air •mixture, the time

history of the heat -inijut into the. fueland: rate of heat losses to the

fuel and environment. We shall'Idiscuss the uncertainties resulting.frow

this simplistic model in a later portion of this chapter.

For additional simplification, we further. model all.fuel elements as

semi-infinite slabs, and neglect heat losses.to the environment via radiation and convection. The solution of the heat conduction equation,

assuming a constant input heat fluxqo" at the slab surface, then-yields 0

an ignition time- t* given by.

=(77/4a)ý [k( 'I"., T )/q"j2 0 0 (4.19)

" -55 -

AL
.41.2 .PARAMETER UNCERTAINTIES.

The .input: parameters. X required by the referenc'e m.del 'described in the

previous sect'ionin-lude .t- only: the% fuel. bed physical piopertie. (e'g.,

-burning rate 'heat- 'of'-combustion, etch.) but btderity, lso".the dim- " bu ni e.-.. j rate-.• d•r. t . etc'. e,

sions and geometry of the :fuel b-ed and "ts loca"tion and orientation

within the:,compartment. .ln.':most-problems, we assume that-cur only input

parameter. uncertainties, are in.the fuel 'bed properti.s,' rhat .the geome-,

try.. of the prOble iS wel n. .

Although the number of' fuel bed descriptors required by our DRM is

quite large (see Reference (l))', we.consider the uncertainties only in.a

limited -number ,.of them. The 'paraa;eters considered are those which govern the.burning rate in Equations. (4.2) ar,Li (4.3), i.e.- C mi, and

. those which gcvern -the heat release per unit fuel volatilized, i.e.

T1, Hf, and 3, and the piloted ýand spontaneous ignition temperatures, r':

p

and 7* respectively, which govern the spread rate of the fire. The re- 5

mainaing pa,-ameters are assumed either to be we.'-known or have a lesser

impact on the calculations; 'the treat"ment can be generalized to include

'these additional factors with little difficulty.

4.2.1 Burning Rate Parameters

The parameters .CV, m", and CS. are important because they determine the

rate at which flammable vapors are made available. for combustion. Of

the three, C5 has perhaps .the least uncertainty, being essentially the

inverse of the fuel's heat of vaporization. Figure 4.2 -shows that C

the slope of the burning rate lines, varies little for a number of

different plastics.

-56-

IWEIV ZU-1
The remaining two factors exhibit a greater degree of variability because, we are modeling the complex burning process with constants. In

;Equation (4.2), we propose a-direct proportionality-between the burning

rate and the compartment ventilation rate, although the former involves

detailed-phenomena at the fuel bed surface and the latter is-a result of

grosser fluid movements at- the compartment level. In Figure 4.7, we

present a frequency diagram for CV obtained from Gross and Robertson's

experiments with scaled down wood-fueled enclosure fires [43].. To allow

for fires which burnother fuels, we can argue that since the combustion

of enough fuel to release 113 Btu. requires approximately 1 cubf c foot of

air, regardless of the actual amount of fuel vaporized, that the burning

rate should be inversely proportional to the fuel heating value, i.e.

= CV Win [ Hfwood/Hfother 1 (4.20)

As for in, we have already discussed reasons for its variability from

fire to fire, and correlations have been presented in Figures 4 4 and

4.5.. The largest uncertaint. es in mo arise when we consider cable insulation fires, there being little experimental data for this quantity.

In fact, for fire-resistant insulation, it may be argued that V' may be

negative, since in some cases it-has been observed that an external heat

flux must be applied before the cable will burn, and that if the external heat source is removed, these insulation fires will self-extinguish.

- 57-
.4

11

0, *05 .15

c k9 -. 5/2 Cv-,.kg/m t -.

Fig. 4.7 - Histogram and Fitted Lognor.al for Cv

(Data- from "43])

- 58 -
4.2.2 Helat Release Parameters

Ofth t~hree parameter~s mentioned in this ,group, i 'and Hk control: thet otal:;amount of heat r el e ,d per unit Mas s of fuel volatilized,, while

determines' the fraction of this heat which is..radiated from the flame.

,Hf is-usually iqite well-known for a variety of fuels; some 0 typical vaSues.s are given in Table 41.2. The combustion efficiency is not cs :well

knon., but' is expected-, to'.be fairly clos.e to.tunit'y for"fires which have

- . - access to largemounts of f-r esh air T- hasý,- been ,measuried .be. in. the..

range.,- .20 to 0.40, and, some, typical eStimat-es. are giveI--'in Table 4.13.

Throughout" this work:, we;,conservatively assume that ;X equals 0.40.

From -the above discussion, wesee'. .that. th,e., :onlI y uncertainties consid,

,eed arising fom .this group, of , parameters: -aare uncertainties in. :the.

.types and amounts of fuels in the fuelP. bed. studied.. ýIf the fue il bed is.

-.... w}ell, specified, these uncertainties.. il bw'-negligwblei 4.2.3 Ignition Temp eratu res'

The ignition: temperatureTl" -is defined to -be the threshold surface temperature of the f uel such that, ithe fueul ignites :wh'en this value-is ex-

. ceeded. However, it was noted earlier that tlhe concept:. of a threshold

igni.ion temperature is somewhat imprecise; while'we postpone a detailed.

discussion- until the next section, we note that the vvalues q6 oted in the

l.iterature. for both piltedand spontaneous ignition generally exhibit

significant:, variations (see Table:f4,%4). Further uncertainties result if

the fuel bed is not well-specified (e.g. an uncertain mixture of cable

-insulations). -We choose only to incorporate tthis-,latter source of uncertainty at the parameter level, leaving the uncertainty, in the ignition.

- 59 -
3/4"• Pine Wd

1/2:'" Pine Dowel

1/ 4.? Oak Do•qe1

Box ýCardbo~ard

N.ewspapqxr

White C a nvas 7 -.Gýiie-nCotton• • Clo ."th'

BakRu'bbecr

:plyur'ethane Foam'

.408

341

351

3.23

-2.83

3.35

3,28-

29,4

P. ii .. d'• s ,• <,

Piloted So~ntaneous

Ige Range :Ave'rage Rn e f rec

345 7'570 - 490 415 ->565 1[46] -3 0 - 3 80 660 -5 735 [46] 285: -405 - 46]1 3,00. 3o. 635 540 - -75 0 [ :235 -340 295 2-70 -365 [46] 300 .[46370 

'265 5 4 440 ;,390 545 [6 315 2 500 665 640. [46 ,]690 155 : 7:4.15 380 360 -400 4[46]

1>

TabIe. 4.'5- Properties ,of Wood Crib ,in137]

b: (stick <thickness)

.7 -.(st ick length)

n,. (no. of sticks per lyer)

N (no,.-, of "layers)

h (height :of. crib)'ý .

S (spa'cifig between sticks)

- (vent.area.x / - )

- exposed surface, area'

3 .-:17 CM

22- .7 cmh'

19.8 cm,

3,o -cni

" 60 -.
temperature for each particular material to be-treated with

ing modeling uncertainties.

4.3 MODELING UNCERTAINTIES

We distinguish two basic classes of modeling uncertainties:

ties in our basic physical modeling, and uncertainties in t

implementation of our reference model.

4.3.1 Uncertainties in Basic Modeling

O'tr uncertainties in the deterministic model for fire growtmarily from uncertainties in the various component models

toesstruct the DRI (see Section 4.1), although there is some ur i"-the synthesis of the component models. As discussed in C]

treat all modeling uncertainties with a single error factor I' simulations; however, we could easily define error factors

• • .... ÷ .•1. 1 ... A •..ot these uncertainties through th•

- -

the remainuncertainhe numerical

result priused to conncertainty in

hapter 3, we

ET in actual

for each comnRM •n, a manýM. -

4

L

I

ner identical to that used to propagate parameter uncertainties.

4.3.1.1 Heat Release Rate Modeling Uncertainties

The burning rate equations (4.2) and (4.3) are the most important sources of uncertainty in our heat generation model. Equation (4.3) is of

special interest, since it introduces non-linearities into the analysis.

Although the forms of both equations are appropriate in a variety of

situations (see Appendix E for two tests of Equation (4.3)), we are not

certain that they are accurate for the wide spectrum of fires which may

be observed in nuclear power plant compartments.

- 61 -

- .~.

______ .- ~ ~ K ~ -

KU ............. ..
The"burning rate equations .used were developed for the combustion of

relatively flammable materials in :'compartments with medium-to high' levels of. ven atiaon".' I f, :the tilat 1v ionm eveis 7 'are very low,.:, the fuel.

may smoIlder -rather .;than: burn,- and--what open flaming exists may oscillate

in intensity .and-.extent [48].

If the fuel bed is of low flammability, as is typically the casewhen

cable insulation is the main substrate, our modeling of the burningý process must be examined morei carefully..

As an example, we consider a.cable insulated with polyvinylchloride

(PVC). The 'decomposition of PVC is a,.highly endothermic reaction.

.Further, HCi forms nearil 60t' of the products of decomposition, HCl being a non-combustible gas. The result is-that PVC is extremely difficult to burn; as soc-. as the external source of heat used to pilot the

flame -is,removed, the fire tends' to self-extinguish. On the other hand,.

for strength and rigidity considerations, PVC is usually impregnated

with various other, polymers before it is used as insulation.. These

plasticizers often increase the flammability of the insulating material.

In order to accurately model the burning of the cable, we must determine

the heat required to decompose the PVC-plasticizer mixture, the composition of the resulting vapors (a composition which usually changes as a

function of .tempera ture and heating rates), and the heat released by the

burning of the combustible fraction of these vapors. Such a detailed

model involving the fuel chemistry is iwell beyond the scope of. this

work,, but we mast recognize the uncertainty resulting from our simplistic modeling of the problem.

- 62 :

7
n n- our- . r. - ra es is -ur

Another sourcefof uncertainty in Our modeling of burning rates is our

neglect of physical, movement of the fuel bed. 'This neglect• is reason-.

-able when dealing-with wood-fueled fires.: Howeverr when plastics (e.g.

cable insulation). are involved, me lting .. nanddrip.ing. _..s..

ten alters the -behavior of the fire., because this movement carries. heataway from the fire. Since heat losses:. decrease the rate of burning,

this neglect clearly is -conservative. I.

4.3.1.2 Ceiling Gas Layer Modeling Uncertainties

'Our uncertainty ,-in the model for the ceiling gas layer is a direct re-.

sult of the crudeness -of theianalysis'used.- More specifically, we have

assumed that the layer will be :strongly separated from the colder. air.

below. Whileý such a* flow configuration is, often observed, especially

for smaller fires,, alternative flow patterns are also possible, as demonstrated by the detailed numerical calculations presented in Reference

(32). Another of.our modeling assumptions leading to uncertainties is

•2. our assumption of a uniform gas temperature throughout the layer. -n

his work on ceiling gas temperature profiles, Alpert (39]. reports not

only variationis -as a function of radial distance from the plume axis but V >. also with distance from the ceilng. In fact, the correlation given in

- Appendix C is actually derived for the maximum gas temperature in the

layer at a particular radius. Alpert mentions that the axial temperature profile falls from thismaximum (usually a few inches away from'the

ceiling) to essentially room temperature within a distance of 5.5 to

12.5% of the ceiling height. While strong mixing and plume interactions

may tend to reduce these variations, nevertheless, our constant tempera-

- 63 -
ture model will lead to optimistic calculations of fire stress o•na• co-

' one in-. some. ,siat i os•' and conservative ca•Iculations in others.

4. 3 3 Heat Transfer Mdln netite

-c f'o'rmulation oft'he: heat tran ser -from a hea.tsourc eto afuel

element neglects.le the prilreflect io of the impining radatieha

fluxes, .as" wl as the) re-,radiation., . c onvection, and oLher lOsses from

theeleen.- erorinroducedb hi5 nelct

, . ,increases with timesincthese mechanisms become more important as -the

f due element heats up-.

.Unceritaintes "q,..

4%.3. 1.4- Fuel,, Eleýment IgnitionMdln netite

.... h' have .... deed )allt fuel elements as-semi"inf inite''slabs in th • ^ thou h we have mode . .. .. "e

!i~~~~~~~~~~d~m -,. .. )- ':1 ':• ,: .f :-- ' "'". . • s not la .to --larg

.'derivation of Eq.a.ion (4.19), .this assumption does* t

.. . . errors especiallyo -fr materials with low thermal dif usii ti es"ch

lec'tr.ical• -"insulati Amoren Source of- uncertainty in- ou

modeling is the use of a threshold-temperature, igi.ition criteria. C-I " ~~mode'lingi le•~ :

'The process -of thermal decomposition (which leads lo burning)igene

- ally begins at temperatures well below. the levels associated- with co

for oyt

- bustion., Figure.A 4:8-showS a',ithe*rmal decoPOs-.t.n. - .:curve. fo P" -

,.-. ee. Whil rapd., volatilization- occurs_ at around. 350C, -rthe f.ue

•.:•'•i-!•.. .. erie . Wh ile _a. . . . . - .- " . ." .. . . : . .: " - "." . ." "C

acual.y producing combustible vapors at temperatures around 250 0C

-. above. Combustion of these gases only requires oxygen and a uiias

netouce Thus, while ignition is far more kelyat hig;er surf

'"- neat sou rce . Thu . .. g 

'" :

temperatureS, it cannOt be ruled out at. lower: levels. We. can see *t

the crucial issue is,-not whether the fuel surface -reaches a cert

- 64 -

.s e

ur

ris

and

ble

ace

IHatm

•a ih

- 3-_
* 0

US

,7d 

50:

ii .~~ 501.

Fig. 4.8 - Thermal Decompositio Co.,Cur v Polstrn [

n pl~yee , rCN e~o [9

0 I 1 C

C >
K A 

,-temperature leeb, ahrif the heat gaiTS. by the -gises are great,

eno'ugh ::to overcome the lo. .ses, .and trigger the combustion reaactions, ahd

if the_-resulting.,-heat of. combustlOrio is •gieat enough to.sustain .the reaction.. Indeed, the, f act, that two different ignition temperatures are

specified for fuels , depending onI whether or not-a fla,:mer is in contact.

with the fuel surface and. pilots the flame, indicates that the ignition

temperature is a function .ofthe environment.,

4.3.2 'Uncertainties in NumericaflImpI6ementation' of DRM

In'Section 4.1, a deterministic reference model for estimating the progress. of a compartment fire was outlined. For realistic; problem configurations, the calculational scheme needed to implement the DRM requires a large number of computations.- In order to arrive, at a

r'easonable computer code which requires limited storage area and execution times, further simplifications and approximations (over those already made in the formulation of the reference model) are required,

sometimes at the expense.of the physics of tie problem. In the following sub-sections, we discuss the uncertainties arising from three of the

more important approximations.

4.3.2. 1 Fuel Bed Discretization

The division of a continous fuel bed' into a set of adjacent but discrete

fuel elements directly affects our modeling of fire growth, since it affects the calculations for the mass burning rate and those for the heat

radirted to other objects.

66 -

- %.Ž . *" '<' V ' It..
The ",,"' "pmass burning a -"hch so-id o"r 1i i id

f uel i"•• :converted i n o b -u it are a of fu e .b ed i s

.- i iiJ-:(•fliux '!in .:tur U is• S!:a, fu ct [ o "-of f lame isi ze ,•<a'.. 1 rge r if l amei l~ead ing :lto iila r-:.• •

.... ~~~~~~~~~~~~ .. .. :..igle,:fr•i~~rae: ea -eparate, ir es:,---:we artiffia-ly. 'in7-:i:'•: •"'." .:

•]'"..!.::ii::'::::c reai• e':'.-thiel- f Peedback6to".0 e::u n~ing.,. s ur~face•.. ab e.: the .I-evel : imp licit ly'-i} •. :: :•5 /"; %

ch- of:, t- '4-"-f am s..t

:,I. '.-,'' .. c ont..ca n dn -ind -mT ' s nien.th-`lu ese-ro e 6i chho .,thelm"%.." ll r-l-me •to a' "" ". ."l •'

Thespeif mss urig ae h r at twihsoi rlqi

fue"ivsný.:fuel celement arecnowm o cnsidered as "externiat heato fuxes d This',

'augmentation can beojstif flxthe partition ofth a cablelb tray, sincea- -

f xis tu gis efnctio of-a flm si ,e.' larger lamie ledntoa

sin .gen isltreaten. aH seral-separat-.e f.a iresl-w aitiy symmetrica i

•.'-,• .... fireuandihe~ he il beongated,:irv ative.vsaie sanme fcniu

crea;`se the atfeedback to chean t burni eng surael abvetele s:lo vets iaffecitedly

.ontaoinsed.. ifnofao s-inc h fl from each-,of s maller the flames er tson al

wivrn fuel elemefnthae s w iderep re das thex•cteralradiatovt heat fluxe this

augmenoetato canl bhe. jushtified for thepartition ofe radcablve hetra, since

fompre), ander thelo amatedfire cn sbeivisudieda. s ao numear hof thigu-

'r :ýdi .i ..... tia.i n l - h .co pu e hea flx....e eraI s tu 'i . .

.4ous pol.2.- Howeatver, if a pool firenuaisonbiie no salrpos

... ...- . . . . .. .... - .

"V":_ ,i.,•::,-,: ;•. . The'• heat flux; inte&rch~ange •bettween fuel elements, is also affected :by ... _' :i !:?:

.'--' ._...•::the .separation "Of a.single. flame into smaller flames,-since, in genera~l,.• -. "' .4'•!1: .• 

" 4 = " -. • -'4-/"",• .- •'"

•.. ... 'ofi..qjl' F0 . # Z'.:o-fli qfl~ i * (4.21) -" ... 

Y'• ~where the ,'left-hand side represents the actual radiative heat flux to

.•!. .. computed after, the flame has-,been subdivided.- It is not clear how this •-i-

• " •' :-"di: zretiz'ation w~ill affect: the computed heat f~lux in general situations. "

> 4.3.2.2. Radiative Heat Flux Attenuation

- As :is discussed-in Reference (5), :the heat fluxes pas~sing through a lay- :

" ~ ~er of hot ga.. or smoke att•enuate> essentially exponentially with dis-":

;,.'• " -.- '-,:•. 67 -

'C -',.. -'4"'-..," •!m . . .•, . - "
tance. We nieglectthis' attenuation in our model .,even though a, burning

omatetmay contain? large quantities of these absorbing mjeidia. _Thus

the heat fluxes from a fire onýone side of the roon. are assumed .?to reach- ."

the other side of the room without losses.:due to the' intervening gases.

.Perhas more -importantly (since . in thc preceding example', -. geometnrTic at-

'tenuation wi 11 probably, be- .large), the heat fluxes from, a, flame to a

'neighborihg flame5 will beunaffected -by the distance it, must travel

:,.,.through eithe-.-.s 

vhtgaes 

and smok h ffect of th su i

seen: somewhat %in the' modeling 'off Huffman.s, experiment [62] (Appendix.E),

.where the path lengths between. elemes of flame and the fuel surface

" -:")change--as faeeing and nekinig effects become :prominent.

4.3.2.3 Ignition of Fuel Elements %

The exact solution of the heat ýconduction equation' in, a semi-infinite

homogeneous medium with a time-varying heat flux at the surface yields

the, following implicitr relation for the, ighition time t* as a function

of the ignition temperature T*, 50]:

1 f E•

0*

this equation reduces to Equation (4.19) onlynif the input heat flux is

constant. In our model, we, define an average square of the input heat

.2 flux as *.. f_____

.0 ft dt'

-- , -q)d E )2

~~(qo) =

00

- 68 -
'1I

S

and;'use thisv-:.alue inEquation (4 ,19).

Whfle a more detailed approach would,. be:: to model .th component using

'..becae finite ................dferenc. f ft i or finite e ement.scheme, we ner Less expect

that this more exact approach would still be subject t6 uncertainties,

because of the controversial nature of 4h ignitiontemperature, concept. .

To-.demonstrate`.:-this:. last-nameds:source. of "uncertainty, Figure`4.9 presents acomparison-.of experimental data Jfor spontaneous.- ignition with a

<.::., theoretical predictin. when., the inc-is' convective, and . .ea

Figure-4.110 is a ploti.of :the .histogram.distribution for the r andm variab le - " .

" - --- experiment .. Lteory

"4.4 REFERENCE MODEL. VERIFiCATION

Since our discussion in';the previous :sections indicates a large number

~~, CC~~of uncertainties -in oui growth Periodrfenemel we is to et

the -model to demonstrate that••it-does- easonabypredict the e ior of

fire. In this. section, we use ýthe DRM constructed to simulate a number

of experiments documented in the -literature.

-4.4.1 Wood Crib- Fire

The first calculation is a simulation of one of a series.of wood crih

fire experiments conducted by Delichatsios [37] The specifications of

the particular. crib studied- are given in Table 4.5., and a view of the

crib is provided in Figure 4.11.

- .-- Two alternative models are developed 'to handle the complex fuel bed

geometry. In each case, we compute the. mass burning rate as a function

In.~~~~~ c9e ~ eac~ te a f n t o

'-';TT --. ~SCC~. -, ,~ -

S- C ~
10-

.0

i~L -c

£

IO"X .i(YI)r~fC(T)

0.3

0.5 0.6 0. Ol. .0.9 1.0

d -T1-'

.T A -TO

Fig. 4;9 - Spontaneous Ignition Data Versus Theoretical Predictions

i 381I

- 70 -

- ~- - ~ -. -7 .- ,--,~.------ 

- -

Wý A-' - Mk" - , -i ', P
f I' I

10 - I.

m i~'iY;:•~ 

T u t-

.1

0. 045 71

,•1.0 1.5

Variationis: ofL Experi mrenta" i...:Spont aneous

.ignition Times [38 .

Fig 4.10

'II ~.

Fig. 4.11 - Diagram of .Wood Crib. Used in [37]

Ll

71 -

77 .~,I Ž _____
•i.7• .:•#•

,.;:''

of -time (in kg/s) ando the average .flame spread velocity o ve r the top of

,the .crib, (the effective radius of the square crib: divided by the time it

takes-the fire to cover"' the ,crib top entirely):'

The ..4 first conf iguration chosen to model the. crib is simply a square

block of wood, divided into 25 fuel cells;., Because our model:treats the

ý:"three-dimensional wood: sticks of the crib as two -dimensional. fuel slabs,

S the fuel slab surface area in the model is enhanced. by,"a "porosity fac7.

tor.",- defined as "

A.

f I fuel surface

P fuel base

-.For example, 'referring to Figure 4.11, the area of the, fuel base is approximately £. Since the actual surface area of fuel available fors

.!burning is about 3Nbf. where N is the total number of sticks, and where

we have discounted the bottom face and ends of each stick from the total

area, the porosity factor is 3Nbi/9?, or about 13.4 for the crib considered. - -

'The second configuration models each individual crib stick. as a separate two-uimensional slab of length E and width b. The porosity factor

f for each slab is taken to be 3.0 (w, once more discount the bottom

face of each stick). .

Delichatsios does not give any specific information' concerning the

enclosure surrounding his experiments, and we assume that the enclosure

effects are small.

The results of the simulations are shown in Figure 4.12. For each

configuration, two trials are made. The first includes a non-zero feedback coefficient C for the specific burning rate, while the second does

S

- 72 -

- -

- - I~.-'.-.
IV-uamxxadxg 1818( [LEI sq~aID snsxaA 

suollOTP~.d ioj 2utuxng a~ju So POOM a~ua"Xal TOPOW -V

JOO q*2 Al.0 

~ off er'a 

qelg ON vl0 0

W 4-71X~mu 

0

*0

"I

V

v~

0

3.

. -¢ 

Y

v

0 0 0

4-
c/c) -x'" 4 - -

~4%

-1

'A

-'54

- 5-'

I.

not. Note that even,:though the ventilationlevel in' the room is assumed

large enog .Iotf"t6 be restrictin t s saigof the crib limits_'.'Iý

-.the amount', of fresh air available: to."-the burning: fuel:.,within the crib.

Thus, the'- trials using: zero feedback r.epresenit the .actual experimental

data much-better than'those using-non-zero feedback. The best simulation, of the experiment is obtadned using a synthesis of these two extremes. The top,.layer, of fuel is al-lowed a non-zero feedback term since

ýit is exposed to the essentially-unlimited quantities of fresh air, while-the remaining layers are considered -to have limited access to air,

and so their burning.rates may.not be-enhanced.

Although the agreem ent between experiment and simulation is encouraging, we must remember that the. models we have used in our synthesis have

generally been developed for wood-fueled fires. In particular, the value of 0.0062 kg/m2 s selected for is a value derived from 'wood crib

data as a best-estimate for surface controlled fires [11]. Thus, the

accuracy of the steady-state burning rate prediction of our simple slab

model without feedback is not too surprising; we have preserved both the

burning rate per unit area (0-) and the actual surface area of the crib

(via the porosity factor f.• .

p,

A less expected result is the accuracy of our simulation's handling

of the transient burning period of the crib. The predicted burning rate

as a function of time and the average velocity of the flame front over

the top of the crib agree well.with the experimental values, giving us

confidence in our simple heat transfer and fuel ignition models.

.74 -

- ~ ~ -- - -~
77 -Ww.7n_ 7 7- 7 7 9 TA9M - 7

:• ,::; !:4A.4 2 Vertical 'P 4MMA ,•Siab Fire -- -:•- = - ,.. --. .. ,. . '.t

The simulation described-above provides a test- forour reference model

•::i;::• .. "":i when. the fuel'.is' divided,-into_ horizontal strips:.. .To-check .the validity"

.of : the model wh'en the fuel. is.:a long, ver-tic-ail strip, We re~fer. to the.',I- ...... • :.!:'.

: .- experiment• of Orloff aet .-a -[51] :where .- a -1 , 57 x:0 41 x .b. lo0-'m .s lab of ...

PMMA was burned. We note. that~our discussion of a vertical. PMMA f ire in

Aped..isfor verification of the: linear• feedback model used for the

.burning rate, and that.:the model used dea~ls :with: the f uel1 as a continu-.

;ous entity. In this 'sectioIn, we. Aiscretize .the' .fuel bed :into fifteen ,

separate •but contigous fuel cel~ls, in order to. apply our reference rho- !:

del. -

4-. .4•

SComparison of our predictions with the experimental values for the :•

flame propagation velocity up. the slab and-the steady-state burning _ii;

.irates (see Figure 4.13) -shows that as in the previous simulation, our--

reference model seems to be quite adequate. •::

4.4.3 Vertical Cable Tray Fires

In both the wood crib and P pMA slab burning experiments, we are dealing

with quite flammable fuels. Sint e we also wish to apply our modei to

Scable tray fires, where the fuel is generally much more difficult to

burn, we would like to check the accuracy of our reference model under

these different conditions.

In these calculationsi o we refer to Przybyla and Christian's test

burnings of 40% filled cable trays, detailed in t52e . The cables burnt

were 0.515 inches in diameter, had a G.050 inch thick PVC cable jacket,

a 0.006 inch nyron conductor jacket, and a 0.22 inch thick PVC insula-

. "•. 75 -

• •~bun we would lik to chc the accuracy . of our- reference" model under*•..=:,,.'•v:, ..:•,,- •=.,•:• •:
AAm

Fij

- .> .. ~ . *...'A 'A: 'A' -- .- -. A

2~ <C..>

10.

'A SC <'A ., 'A

C'

-~ 7±.

0

0

0

0

CV<'A~,~

'A-i

9.

.8'

7

S

: 0 '

W

a

0

0

* modal 0.40

0O. a

5 I I II Z . I120

0 *40 80 1I20 160

z crm

.. 4.13 - Reference Model Predictions of Burning Rate of PMMA

Vertical Slab Versus Experimental Data [511]

76 -

- .-- ~- - -~ - -~'A
tion layer Th. e trays were-8 feet long, 12 inches wide-, and 3.375 inches deep, and were,.of open ladder construction. The pilot f"lame us-d to

initiate and sustain the fire was provided by a 10 inch wide ribbon type

propane.burner, iand was located at vary ng distances from the `cab1tray,.

and at varyingh -heights, above: -the cable, tray bottow. The test-seti-up is ..

Before we examine the results of our simulation, We0note that.the sel ection ofoparameter values to be used in the model-ing 'is a major` coan-

. cern. As ýnoted, earlier,.. the primary." fuel;. PVC,` is u'sually irmpregnated

with a, wide variety of substances which nay drastically, affect he burn.- .

.ing characteristics of the main fuel. These effects are generally, not

quantified, and -so many_ of' the parameter values-used. in th• sii ifiation -

are assumed (see Table 4.6).

In -particular, the value for. is not known. This quantity. may . e.

positive or negative, depending on the flammability of.the:. plaist icizers

added to the PVC. Since we are only interested in a reasonable reference model simulation of the cable tray experiment, we. assum.e that the

threshold takes on a tew different values, and then compare the ~resulting predictions with the experimental data.

We plot the measured heat flux profile along the tray due to the burner only (no cables are burning) in Figure 4.15, to determine the accuracy of our piediction of the external heat flux impinging on the cables, anid also compare the predicted average. flame front propagation

velocity up the tray with Lhe actual range of values in Table 4.7.

We-note that in our model, the entire portion of cable tray abcve the

pilot flame eventually becomes involvedin the fire. 0'.the other hand,,

77-

?..- 

.;..
_-fuel.-spe~cific heat

fuel thermal conduactivity k

fuel heating value . " .. f"

-fuel'p~iloted ignition tempera $ ',--.

Sfuel Usnan s, igntionit.ý-tr

.lspecificbu l ingrate constant . -0

fuel burning:-rate -augmentation 'constant,

flfrtinof, heat released as radiat~ion,, Y

burner fuel he~ating value Hf

burner fuel -specific unn rt n

'bu'rner: fract-ion of heat released as ra dia~tion y

, burner area A.f.

com bust ion- e1fficien~cy I

1 .8 x 10

75

ý78

002" to. 0.0 9k

4-.'13x:: x 10 -7

:0.30

5.05 x 10-,

0. 0694` k

.0.39,

0. 0025 iff

- 0.40

kg/ J

/kg

g/mns

- 78 -

- - - . . - --.

-.-----.~-'-
:0.00 1 l.8 ,x A1 "',

Exper-ibnent (52]

Properties -of Wcod Cr ib in. [53]

bl(stick thickness)

i (stick. length)"

n (no. of sticks..per.layer)

5.11-cm -;36c-"•:6 '

-"4 (top 61 layers)

(bottom 2 layers)

Nh

s

m

(no. of layers) .

(height of.- crib):

(spacing between sticks)

(mass of crib)'

*8.

30

6.35-

cm

cm

kg,

(top 6 layers)

- 79 -
I

I1

-4'

A C- - -, - -

- - - 'C 3⁄4 A'

A. - I

I.

8!b

Fig. 4.14 -Cable Tray Experimental Set~up [521

- 80,
- t

2.

•- - EXPERIMENT

3.0-

D

0

M 0 OOEIL

E

x 2.0-

1.0 J,

00

0

0 -4 0 I.

0.5 0 1.0 1.5

Height Above Burner, -

Fig. 4.15 - Comparison of Experimental and Predicted Burner

Heat Fluxes Along Cable Tray [52]

- 81 -

. ..----
t1

espoc ia,11ly- insulation mel~ting;,and.dripping..

4.4.4 Compartment F ire Simul'ation.

Our final simulation invol~ves asý•-mall wood crib fir i•nan.enc losure repotdbyFn[53] The:experiment is one of a series of,' furniture -and:

wood c"ri. fi lres covered in: that work,- and th heý . p r6blem -parameters of,,- the

particular experiment-modeled .3re-:given in Table 4. 8..

Fang reports.-,an :-impressive- amount of data from-.his experiments, including, fuel mass -burning, rat'es, heat fluxes, and fire: pume temperatures as functions.- of -time. Our reference -model predictions of these

quantities are compared to the data in Figures 4.16, 4.17, and 4: 18.

'The observed 'ýdiffe.r-ences in-behavior of our =reference model predictions -and the experimental: data can be largely attributed to..the coarse-

,ness'..of-;* the mod•l-,used. ;ur-mode .for ZI th cb ils-aSab.s imi lar -in

type' to the slab usedd in ,the simulation discussed in Section 4.4.1, i.e.

it is composed of 25 fuel cells whose surface areas are enhanced by 6

porosity factor fP' to. account for the actual area. available for burning. This discretization of the fuel bed is too coarse to allow the

partial feedback modeling employed earlier, and so no feedback to the

fuel bed is allowed. :Consequently, our predicted burning rates and heat

release rates are somewhat lower than expected. :Another result of our

use of relatively- few fuel cells is that fuel burn-up. can only be modeled rather crudely. We thus observe our model's prediction of a cons4'

- 82

-.- ~ ~
gp}

Vii

6

E

E

C

S2,

0

E Y ER L ME 

- EYPE .GUE 0

0 .5 10

Time , min

Fig. 4. ! - Experimental and Predicted Burning Rates [531] ft

I!•
.2,...

A

A 

E

E

I.0

0 .~

0

0

Time minFig. 4.7 -lixperimentn!d .Mcl Predicted
q

t

MODEL (Avol

EXMPERI MEINT

I Above PFit

U

a

200-

100

kA

e g . . . . . 0

0 0

4 a1 Iu f

Room, -Average

IEpo)

0

. 5 10 15

I , min

Fig. 4.18 -'Experimental and Predicted Gas Temperatures

20.,.

5 3]
i -. - .. ::'.-e x•e r i m en"t. a c t ua i li -¥ý s e e m s s o m e w h a t ., a n o m a l oi u' s ., w h e n c - o m p a r e d • -w i t h 'r e s u l ~t s - . ' , ' - .. e - .t, -. '- 4'-" .-

.... :•ments i-xlre od crbý

inexpel~h nen't appreelabýý•,`- ireater• .,.thlan.••,. . .,•,•.

'e- wood Aml~ -i

L-:t-1wood,"crib case-. We-thus.-may h-have ,reason t-fr 

V - notbelieve, that- the h enc data fluxm.at pe n d, ev n, bett -

of -his othe exeiet1i-rpre heat flux pes-tiný for lre wdd ri

th~~~~ -sof.:n,

i.......- :-.-- re "- .']-:, 6 --... •.n.X -,,•....... -"",:... 1 •-. . -- •.'":" , i.

S. ..-. A " ". _ .. fie"maue ydtetr9ic&1srt th f" arnest" an ths used "]•"-[-"- ].

- - '-.c ft -- 'Ak' - -X 

• ' 4. if i. ".c, ' :

in- .. . ki# -5- A> ae zb {tpea re.~~u~

.A ., 8 6 -; 

- 86 -,.: ,•

.. . . - . - , .-.. ..-. .- -- ;;••:..;;; • -,,1,. •,<•• • •-••--' :•• "•+••' .-. • • • t -•- r r ': -. ',i ]' <••)f•

7 1 v-
,72APPLICATION-OF. GROWTHPRO MDE

In this chapter, we,.-apply our model• for7:the thermal hazards -during the

growth period to-ý:two' f-ire --scenarios .. of--interest in our' i:nvestigation .of-.-

theý, risk to nuclear power plants due. ,to...'fire. The f-irst :scenaro:-o con-.

terns the' ex~posu fcti oafixed ffa flme~,j hl h

second: deals with fire, spread& -betw6een two v6rtically separated parallel

-cable trays. -

5 . ELECJTRIC CABINET:: FIRE

..- Steel, cabinets are often: usedý in nuclear power plants. to hold, relays and

ogic circuits associated with plant, control and instrumentation. These

electrical ComponentS are generally. quite- sensitive to the temperature

of;the-surroundingý-aair if 'the temperaturre.ýrises- more than .a.,certain am-.

ount, e.g. 20 0 F or more, ;. the --components can be damaged and faulty sigrials may. result.. Thus,., one question in a nuclear power pJant risk analysis. is what size fire is required to cause this temperature rise. In

this section,-we shall construct-a crudephysical model for the exposure

f ire and for the steel '..cabinet-!s:-.thermal respofise. We cannot use our

growth period reference-model directly, since our concernlies primarily

with determining a response to a thermal stress, rather than with the.

stress itself. However,. the construction of the cabinet model will pr',-

vide a simple illustration of- thc *process used to construct the more

- 87 -
gee'l grOi ITdl e jrCertaintrA~es-Shi.1' not- be 'Ici~pore notekc

.antal sis bec s e we re ..00•Jng for-- te -broad C-haracteristc of t6he -

crita al1 expr'sý ire fir-e

5.1.1, Expos ire Ftre Model

A f' mn ,d; A diane 3 r: poc:i: of ruel is :assure•d .• uo & en e compaL.ia-n fl oor

anc6ý'. ~~~ -tomoe---45't-t-;-3iE:t ~ ~ ~ ~ 

-"Sn. e., the a5s- 1 sep.sraeing Lows -of these _i -- :.anets sxpyc1 6 . m

this s: t'e ,fi.-mum separation censiderid.- fhe 1 am'r height.cof the fire

is given y -Fuadtion (2. 1)

N-- 'r .'"

--" 42 D I /U, - (I5.1)

.wh6re in" is replaced by mn siince hier' are no other iU.f-- lt heat

sources in tle. scenario, in( ., ince thue'.ire-. is asiam,.d to: 4 -tol sur- -

Sface -:ontrol.ltd. t- mede the flan' •s a--1 hlinder of diame-t,.r D and

height Z f . he rad ative-hcat f lux.ea--theu flaame.boun.a k, .- . •.

prox imateIv givven by

= ;.0 H (1 IfI +.4 Z i,'Df)- (5.2)

and the. he-at" .u I x impinging on the cab itet.;face is given by.

car cab-fl f (

We conservati-:ely a)sstvm, that the heat flux rcered by a differential

surface lying on the flame-cabinet axis and a distance Z up from the

floor is repri-sentative o!h. average heat flux received by the entire

-cabin.t face, and so 1 caFI can be calculated -:sing Equaticn- (A.6).

[ i I ' --I8 8 --

caint 1ý3L ac, nd .)

~j~7--------- -&c,.,.-'~~ rN- f
I ~ sp~<

evt!,T.1T

PC

h I T..4 To I

Ty~-' -T.

I

00

Hot Wall

Fig. 5.1 - Exposure Fire for Cabinet Fig. 5.2 - Heat Fluxt:s :for Heat• Balar

Each Node :

at

N'

.1

- - - .- A. t
-.. y CULJIIUL dle lis~ea' in1 iaoiL e .i. -'

_- 2 hrmrale Responese Mo.de.l.

Inorder to. calculate te --a ir. -t em p er ature wihnte c~abint we repre_

'.sent t .b.n:

,~ ,....(corr sponding to the "hot, wall" 1 facing the fire, thecold wall" .epre

.senting all the -remaininga•b•ineit faces, and the cabinet air ree itself), * • ,r : 3

..... . .... ..

.rform_-Aý_heAt balance. ,to' obtaifni the overnng-eu

R.efer.ring. .,, Figur there results,

: (( h T." T cab " Cab . +. _'17. -. o HW e HW e +h. (T• TA), +o(T:wT4 T.

b) Co0ldWall:

C,iA (Th CTA -T' .+ Eo(-T ') = .'A CW

Co W e. C _e, ~ .... .. C,o Tcw - Te) + £o(T : - .... (5.5)4 :•.•;•:•:.';,:•;•::...:"-_,............................L :... . .•-.. -. . ..... ........- ' ' ::: -

')- Cabinet Air:

MA cp A AH [hHAi(T,T TA) + ao(T• - T)]

- A w[h (T A T ) . ro(T' - T' )] C Ci CW A (5.6) CW :

The heat transfer .coefficients can be calculated using.standard natural

'convection correlations. For example, one formulation for a vertical

wall of height L:is [54]:

n [497: 9 + 0.0182a'"(L - ) (W/m 2 OK) " (5.7) La

where

- 90 -

.......................,..,~ .,~,....... <zi. <~Z~ t;x~r.X.roSzt.Z4~~~ZC ~.
-0,25

050

1.00 0:.028 0.14 0.39 • .%. Z " - " .' •

*ExYpos -ure- Fire 1:

Exposure, Fire 2:

E~sre, Fi re-3

15.cm radius :paper f ire

T.~5 on 2 isolfire,

15 cm riid oi'ie.

Table-5.,2 - Cabinet Air Temperature ( 0

F)

Exposure fire Distance (m)

Time (sec)

0 20 40 60 80

.I.

2,

3

1.0

1.0

100 100

:100 111

100 141

99 99 99

115 118 119

15 6 162 165

- 91 -
-t -.- -

and allI-barred qua-tit~ies :'are' e6Valuated at some 'ý.referenýce tmea~

Athou~gh"thesecoef fi~ienis are-: tempeiatu reidepebdent (hence our d S . -1 o d -d h -', 66 -- ,Au s) 'te e

hot'Wal, outer ea'nd inner, alues the de pendeice

is fairl weak.. 

., ,,. The unknowns to be, solvedeforin Equations (5.4) through ý.(5ý. 6) ae. re thehotwa"ll temperature T the 'cold. •all temperatureT• T and' the"ca

- Hx4~ -,, CW.'-

binet air temperature TA. Note 'that. Equations (5.4) and (5.5) do not

contain time deriva'tives. Thisý is due .to the: low mass and hence. small

- thermal -inertia- -of the: cabinet walls ..We:have.alsoý neglected the temp- erature drop across the cabinet ýwalls, sirce they are thin and their

thermal1 conductivity is high.

A simple 'numerical scheme to solve this set 'of Tionlinear equations

begins by -assuming an-'-initial cabinet air temperature T , which is about

A 100 0 F. -We solve, for THW and in Equatior°s .(5.4) and (5.5), and use

-teenew-!.'-va'1ues8-to.'solýeý:fo-' an-upd ated T ni~ f'idfeenefr

A

mulation of Equation (5.6). The r'sults of this solution scheme, when

the exposure fires are those listed in Table 5.1, are given in Table

5.2. Some relevant. problem parameters are given in Table 5.3.

It can be seen that in one calculation, the cabinet air temperature

-actually drops. -This non-physical result is due to the neglect of the

heat generated within: the cabinet during normal operation, heat which

causes the initial cabinet temperature to be. 1000 F. We note that calculations which roughly account for the heat sink effect of the componIents

within the cabinet indicate that the components can usually be neglected.

- 92 -
cabine't.--lengt h

ca-binet 'width

-Calbinev.d6prh?

cabnetw,.fl m1ssivity., Ca~binet a,,,r v7olume,

a

~'0 4% -

I $2..

- 93 -
Th e Ipower,,. control, an'd instrument -t ion' signals for. each safety division

in a nuclear power plant are genera.Jy pse hog lcrclcbe

lying-in' .cable ýtraysý dediae toi fr ahd

sebarted,_by: h''of.

vis ion are b aditance" des igneýd- to preveni-ýt tesredo

fire. Ifrom" one division , to another; if a fire does. spread :.between diviions, serious consequences"may resul.t.

A typical: arrangement- •f' cabl:e :"trays :within the cable spre'ading room is

shown in Figure, 5:.3.- The -three cable .trays.;ný.each horizontal layer-..

carry the cables, for one division; thus there-are four divisions passing

through this. location (not all are safety-related). The trays are 12

inches wide, and are separated from their neighbors by 2 inches horizontally and 4 feet vertically. Th, trays are open and are presumed to be

filled to capacity with cables.

Preliminary calculations indicate that,,the time for the fire to

spread from the iniiailly"-buring tray to the tray immediately above

differs from the time to spread to a horizontally adjacent tray by a

small amount. Further, once the fire has spread to another tray, the

spread to the remaining trays is very .rapid.- we.therefore concentrate.

on predicting the spread from the initiating: tray..to, the next higher

tray, %here the lower tT.ay is called tray 1 and the upper tray.-is called

- tray.2. We denote the characteristic time for fire spread between the

trays by "

(~j~ ~

V -

-A

hI.- * - -

- 94 -
Al. -~~

K *. 

-

I 1.. 4 I •• o

Fi.53-- Cale-rayFir-Se-u

Fi -5 3 C b e T a i e S t u

- 95 -
.'•-:-z:- deerm'inistic eference: modelfor this it ationis ..elc.muter -

cod pBRN escribed in Reference (1)-,withinput su"ted to model the

given f ire. In Table 5.4, we list some important fixed parameters and

their. assigned values.

SSome commentspertain to our modeling of the fire.

_T•The cable trays are 5 fqet -long, and are divided into 5

discrete fuel cells, each being 1 foot squate.

* The-fire starts on the widdlefuel cell of .ray 1, and additional piloting fuel (e.g. oil, trash) may be present, in

thisi) ell. The pilot fuel is discussed in the next section. The remainirg fuel is cable insulation (4.66 kg/

tray).

* The porosity factor fp for each tray is 3.14 (i.e. the actual surface area available for burning, per unit tray

length,.is 3.14 ft 2 /ft).

* No credit is given to the shielding effect from the bottom

of tray 2; the flames on tray I are assumed to transmit

heat directly to the fuel in tray 2 with no attenuation by

the intervening ma]terial. This, of course, is a conservative assumption.

• Enclosure effects are neglected. The fire is considered to

be too small to cause a significant hot gas layer near the

ceiling. We note that this neglect may be non-conservative

when the trays are near the room walls, since reflection

from the walls may be important.

- 96 -
ition

a]

cab~ies

mass ,o

t ray ' p

2

tray i 5

1.52 M

0 0. in'

-4.66 kgj,

z 3.14ýT

298'0 K~

0.90

60s

1715 kg/rn

1045 J /kgK

Y 0.1092 W/m K

mperature Týr- 40 0K,

tation constant 1.86 x 10'- kg/J

d 0.40

SC"I,,Uo uLion;-, ext iciency.

etime incre'me'nt fuel density

fuel't hermal .:condu'ctivit

.+.fueVi.ed-: -ignitionn.te

-fuel burn ing,..r'-ae':-augmen

.fraction ,of heat radiate

1.'

2...• -.. ,

L.4 .d.: :; .•,+

97 -
5.2.2 Parameter Uncertainties

Chapter - T o f [ -a d [to model the initial conditions of

te4. ' The pilot fuels aeus

-"the fire, and require further.explanation.

As has been noted previously, cable insulation is generally a difficult material to burn. The type and amount of pilot fuel involved in

the initial fire is thus an important factor in determining the rate of

.fire growth. If there is a large Amount of oil, rapid growth is nearly

assured. If negligible amounts of additional fuel are present, fire

growth may not. occur at all. We model the pilot fuels with a single

parameter Q p, the amount of heat released when the pilot fuel is consumed:

Qp = Hfp (5.8)

The use of this single parameter, while crude, is not entirely unreasonable, since the ignition of a fuel element is governed by the total amount of heat absorbed. A more sophisticated analysis may take the

different burning ratcs of the pilot fuels into account, since the more

intense fires, e.g. oil fires, have high flames, and thus transfer

greater amounts of heat to the target cable.

Before we propagate the uncertainties in these parameters through the

DRM to determine the resulting uncertainty in our prediction of xG' we

construct a response surface for the DRM to lower computational costs.

The Latin Hypercube Sampling (LHS) method described in Appendix B requires us to first select upper and lower bounds for our continuous par-

-98-
A F

that two seLs ,or triais were, use toconstruct Lne repunses bule.-.- r .

second17 set of trialIs usedý;tighter 'boobnds' on the ,parameters,.'than the,,-

fikrst', in order to more accurately r~epr'esent the' cable' intulat ion' prop-.

-erties' ýactually expect~ed.-7 The bounds ýfor : are;, iicj6t-ive estimates-,-

w hiIe' ' the bounds for"Hf,:andT result from a synthesis'of data fo r p l as:-7

tics and data from Reference (45). The pilot fuel variation requires a

somewhat -d ifferent 'treatmnent i -

. e...4 -Pilot .. f..els". whose burning characteristics' ar'egiven.':in-.

Table 5.6'. .As seen. ."in: itheT, la'st entry-: of Table -5.6, each "fuel. corresý-

-ponds to a, fixed. amount of heat reeased when the pilot fuel -is consumedt. This .quantiity ,:is the' previously discussed we use to define ou . piiot.:fue. in our LMS' simulations, therefore', we randomly

assign one' -pilot" fuel' (one Q)) to, the input parameter' vector -which is

-- then prooce'ss-ed.- by- the R "RM.

To provide some physical basis- for the choice of Q s, we note that

400 Btu from..a.l `ot.fuel. 1• packevt-•rughly::coirsponds o abl.

insulation burning for 'one .minute '(i.e.. -.no -additional pilot- fuel),

2,000 Btu roughly corresponds to wood chips or paper, 10,000 Btu can" be

associated with 6ily rags,..and 40,000 Btu roughly.corresponds to, a quantity of oil (apprOximately one' quart): burning for -.two minutes.

To correlate- theTCOMPBRN pred ictions ,'.tf" , we can define a characteristic time' before self-extinguishment vse, as

Mf

-se r"

and a characteristic ignition time x as (see Eq. 4.19)

99

- 99 -

p.
Trial Set ,••-f'•d:!' :2•••:•.,"' =• '.,\ • ¢M '".• •'•777 •

• •. •••:•.;• 7. -r• • ,••.• :.;•.••1. Parameter

(J.kg/M2 s)

)u,.C

7 7.-77 ..

u.u

.

U; U

( (K)9 

2 r~(kg/2s

.1 v8 5 . ' 1 0. 2 . 7 0 -x 1 0 7

710 850-

1x -10- -0ý.003ý

•.1. 8 O 107 . 2 707 - ..

7

ý850. 7 7777 777 77 7 77

:72

7:

A'H (J/kg

S

Note:,, 28 .samples were se'lected fromrial set'. 1,, 'and

32 were selected.-from trial set-2.

7~7.7 7777 .7 7

~ 7774 777777377.7

7777 77~7777 777 777777777 *~7A

773777

77777 3 7 37777 ~r

-7.7~7777 7777 7

7777777777777 7

77777. 7

77%37777777 37

7 .773~7~37.777~74

~777777;3737. 3777

77747777 7

~77777777777 7 77 ~777777

Table 5.6 - Burning Parameters for Pilot Fuels

'Pilot n (kg/m.s)

* 1 0.001

2 0.0062

3 0.008

4 0.0231

Hf (J/kg) mass (kg)

2.32 X! 107. 0.02.

J1.86 X 101' 0.11

3.77 x 107 0.28

4.67 x 107 0.81

Qp .(Btu)

.400

2,000

10,000

40,000

- 100 -
denor

e tai

or within the brackets re i' heat

i,,i$

y AkI Ius i t..i IaI- LCi- 2 11 L IC par

ameter 0, which..upon-,substitution for constant yvalu•eus is .givenIb y .

3.6 "o~T* 298]2*

I H,.

Th re " .for _,,ou. ralc;lat:ons c then_ ind io tc tt elie"characteristicý t ime f or f ire", sp read -f rom tray. 1ý: to tray 2,is well correlated by-

.A+ B~

'C

(S.9)

"WA', - - "K'- U'

where A and B are given in Table 5..7.

difficulty fne with the above correlation is that occasiona-lly, the

DRM-predicted- value of XG.becomes infinite, even for relatively moderate

values of 8. The physical,. situation is -that the flames on tray 1 are

moving~to the ends of the tray as the fuel in the centler of the tray is

comCpetely consumed. Occaionally,, the- middl1e, f uel.:.-i: emen"I i i nrtetr a' 'y' 

(the portion -closest to the initial fire location, and the recipient of

the greatest amount of heat ) has not absorbed enough heat to ignite before the flames on tray 1 leave the center of that tray. As a result,

the fire on tray 1 exhausts :all its fuel and self-extinguishes before

tray' 2: can ignite.

We define a further parameter 3.

t

(T* - TA)

• •..- 2P1.

L .Y I H - .. . .. •~ f

- 101 -
Dle 5.7 -Response Surface Co-efficient

aA(min)Z B(min),

s (Eq. 5 9)

S2

. . . 3 3 1.5

:2,0

14. 7

-8. 0

0.

10,000

40'•:-, 000:i b o:.:• •:

4

T able 5.8 - Histogram, for. Verticl.' Spread Time

• " • • • i . ".

- (min) G Pr (T)

30

50

70

90

'110

> .120

0.26

0.21

0.12

0.08

0.04

0.03

0.26

k

- 102 -
V

and our response suirface is Equation,5.9) 'with the following conditi'on

-- -with j frequency of 87.5%

when

-..Q 400Btu, a > 09,_ a nd .; 0.06,- or- .. 

Q_ 

p

2,000 Btu,i. 8 > :1.9, and tp > 0.06.

it.must. be.emphasized that-, this response surface ,:is .ap'plia loniy, for the -given-. p'robl~em; if -the ý,problem geometry is 9altered, he ýresponsesu face coefficients will a1so h'ange.

In order to-propagate.our state of: knowledge uncertainties inmon:, . T*, and. % through:the response surface, we must const.uct distributions for these parameters. We use the: following subjective distributions:

1ognorma.1ly 

(median 0.001 

distributed with v,.= .- 6.91, a = 0.865 kg/m2 s; error -factor 4)

Hf uniformly distributed betweeen 2.7 x 107 J/kg l.85' x 167 J/kg and

T* = uniformly. distributed between.8009K 'and 850"K

and

Pr(Q 400 Btu) 0.10

Pr(% 2,000 Btu) =0.44

Pr.(% 10,000 Btu). = 0.44

Pr(Q = 40,000 Btu) = 0.02

The histogram for Q should be interpreted as follows. If we examine a large number of fizes involving the configuration studied, we will re- veal a frequency- distribution for the heat. released by The pilot fuel. We consider to be a characteristic parameter (e.g. the mean) of this frequency distribution and our uncertainty in the pi lot fuel is modeled

- 103 -
quency distribution- Thus,,_ our' state of knowledge histogram sa.ys tha,

weare .0% -confident -that-the characteristic parameter ise400 Btu 44%

.confide t that, it is 2' O00, BPtu, etc.;.-• ...This;. distribution.-quantifies our

b'elief .that. the cable tr fire will .. probably invo ve additional fuel as

wel ,aslthe cable 'insulat ion. -

Based on the above. distributions, the probability: histogr am for

reflecting our uncertainties due to parameter uncertainties ::is shown: in

Figur":e5.4. This. is .a.state-of•-•know.ledge ,distr-ibution ..ffor the 'variable'

- >'~"'~he lageti1 l6 20 intiutes.. is mainly du _oour model's.

'prediction of an infinite propagation time for certain values of 8. We

choose to conservatively tinorprate 'this information using a smooth,

finite tail, noting that either treatment (i. e. incorporation or neglect) will make little difference when suppresion effects are included

in the analysis.

i~f

5.2.3 Modeling Unceri

Our distribution for. E.

constructed not only fr

ter 4 concerning the'd

also from some rough •

-,with published values.

There currently is

times between non-contz

mation on the average

if.we define-i to be a

vexperi

VDRM

tainties

', the error factor ýfor the spread time T,- .is

om the qualitative info-rmation presented in Chap- ., .

ifferent sources of uncertainty in the DRM, but

comparisons of predicted flame spread velocities

no-..detailed data availableý forf the, fire spread

iguous cable trays. However, we do possess inforflame spread velocity V over a single cable tray.

characteristic spread time, sicY that

nent T DRPFM i

texperiment

- 104 -

. •-ý

.2,.., .... . . ::::! __ . - - I I . a -

Z:7 - - WWWMA
K"

'I;

t

.20

0

.0 .10 0

CL

4

0 r-- - - I-p I I

0 20 0 60 i0 100 120;; 140 160

"*7o* .(min)

Fig. 5.4 " Distribution of Propagation Time, Due to Paramieter uncertaintie'sji".

. "•, , . .- • -. . . ::. 
6. o8 V-< C -nis,: .

ernf p iment --- " "

ýDRM,

and so

1.6,< E < 16

The cons ervat-ism ýof the DRM s Prediction i xetdfrtevria

tray cniuration' since7ý fuel meltn ad dipn r goe

For= horizonl .. xayfir r es P '"nel quotes an. av .ragep roagatio velocityl" of one niich/minute, "or"1.0 cm/s (3]. C u o f the", f lam

.front velocit along:.tray , in our simulations, whereno'addi-tiona Pilo"t-" fu el is added indi1cates that

.00 - 0.14 cm/s . DRM

e var•iability e the is .due, toý parameter- uncertainties The nean of

this average velocity-distribution is 0.06 cm/s, and the iiedian is 0:04

cm/s.

If we use Pinkel's estimate.as a reference-point,

.5 • " .. .. 0 5 ' < < E "1 < 3 . 5 " .

On the. basis of this evidence, as well as -our critical. assessment of the

conservatism of-'the various physica. models used in our reference model

code-COMPBRN, we construct a lognormal distribution for E with -the folT

- -106-

A'
A ~iii

I Iowjing characteristics'.

P 0.582

o 0.489

-ean M 2.0

Median = 1.8

E =4.0 r,95

i E =0.8 ~T,.05

It is important to note that this quantification of our uncertainty in

the predictions of the growth period DRM reduces the conservatism of our

analysis.

5.2.4 Combined Probability Distribution for Spread Time

The average time required for fire spread from the initially burning

tray to the tray immediately above, TG' is given by

• = E T (5.10) T " DRM

The first factor on the right-hand side:.is an error factor representing

our modeling uncertainty, while the second factor is a prediction of our

DRM which also incorporates input parameter uncertainties. Both factors

have associated probability distributions.

In order to determine the distribution for TGP we choose to discretize the distribution for ET, and combine the two right-hand side distributions using the discrete probability distribution arithmetic described in Reference (55). In essence, if Er ,i represents the ith block

in the F. histogram (i=1,...,n), T represents the jth block in the

- 107 -

-- L.
k kDM G f LC

-', i M~ i•" - -

.. ..... :: C-"'v... .. . .

Although in general the modeling accuracy is.somewhat dependent on..,.the . rob Iem. parlameters, thi probl emsparameters d- rot-vary ,a reat deAl .- . . ... . -: . • ... - - . . ., . - . .. m -a . -. . . ." . - - _ -. ý g• r e t. .a 

since we are, looking at, one, particular-. type.i oft-fire •.and' so. we, .have.. as:-

sum-edindependence, between -'E and -

t ,--inEquation(.5 i)

significant fracti0on. of fires takes longer than two hours to. propagate

to tray 2.- This reflects our uncertainty. that the-cables are not. flammable enough to allow such propagation.

. .. ,•

5.3 FIRE:SPREAD FREQUENCY INCLUDINGSUPPRESSION

The dlstribution for T derived- in the last section is a probability

distribution 'for the event "fire spreads to second division .in t minutes, given an initial fire on tray I and ignoring suppression ef-

-forts." An-, order.-.to.compute the conditional frequency that two cable

trays, each-in a different division, are involved in a:fire (given a

fire in the cable spreading room), we must include a model for fire suppression.

Examination of the data reported by Fleming et al (23) leads to the

following 'dist'ribution for TS the mean--time between fire ignition and

suppression. We note that the suppression times for fires occurring

during.plant construction have not been included in the distribution.

Pr(tS = 5 min) = 0.40

Pr(-S 15 min) 0.30

108 -
Pr r .-.O min) 0 20

The -justification fr: thiýs, distribution is .as follows. The estimates

reported1 n 'Reference (12) a~ the te Thereofekp~ 

are 17 cable insulation-fue'ld fires studied which occurred during plant oj-.ratio...'Thex pt e t .:se'ven :(-40.) of -.them were: extinguished Wiithin- 5 minutes; -Four of the ]seventeen fires were estimated to

.have:•::been ,ext ingu ished between .'•a•nd 5 '25-, m inutes.. a fter. init i'ati0on:, and

*aoher, group. *of .four had extingýuishment:ý tmsbewn 30an 5. inutes. Finally, there, were two. fires -that were extinguished after 60

,-minutes (but. befo re 85 minutes) Of course, these estimates are not

statistical data; we judge that -our discrete model is fairly consistent

with the given information. Furthermore, it conforms with our belief

Sthat it is very likely that most of the fires in the cable spreading

room will be extinguished .fairly quickl.y by the perscnel who started

them. It•t is important to note-that the. distribution does not include

the Browns Ferry cable spreading room fire as part of the data. base.

The seven, hour duration of that fire was largely due to the hesitation

of plant, personnel to apply water to the flames, a procedure which has

been much addressed since that incident. The neglect of this data point

may be an important non-conservatism in the analysis.

Let us denote 'the conditional- frequency of fire spread to a second

tray, given a fire, bby f .f is :given by c c.

c exp(-T /-) (5.12) (" •g /S

- 109 -
I where eumed'-thiat-, have ass the •time tosuippression is exponentx ally dis-,

,,j- -tributed.with mean . ,WeS note that the., frequency of fire spread by

tine t is also the frequency of, non-suppressinby. hat time. Thedis- c , G-' .- '.. :.: .:•',. -. tributiOn of-f t: :given.:by .':. i: -. "" " ": : " . -::. . .-. ''.. .... ..

where •the.iedistr.ibuti-ins:.of•: and'- -were:, given, .previous ly`.

It sho•uld- be:noted that.in-his.simple eW ana lys1is ignore-any' depn-

"denc between -. -and.:,'although-one certainly- expects' "acorrelation.

":' ' - ' "- 15 . . b: ... .

0* S

between the suppression: time and the size of the fire ka large fire implying: .a large. x). :

The distribution for f. obtained is listed in Table 5.9.

C

5.4 COMMENTS ON ANALYSIS

Refinements of the proposed mode] are certainly possible and desirable.

For example, it may be necessary to determine the distribution of the

frequency of fire spread to trays other than the ;ones' immediately adja-

:cent to the initially burning tray; to give stronger information for the

loss of two electrical, divisions. Another concern is and-the neglect of

flame front slowing-down caused by-suppression efforts.

In our analysis of fire growth and suppression, no credit has 'been

- -taken for the inhibiting effects suppression efforts have on the growth

of fire. Agents such :as Halon, CO , and water can markedly slow down

2

the rate of growth. Indeed, the flame front speed during much of the

Browns Ferry fire was re-latively low [55]: rough .modeling of the portion of the Browns Ferry fire in the Reactor Building indicates veloci

,2

4

- 110 -
T~b.'5~9Hftogam:orth6Cnditionial- Frequency, of Firez,

SpradtoanAdjcn Try Giveni a Fre,

Frequency

3.4 x 1..4

3.8 x 10,"4

4.3 x 10-'

4.9 x 10-4

s-;5 X 10"

6.2 x" l0-•.

7.0 x 10-i

4.4 x 10-2

1.4 x 10-1

3.0 x 10-1

5.1 x 10-1

6.9 x 10-'

Probability

0.16

.0.02

0.03

0.03

0.06-

0.08

0.12

0.10

0.10

0.10

0.10

0.10

- ill -
ties nearly an order- of mantd agrth~nn those reportd_-$ Thus, the,'--_

n-eglect-of-,f.regrowth inhibitionby -supprepsionefforts -adds .trong .

conservatIsm',to'the analysis.

Sinally, we comment on the problem of obtaining the uncondit ional

frequency.of fires which involve two, cable trays., f is, as 'defined.

above,ptecndtal frpquenc 'of fires i'nvo~lvingz,two 'trays -'in-the caT6ble- spreadinfig 'rooMi of- 'a hypothetical. plant, given a: fire. To make- this

Sfrequency: uncondit ional ,: we. must incorporate the"frequency. of fires in.

tit.ha particular ro m. While Ref. (19) does provlide. a. dis tribution- for, .

. .,7',ýý-this quantity-•,soýme..ecaremust,.:e -exerc'.ed--ix usingthisinformati~n.

The initial fir:eassumed in thi:; analysis is' moderately sized: (1, foot

by 1 foot). Before we'utilize any fire frequency-distribution, we. must by 1,,o)ý ,B re 'we, .t i..

decide if that distribution represents the frequency of fires which

reach a comparable size (and thus are considered to be serious), or if

it also includes the potentially large number of small fires which either self-extinguish or are put out immediately. We note that prior to

the Browns Ferry fire, a number of small fires were initiated by the

same leak test procedure which caused the large fire, but none of these

were incorporated into the fire frequency data because they were small

and were prc.ptly extinguished .8].

- 112 -

" ' - - :: L•::q :~.'i .:A . _ ..... ' . ... . ..... / ..
>-•',k ,'•"•' '• '":';• > . % k•

:4 ..• ..-.. , ] " " " .... .. ' 

Chapter, 6

-,FU•LLY .DEVELOPED' BURN ING- PERIOD. MODEL-- i

DESCRIPTION-. V

During the period of fully-developed burning, the spatial distribution

of the compartment air temperature is assumed to be uniform, and so the

"thermal stressexperienced byacomponent is independent of the compo-,

.nents~position. The analysis of this period, which follows..that-of Ka--

wagoe and Sekine [10], is therefore computationally much simpler than

the period analysis.

It should be mentioned that the likelihood of fully-developed burning

within most of the compartments in a nuclear power plant is considered

to be small. One of the inhibiting factors is the large size of the average room, larger rooms being better heat sinks. Another factor is the

low combustibility of cable insulation, which forms a major portion of

the fuel bed in many rooms The slow burning rate means a low rate of

* "-*-heat release into the compartment, a slow fire growth rate, and thus a

higher likelihood of fire suppression before fully-developed burning occurs. Finally, we note the low overall amount of fuel in most compartments; even if .'uppression efforts are not initiated, the local fuel

*bed may be exhausted during the growth period, preventing further burning.

113-3- 
- -4-: --

* - "• "•7

6.1 ---THE, DETERMINISTIC REFERENCE MODE

The compartment. is modeled with two6nodes: the compartment walls and the

compartment air. Heat balances .for.each node yield the governing equations for the compartment air temperature.- -

"a) Compartment Wall (of thickness L):

•,?: ! DT D2T • at aC (6.1)

Initial Condition: T(x,O) = TA

i•:i :,Boundary• .. •.•i. ~dtois•""'°

o r o[Ty(t) -T(0,t)J + h[TG(t) -T(0,t)] 421 0

.h[T(L,t) TA =-Lkk] 

The radiative term in the boundary condition at x equals 0 is generally

neglected in most analyses.

b) Compartment Air

llnmHf ACo[TG(t)-TA] + mc [TG(t)-TAJ

+ Aw CoT'(t)-T'(0,t)J + hA•jTG(t)-T(0,t)] (6.2)

7- As opposed to our cabinet air model, the time derivative term is carried

in the compartment wall beat balance, rather than that for the air mass.

This is because the concrete walls have large heat capacity and low

thermal conductivity (i.e. a low thermal diffusivity), and so the thermal wave penetrates the wall very slowly. As a result, the wall's tran- F

sient behavior is important. On the other hand, the temperature rise in

the room is expected to be slow enough that the time derivative term can

be neglected with respect to L., L isaer tei".

-114
4,'4'

Equation (6 1) istandard trans.ient heat. conduction equation for slab.• geometry, with appropriate' heat balances at the boundaries. Examining Equation:(6 2)," the left--and side represents the heat

source, while the right -hand side represents the heat losses. The first

term on the RHS represents radiative losses out of the room openings

(windows or doors), while the second represents the gross movement of

hot gases out of the openings. The third and fourth terms represent the

radiative and convective losses, respectively,:to the room walls.

To put the compartment air heat balance into a more conventional

form, we divide both sides of Equation (6.2) by AW. There results

(T AO

(Of -. TA-) (TG - T•) +

e oFI - T'(o,t)].+ h[TG - T(O,t)] (6.3)

As we saw in Chapter 4, the mass burning' rate may be either fuel surface controlled or ventilation controlled. If we assume that the room

ventilation is the limiting factor (as is usually the case), Equation

(4.2) shows that

m = CV W (kg/s) (6.4)

If there is no forced ventilation, Reference (11) states that

Win 0.145 pA gA0 07 0 (kg/s) (6.5)

- 0.54 A /Z- 0 0

Thus,

A 0.54 Cv (AO rZAW) (kg/m's) (6.6) ~w

and the geometry of the compartment is contained in the ratios (m/IA)

and (AOIAW). The grouping (Ao40 o/AW) is frequently called the ventila-

- 115 -

- &tM -.... ~ - -
M.. -~ 1ý

F 3V -7-.

tion 'factr -F. 

0 To :accoun for forced ventIilation, we define a modified

ventilation-factor Fo such that 0.

.. FF . " - F 2.20(V/AW) . (ma) (6.7) .145 rgzA~

Thus, our general expression for m is given by

- 0.54 C o (6.8)

Equations (6.1) and (6.3) form a relatively simple: set of equations.

One possible numerical solution scheme would be to solve Equation (6.3)

for T 

G (t), with T(O,t) fixed at its initial value, use this new value of

T G to solve Equation (6.1I) for T(x,t), increment time, and repeat this

process until a set fraction of the fuel is consumed.

As discussed in Chapter 2, the results of these calculations are typically presented in the form of time-temperature curves. Since it turns

out that varying the ratio (A /A w) has little effect on the calculations, one curve is calculated for each value of F'.

6.2 PARAMETER UNCERTAINTIES

For a fire in a given nuclear power plant room, the only parameters possibly subject to significant variations are the combustion efficiency n,

the fuel heating value Hf, and the burning rate constant Cv. The room

geometry, the physical properties of the concrete walls, and the heat

transfer coefficients are either known or, in the last case, can be calculated with reasonable accuracy. The combustion efficiicy -s actuAlly somewhir of an unknown for fully-developed compartment fires. Investigators often use this term as an

- 116-
•. "• . .• . -- • , .y•, ,,, - . ,• ,. - • -, J ; •• I • -,, .- ,.'. . . .. .. * ... _ _ _

empirical adjustmentV factor to bring the results of c l u aions in be

ter agreeinent with experimental data. Kawagoe and Sekine use an "effective :heating, value". for wood, which !is 600%/ of the' nominal heating ,valu

for wood (i.e. TIi .60) in deriving the. theoretical time-temperature

curves see -n. in Figure '2.2. .n rea ,: son for the adjustment .. . -is t .ha t in

some enclosure fires, combustion gases are burnt outside of the enclosre in. flames which extend out of the com'part~ment windows, and so *some

energy is lost f rom the room [11]1..:

The heating value H fis generally well-known for most fuels. As in

the growth period modeling, our uncertainty in this factor arises primarily. from our uncertainty in the actual mixture of fuel types within

the room. Since wi and Np do not appear inwivoe uan l in Equation.3)

but rather as a product, we choose to fix H at its nominal value for

f

wood, i.e. 1.85 x J/kg, and nodel our uncertainty in ites value using a modified efficiency factor nin' where

H

= T HI wo (6.9)

f,fuel,

As for CV we have presented a frequency distribution for this factor L

(derived from wood-fueled fire data) in Figure 4.7. We adopt this distrbution directly as our probability distribution for Cmbecaue se o

the large amount of data used to construct the distribution and because

best-estimates of C Vfrom other sources [10,11,57] are generally clustered around the mode, median, and mean of this distribution. However,

we note that the experimentally derived frequency distribution and the

idenL-calt y shaped probability distribution for C represont two diffe- V

rent notions. The first distribution presents the measured values of C

V

117
7f a i~ i t ~i : t t"d e . 6: :6 f . : d iZ ff e r t r i d n t . t he n sec r p e e n t -: o u

" unCertanty in. the value of C fOra prtcario . ~ ~ ~ ~ ~ rm*~ ~ ~ ~ ~ ~ -- V . 2 mlttd fdfeete pertiments the . ;secn e rent our

$-i~~ . 6.3 MODELINGUNETITE

The two primary. sources oi uncertainties in ýour baslic modeling are the e assumption of :spatial uniformity of thermal stresses and the burning

rate celation Equation (6.4).

In computing the. stress experienced by' a i- is iomponent, assumed

that, the stress is only a- function timethat.varyingi: ,of th`eý-,locationJof

-the component will not vary the heat it absorbs. While the assumption

of a uniform air temperature may be reasonable, the radiative .heat flux

will still vary with position, unless the fire's flames occupy the entire compartment. This latter condition will not hold if the fire is

.strongly limited by the available ventilation.

Somewhat more importantly., the direct proportionality of the burning

rate m and the ventilation rate Win in Equation (6.4) is an observed relationship for ventilation controlled fires burning in wood-fueled rooms

with moderate ventilation. However, this proportionality may not hold

for fires with very low levels of ventilation. If the amount of. fresh

air available is too low tosuppor- open combustion, smoldering of the

fuel may occur instead, with the consequent -release of toxic Sises and

unburnt fuel vapors. Thomas [131 notes sources which state tnat. fires

burning in compartments with very low levels of ventilation may oscil-

"Lte in size and intensity (as opposed to the monotonically increasing

fire severity predicted). One nan even visualize explosions within the

compartment resulting from the sudden availability of fresh air, perhaps

-118- •..----------------------------------s-----,-
.1:4w! - tiC. ":~.. S~~cs.¶V~r4.b.;.'rmsr¶4y:.,rA.. *"-~ 3

-- .. .t-*_ r~.r~J ra~* :A. t-. .±wtJ~Fev,4qw**-*~4%. .. "'rrp..._*'~ _ *Aw~ - --

1

4-

44

I

p

L

due;to•the- openinrgf 'of a ddor or i.breakino° window, t o a. r• roomfil•

• . . ;i with 

ii% F !• 

hot; 

"c f : £' 

combustible t a.n4 t e , • ''•u . . . o e . t " ' .. ." . . . . ,vapors.

Further uncertainties in the burning rate model arise if the' fire is

u* ele'' /"n5-

not uh etnel:.tio:` allow for non-cellu-

.-osic fuel s i. n ._our:'analysis (see Equationý (4.'20)),,such an extension is

..nothneessarily a t..rivial: matter;: 'Harmathy states that fires, burning li.-

quid-"fuels' aan d_. some plastics' -may. ,involve rather different. burning mechanisms.

_ /6.4 RkE FERENCE MODEL VER'iIFICATION

'The -bcuajof the. DRM approach was indicated earlier"in- Figure 2.2.

However, .the fires simulated "in that-"figure are wood-fueled and are relatively well-ventilated, and -thus are more representative of residen-

''tial fires than they are of nuclear power plant fires.

The ventilation factor for a residential fire is typically about 0.1

mo " while Figure 6.1 presents a histogram for F' for rooms in two nu- 0

clear power plants. It should: be noted that these latter values are estimated- on the basis of the rated .volumetric flow rate of the forced

ventilation systems for each room, that the effects of doors have not

been accounted for.

To check the accuracy of the simple reference model approach used for

fires with.low ventilation levels we consider Gross and .Robertson's experiments on fires in scaled-down compartments. The time-averaged compartment air temperatures measured for a given value of F typically ex-

•hibit a fairly wide range of values. Figure 6.2 shows a p.lot of 'the

upper and lower values measured for a ventilation factor of 0.0022 mih,

tg

- 119 -
4.

14 31. up'oer,

600"

400-

t-a

0

1

JU

(43 1, lower

200-

n i

0 2.0

0.0022

' •" ' ..• , •'. "•",I' 

4.0 6.0

t, hours

h Experimental Data. from Ref. (43).-

,'L

Fig. 6.1 - Comparison of DRM Prediction wit
f

20

. I 10- I

II

0 0.01 0.02 0.03,

F., ,m .. 0

Fig. 6.2 - Distribution of the Ventilation Factor for Two Nuclear Power Plants.. [58,59]
as wel•lasa' time-tempe r atiure curve deve16oped, usirng theDRM.- If we assume' thatý_ .the~ experimentar -:n irdes.va have abou tthe,: u same, duration, -wed.- can

see, "that tere is,.a. cons~iderable,_ variation about-ýý the predicted.' time-.

temperature curve. 9.' 

c.

-9.9~

;6 .5 A SIMPLE APPLICATION"

To illustrate the fully-developed burning period model, we consider a

ventilation.controlled fire ..in a',ro6mo whose.characteristics are given in

Table 6.1. We note that the:'roomr is ve'ryl..Jarge, has ;a fairlyo lW vent i-.

S latio~n factor .!:(we assume that.one of the doors. leading, into the room -s

open), and has a fuel loading consisting of cable insulation.

-The twolparameters varied because of state-of-know0ledge uncertainties

are C and i' The distribution for C is as shown in Figure 4.7,

V. . , •V

i.e. CV is lognormally distributed with j = -2.12 and o= 0.37. We

choose Ti' to be uniformly' di.stributed between 1.0 and 1.46, which. dorresponds .to a combustion efficiency of '1*.' and aheating value H rang7

ing. from .1.86 x 107 J/kg-to 2.7 x i10 J/kg.

The response surface fitted. to ',the .time-temperature curves computed

using the DRM in a 20 trial Latin.Hypercube Sampling scheme is

T (t) = A + Bt (°C) DRt (0 (6.10)

where t is measured in minutes, and the coefficients are given by

142 + 832C -229GC4 + 8211'C V V V

(6.11)

B 0.480 + 0.153C V - 0 *.1 l 2 + 0.0394TI'CV

The time-averaged temperature after one hour is given by

- 122 -
4-. 4-

2. ~:&~

T ablIe 61 Fixed Parameters Used in FlyDelod

Fre Sim~ulait'ion

Parameter Symbol ValueJ

ventilation factor F 0.0083 'm/2

-door to wall area ratiOs . o0111..

gas emissivity. .

1.0

fir c6560 J/kgOK

.gas speci.fic heat

ambientý temperature Ai'- 9

wall emissivitY 

.

wall surface heat transfer

coef f icient 

h 10 W/m2 0 K

wall thermal conductiVitY k 1.16 W/m0 K

wall mesh thickness Lx 0.01 m

H .1.86 x io7 .1/kg

fuel heating value f

does not include floor area

123 -
'. . .'" .-

" 

..- .- .

-. ....

.... .:• .. ;, ... . ... :: . ,Dm•: = ,,-::-.• .. . ..- , .. .. . .. .c ° )" [ . •

157 52Cv 30qq, 82OiTC. C)(12

... -.

We remark that this reponse surface's predictions vary from the solu

ti-ons of Equations (6. 1) and(61 3)only by a few degrees.'in most cases.

.The-largest errors(typically around 10°C) occur at the beginning of the

fire, when the fire is growing.rapildy, and the assumption of fully-developed burning is not necessarily a good one.

Since the'temperature rise is quite linear and fairly slow, we consider the average temperature predicted by the DRM, TDRH as the representative m.easure of fire severity in this analysis.

The propagation of the uncertainties in C V and n through the given

response surface leads to the histogram in Figure 6.3. The lognormal

distribution superimposed on this histogram has the following characteristics:

ii = 5,81

a = 0.157

Mean = 338 °C

Median = 334 0C

T DR,.95= 432 °C

T-DRM,.05= 258 0C

With regard to the modeling uncertainties in Equations (6.1) and

(6.3), it is felt that the DRM is somewhat conservative in its neglect

of heat tr&nsfer to the compartment floor and contents. Furthermore,

conservative values were generally used for the constant input parameter

values used in the response surface construction. On the other hand,

Figure 6.2 seems to indicate that the DRM may be rather non-conserva-

- 124-
.20-

PI

.5

0

A

~ .10 0.

0

200 260 380 440

TORM (°CI

Fig. 6.3 - Distribution of the Average Compartment Temperature (Uncertainties Due to Input Parameter Uncertainties)
.. ..... . .......... • ,..•. •, • • ,• ,• . y.•L.: ••.:••,:- ,•.:.• •

~~AT,

i-ve:./ Fo Wh ilewedoep pssei use, of this la'ttrdswe do

g ýie t.overwhlnicing i.eigt Gos ndRbronsexrime~nts were

conducted in scaleddown~~ compartments, There we intivey xpc th

behavior of a 'lully-developed fire to Aiffer from the behavior of a fuled'iewt . e is :a so, well now tha the2

*. scaling of-a fire experime is a dficul t task, 'since many of thep~ortanthf l :phenomena' 06 .Iot: ,olwthesamne s'Calin laws. 4

Fo he purposes o f, thi's -analy's i s o .ur ditiuinfrE te, e'rrior

..fator ,epeetn ou-luncertain~ty:in the 'DR.s predic'tions, is ,ade

o : •' " '.. .. . ea = 0 . 2 K, .

hVRognormazistribu t on _withNýikth~e-rfoý 9i-JP

amete~rs -

aA

p -. 44

o 0.'334

MIean =0.92ý

"ied i an. 0.T7

-E ,1.5,

T,0.,95: p-

'E =0.5 T,0.05

The bulk.of the distribution lies 'to .the left of unity, indicating our

belief in the conserVatism of the model.. However, a substantial tail.. -.....-..

lies to the right of this point., ýal~lowing for the possible non-conserva- . "

tism seen inFigure 6.2.. -

ion! for T'R and out distribution,.

t'•'•:. ., lognormal, our -final state-of-knowledge distribution for T-, the average . -

.compartment airtemperature after one hour'sl.duration of fully.developed . .

.. burning in the. given room, is also lognormal, and has the following par .. :

126.
I ameter

-e ~ ,.-.' 2...-. '--Thz- ~ 'it ?c.XAYJ¶c- ~ ~ -- A

'~-~> ~ -'

S:.

I

I

I

I

* p5.67

a - 0.369

Mean =309C

Median = 289 °C

T1,0.95= 530 °C

T . 157 OC

- 127 -
-~ -~~4 ---

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- 133 -
Appendix ýA

SHAPE FACTORS.

-, ~ ~.

Al<

.In,: general, ,the radiative.• h••t ransfer from a source to a receiver: can

be separated into two factors:, one represents-the-strength of the source,--

* and the other represents,* the 'fraction of-theisource output,:intercepted

bv.-h':t e recieve.. .This- sst. fas raction. is.-- cale- th-sap fa It orfrom the

• source to the receiver." Thus,: if the .source. strength is watts,* the

receiver sees QR watts, where

QR= FS-R QS (W) (A. 1)

If we define an average source heat flux S as

qS Qý/AS (W/M2) (A;2)

and an average receiver heat flux qj in a similar manner., then

AR", = AsF " S -

or

q R (A /AR) q" (WIM2 S F S-R S (A.3)

This is the basic equation. needed to calculate the *heat flux impinging on an object's surface, given a heat flux source of strength q" We

see that the modification term is purely geometry dependent (we are neglecting any attenuation of the radiation between the source and the receiver)

- 134 -
In orderto compue Fs~ Rwe 'first -look atth sapfcor ore

change:betteen diffe rential surface elements. Given the situation shown

.,.in Figure A.1 it can be-shown :that

'. . .- . . oS 1 cost 2 dA1 dA 2

l1dS --dS 2 1 2 ,i7r

Clearly, if we wish to compute the shape factor between arbitrarily

shaped and oriented surfaces, we must, perform a quadruple integration,

i; e.

- f o~1 col€s•2

F f . dAi1 dA2 s (A.4) .-2 A1 J J 2r

A A

12

- Analytical solutions of the above integral are fey•'. Moreover, in the

* context of our problem, we rarely deal with traisfer between exactly defined surfaces. In order to reduce the complexity of our problem, we

assume -that our receiver can be adequately characterized by a strategically located differential surface element, that the heat flux received

by this element is the average heat flux received by the receiver over

* its entire surface. We now have

R= (As F s-R/dA ) q""

Employing the reciprocity theorem for shape factors,

i i-2j jF (A.5)

we obtain

qR = (dA RF dR-s/dAR)

- FdR-S qs

- 135 -
& ~

.7

S

t

Fi g. A.l Transfer(Between Two Differeniial Surfaces-

'1<1~.dR.

''Fig. A.2 -ransfer Between a Cylinder and a Parallel Surfa

b

0 S 

E7dR

Fig. A.3 -Transfer. Between a Cylinder and a Perpendicul-arSurface

"136-

....

. ~I IjUJIihd 

dji jH
Since the evaluation of FdR S only requires a double integration,

many more analytical solutions are available than for the earlier case

of two finite surfaces. Some representative 'formulae for transfer from

*.. a cylinder to a surface.element and transfer from a rectangle to a surface element are as follows [5,6].

1. Transfer. from -a cylinder to a surface element parallel to the

cylinder axis (see Fig. A.2):

F -~~- can 1( )

dR-S - .Y

X [A-2Y -1 i A(+1) 1L - :1

y tan B(+1 I tan (A.6)

where

X a=''

Y = c/b

A = (I+Y) 2 + X

2

B = (l-Y) 2 + X2

2. Transfer from a cyli-der to a surface element perpendicular to

the cylinder axis (see Fig. A.3):

FdRS 1 [an-- V - - tan' Nt ] (A.7)

where

X = c/a

R = b/a

A = 1 + X- R

B = 1 + X2 + R2

- 137 -
" , .V T:""':

3 Trans 'I f er.f rom a e retngen e to- a -,surface-tae eme --parllel ut) .hep.

S1 ahe eof th e Ieca n (see Fig. A 4 .

1 X 1Y _I

F -'- tan (Y/A) + AYtan (X/B) : (A.8)

dS-R . 2-q A"'... -

X a

*Y b/c

A I/l T -

B n _7

.4 ~rfrma Ta etage5 to a s urface eerne perpendicular t

the plane of -.the retnle.e Fg .

s-e- -- ' .

~dSR 27 (t an (/) Atn ()I (

i:

'r;p,.::5 .:: 

where'

X =a/c

A c/b

'A 1.1 \2 .

. If the -receiving Object is

"ures we apply another shape f

ing . integrals)

A1 F1

For example, if we have the si

A F

(1+1' ( -+2')-

and so

"A A.F =F A

1 1-2 1+1')

i- .-. 

4 -,.j

not. conven-ient ....

• not. conveniently aligned as in the fig

actor• re 'ation (derivable from the definZ (Ak Fkj

tuation shown in Figure A.6,

A.IF .•2. + .- iF i,- 2

(1+1')-2 . A,

-. 138 -

(A. 10)
. - , .'. -, ,,., - -r..... .";. -.-.. 

;- :-... ; -- - -:. -I- - "-

dRA6

I Fig. A.4 - Transfer Between a Rectangle and a Parallel Surface

S b

dRo

I- a I Fig. A.S - Transfer Between a Rectangle and a Perpendicular Surface

- 139 -
I .•r•-•••*-,- .: .-o' • :F:..•d;'2<:'

.2

. . . ,.

ft

Fig. A.6 - Transfer Between Combinations of Surfaces

- 140 -

.----------. .... - : . .., . . ...r . .- .
N

-- 2 - ' -

I- --

2,

4

i

- 141 -

aj

NOMA" ikikiýL
Appendix B

-LATI1N 'HYPERCUBE .SAMPLING

•p: .•- •: • ,-•• •v.• • ;,•! . .* ,<,v. .%.• "- a %%T.,.: --- '---,•.:,•,•-

ih5 scesof a respos s'urface- simulation depends. strongly on the, accuracy of-the function (rsose s ur fa'ce used to approximate-the atua,

phys-ica•l• 'm•d in-questioni. Since.:.the- response: surface is based..on..the

physica moe' rdcin-or aý_,ýlimiteds number o poi-n ts'; t~pyi

Siiis

. -t.parameter cal mode.'.. s .hese vectors r.equied in the.- ..t.. 

poin ts-.m I ust bee. carefully:': chosen. One sidccessful scheme:- foro. selecting

iptecosis-the Latin Hypercube_, S amp ling, (LHS) technique,,described,

in Reference':(31)

To illustrate the.LHS approach, we consider a simple example. -Let us

' "passume that our psia moel only requores three input parameters,

XX

adthat we wish to construct, our, response surface from. only five data

'poins. -nodr to slc thdaapns w proceed7 as follows' '(see

Figure BAl).

1. Define the highest and lovest values of each parameter expected

Iin the problem, i.e. find 4 and x* j. such that

x0 <,X < , i=1,2,3( i A i

2. Divide each of the parameter ranges so defined into five sections

(since-we desire five data points). The widths of the intervals

need not be equal, but often are.

142 -

- -~ - ~-Z~ ~*~-- ~ . - ~ - - -~
/

- H'

* U.)

2 ~

x xCX~d X I0

'.3

5 '.3

V Cft0r xi

3 .3b 1

3 3d d,-1 d- b

2 d

7".' 3

12

.X2,

t

x X3

Fig. B.1 - Illustration of Example Latin Hypercube

Sampling Scheme

.C a a

5- e b C.

' ' 33 '"3'. . . "'
-' ameter 1•.-, ý'Assign 'this interva-l one. : -arank-of IiilFigur. 'B -',

thisithe .interval -ith acircled one. -Randomly, select, one: of

the remaining interval and,:assign- to", it a rank'oft,,Y'Cot inue

in a similar .manner until all five of the intervalss have an associated raking.i Repeat -the'entir'e,'.process, for. 'pa•rameters'. 2 and

" " " ~~~... 3' ::.

4 Randomly select. a point within the first :intermval`"of'.parameter .

'Th:is•,.point', labeledl,-ill represent the range ofvalues-withthisinte'rv7al hin when i'w-.- construct a response surface- We-note

that the selection process need not be hased. on a uniform distribution over the interval. Repeat the process'forthe remaining

intervals of parameter 1, and for all of the intervals-6f parame-

- ters 2 and,3. By followink.this procedure, we generate five va.-

lues of each parameter which represent the entire range of values

" for- that parameter. Moreover, each of the five .values.has.-an. associated ranking, derived in step 3 above. Thus, x has a rank

la

of three, x3e has a rank of four, etc.

5. Form the first- input vector by -grouping the value of parameter I

whose rank-is one with the values of parameters 2 and 3 whose

ranks also are one. In Figure. B.1, this vector is the triplet

(xId,x2d,x3b). Next, form the second input vector from the parameter values whose ranks are two, and so forth The inset table

in Figure B. 1 shows the -five input vectors generated in this simple example.

- 144 -
Among the many advantages of the LHS, one of the most important ones is

that the full range of each parameter is represented in the sample,: and

So extrapolation of the response surface constructed from the s'ampletis

not necessary. This can be contrasted with an ordinary random sampling

technique where sample parameter values are randomly selected from the

entire parameter range, and so certain ranges of the parameter values

may be over-represented, while others may be under-represented in the

construction:of the response surface. . -

Once the input vectors have been selected, we can calculate the response of the physical model to these inputs. We next fit an appropriate

approximating function to the results of these. calculations, typically

using least-squares regression, and our response surface is complete.

Conceptually, the above procedure is not difficult. In practice,

however, the situation may be different, since the determination of an

appropriate functional form for the response surface's dependence on the

input parameters is often a non-trivial problem.

4

I

- 145 -
* 

7

-,Appendix C

CONVECTIVE H.T-DR G REGROWTH

.. The:convective heat fluxesto an- objectduring theiperiod"of fire growth

rare assumed to be non-zero. only when, the object:is i.within the- fires -

flames, the hot buoyant plume above the flames, or when-the object is

within telayer of -hot. gases, accumulating, near, _tne ceiling. Nea

.-transfer- wlihin-the-flamres is.%-tre:ated in Chapter'",. Inthe remaining

two cases, we need models for the temperature of the hot gases and forthethe heat transfer coefficient between the gases and the:object's sur- r

face., since

q:h [ T T I (W/m) - (C.1)

o,c e 0

-C.-I HOT GAS TEMPERATURE

'A -number of researchers have studied the characteristics of the plume

caused by turbulent flames, and have arrived at similai correlations for

.... 1 the temperature r rofile The correlation we shall use is reported by

Alpert 1391:

-

.

": ! ~ ~~(Q Jr)23 2/3 , i:•

0.,0538 cc, r > .18z z

T p T =2/3(C.2 . pl A ý/

0.169 C, r < .18z :•".. 0.169 .5/3,

W.;here z

QC = (1 ,- '.)Q (W)

z = axial distance from flame base (m)

-146-

• ' -;... N: • . .
.r radial distance from plume. centerline (i)

Note' that 0r=O.18Z canbbetaken to •define an effective :radius of the

buoyanftpume.

Strictly speaking,.the above correlation is good only when the fire

is small couioared to the room dimensions and is far away. from any walls.

However., if the fire is very close to a wall or is in a corner of the

room, Alpert claims that the correlation also works well if Q is dou- c

bled or quadrupled, respectively.

As for the temperature within the ceiling hot gas layer, Section

4.1.2 describes a simple model for this quantity. If a small. fire under

a high ceiling is considered, Equation (C.2) can also be used, where z

is replaced by the height of the ceiling above the flame base.

C.2 HEAT TRANSFER. COEFFICIENTS

The convective heat transfer also depends on the surface heat transfer

coefficient h. For object surfaces parall6l to the plume flow, this

parameter does not vary strongly. Experimentally derived values for

heat transfer within a flame are of the order of 23 W/'°K [601, and

this value can be used conservatively for transfer within the buoyant

plume and in the ceiling gas layer.

When t?.e object is perpendicular to the plu•,e flow and is large with

respect to the plume radius, Veldman et al. [611 give

0 .Ol(r/z)- ' < h' < O.02(r/z)- • (r/z) > 0.1 (C.3)

- 147 -
- o-imensionalize-d he at tan sfer coefficient

- nori-dimensionalized heat production r at e

- Q'/~Ac z~TA. K2

-If R is -the object'S effective- radius, ýwe 'compute the

transfer coefficient to be "

h(r) [T 1 (r). - r dr

T (r) T ]rdr

p • . A. .

average-heat

or

h=

0.64 ( /

2.06 (Qh/z) c

'W/m2,K, (R/z) > .18

W/m2°K, (R/z) < .18

(C.4)

if Equation (C.2) isused for T p(r)..

- 148 -
Appenhdix D

H EAT- FLUX FROM CEILING, GAS LA,.YER: -

-n; the following derivation of the expression 4for-the .heat flux from the

ceiling gas, layeri.(Eq-. 4.15), we follow a more ge-neral and':.detailed description-found in Ref. (5).

.Let us. assume that: the..gas layer can be modeled as. an 'infinite slab

7of.:.: thicknessl D, "witfih an. isotropically reflecting .and emitting boundary

at z = 0. and a vaccum boundary at z = D (see Fig. D.l)

Consider a slice of the gas layer of thickness dz. The wavelength

dependent .emission of the slab per unit solid angle per unit surface

area is

a (eb(z)dz

IT

a = absorption coefficient

e, blackbody emissive power

The subscript X denotes the wavelength dependencies of the q'lantities. The emission per unit solid angle per unit surface area in the

direction of the angle dependent flux is

a Aeb(z)dz

IT Cosa

The attenuation of the rays passing through the element is

$,J)z)dz

cosa

- 149 -
-- zoo

II - D

;A d IA

Fig. D.I - Model of Ceiling Hot Gas Layer

~ pqo

Fig. D.2 - Model of Ceiling

- 150 -

.4
:" -.- scatte cingoefficient,,..

Ifi we assume tViat our gas is•,Ia pure absorber, .- = .0.0. We further

assume, that our Ebsorption coefficient is wavelength independent, i.e.

a = a.' This is known as the "gray, approximation.",. Defining

z

3ý

V

-a dz,, TD D T (D), and p cosO,

dl- + I e (T)

or

I)(0,.)e + ebx (t)e .dt , 0 < v < I -0/ - , -(t 1)jd

I x Cr, 1j)

•D

IA "(TD-vi)e/- l+ -f (t)e-(t-T)/•dt_ ,

T

-I < vi.< 0

Integrating over all X, and noting that

fbdX = oT4 and fId) = I

0 0

(T, 11)

f T

I(0,0)eq/1i + Jf OT4(ý)e-(T-t)/"dt

0

I(TD p)e (TD +- iOT.4(t)e-(t-T) T D /vl dt

vi

-1 < P < 0

T

Integrating over the ranges 0 < p-< 1 and -1 < p < 0, and defining

- 151 -
;•i• .••.-q , I ., ,- .:• •;,•• • : *••.) __" . • •• .. . . ... . .•.; I 

;•:i • " "there obtains " I

:q (T) - 2,ff I (O.I)*e- /]pdjj 4+ 2 oT4(t) (t-t)dt 0 < 11 < I

2 

2

0 .I . ..

and

q (T) ,, 2rf hj+2,h)+ CJT(t)E 2 (t-T)dt, -1 < Ij < 0

To find I(O,u), 0 < < 1 1, we assume that, the ceiling is an isotropic

reflector and emitter of radiation. Thus, as shown in Fig. D.2,

=[coT [(O,p) + pcq-(O)]/r 0 < V < 1

where

C c eiling emissivity

PC= ce.[7ling reflectivity

T = ceiling surface temperature. C

To find I(TD,IJ), -1 < V < 0, we further assume that any incoming radiation at the vacuum boundary is isotropic, i.e.

I(D,p) qD/1r -l < V < 0

Thus

21f Ll(0,v)e 1

-/Illdp 21rEoTL• + pcq-(0)IE3()

and

-1

21f( I (T D, p)e - ('D-'•)/P)Jdp = qE( D1

where

- 152 -

- . . -.
E3 (x). 7.I

Evaluating q-7 ) at "r 0 "we-obtai •E~ 0. e,.:(. - . .. n

and so 2 2E .., 0 ,, . (, ..I IpE 

"+'2PC OT'(t)E2 t)dt]-.or + 2 t t LD-t)dt

. .. D..I ý I• - :- 2

f0

.If *the gas t~emperature is uniform and equal..to TG throughout the. entire

slab,-and if-the absorption coefficient is independent of z (.0.aD),

q+(aD) = 22coT[E 3 (aD) + [2E3 (aD) pcqD

3 . C'-D .D 3., D

aD aD

2pT (2•cE3 (aD)f E2 (t)dt + E2 (aD-t)dt]

0 0

.We now approximate the exponential integrals E3(t) and E()by

equating the zeroth and first central moments wit. thhose of the function

A exp(-Bt) (a good approximation in most cases), 'd.n it turns out that

• .- E3(t) -0.50 exp(--1.5t.)

E (t) " 0.75 exp(- 1 .5t)

2

.Finally, the outgoing *heat flux at the vacuum boundary is

Sq+(aD)

+ + oT(3-X)(I+2cX)

where

X = exp(-l.3aD)

- 153 -
..... :.

* J

A

Ak

- . - , :: .... " : - :==== == == ==

- - Appendix .--.*.- -. .. - : . ... E,..

VEIICT ON F EQUAT1ON (43)

A The equation, for the6 fuel-surface--controlle'd burning rae per unit -area,

Eq. (4.3), introduces positive feedback to Our burning rate model. Be

_c-aue ethi e . an imortt in predicting -t.he accelerat ot

careph resents. an',pq nfactor.i

-ofiegotwe. simulate -two expermet iorer -to check i t he" va-li dof-fir ~.- imn-: n

ity ofthe equation.

"'.The first. experiment,, reported by- Huffman et. al. [62], investigates

the burninig-.-rate of cyclohexane. in pans arranged in a hexagon surround-

" ing a central pan of cyclohexane (also burning) as a function of pan

diameter and separation (see Fig.l). The fuel parameter values assumed in' our..simulation.are given'.in Table E.I.

.If the feedback to the. fuel from the flames "rc-i i.ct mportant, we

would expect to see little change in m"-with changes in pan separation.

As-seen in .Fig. E.2 -howeyer-, .-there: 'are ;dramatic changes as the- pans are e.

brought, closer together;

Our reference model predictions' are also shown in Eig.E.2 'While,

the agreement with experimental data is not outstanding, -especially ;for

the smaller separation distances (where. flame interaction effects such

as mergin become important), nive'rtheless, the predictions are reasonably

close in magnitude, considering the simple nature of the model."

The second simulation, is of Orloff, .1odak, and Alpert's experiment,.

where a-3.56 x, 0.914 x 0.064 m slab of polymethylmethacrylate (PrIMA) was

- -d

- 154 -

~~~~~~~. ;. . -•. . . ,;-J........ ....... *. <4.-:•• ,J• • • • m• •:* . . _4
•iilJ.

,o 0 .• . .• •.-: • -: 0t;0231 k

c:: Fs:2.56 x 10o : k

,Hf 4.67. x':10 j

0Y 0.035

'1 1.00

Table E.2 - Parameters Used in PM'IA Simulation

g/mi2

s

g/j

/kg

1190 kg/rn

Cp 2090 J/kg0 K'

Vk 0.27 W/m0 K

Hf 2 67 x i0o J/kg

T*" 636 0 K

p

T-: 743 0 K . S

" mo 0.006 kg/m 2

s

CS 6.21- x 10-7 kg/J

y 0.39

TI 0.90

- 155 -

.. - ~ ~
!

rv

0.10 -

0@

I

0",

E

00

'• 0,05

Coentra iSjDrmor

P~v~peroIBurner

b- 0. Mbd.Il

0

0. 6.0 8.0

I

0 ! . .

0 2.0 4

f:~ g.1: - Modc I o f 1: xper-irnien t (6 21 Pig E. 2 - Comparisonof OPredictedResults Witth

ýexpcrimerital Data 162-
u d[63] - Some, petiln,ýpyia properties o~f, theý slab are given'_

in-Table E.2.

Since the hot gases from the fire on the lower portions of the slab

will pass through the •upper' burning zones, causingthe flames to grow- in

_thi~c1knes and intensity,- we ,expect the fnraighetfue. rm'h

" .- flames- /;to `-the burning fuel ,,bed to.clause6 an. increase'. in' the 6jlocal •burning

rate" witth• height, thus providing,-"a good, -test for Eq. (4.3). We note

:that although the• flux' to .the fuel bed is-not an external heat" flux as

we have defined earlier, we model it 6as such for convenience.

" "R'e'f ingi{ F'igure E 3,"it 'is:assumned':that the''flTami -height Zf, -is

m!,ch greater than the flame width W. Ifn order to detrmine: the heat

f.ux. from, the flame to a particular--differential element. at height Z on

the fuel bed, we model the flame as a rectangular slab of height Zfl+-,

width W, and' thickness d. The flame height Zfl is given by Thomas' correlation for a line source fire 17]:

Zfl = Cft(") I(E. )

' :•re

burning rate per unit length of line source (kg/mi s)

Z m"(z' )dz' f

0

Cfl, 29.7 m.hsý//kgO for flames adjacent to walls.

For our problem, we assume that m" is :oughly constant, and so

m zm" (z) (kg/m s

1 5•
.1 , - }4I<S~7~2~ ~J3 v:

L 4

1C

4-~10

Fig.

•=" 2C

;E" 10

a

/1FL?

Fig. E.-• Mode ofExperiment

CA2

[63]

E.4 - Transfe-r Betw'een' a Differential Volume

and a :Differential Surface:..

z cm

Fig. E.5 - Comparison of. Predicted Results With Experimental Data [63]

- 158

........
As forý the flame thickness,'Orloff et al. use experimental-vidence to

A'lthough" the heat. flux' the fuel bed has both convect and r adi -

tive-components, we- assume that)- the radiative component.d.ominates. it

-is hoped that the resulting model will reasonably predict :the desired

burning rate, evenwhen this assumption is not good.

The-heat flux received by a differential surface element is (assuming

100% combustion efficiency) . •. ' ' '' - ' 7

dS dS-fl qfl-- is '

For the problem at hand, where, the radiative exchange,.' between. a

* differential gas volume and a surface (see Fig. E.4),

dV[a coso T(r)/irrr] (E.2) dS- f,

V where. fl.

T(r) = 'transmissivity of gas

.. exp(-ar) for constant property gas.

a attenuation constant for flame gases (m-) 1 .' If the differential surface is located along the centerline of the

flame (W/2,y,O), it Lurns out that

FdSf = 1 - (aD) - dO[E 3 (api) - E3 (a )] .. ,j T... 3i .-•i3(a

where

P1 = Z /cos°(/2 + 8) 83 -T/2

P2 = W/2cosO 82 = tan-'(-2Z0 /W)

P3 = (L+Zfl-Z )/cos(r/2 - 8) 03 = tan-'V'L+Z 1 -Zo)/W]

84 = 7, 2

-159 -

•i............................................, "
a, ... ~:

.A5". 4 -4 :4

an . : . .: -. ,-

Y F j1

e A- fl 2 -fL) fl, (E 3) fi El

and

q n + C q - 0 S e'

in =[ ~ CSFdS flyH~f ]

Asumn. I 0.39 [40Jý and Assuming. 2"=0.0 - 0 gms. [351, the predictions of

this model are compared with the experimental results in Fig. E.5.

Again, the predictions are reasonably cl.se in magnitude to the experimental data.

- 160-

MENEM
LIST- OF. VA RIABL-S-K' ::.:

Def in~ition

44~4 A

C

C

H.

Q

Q

X'1

'1

Aur1.aea (int ole,)rqn rat - consfi~nt (kg'/J)

V.:entilation control-ledbu'rning-irate constant

S.Shape -.factor f-rom object.. to flame:

Heating.value of --fuel -(il/kg)

Rae fhat productio. W

Rate heat is. radiated away from fire (W.)

Temperatu're ('K),

.Ignitiontemperature: ()K.;' '

,Rae of"airf 'flow. in'to. comp rtmen•t ig/s)

Height m): - .-- . ' h... ;j?..:.- e•. -: ! i: . ,i: -.-:: '" .

44'.

C

h

k

m

q

t

a

T1

ai

Specific heat"-(J/k gK)

Heat transfer coefficient (W/m2 0 K)

Thermal.conductivity (.W/m.K)

Burning rate of fuodi (kg/S) '

Specific burning rate constant (kg/m 2 s)

Heat flux (W/m 2 )-

Time (s)

Time to ignition (s)

Thermal diffusivity (m2 /s)

Emissivity

Fraction of Q radiated away from fire

.Combustion efficiency

Stefan-Boltzmann constant. (5.6697 x 108 W/m2 0 K

4 )

elected Subscripts

e

ext

fl

f

G

0

0

W

Definition

Environment

External to fuel bed

Flame

Fuel

Gas

Opening (e.g. window)

Object

Wall

'4.

"N

/4 1

rj

NRC FoRm335~ (777 US. NUCLEAR R'EGULATORY COMMISSI

8I1BLIOGRAPHIC.DATA -SHEE

4.:TITLE '%AND SUBTITLE (Adl ;V'ourne No. ,t ifopr?,oatel

Pro_,abi istic Model's for the. Behavior of C

7.-AUTHORISl

NathainO.. 0 Siu

--1. 9. PERFORMING ORGANIZATION NAME AND MAILING ADDRE'

UCLA California.90024

Los Angeles, California : 90024

School of Engineering and Applied Science

7;1• SP,-NSORING ORGANIZATIONNAME AND MAILING ADORE

U. S. Nuclear Requlatoi-yCommission

.'r . .Office of: Nuclear.Regulatory Research

JDivisonf Risk Analysis

Risk.Methodologyn an'd Data Branch

13. TYPE OF REPOP?'

Technical

15. SUPPLEMENTARY NOTES.

" 6. ABSTRACT 1200 words or less)

Physical models which predict the thermal

_i ' .. fluxes) duhing'a- compartment, fire as. funct

Sincel.arge uncertainties-are inherent .to

General models are constructed for the per

burning. These models are used in sample

particular .compartments.

The overall methodology requires the synth

information available in the literature. I

by the deterministic model are addressed ai

the model itself, to form state of knowled

17. KEY WORDS AND DOCUMENT ANALYSIS

Fires

T hermal Hazards

Fire Growth Model

Heat Flux

" l7b. IDENTI FIE RS/OPEN-ENDE D TERMS

18. AVAILABILITY STATEMENT

i#•.:::1{n1 irni tf rl

A '~'

1. REPORT NUMBER (A ssgned by DOC] IONJ

T NUREG/CR-Z269_________

2ý (keae blank,k

ompartment Fires 3 .E-CIPIENT'S ACCESSION NO.

5. DATE REPORT COMPLETED

November i. 1980

SS (Include Z2p Code) - DATE. REPCRT ISSUED

" MONTHl. .. ..'

August 1980

6. (Leave blank)

IS.' (Leave blank) J

SS (inc/ .ute Zip Code) 10. PROJECTITASK/WORK UNIT NO.

1, CONTRACT NO.

NRC FIN B62,4

PERIOD COVERED (inclu$ive dar•s]

:1.0/78.- 11/80

14, [L c-v olAk) -

hazards (including temperatures and heat

ions of space and time-are developed.

the analysis, the models are probabilistic..

iods of fire growth.and fully-developed

analyses to estimate the fire hazard in

esis of'a deterministic physical. model from

Uncertainties in the input parameters required

nd. 3re incorporated with the uncertainties in

ge distributions for 'the thermal hazards.

17a DESCRIPTORS

19 SECURITY CLASS IT"¢s Aeoorrj

Unclassified

20 'e TY Cl.&SS Th,s P,9eI Uric S( 1 e(

21 %0O OFP L

22 PRICE.

S

NAC F'ORM 335 17-77)

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