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Fire Growth in Combat Ships

John A. Rockett

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Fire Research

Gaithersburg, MD 20899

September 1986

QC

100

. U 56

86-3451

1986

d in part by:

tvid Taylor Naval Ship

sh and Development Center

gton, DC

Research Information Center

National Bureau of Standards

Gaithersburg, Maryland 20899

NBSIR 86-3451

FIRE GROWTH IN COMBAT SHIPS

John A. Rockett

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Fire Research

Gaithersburg, MD 20899

September 1986

Supported in part by:

U.S. David Taylor Naval Ship

Research and Development Center

Washington, DC

*

1

C

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

TABLE OF CONTENTS

P age

LIST OF FIGURES iv

LIST OF SYMBOLS v

Abstract 1

EXECUTIVE SUMMARY AND RECOMMENDATIONS 1

INTRODUCTION 2

FIRES VENTILATED ONLY FROM ABOVE 2

Steady State Model 2 Transient Model 3

PREDICTION OF FIRE GROWTH RATES 5

Exponentially Growing Fires 5

SPREAD OF SMOKE AND TOXIC GAS 9

Recommendation 11

REFERENCES 11

APPENDIX 1. FIRE PLUME FLOW THROUGH A HOLE IN THE TOP OF AN OTHERWISE

UNVENTILATED COMPARTMENT 15

APPENDIX 2. FIRE PLUME IMPINGING ON THE CEILING OF A COMPARTMENT

VENTILATED ONLY BY A HOLE IN THE CEILING DISTANT FROM

THE FIRE 19

APPENDIX 3. FIRE GROWTH RATE MODEL 21

iii
LIST OF FIGURES

Page

Figure 1. Fires ventilated from above: steady state model 28

la. Unconfined plume

lb. Plume flowing through a hole with simultaneous inflow

Figure 2. Fires ventilated from above: unsteady model 29

2a. Initial stages of outflow cycle

2b. Inflow cycle (or possible steady state)

Figure 3. Relation of fire development to fire control capability 30
LIST OF SYMBOLS

b plume half width

C constant in eq 5 c specific heat

D equation subgroup in Al .

7

f oxy gen mass fraction

G h/Jkpc/3

h heat transfer coefficient

1 integral

K kernel function

k thermal conductivity

m mass flow

N number, equation subgroup in Al .

7

02 oxygen mass flow

p dummy variable

Q heat flux

q heat flux R JR2 + x

2 r fire radius

s dummy time variable

T temperature

t time, dummy variable

u temperature function

V flame spread velocity

v win/wo

> temperature function

w vertical velocity

x (radial) distance from flame

Greek letters

a thermal diffusivity

/3 exponential flame spread time constant

£ dummy variable, rh /b

p density

p dummy variable

r) dummy variable

C dummy variable, ro /b

a Boltzmann's constant

Subscripts

s surface

i ignition

o initial

,

f flame

a ambient

in inflow

out outflow

con consumed

centerline

,

radius where w = 0

v
Superscripts

per unit time

per unit area
Fire Growth in Combat Ships

by

John A. Rockett

ABSTRACT

Several enrichments to FIRMOD , the Navy's ship battle damage estimation

computer program, are considered. Some proposed enrichments could now be made

to good effect, others should only be considered after further experimental

work has been done to better define the phenomena involved and the algorithms

to be used.

Enrichments recommended for immediate consideration are exponentially

growing fires, ingestion by one ventilation system of smoke issuing from other

ventilation system exhausts, and smoke transport between spaces served by the same ventilation system. Areas where further experimental work is recommended

before FIRMOD enrichments are considered include fires ventilated primarily

from above and buoyant smoke transport up shafts and ladderways

.

EXECUTIVE SUMMARY and RECOMMENDATIONS

Three issues raised in an earlier study of fire in combat ships [1] are addressed: (1) Fires ventilated principally from above, (2) Prediction of fire

growth rates, and (3) Spread of smoke and toxic gas.

In the case of the first, fires vented from above, model equations are developed for the steady burning case, the one to be expected where the fire is

relatively small compared to air supply capability of the ventilating hole.

The equations are a straight forward extension of conventional buoyant plume

theory, but are intricate and involved. The unsteady case - larger fire and/or

smaller hole - is very likely to occur and will present significantly more

difficult firefighting problems. For this the situation is sufficiently

complex and poorly understood that experiments should be undertaken before

analytic work is attempted. (The experiments might also suggest a simpler

model for the steady state case.) Experiments would identify the important

elements of the problem allowing a sound, efficient model to be built. The

main text of this report suggests that three elements of the experimental

design will be important: the ratio of the ventilation opening area to the fire

size, the area and heat transfer properties of the walls, floor and ceiling,

and the mixing of incoming air and vaporized fuel within the room - i.e., the

location of the fire relative to the ventilating hole.

The study of fire growth rates was undertaken because, although it would

be desirable to include fire growth in FIREMOD, a way to get suitable input

data is needed. The study establishes a cross walk between small scale test

data and the needed model inputs. The model assumes that transient heating of

the fuel ahead of the flame controls flame spread. It deals specifically with

1
exponentially growing fires, and fire conditions likely to result in this type

of growth are considered (some conditions which would cause deviations from

this type of growth are also identified) . Experiments are recommended to

verify the fire growth formula developed and to define more precisely the

relation between the exponential growth parameters and the data produced by the

IMCO, or similar small scale test apparatus.

The ability to predict the spread of smoke and toxic and/or corrosive

fire gases is very important to estimating ship survivability. One fairly

simple improvement to FIREMOD is suggested: inclusion of the possible ingestion

through the ship's forced ventilation system inlets of smoke exiting from

exhaust vents of the ventilating system. Existing smoke movement algorithms,

developed for use with building HVAC system analysis, could be added to

FIREMOD. These would allow better estimates of the efficacy of fire dampers to

be made and would give FIREMOD state of the art capability relative to the

transport of smoke by the forced ventilating system. It is noted that the

vertical, buoyant transport of gases in shafts and ladderways is not well

understood and the horizontal transport in long corridors is only just

beginning to be systematically studied. For these two important aspects of

smoke transport additional experimental work is recommended as an adjunct to

modeling efforts.

INTRODUCTION

This is a follow-on to the work of Quintiere , Baum and Lawson [1], Three

specific elements for possible enrichment of the fire growth model for use in

combat ship damage control (FIREMOD [2]) have been studied. One element is the

burning of a fire in a compartment whose only ventilation is via a relatively

small hole in its upper surface. The second is the prediction of fire growth

rates, and the third is the transport of smoke and toxic fire gases to non- fire

involved parts of the ship.

FIRES VENTILATED ONLY FROM ABOVE

As pointed out in [1], many shipboard fires are likely to be ventilated

primarily from above. The water- tightness of the ship's hull, if nothing else,

leads to this. Such a fire can be expected to be "ventilation limited" at

quite small fire sizes. To understand this, consider the simple, idealized

case of a compartment whose only opening is a circular hole in its top, well

away from any wall. Two cases will be considered, one which leads to steady

burning, and a second which leads to non- steady, pulsating burning.

Steady state model

The steady state case arises when the fire is relatively small compared

to the hole size. For this, suppose that a circular fire of known radius

originates and grows on the floor of the compartment directly under the center

of the hole. A plume of flame and hot gases will rise from the fire and try to

exit via the hole. This plume will be turbulent and oscillating in size and

2
strength. However, on the average, the highest velocity flow will occur along

the fire centerline. The average velocity, temperature and combustion product

concentrations will decrease exponentially with the square of the radial

distance from the axis (Gausian distribution [3]). But, except for transient

situations, gas cannot flow out the hole unless an equal mass of gas flows in

and, since there is only one hole in the compartment, the outflow and inflow

will compete for its use. Under quasi steady- state conditions, we can imagine

that the actual flow through the hole is a superposition of the Gausian outflow

and a uniform inflow (see figure 1). To have a mass balance, the inflow must

adjust itself until the net outflow of the hot gases from the center of the

plume is just balanced by the inflow of cool, ambient air around the plume's

perimeter. As this ambient air enters it will entrain and carry back into the

room the combustion products and enthalpy of the gas in the outer part of the

fire plume. Thus, after entering, the incoming gas will no longer be "fresh

air" but air contaminated with combustion products. For larger fires, with

larger diameter plumes, pollution of the incoming air by recirculated

combustion products will become more severe as a larger fraction of the plume

outflow is entrained and returned to the room by the inflow. This model is

developed in Appendix 1. It is to be expected that it will be valid for small

fires aligned directly under a relatively large hole. Steady- state burning can

be expected for this case provided the ventilation requirements of the fire are small compared to the hole's carrying capacity. A common example of a steady

burning fire ventilated only from above is the candle in a "brandy sniffer"

often found on restaurant tables.

Although fairly straight forward, the equations of this model are involved and non-linear, requiring iterative, numerical solution. In view of

the dearth of experimental data on fires ventilated only from above, it is

recommended that these ideas not be pursued further until some experimental

verification is obtained. Initial tests of this model could be done with

"fresh-water, salt-water" experiments. These would, of course, have to be

followed by actual fire experiments to determine the qualitative behavior as the fire became ventilation limited. Small scale (room about 1 m height) fire

tests should be used for the initial part of the fire test program with full

scale tests only after a satisfactory analytic model had been assembled.

Transient model

An alternate, more complex model can be used to describe the above

phenomena under transient conditions. The discussion is simplified if we consider that the fire is laterally well displaced from the hole in the

ceiling. In this case, hot gas will be delivered by the fire plume to the

vicinity of the ceiling at a rate depending, initially, on the fire to ceiling

distance but only weakly on the heat release rate of the fire. The hot gas

spreads under the ceiling and some of it, upon reaching the hole, will flow

out. Hot gas outflow occurs because buoyancy has brought the hot gas to the

hole, but, more importantly, gas is driven out because the fire releases energy

within the room. Some of this energy is transferred to the surfaces of the

room, but much of it remains in the fire gases. It is a consequence of the

perfect gas law, which is a good approximation for fire situations, that the

total energy of the gas in a vented room must remain constant. Thus, if energy

is added to the gas in the room, there will be a transient pressure rise

3
causing gas to be expelled until the rate of energy removal from the

compartment (enthalpy flow of the expelled gas) balances the net rate of energy

addition by the fire to the gas remaining in the room (rate of heat release

less that lost to the room surfaces)

.

Fire plumes entrain a great deal more air between the fire base and the

height of the visible flame tip than they need for stoichiometric burning [3],

If enough gas is delivered by the plume to the ceiling, not all of it will be

driven out the hole; the excess accumulates below the ceiling as a growing hot

layer. As this layer thickens, less flow will be delivered to it by the plume

since the mass flow into the layer ultimately depends on the distance from the

fire to the base of the hot layer. At some thickness of the layer it would no longer grow, provided a source of fresh air were available from below.

However, no such source is presumed to be available. Therefore, as the initial

charge of fresh air is consumed by the fire, the layer continues to sink.

Ultimately the descending hot layer will submerge the burning surface in oxygen

depleted fire gases. The result is a diminution of the burning rate which

further decreases as the air becomes more severely depleted of oxygen. Since

heat will still be lost to the room surfaces, at some point the fire will

release less energy than is being lost by the hot layer gas to the room surfaces; the net rate of energy deposition in the gas will become negative

and, just as heat addition caused gas to be expelled, heat removal will cause air to be drawn in, to keep the total energy of the room gas constant. The

fresh air flowing in the ceiling hole, being cooler and denser than the gas in

the room, will plunge through the hot layer (see figure 2) . In doing so it

entrains hot gas, forming an image to the fire plume - a falling plume with a dense, cold core, trailing off in a Gausian decay to the properties of the

combustion products found in the hot layer. Thus the gas reaching the lower

part of the room will no longer be pristine, cool, fresh air, but air

contaminated with combustion products. Just as the hot gas from the fire plume

formed a layer which spread under the ceiling, the plume of cool, descending

air will form a cool layer spreading across the floor. This will eventually

reach the fire base. Although partially vitiated by entrainment of hot layer

gases, the oxygen of the air which entered the room will become available to

the fire and the fire will increase in intensity. The total energy in the room will increase and a second puff of hot gas will be expelled from the hole,

blocking inflow and setting the stage for another die -back due to oxygen

starvation. The puffing frequency depends on the time it takes for the fire to

become oxygen limited, the time it takes for the gases to cool down, the time

for a fresh charge of air to enter, sink to the floor and spread to the fire

base, and the time for the fire to form a new rising plume.

Note that, in this model, flow is either wholly in or wholly out the

vent. Bi-directional flow through the ceiling hole does not occur,

reality, there will be periods of bi-directional flow around the time of flow

reversal. Unfortunately there is no satisfactory model for such transient

flow. The assumption of the present model is that the periods of

bi-directional flow are short enough that they will have little effect on the

total cycle. One purpose of an experimental program would be to ascertain if this assumption is viable.

4
This model is discussed more quantitatively in Appendix 2, however, a complete, consistent set of equations have not been developed. The work of

Porter [29] on the burning of alcohol fires ventilated only from above supports

the qualitative features of this model. Scaling laws based on his analysis

suggest that the "puffing" of fires, as postulated here, would be a maximum for

holes in the rang of 1/2 to 1 m in diameter for a typical ship compartment.

The complexity of this model and the uncertainties associated with the

analysis suggest, as in the case of the steady state model, that experimental

verification of the model should be sought before an effort is made to develop

its equations and incorporate it in FIREMOD. This model cannot be tested with

"fresh-water, salt-water" experiments as it depends on the thermally induced

expansion of the gas and heat transfer to the compartment walls. Small scale

fire tests should be used. To aid flow visualization, normal smoke production

might be augmented with artificial aerosol. If a methane gas burner were used

for the fire source, the aerosol could be added alternately to the fire gas, or the air entering, so that the two parts of the flow system could be observed.

Again, small scale fire tests (room about 1 m high) should precede any full

scale tests. The fire size should be large enough to guarantee turbulent

plumes, but otherwise as small scale a test as possible would reduce costs and

facilitate variations in the fire-room-hole geometry and changes in

instrumentation. Once suitable instrumentation and data sampling locations had

been determined and a viable model selected, larger scale tests could be

contemplated for confirmation of the model and development of a data base for

FIRMOD computations.

PREDICTION OF FIRE GROWTH RATES

Quintiere et al have laid the ground-work for this study in [1] . They

point out (their Appendix B) that the heat release rate of many fires increases

exponentially with time. In their main text, equation 9b, they present a case whose analysis leads to this result - fire spreading through a porous fuel bed

as a result of radiant heating from the fire itself. This result is more general than the text suggests. It's generality depends primarily on the fact

that the radiant flux, which heats the uninvolved fuel, is a simple function of

the fire size. There are a number of other, important cases where this same type of behavior is found.

Exponentially growing fires

A number of workers have noted that fires grow exponentially with time.

[5,6,7]. This is used in the Harvard/NBS fire models to calculate the rate of

growth of fire radius and burning intensity, [5]. At present there is no available model for predicting the characteristic time for this growth. Thus,

in the Harvard/NBS models, the fire growth rate is an input parameter derived

from experimental data for the fuel type and geometry in question. Obviously

some more general approach is desirable as it may not be feasible to obtain

suitable experimental data for all fuels of importance. As stated above,

thermal flame spread theory suggests an approach.

If heat is applied uniformly to the surface of a flat material (fuel) we need consider only one -dimensional heat conduction in calculating the

5
material's surface temperature. If the heat is applied non-uniformly , but in

such a way that temperature gradients along the surface are small compared to

those normal to the surface, lateral heat conduction in the fuel will be small

compared to that normal to its surface and a one -dimensional heat transfer

model may still be adequate. For many cases this approach is, indeed,

suitable. The principal exception is near extinction (very slow flame spread),

where the flame spread rate is comparable to or less than the surface

regression rate. Assume that extinction is not our present concern, and,

therefore, the one -dimensional approach may be adopted. Let the thermal

response of the fuel surface at x to a unit heat pulse be K(t) . Then the

surface temperature at this point as result of heating by a time varying heat

flux q" (x,t) per unit area (incident at x) is

t

Ts (x,t) - To =

j* K(s) q" (x , t- s) ds (1)

o

An expression for K(s) is developed in Appendix 3. Note that setting Ts - Tt , the material's ignition temperature, in equation 1 results in an implicit

equation for t(x) , the time necessary for the fuel at x to be heated to

ignition. The flame spread velocity is then related to the inverse of this,

x(t) , by V = dx/dt. We defer discussion of the use of equation 1 and focus our attention next on q"(x,t).

Modak [8] shows that the radiation from a flame to an upward facing

target in the plane of the fire base can be approximated as coming from a point source on the fire axis, one fire radius above its base.

q" = Qf cos(0)/(4 n R

2

) (2)

where

Qf is the radiated portion of the fire's energy release rate

R is the distance from the virtual point source (r above

the c enter o f the fuel) to the point being heated,

R = /x2 + r

2

cos(0) is the radiation view factor = r/R

This approximation increases in accuracy as r/R decreases, but is approximately

true for all points outside the flame.

For a fire spreading across a horizontal surface, Qf will be related

directly to the average heat release rate of the flame per unit of fuel burning

area

.

Qf = Qf

" 7T r

2 ( ’

)

deRis [9] cites data which indicate that, for fires above about 1 meter in

diameter, Qf

" is independent of fire size for any given fuel. For smaller

fires this is approximately true. Further, and consistent with this, for files

6
above about 1 m in diameter, flame radiation dominates the forward heat

transfer process (convection may be neglected). Thus, Qf does imply a length

scale, but it is the fire radius.

From this we see that, for fires spreading on a horizontal surface, it is

approximately true that

q"(x,t) = Qf cos(0)/(4?r R

2

)

= Qf

n [* r2 /(4 7T R2 )] (r/R)

= Qf

n /4 (r/R) 3 (4)

Note that, in this, Qf

" is a constant and r = r(t) . In Appendix 3 this

expression, together with the K(s) developed there, is used to derive the flame

spread velocity. The result is

V = C [x/(kpc)] { [Qf / (T±

- T0 )] - N*h} 2 (5)

where x = distance from the fire center to its edge

n Qf = radiated portion of the flame's heat release rate per unit

burning area

T; = material ignition temperature

T0 = material initial temperature

h = convective plus linearized radiative surface heat transfer

coefficient

N = a numerical factor, approximately 20 (see Appendix)

C = a numerical constant approximately = 1/247

This is shown in the Appendix to be consistent with an exponential growth of

fire radius with time.

How good is this formula? One question, assuming that the formula is

adequate, is are we any better off with it than we were before? Although, with

this formula, horizontal slabs of fuel may not have to be burned to determine

the fire spread rate, it may not be easy to get data on the thermal inertia

(kpc) and ignition temperature, Ti , so that the formula can be used. Thus a study of the burning rate results cited in [5,6,7] yielded only enough data for

two materials, out of a total of 15, to make comparisons with the formula.

These two were qualitatively ranked by the formula (see below). Gross and

Loftus [10] found good agreement for a similar formula applied to flame spread

data from the ASTM E-162 flame spread apparatus. Their results applied only to

the effect of (kpc). Rockett [11] extended the thermal flame spread concept to

a more complete analysis of the E-162 apparatus. He included the competing

effects of convective cooling and radiative heating by the radiant panel and

the approaching flame. The resultant analysis gave good results both in

7
calculating the flame spread velocity and also in predicting other aspects of

the E-162 test. Quintiere et al . [12] and Atreya et al . [33] have used similar

concepts to correlate flame spread data from the proposed IMO apparatus

(horizontal spread on a vertical surface) . This suggests that the thermal

spread concept is promising.

Calculated values for the exponential growth time constant, neglecting

fuel surface re-radiation and convective heat loss, and using the C value just

refereed to, were 281 sec. for PMMA (compared to experimental values of 120 to

444 sec.) and 7.95 sec. for urethane foam (compared with experimental values of

6.6 to 100 sec.) Considering the approximations in the model and the scatter of

the experimental data this is quite encouraging.

Gross [13] has collected thermal property data for a number of materials.

His work shows that thermal properties of many materials of technical interest

vary with temperature sufficiently that this variation should be taken into

account when evaluating equations like 5. However, for a use such as FIREMOD,

where many other, severe approximations must, of necessity, be made, it is not

clear that the additional accuracy would justify the added complication of

temperature dependent material properties.

With exponential fire growth models, there is always the question of

what initial fire size to pick. A decrease of an order of magnitude in initial

fire area results in an increase of about 1.5 characteristic times for the fire

to reach a given size 1

. If the characteristic time for fire growth is short

compared to the time for other critical events, such as damage control crew response, then initial size is not particularly important. Flashover will

probably have occurred, if it is going to, before intervention is possible,

whether the number of characteristic times is increased by one or two or not.

The exponential growth model could, in general, be used as follows: The

response time and fire control capability of the damage control parties must be

known. Control capability should be expressed in terms consistent with the

fire growth model, i.e., energy release rate, fire area (or radius), etc. (as

functions of time) for which control is possible. At least two levels of

control may be specified: the ability to extinguish and the (lesser) ability to

prevent extension beyond some defined fire barrier. The initial fire size,

estimated by methods to be discussed below, and characteristic fire growth time are computed. Then FIRMOD looks to see if, after the response delay, the fire

control capability exceeds the fire challenge for a sufficient time to allow

extinguishment, see figure 3. If the answer were that control capability did

exceed fire challenge then control would be assumed, if not, uncontrolled

growth to the limit size (ventilation or fuel area limit) would be assumed.

With weaker control capability alternative "end points" would, of course,

apply.

When fire growth time and other times are comparable, the ability to

estimate initial fire size is important. Obviously this is highly incident

1 The characteristic time is 1//3 where (3 is given by A3 . 3

8
dependent, no general rule for estimating initial fire size can be given. In

the next paragraphs a few guide-lines are suggested.

If the fire is due to a flammable liquid spill, fire size, for most

liquids, will grow very rapidly to cover the entire spill area, after that,

fire area will most likely depend on the rate of supply of additional liquid

and physical bounds to the liquid's spread. The fire will not grow

exponentially. Burning intensity can be calculated using an algorithm similar

to that used in Harvard V for pool fires. The fire area would be considered

quasi -static at the current spill area.

If there is no liquid spill, and the impacting object is not itself the

initial fire, assume, first, that the fuel present in the space has been

effectively homogenized by the initial damage and that multiple fires are started by impact of high kinetic energy fragments. At this point a decision

must be made whether to assume exponential growth as given by the present

model, equation A3 . 3 , or some other model such as [1], equation 9. Assuming

that the present model fits the supposed circumstances, assume that each of

these fires grows independently with the same characteristic time. (Because

each ignition in the space is considered to involve the same homogenized fuel.)

If there were n fragments, causing n equal, initial fires of radius rQ , then,

after these have grown and merged to form a single fire, the equivalent initial

single fire radius would have been ri = Jn ro

. Thus, if n=100 and ro is 2 cm,

the equivalent initial fire, after complete merging of the separate fires,

would have had a radius of 20 cm. A suitable guess for the initial fire size

for the separate fires would be the fragment size. More complex expressions

can be written for a distribution of initial fire sizes and non-uniform fire

spacing, but the added accuracy is probably not justified due to the

uncertainty in modeling the initial event.

Where the initial fire is the impacting object, the size of the object

itself can be assumed to be the fire size. Growth may or may not be

exponential. An expert on the particular device should be consulted for an estimate

.

SPREAD OF SMOKE AND TOXIC GAS

The ability to predict the spread of smoke and toxic gas associated with

a fire has become a significant issue. The following is a discussion of

improvements which might be considered for the FIRE DAMAGE SIMULATION MODEL

(FIRMOD)

.

Two mechanisms of smoke transport are important - smoke entering the

ship's ventilating system and being carried by the system to non- fire involved

spaces, and smoke transported, by natural convection generated by the fire,

through openings of the fire compartment into adjacent spaces and thence to more distant locations.

Fire gases can enter the ventilating system by two routes: fire gases

exiting through holes in the deck or ventilation system exhausts may pass near ventilation inlets and be drawn into these adjacent systems, or they can directly enter the ventilation system of the fire involved space. Where this

9
ventilation system serves several spaces, there is the likelihood that the fire

gases will be carried by the system to these. An unpublished review of the

ventilation system of one frigate [30] suggests that both mechanisms may be

important. FIRMOD already accounts for the first mechanism where the hole is

weapons caused, but the frigate's ventilation system showed numerous cases where the exhaust from one space was located near enough the intake for another

that, even without battle damage caused smoke vents, ingestion could be a problem. Where ventilation systems are not provided with fire dampers, gases

from a fire in one space would be exhausted by its ventilation system and could

be ingested by a neighboring system. To include smoke ingestion, the relative

locations of the various vent system exhausts and inlets must be considered as well as the ship's heading and speed relative to the wind. FIREMOD already has

provision for the effects of heading and speed, so the only new element would

be recording the relative locations of ventilation system exhausts and

calculating the smoke discharge from the exhaust vents. The present

calculation, or an improved version of it, could then treat a smoke emitting

exhaust just as it now treats a smoke emitting battle damage caused hole with

respect to ingestion of smoke by other systems.

On this same frigate a number of ventilation systems served multiple

spaces. Fire in one space could result in fire gases reaching other

compartments on the same system. This would be especially true if a high

recirculation ratio were used with the ventilation system and fire dampers were not provided in the system's ducts. Present computer models for smoke control

[14,15] assume that the ventilation system air flows are known and unaffected

by the fire, except where automatic fire dampers are provided. Thus the

computation of smoke transport via the ventilation system is a matter of

identifying the "nodes and flows" of the system. Smoke transport would be

considered as "slug flow" ingested by the exhaust from the fire room, diluted

by mixing at the nodes, and supplied to other spaces as the system design

dictates .

Fire induced natural convection is often the most important mechanism for

smoke migration from the fire to other areas. This is not currently modeled in

any detail by FIREMOD. Modeling of buoyancy induced flows on a single level

has been well developed in the past decade [16,17]. However, recent studies

[31,32] suggest that present, widely used two zone models may not be appropiate

for long corridors. Computer fire models have been published recently which

treat multi-level structures [18,19]. They model well the (Japanese) vertical

transport data available at the modelers own laboratory. However, the

algorithms used in these multi-level programs may not correctly account for

interference between the rising and expanding gas plume and the walls of the

vertical channel. Other experimental data relating to buoyant flow in vertical

shafts is the work of Cannon and Zukoski [20]. Their work is based on "salt

water analogue" experiments which, though qualitatively correct and

semi - quantitative , may not give sufficiently accurate results for correct

modeling. It suggests interference effects not included in the Japanese model

The techniques used in [16,17] are probably as detailed as can presently be

justified for smoke transport on the level of the fire. The techniques of

[18,19] are open to question but are as good as currently available knowledge

Their predictions should not be considered always quantitatively accurate and

might, occasionally, be qualitative misleading without further analysis.

10
Recommendation

As indicated above, a relatively minor modification to FIRMOD would

account for the possible ingestion of smoke disgorged by ventilation system

exhausts from burning spaces. This would seem to require relatively minor

program changes and could be a significant improvement. A more extensive

change would be inclusion of the transport of smoke by the (undamaged)

ventilation system from a fire involved space to other spaces served by the same system. This would only be a significant improvement if the ship's

ventilation system were without fire dampers (apparently rarely used on present

ships) and if relatively high recirculation ratios were used (less common)

.

The computing techniques for analysis of forced ventilation systems are well

established. The value of these additions would, as indicated, depend to some extent on the design of present fleet ventilation systems. Buoyancy driven

smoke transport as treated by present "two layer" room fire models is a more ambitious addition to FIRMOD as it requires all the computational features

needed to track the energy and mass content of the two layers. However,

various levels of modeling detail can be adopted with corresponding levels of

numerical complexity. Thus ASET [27] is very much simpler than NBS/Harvard VI

[16] or FAST [17] but captures some essential features of the two layer model.

For vertical transport the Japanese modeling technique [18,19] could be used,

but the methodology for buoyant transport up shafts and ladderways is not as well developed as horizontal transport. Vertical buoyant transport is likely

to be at least as important, if not more important than buoyant transport on

the same level as the fire. Therefore, it seems advisable to begin an experimental and analytical program to document and understand the various

regimes of vertical transport, the conditions under which they occur and how

they should be modeled. Concurrent with the experimental program the

feasibility of including a two layer buoyant gas model for the level on which

the fire occurs could be explored. If this were added to FIRMOD it would

complement the inclusion of exponentially growing fires, but the exponentially

growing fire addition does not, of itself, require the two layer model (see

discussion above)

.

REFERENCES

[1] Quintiere, J.G., Baum, H.R. and Lawson, J.R., "Fire Growth in Combat

Ships", NBSIR 85-3159, U.S. National Bureau of Standards, Gaithersburg,

MD 20899, May 1985

[2] Tinker, C.W. and Hackett, O.F., "FIREMOD user's guide", TN 174-49, David

W. Taylor Naval Ship Research and Development Center, 1985 (Distribution:

"For Official Use Only")

[3] McCaffrey, B.J., "Purely Buoyant Diffusion Flames: Some Experimental

Results", NBSIR 79-1910, U.S. National bureau of Standards, Gaithersburg,

MD, 20899, (1979)

[4] Rockett, J.A., "Fire Induced Gas Flow in an Enclosure", Combustion

Science and Technology, 1976, Vol 12, pp 165-175

11
[5] Mitler, H.E., "The physical basis for the Harvard computer fire code",

Home fire project technical report No. 34, Harvard University, Cambridge,

MA (Oct. 1978)

[6] Pape, R. , Mavec , J., Kalkbrenner, D. and Waterman, T. , "Program

documentation and users guide, Semistochastic approach to predicting the

development of a fire in a room from ignition to flashover, RFIRES",

Engineering Mechanics Division, 111. Inst, of Tech., 10 West 35th St.,

Chicago, 111 60616 (June 1976)

[7] Huggett, C., "Time -dependent fire behavior of aircraft cabin materials",

U.S. Dept, of Transportation, Federal Aviation Adm. , Systems Research &

Development Service, Washington DC 20590 (Dec. 1977)

[8] Modak, A.T., "Thermal radiation form pool fires", Combustion and Flame

29, 177-192 (1977)

[9] deRis, J., "Fire radiation - a review", Factory Mutual Research Corp

.

Norwood MA, 02062 (Dec. 1978)

[10] Gross, D. and Loftus , J., "Surface flame propagation in cellulosic

materials exposed to thermal radiation", Journal or Research, 67C, no 3 (July- { September 1963) pp 251-258, U.S. Nat. Bur. of Stds., Washington,

DC

[11] Rockett, J.A., "Mathematical modeling of radiant panel test methods",

Fire Safety Research, Proceedings of a symposium held at NBS

,

Gaithersburg, MD, August 22, 1973, pp 909-96, U.S. Nat. Bur. of Stds.,

special publication 411 (Nov. 1974)

[12] Quintiere, J., Harkleroad, M. , and Walton, D. , "Measurements of material

flame spread properties", Combustion Science and Technology, 1983, Vol

32, pp 67-89, 29, 177-192 (1977)

[13] Gross, D., "Data sources for parameters used in predictive modeling of

fire growth and smoke spread", U.S. Nat. Bur. of Stds., (to be published)

[14] Klote, J.H., "Computer modeling for smoke control design", Fire Safety

Journal, 9 (1985) 181-188

[15] Klote, J.H., "Smoke movement and smoke control on merchant ships", NBSIR

81-2433, U.S. Nat. Bur. of Stds., Gaithersburgh , MD December 1981

[16] Gahm, J.B., "Computer fire code VI - Volume 1", NBS-GCR 83-451, U.S. Nat

Bur. of Stds., Gaithersburgh, MD (December 1983).

[17] Jones, W.W. , "A multicompartment model for the spread of fire, smoke and

toxic gases", Fire Safety Journal, 9 (1985) 55-79.

[18] Tanaka, T. , "A model of fire spread in small scale buildings", Building

Research Institute (Japan), Research paper #79 (1978) and #84 (1980)

12
[19] Tanaka, T., "A model of multiroom fire spread", NBSIR 83-2718, U.S. Nat.

Bur. of Stds, Gaithersburh , MD, (August 1983).

[20] Cannon, J.B. and Zukoski, E.E., "Turbulent mixing in vertical shafts

under conditions applicable to fires in high rise buildings", Daniel and

Florence Guggenheim Jet Propulsion Center, Calif. Inst. of Tech.,

Pasadena, CA, Technical Report No. 1, National Science Foundation Grant

No. GI 31892 X, (January 1976).

[21] McCaffrey, B.J., "Momentum implications for buoyant diffusion flames",

Combustion and Flame 52: 149-167 (1983)

[22] Morton, B.R., Taylor, G.I. and Turner, J.S., "Turbulent gravitational

convection from maintained and instantaneous sources", Proc . of the Royal

Soc., London, A, 234, 1-23 (1956)

[23] Turner, J.S., "Buoyancy effects in fluids", Cambridge University Press,

1973

[24] Torrance, K.E., Orloff, L. and Rockett, J.A., "Experiments on natural

convection in enclosures with localized heating from below", J. Fluid

Mech. (1969), vol 36, part 1, pp 21-31

[25] Torrance, K.E. and Rockett, J.A., "Numerical study of natural convection

in an enclosure with localized heating from below - creeping flow to the

onset of laminar instability", J. Fluid Mech., (1969), vol 36, part 1, pp

33-54

[26] Alpert, R.L., "Fire induced turbulent ceiling- j et" , FMRC Technical Report

19722-2, Factory Mutual Research Corp. (1971)

[27] Cooper, L.Y. and Stroup, D.W., "Calculating available safe egress time

(ASET) - A computer program and user's guide", NBSIR 82-2578, U.S.

National Bureau of Standards, Washington, D.C. 20234, Sept. 1982

[28] Carslaw, H.S. and Jaeger, J.C., "Conduction of heat in solids", Second

Edition, Oxford, 1959

[29] Porter, J.W., "Pulsating Combustion of Liquid Fuel in Partially Closed

Vessels", Combustion and Flame, Vol 11, No 6, Dec 1967

[30] Jones, W.E., private communication

[31] Zukoski. E.E., Kubota, T., and Lin, C.S., "Experimental study of

environment and heat transfer in a room fire. Mixing in doorway flows and

entrainment in fire plumes", NBS-GCR-85-493 , U.S. Nat. Bur. of Stds,

Gaithersburg, MD 20899, May 1985

[32] Baum, H. , Seminar lecture at the NBS Center for Fire Research, 1986 and

Private communication

13
[ 33 ] Atreya, A., Carpentier, C., and Harkleroad, M. , "Effect of sample

orientation on piloted ignition and flame spread", NBS-IR (in

preparation, 1986), U.S. Nat. Bur. of Stds., Gaithersburg, MD

[34] Quintiere, J.G., private communication and unedited video tape (copy)

brought from Japan

[35] Takeda, H. , "Experimental Investigation of PMMA Compartment Fire", Fire

Science and Technology, Vol. 5, No. 1 (1985) pp. 11-20.

14
APPENDIX 1: FIRE PLUME FLOW THROUGH A HOLE IN THE TOP

OF AN OTHERWISE UNVENTILATED COMPARTMENT

The strategy of this appendix is as follows: The "standard, point source buoyant plume solution" [22] is used to compute the upward flow of mass and

oxygen as a function of radius and height. Next, similar calculations for a point plume, but with a different half width, and with a superimposed, uniform

downward velocity are carried out. Third, the inflow through the ceiling hole,

in the presence of the retarded plume outflow, is found. Fourth, the mass inflow and outflow are equated. This yields the value for the inflow velocity

and, from the plume equations, the dimensionless radius of the zero velocity

location. Fifth, the mass and oxygen flow in the core of the "free plume" of

the first calculation is equated to the corresponding flows in the retarded

plume. This determines the dimensionless radii in the two plumes where the

flows are equal, which, using the already determined dimensionless radius of

the zero velocity point, fixes the radius of the free plume core. The second

set of equalities is used to set the ratio of the plume half widths. It is

presumed that the free plume half width is known from the burning conditions

and the plume heat release rate. Note that, in this, it is necessary to choose

an upward velocity on the plume axis for both plumes. It is assumed that this

is the same for both and that of the free plume will be known. Finally, the

outer mass and oxygen transport of the free plume is found. This is the

recirculating mass and oxygen. Using this and the mass and oxygen inflow, the

oxygen mass fraction in the room can be found. This allows the burning

conditions for the free plume to be found. The oxygen consumption is the

difference between the oxygen inflow and outflow. This should be consistent

with the plume burning conditions within the room, but does not necessarily

represent the total heat released since additional heat may be released above

the hole (i.e. in a flame extending through the hole).

In the absence of any obstruction to the fire plume's upward flow, the mass flow, several fire diameters above a pool fire, inside a radius, r, is

given by

r m = P

owo J

o Here it has been assumed that the temperature excess and upward axial velocity

are both Gausian distributed with a half width b.

AT -(r/b

)' T

e

_

( r/b ) 2 „rdr (Al.l)

w

-(r/b) 2 w„ e

and

! . e

' (r/b)

= P o

1

AT -(r/b)

2 T

15
The gas density far from the plume axis is pQ , the centerline velocity is WQ and the centerline temperature excess is AT - (T - T0

) . The gas temperature

far from the plume axis is TQ

. Similarly, the vertical mass transport of

oxygen is given by

m p w

o o

l

. e'

(r/b)2 g-tr/h)

2

1 . |^ e -(r/b)

2 T f

o

2?rrdr (A1.2)

We now superimpose a uniform downward velocity, Win , on this upward flow.

The down flowing gas has a density p a

. If the hole radius is rh and rQ is the

radius at which the superimposed down flow velocity equals the free plume

upward velocity (i.e., the radius where the vertical velocity changes sign)

W; = w_ -<r„/b )

:

Then the net mass flow will be

m = m

out

m.m

(r /b) 2

i 2 r , -t

p w 7rb (e o o

J V)(l AT

T

e

t

) dt

(A1.3)

<rh/b) 2

- p w 7rb^ f (v - e

t

) dt

a ° (r

Q

/b) 2

For steady state conditions this must be zero, giving an implicit

for v = win /wo .

(A1.4)

expression

p a (v[(rb /b) 2 - (rQ /b) 2] - [v - e

' (r° /b)2

]) -

9 AT 9 p [- v(r /b)

Z + (1 - v) - -T=r (1 - v) ] (A1.5)

O O

which, with Al.2, implies values for both win and ro

. The equations, being

transcendental, can only be solved iteratively. Similarly, the steady state

oxygen consumption will be the difference between the oxygen entering and th.it leaving

.

con i n ^2 out wo 7rb 2 (p a (fa

- f0 )[v(f - O - (v - e )]

f r o o f 2

r\ • + P (1 - v)

2

- i- (2 - 3v + v

3

) (A1 .6)

16
where f = the volume fraction of 02 in the ambient air

f = the volume fraction of 0, in the air in the enclosure O u £ = rh /b dimensionless hole radius

f = rQ /b dimensionless radius at which vertical velocity = 0

The fractional amount of oxygen in the gas available within the room, at

steady state, is the ratio of the oxygen entering plus the oxygen recirculated

from the plume's outer sheath to the mass entering plus the mass recirculated.

f ^9 P~ ^9 1 f /f = l/{l + — 7^-}

o a ' { D

x P a N

x

J where

N, = (£ - O - (v - e' 1

)

, 1 AT 2,

°2 = ~p~ (v ‘ 2 T v

XT 1 .AT Af . 2 1 AT Af 3 N

2 = v - F ( — + —)v + r — — v

(A1.7)

Here m. and 0,. are as given above and in and 0, are calculated from in Zin c? rec Zr ec equation Al . 1 and Al . 2 integrated from r to infinity.

A refinement to this model has been considered which adjusts the plume

characteristics at the height of the hole for the deceleration caused by the

inflow. Note, however, that this correction would only be significant where

the inflow velocity was a major fraction of the free plume centerline velocity, wQ

. In this case there would be high shear stress on the plume caused by the

inflow. It is questionable if the present model would be valid were this the case. Thus the added complication does not seem justified.

The analysis presented here is incomplete in the sense that it does not

include equations for the calculation of the characteristic plume width, b, or upward velocity, wQ

. These calculations, however, follow directly from

published work. For b and wo there is the work of McCaffrey [3,21]. To use his results the heat release of the fire is needed. This is related to the

oxygen consumption per equation Al . 6 (using Al . 7 to get f ) . Since all these

equations are highly non-linear, an iterative solution scheme would be needed.

Thus some management decision is needed relative to the amount of additional

calculation permitted versus the added flexibility of the more complex model

achieves

.

Critique

The Gausian decay of plume properties is not the only way to interpret

experimental plume data. If the plume properties show a "square hat"

distribution, but the plume axis wanders with a Gausian probability, very

17
similar data would be observed. For a "square hat" distribution, the above

equations are slightly altered. The most significant difference is that, since

there is no "tail" to the distribution, there is no vitiation of the incoming

air. Since vitiation is expected, shear induced mixing has to be invoked in

the locally high shear region around the vertical position of the hole. No

model for this type of mixing has been proposed. In view of the total lack of

experimental data for fires ventilated solely from above, it is recommended

that, before this analysis is perused further, the ideas advanced be tested by

experiment. They may then be accepted, altered, or rejected. To proceed

without an experimental base seems unwise to the author.

18
APPENDIX 2: FIRE PLUME IMPINGING ON THE CEILING OF A

COMPARTMENT VENTILATED ONLY BY A HOLE IN

THE CEILING DISTANT FROM THE FIRE.

The model assumes a single gas layer in the room. It is partially

vitiated by the admixture of combustion products and warmed by these combustion

products, but its primary characteristic is that its oxygen content is at the

limiting value for combustion of the fuel in question and for a gas at the

current temperature in the room. There are alternately two plumes in the room. A hot gas plume rising from the fire to the ceiling, when present, spreads

under the ceiling and some gas flows out of the room via the hole. For

simplicity assume the fire is flush with the floor of the room. The other

plume appears as the fire dies down. It is formed of cold gas which descends

from the ceiling hole through the warm gas layer to the floor. At the floor

the cool plume gasses spread and some is entrained by the fire supplying fresh

oxygen for combustion. Thus the fire is rejuvenated and the accompanying

pressure build-up cuts off the descending plume as the ascending fire plume

becomes re-established.

Turner tells us that the head of a buoyant plume advances at 61% of the

plume velocity which would be found at that height after the plume has become

established [23]. Experiments and numerical calculations by Torrance, Orloff

and Rockett [24,25] show that the head of the starting plume is a ring vortex.

The vortex advances into essentially stagnant fluid, behind it the steady- state

plume flow is fully developed. When the progress of the vortex is blocked by a ceiling (for a rising plume) or a floor (for a falling plume) it grows in

diameter but still the ceiling jet behind the vortex is the fully developed

flow and the fluid ahead of it is stagnant. Alpert [26] has studied the

ceiling jet. Recent work [27-29] has focused on flow in corridors. Together

these papers give a means of calculating the thickness of the ceiling (floor)

jet; typically it will be about 20 cm. What concerns us is that the ceiling

jet has a definite, calculable thickness and volume and that the flow behind

the front is essentially steady state (for a constant strength plume source)

.

Using the plume relations of McCaffrey [21], we see that a jet of undiluted air

20 cm thick could supply enough air to the base of a fire for stoichiometric

burning of about 4 1/2 megawatts. Thus, even allowing that the descending

plume will be vitiated by entrainment of room air, contaminated with combustion

products, a very substantial fire can develope feeding on the floor jet created

by the descending, fresh air plume.

The question may be raised, can this lead to steady-state burning rather

than the cyclic condition described here and in the main text. If the carrying

capacity of the hole is large compared to the needs of the fire, a steady- state

solution certainly exists. The limiting situation will be related to the

amount of shear which develops between the incoming and outgoing flows at the

hole. Recall that, for steady state, just as in the case of Appendix 1, the

inflow and outflow must balance. At present, fluid dynamics cannot predict the

conditions which will allow steady-state, nor the flow at the hole which will

result. We can be reasonably sure, however, that for a small hole and a fire

that is able to rebuild quickly, for example a flammable liquid spill, a pulsating fire should be expected. The exact conditions would have to be

determined by experiment.

19
Some times can be estimated. The plume velocity can be obtained from

McCaffrey [3] as a function of fire size. For a 100 kw fire the peak velocity

would be about 5 m/s. Thus, from Turner [23], the head of the plume would

travel at 3 m/s . It would take about a second for the fire plume to traverse a typical 2.5 to 3 m room height. The descending plume would be somewhat slower.

It would take about two additional seconds to build a ceiling layer out to a radius of 2 meters. Thus the plumes can form in about 3 seconds (perhaps 5 for

the descending plume). After the floor jet formed, the fire would continue to

build even after the heat release had so built the pressure in the room that

the entry of further fresh air had been cut off. The volume of fresh air

available to feed the fire would depend on the room dimensions, the location of

the fire relative to the hole, and the fire's build-up dynamics. Nevertheless,

the longest time interval for this model seems to be either this, the time to use the fresh air charge, or the cool down time discussed next. The cycle time

would also depend on the time for the walls to extract heat from the fire gas

left when the fire died back. Porter [29] provided an expression only for this

part of the pulsating burning cycle but got good agreement with the pertinent

part of his experimential data. This too will depend on the room dimensions

and also the thermal properties of the wall lining materials. If the walls are unprotected steel, heat will be absorbed as fast as it can be transferred to

the walls. Even so, the heat transfer time can be comparable to the air

burnout time. Time for a complete cycle is obviously a complex set of events,

quite case dependent. It is estimated to be in the order of 20-30 seconds for

a room about 3 m high and 4-5 meters in each lateral dimension. Experimental

experience2 could change this estimate by a considerable amount, though

probably not an order of magnitude.

2 Japanese experiments with a room lined with plywood covered by a thin layer of

steel and ventilated by a door, developed pulsating burning. This may be

related to a somewhat different set of circumstances within the room but the

pulsation frequency was higher than our estimate for the fire vented from

above. They observed about one pulse every one to two seconds. Successive

pulses had a growing severity. The fifth or sixth pulse was a violent backdraft

sending a large fire ball some distance out the door. The backdraft was followed by a short quiescent period and then another succession of gradually

increasing pulses. [34,35]

20
APPENDIX 3: FIRE GROWTH RATE MODEL

This appendix has three parts. In the first, the kernel function, K(s)

,

of equation 1 is developed. In the second, this kernel is combined with the

fire flux expression, equation 4, and the resulting version of equation 1 studied. The assumption is made that the fire growth is exponential in time.

This is shown to be consistent with the expanded equation 1 and the exponential

growth parameter, /3, is obtained. In the third part, several variations on the

heat flux model are discussed and an expression for the fire spread velocity is

generated by further approximations. The result is equation 5 of the main

text

.

Section A3 . 1 : Development of K(s)

The kernel function desired is the one -dimensional response of a semi- inf inite solid to a (unit) radiant heat flux (per unit area) imposed at

the surface of the solid (z = 0). The surface re-radiates to an ambient at

temperature To and is convectively cooled by the flow, toward the fire, of gas

at ambient temperature. The formulation follows [28] chapter 2. Thus we want

a solution of

dT/dt = ctd 2 T/dz2

subject to the boundary conditions

T = To for t = 0 and all z > 0

-k3T/3z = q" - h(T - TQ ) for x = 0 and all t > 0

where

T = Temperature of the solid (above ambient)

TQ = ambient temperature, TQ = 0 assumed

z = coordinate normal to the fuel surface

t = time

a = the material thermal diffusivity, k/pc

k = the material thermal conductivity, assumed a constant

p = material density, assumed constant

c = material specific heat, assumed constant

h = linearized radiative plus convective heat transfer

coefficient

,

h - h0 + ea [<T,+T.)‘ - (T.+T.

)

4 ]/[ <T, +T„ )

- (T„+T.>)

hc = convective heat transfer coefficient

21
Ta = ambient temperature in Kelvin

Ts = temperature of the material surface.

e = material surface emissivity

a = Boltzman's constant, a = 5.67 * 10' 11 Kw/m2 -K4

q" = radiative heat flux per unit area applied to the surface,

q" = 1 kW/m2 for unit heat flux per unit area.

Following [28] define

u = - 3T/3z + (h/k)T (A3.1)

u must satisfy the boundary conditions

u = 0 for t= 0 and all z >0

u = q"/k for z= 0 and all t >0

u may be written directly from [28], section 2.5, equation 1.

00 2 12 f

• „ i

.

2.. 2, u = £

q" (t - x /4afi )e d/x ^

x/2

Equation A3 . 1 is now solved for T. We note first the adjoint equation (u - 0

in A3.1) has the solution

v = exp(hz/k)

Then

-v3T/3z + (h/k)vT + T3v/3z - (h/k)Tv - vu

- 3(T/v)/3z = u/v

T = C2[C 1 exp(hz/k)] + J* u(z+r/) exp-(hfj/k) dr; o

The constant C2 must be zero if the solution is to be regular as z becomes very

large. After substituting the previously obtained value for u, considerable

algebra yields

T

1TiT

q" (t-s) [1 - f (z + /Tsh/k)]e' (z /4qs)

ds

Js~

(A3. 2)

22
where

^ 2 f(p) = y*~p exp(p ) erfc(p)

The terms under the integral in A3 .

2

have the following meanings. The first

term, q"(t-s), is the radiant heat input to the surface. The second group of

terms [1 - f ] , are, the 1: the result of heating in the absence of surface

heat loss, and the f: the decreased rate of heating occasioned by this loss.

The exponential term represents the heating in depth of the solid. When

calculating the surface temperature (our present interest) , z = 0 and the

exponential term is 1. Finally, note that, if q" is constant, the result is

the well known square -root behavior of a solid impulsively heated.

Comparing A3 .

2

with 1 we see that

Section A3 . 2

:

Further development of equation 1

Next equation 4 of the main text is to be substituted in equation A3 .

2

(with z = 0), but first, anticipating exponential growth, the following

substitutions are made

R = x Jl + exp(-2j3s)

Here /3 is the exponential growth rate. We note in passing that, if the

exponential assumption is consistent, the resulting expression, to be presented

below, will be an algebraic equation for (3 with coefficients independent of x and t

.

The result of these substitutions for the q"(t-s) of equation 4 is

Upon substituting this in A3 .

2

and observing that, if both sides of the

equation are multiplied by h, the result can be written

r = ro exp [@ (t-s)]

x = ro exp [/3t]

and

q"(t-s) = (Qf "/4) [1/71 + exp (2/3s) ] 3

A - G 1

1

- G

2 I

2 (A3. 3)

where

G = h/JkpcP

23
and

ln(x/r )

( 1/71 + exp (2s) )^ d.s/Js~

o

ln(x/r

Q

) Jn~ (l/7l + exp (2s)

S erfc (gJs ~

) ds

o

A = Ujn h(Ti

- To)/Qf

"

Examination of I

1 suggests that, for large enough values of its upper limit, it

should be a constant. Numerical integration of I

x (and I2

) shows that I

: is

within 3% of its asymptotic (large upper limit) value when ln(x/ro

) - 1. That

is, for fires which have grown to 2.7 times their initial radius. Since, for

example, the NBS/Harvard simulation takes the initial radius for a growing fire

as about ro = 3 cm, the integrals would be essentially constant for x - 8 cm or more. I

2 , however, is somewhat more complex as it depends both on its upper

limit and on the value of G. G is the ratio of the heat transfer capability of

the surface toward the gas to its capability into the solid. For most

materials G is a small number because the gas, even with (linearized)

radiation, is a poor heat transfer medium compared to the solid. (For

hardboard G ranges from 0.05 to 0.005 [11].) With the upper limit of

integration 2.5, I

2 varies 5.3% as G varies from 0.1 to 0.001 and, with G - 0.01, I

2 increases 4% when the upper limit is increased indefinitely from 2.5.

The change is 8.3% for an upper limit of 2. Thus, for the expected range of

variation of G, I

2 may be assumed essentially constant for fires at least an order of magnitude larger than their initial size. Finally, if Ij and I

2 are constants, then our assumption of exponential growth is justified since G

values calculated from A3 . 3 are independent of time and, therefore, (3 values

are constant. To the extent that and I

2 deviate from constants, so the fire

growth will deviate from exponential. In fact, exponential growth is not to be

expected for the early phases of fire growth. 3

3 Unpublished data of R. Land, Harvard University, gives very detailed

measurements of r(t) for fires spreading on urethane foam. His data shows that

r(t) varies exponentially with time, but only after an initial growth period

has past. Although his data is consistent with the small time behavior of the

present model, this is fortuitous as the model assumes a turbulent flame,

among other things. For Land's small initial fires, laminar, not turbulent

flames would be expected.

24
The numerical integrations with upper limit 2.5 and G = 0.01 are

1 1 = 0.4509

1

2 = 0.5866

When equation A3 . 3 is solved for G, a real G is obtained only if A, the

left hand side of A3. 3, is less than I

1 2 /(4*I2 ). This reflects the fact that,

if the surface cooling is sufficient, relative to the fire size, the fuel will

never be heated to its ignition temperature and flame spread will not be

possible. In fact, we know that flames will not spread on a horizontal piece

of wood without more heat than the wood's own flames can supply. In this case,

the pyrolysis temperature is about 420° C and radiation loss from the surface

is quite large. If, however, A3 . 3 has a real root, then our initial assumption

of exponential flame spread is bom out except for the initial stages of

growth. Note that, at the limit of real flame spread the fire still grows at a finite rate. This is experimentally observed, but the limit flame spread rate

given by our formula should not be expected to be correct. Recall that, near

extinction, the assumption of one -dimensional heat transfer breaks down.

Further, the surface regression behind the slowly moving flame causes the fuel

surface to deviate significantly from a horizontal plane. These effects and

the local heating around the flame foot invalidate the assumptions of the

present model.

The limitations on the flame spread model discussed above suggest that a suitable approximate solution to equation A3 . 3 would be to assume that the

surface heat transfer coefficient, h, must not be too large. The result is

P =

1

yicpc

(A3. 4)

where

N = 4A I

z /l\ = 20.46

This is valid provided

N*h*Ts « Qf

"

Several improvements on the above model should be considered. In general

these lead to fire growth that is not strictly exponential. As a result, more complex analytic methods are needed to find the flame spread velocity. The

simplest is based on the observation that the local heat flux just before the

flame reaches any given location, x, is more important than the behavior at

earlier times. Accordingly the integrand of A3 . 2 is expanded in a Taylor

series in the flame spread velocity, V. The result of such an expansion will

not be discussed here, but some of the additional effects, not included in the

above "exponential growth" analysis, will be mentioned.

25
In a room fire situation, the heating of non-burning fuel is influenced

by radiation from the hot gases trapped in the upper part of the room and from

the heated upper walls and ceiling. This radiation is, to a first

approximation, independent of x but depends on time. If the spreading fire

under consideration is the only fire in the room, this will be heavily

dependent on the fire size (and will tend to support exponential growth as a result) but, in general the connection may be quite otherwise. This effect is

clearly important to accurate room fire analysis and should be included. It is

considered in [11] where its effect suggests that, for our case, it would

appear as a correction to TQ in A3. 4.

Also considered in [11] is the effect of the local heating around the

flame foot (leading edge). In fact, this is the principal mode of forward heat

transfer considered in [11] where it is shown to dominate up-wind flame spread

(on a solid). Using approximations similar to [11] this term could also be

included. For large "pool" fires it would be a correction, not the major

(controlling) term.

A slightly more complex expression than equation 4 is obtained if the

approach of deRis in [9] is followed. The flame radiant flux is explicitly

expressed as

Q" = ctT* [1 + Jl + (h/r) 2

]

[ 1 - exp(-kl)]

where crT* = flame radiative power per unit flame surface area

[1+ 7l+(h/r) 2

] = a flame shape factor (deRis assumes a conical flame)

[l-exp(-kl)] = emissivity. Fo r the as sumed conical flame

Ji/r = 1.2/[1+ 7l+(h/r) 2

]

R = yx2 + z

2 where z is not necessarily equal to r

If it is assumed that the flame is adequately represented by a cone and that

the cone angle is constant (as is assumed in the Harvard fire models, except

for special cases) the area factor would be outside the integral and z would be

proportional to r. The emissivity term should be expected inside the integral

as the flame emissivity will change as it grows, for the sizes of most

interest. (The equations given earlier would be correct if the emissivity were constant and a good approximation if it were nearly constant during the most

recent past) . The effect of these changes are that the integral is no longer

expressible as a number (for /3t large, as noted above). Its value depends on the choice of the flame extinction coefficient, k. The deviation from

exponential growth is not great even when the full deRis analysis is used

inside the integral.

For our present purposes, all of the above improvements are somewli.it

irrelevant (although they would be desirable for inclusion in an improved f 1 .une spread expression in the room fire models). Recall that our purpose was to

find a more detailed expression for the exponential growth rate of fires,

given, a fortiori, that exponential growth was in agreement with observation

26
We have such an expression in A3. 4. In this the material dependency is in the

"thermal inertia", k/?c. Even with all of the above improvements, this will

remain, not only as the major material property of interest, but also its

functional relation to the entire expression will not change significantly.

Thus A3.

4

and 5 in the main text, provide a valid basis for correlating

experimental data for thermal inertia. In this sense, our purpose is

accomplished.

27
28

Figure

1.

Fires

ventilated

from

above:

steady

state

model
Fresh

air

Hot

gas

29

Figure

2.

Fires

ventilated

from

above:

unsteady

model
Ventilation

or

area

of

limited

burning

-

31Vd A0U3N3

30

TIME

Figure

3.

Relation

of

fire

development

to

fire

control

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

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-86/ 3451

2. Performing Organ. Report NoJ 3.

4.

TITLE AND SUBTITLE

Publication Date

September 1986

FIRE GROWTH IN COMBAT SHIPS

5. AUTHOR(S)

John A. Rockett

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) J. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE *. Type of Report & Period Covered

Gaithersburg, Md. 20899

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

Naval Ship Research & Development Center

Washington, DC

10. SUPPLEMENTARY NOTES

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

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

bi bliography or literature survey, mention it here)

Several enrichments to FIRMOD, the Navy's ship battle damage estimation computer

program, are considered. Some proposed enrichments could now be made to good effect,

others should only be considered after further experimental work has been done to

better define the phenomena involved and the algorithms to be used.

Enrichments recommended for immediate consideration are exponentially growing fires,

ingestion by one ventilation system of smoke issuing from other ventilation system

exhausts, and smoke transport between spaces served by the same ventilation system.

Areas where further experimental work is recommeded before FIRMOD enrichments are considered include fires ventilated primarily from above and buoyant smoke transport

up shafts and ladderways.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) fire control; fire growth; fire models; shipboard fires; smoke control; ventilation

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

Qjr] Unlimited

| | For Official Distribution. Do Not Release to NTIS 37

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

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