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Prog. Eneroy Combust. Sci. 1982, Vol. 8, pp. 317-354. 0360-1285/82/040317 38519.00/0 Printed in Great Britain. All rights reserved. Copyright © Pergamon Press Ltd.

URBAN AND WILDLAND FIRE PHENOMENOLOGY

F. A. WILLIAMS Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, U.S.A.

1. I N T R O D U C T I O N

Mankind's concern with unwanted fires likely pre- dates the first practical use of combustion in unre- corded history. Yet the science of fire protection has progressed more slowly than other aspects of combustion science. This state of affairs is due partially to the complexity of the problem and partially to the fact that relatively large technological payoffs generally are not anticipated to be obtained from scientific investigations of fires. Numerous promises and problems in the use of combustion in heat and power production, in locomotion and in industrial endeavors have generated intensive scientific efforts. By way of contrast, the ever-present fire problems have attracted fluctuating interest with a relatively low average level of concern.

Although periodic disasters engender beliefs that more should be done in fire science, these beliefs often are short-lived and are replaced by more immediate concerns. Rarely has a disaster or a series of disasters provoked a program of scientific study of fire phenomena. The long-term efforts that have been mounted in the field instead have been motivated mainly by detailed comparative evaluations of the magnitude of the fire problem.

There is reason to believe that today's rapid technological advances intensify problems of unwanted fires. Hosts of new processes and new combustible materials are emerging and finding their way into widespread use. Too often these innovations become common- place before their fire hazards are properly understood. Therefore there seems to be justification for more extensive development and dissemination of knowledge in fire science.

The present article has been prepared with the objective of bringing together many aspects of fire science for a nonspecialized audience. It is based on an undergraduate course by the same name, given only once, at the University of California, San Diego. It focuses on basic aspects of fire as a phenomenon, presented in an elementary but unified manner. It is more restricted than other treatments of the subject in that physiological, social, economic, organizational and operational aspects are not covered. However, it is broader in that both urban and wildland fires are considered equally; most presentations are slanted toward one or the other of these classes of fire problems. I hope that this article will help to introduce nonspecialists to the scientific phenomenology of fires.

1.1. Books on Fire Science

A number of books have been devoted specifically

to fire science. References 1 through 5 are examples of the type of material that is available. These references are directed toward urban fires; fewer books are available concerning wildland fires. Reference 6 is a collec- tion of articles concerning both urban and wildland fires. Reference 7 often is cited as a forest-fire text. Reference 8 is a detailed documentation of many forest fires that have occurred in North America. Reference 9 is a fictional novel which nevertheless is rather accurate technically, concerning a forest fire.

With the exception of the material in Ref. 6, the level of scientific description in the books cited is not very advanced. Typical undergraduates in engineering have backgrounds in physics, chemistry, thermo- dynamics, fluid mechanics and heat and mass transfer that would enable them to appreciate a more sophisticated treatment. No book exists giving a unified exposition of fire science at a more advanced level. The variety of authors of Ref. 6 provide material at an advanced level with uneven coverage.

A source of references on continuing research in fire science is provided by Ref. 10. This periodic publi- cation contains numerous good review articles on various aspects of fire problems. Much of the information underlying the present article has been obtained from Ref. 10. The biennial combustion symposia, starting approximately with the tenth,11 contain many original research papers in the field.

1.2. Magnitudes of Fire Problems

There have been many compilations of losses attributable to unwanted fires. The accuracy of such information always is open to question because of uncertainties in reporting and possible errors in collec- tion and tabulation of data. Nevertheless, published numbers are at least roughly indicative of reality. Table 1 lists some loss information obtained in the middle of the decade 1970. Some of the numbers given have been rounded here in an effort to reflect uncertainties.

Annual fire deaths per capita depend on many factors, such as living conditions and degree of indus- trialization; they may vary over an order of magnitude. The relative significance of deaths and of property loss is difficult to assess rationally. In the United States, most of the fires are residential fires, and most of the fire deaths occur in residential fires, but most of the property loss occurs in industrial and commercial fires. In the figures for 1972, obtained by the Presi- dent's Commission on Fire Protection and Control, the total cost is calculated as roughly equal contributions from property loss, fire-related expenses in build-

317

318 F. A. WILLIAMS

TAnLE 1. Annual fire losses

US UK USSR Japan

Deaths 10,000 1,000 Deaths per million 40 12.6 1.2 Serious injuries 300,000 1972 property loss $3 X 10 9 Property loss capita $20 $2.5 $0.2 1972 total cost $10 x 109 Number of fires 5 x 106 Fires fought by fire service per 1000

population 4.5 3.1 0.16 Forest acres burned (1973-78 average) 3.8 x 106

$2.6

ing construction, costs of fire departments and fire insurance costs. The forest acres burned are approximately the size of the land area of the state of New Jersey. It appears from these figures that fire problems are significant, particularly in the United States.

1.3. Historic Fires

There are many well-known fires in recorded history. A few of these are listed in Table 2, along with some other fires that are not so well known. It is seen that the famous fire of London burned over a relatively small area. Fires often are associated with military events; the Moscow fire coincided with Napoleon's occupation of the city. It is not generally known that in October, 1871, on the same day as the famous Chicago fire, fires began relatively nearby, in the Peshtigo area of Wisconsin and in central Michigan, that burned through 17 towns and 5000 square miles, killing nearly four times the number of people who perished in the Chicago fire; the coincidence likely reflects the occurrence of optimal weather conditions for burning in the area.

Although the San Francisco fire, associated with an earthquake, is well-known near the turn of the century, the Baltimore fire was considerably more instructive in revealing fallacious fire-fighting practice, x2 The fires in Hamburg, Tokyo and Dresden, during World War II, were caused intentionally by incendiary bomb- ing; fire storms were established, and the loss of life

was record-breaking, but nevertheless about 80~o of the population in the fire area survived in Hamburg. Large forest fires continue to occur when atmospheric conditions favor them; the last entry in Table 2 represents 1260 separate fires in a two-month period, with total losses placed in excess of $2 x 1 0 6.

1.4. Definitions of Fires

Certain terms peculiar to fire studies deserve definition at the outset. As a general definition, a fire may be taken to be a chemical reaction of fuel with oxygen to produce heat, thereby involving heat transfer and fluid flow. This definition is intended to exclude very slow oxidations, such as rusting, but to allow for gaseous, liquid or solid fuels, polymers or metals, burning under controlled or uncontrolled conditions.

A mass fire may be defined as any large fire involving more than one sizeable structure and taxing resources of fire-fighting agencies. Mass fires may be divided into subcategories, depending on their characteristics. For example, a conflagration is a large propagating fire; spread of the fire is a key aspect of this definition. Large forest fires are typical examples of conflagrations, but conflagrations also may occur in cities. A fire storm may be defined as a large, intense, localized fire, usually with a single convection column above it, nonspreading and having high-velocity fire- induced winds. Specific definitions of fire storms often require the velocity somewhere to exceed a specified

TABLE 2. Some historic fires

Location Date Acres burned Homes lost Deaths

London 1666 336 13,200 Moscow 1812 Chicago 10/8/1871 2,124 300 Peshtigo 10/8/1871 3,600,000 1,000 Baltimore 1904 San Francisco 1906 Idaho and Montana 8/1910 3,000,000 Tokyo 1923 1,200 Hamburg 1943 2,500 40,000 Tokyo 1945 9,600 85,000 Dresden 1945 150,000 Hiroshima 1945 3,000 Ft. Yukon, Alaska 1950 2,000,000 Laguna, California 10/1970 175,000 382 3 California 9/15 11/15/70 600,000 885 14

Urban and wildland fire phenomenology

Table 3. A few combustible materials

319

Material Formula Flame color

Heat of combustion

kcal/mole of fuel cal/g of fuel

Gases

Hydrogen H2 Carbon monoxide CO Natural gas (methane) CH 4 Propane C3 H8 Ethylene C2 H4

liquids

Heptane C7H16 Octane C s H t s Benzene C6H 6 Gasoline HC Kerosene HC Methyl alcohol CH3 OH Styrene Cs Hs

Solids Carbon (graphite) C Sugar (sucrose) C12H22011 Urethane C3H7NO 2 Cellulose (~ glucosan) C 6 H 10 O 5 Wood (birch, oak, etc. under

average conditions in nature) Charcoal CH~(c~ < 1) Steel (iron) Fe Magnesium Mg

invisible 68.3 34,150 blue 67.6 2,410 blue 210.8 13,180 blue yellow 526.3 11,960 blue-yellow 337.3 12,050

blue-yellow 1149.9 11,500 blue yellow 1302.7 11,430 yellow green 782.3 10,030

11,530 11,000

blue 170.9 5,340 yellow 1047.1 10,070

yellow 93.9 7,830 1349.6 4,000 397.2 4,460

4,200

4,000 7,260 1,580 6,080

value, e.g. 75 mph. Fire storms with well-developed convection columns may generate clouds of water droplets from condensation of cooled reaction products; in extreme cases rain may fall from the clouds.

A fire whirl or a fire vortex may be defined as a fluid-mechanical vortex with fire in it, generated or intensified at least partially as a consequence of the fire. Fire whirls typically may be elements of fire storms or of mass fires in general; they have been suggested as small-scale models for some types of fire storms. Intense whirls, sometimes called fire tornadoes, can be destructive.

1.5. The Fire Triangle

Books on fire science often employ a triangle to represent the key elements of a fire. The triangle has three legs, representing heat, air and fuel. Strategies for fire suppression through flame extinguishment often are viewed as attempts to remove one of these three elements. The fire triangle is intended to provide an intuitive feeling for essentials of fire at an elementary level.

2. COMBUSTIBLE MATERIALS

Of basic concern in fire problems is the identification of materials that can serve as fuel. Most things, even steel, will burn under suitable conditions; carbon dioxide, water and sand are examples of materials that cannot burn. Table 3 lists some common combustible materials and gives some of their combustion properties, notably the energy released when they burn.

2.1. Hazard Aspects

Many different properties of fuels have bearing on their fire hazards. One is their ease of ignition; even if its heat release is low, a material that can be ignited easily may pose a severe fire hazard. Another relevant property is the heat of combustion, listed in Table 3; materials with high heats of combustion can be relatively more effective in sustaining fires. Flame spread is a third aspect of fire hazards; materials that are difficult to ignite and that have low heats of combustion may nevertheless spread flames relatively rapidly and thereby be dangerous.

Subsidiary aspects of behaviors of materials in fires also influence their fire hazards. Smoke can cause damage and also can interfere with escape from fires and with fire fighting; propensity of a material for smoke production therefore is relevant in assessing its fire hazard. Materials capable of generating toxic products in fires are of particular concern.

Finally, ease of extinction is a significant aspect of a material 's fire hazard. An otherwise dangerous material may be acceptable if its flames can be extin- guished readily. There are many tricky aspects to the evaluation of fire hazards. Some will be considered later in connection with estimates of flammability.

2.2. Fire Categories

There is a partial correspondence of the states listed in Table 3 with the categories of fires employed in fire protection. Fire classes are: Class A, Solid; Class B, Liquid; Class C, Electrical. These classes are defined

320 F.A. WILLIAMS

roughly in increasing order of the danger associated with the fire and call for different techniques of fire fighting. There is no class corresponding to gaseous fuels because fires with such fuels are encountered relatively infrequently and because when they do occur the duration of the fire usually is too short for counter-measures to be taken. This does not mean that gaseous fuels are less dangerous; on the contrary, rapid flame spread through gases often generates pressure increases characteristic of explosions or buoy- antly rising clouds of burning gases with damaging levels of radiant energy transfer.

In all three fire classes usually gases actually burn. These gases are secondary, not primary fuels and are liberated from the liquid or solid fuels in the fire environment. There are a few exceptional cases in which the liquid or solid fuels burn directly without previous liberation of combustible gases. Carbon, some explosives and solid propellants, and certain metals are examples of fuels that burn directly, and glowing combustion of wood or tobacco is a burning process that does not involve a gaseous combustible intermediary.

2.3. Burning Mechanisms of Solid and Liquid Fuels

It is important to understand the usual mechanism, alluded to above, by which condensed-phase fuels (solids and liquids) burn. The heat is released in the gas phase by the exothermic combustion of the secondary gaseous fuels. Some of this heat is transferred back to the condensed phase to cause gasification of the primary fuel. This gasification usually is an endothermic process (requiring heat) which releases the gaseous combustibles to burn. Thus, feedback of heat from the gas-phase flames to the condensed-phase fuels usually is an essential aspect in maintaining a fire.

2.4. Gasification Processes

Two fundamentally different types of gasification processes occur in fire. One, encountered most often for liquid fuels, is equilibrium evaporation, and the other, encountered most often for solid fuels, is finite- rate pyrolysis.

In equilibrium evaporation, interphase equilibrium is maintained at the surface of the fuel. This equilibrium may be described by a useful approximate formula for the mole fraction X~ of fuel vapor in the gas at the surface of the condensed fuel. 13 If T(K) is the surface temperature, Tb(K ) the normal boiling temperature, R ~ 2 cal/mol K the universal gas constant and L the latent heat of vaporization, then

L 1 1

Tables of L and Tb are available; 13'14 for example L ~ 10kcal/mole and T b = 373K for water. Equa- tion (1) thus provides a relationship between Xe and T. This relationship must be used in formulas for transfer rates to obtain gasification rates under conditions of equilibrium evaporation.

Pyrolysis means a chemical transformation produced by application of heat. In finite-rate pyrolysis, molecules from the gas that strike the surface of the fuel enter the condensed phase to a negligible extent, and approximate formulas may be written directly for the rate of gasification as a function of T. 15 A useful approximate formula for the gasification rate m, the mass per unit area per second of vapors leaving the surface of the fuel, is

m -- msex p [ -Es / (RT)] , (2)

where ms and Es are constants, the latter being an effective overall energy of activation for surface pyrolysis. Values of m s and E~ are difficult to find in the literature, although some information is available. 6

All of the gasification parameters that have been introduced here are properties of fuels. Understanding of these properties and of pyrolysis processes requires knowledge of chemical bonding, chemical conversion and heat liberation.

3. CHEMICAL CONVERSION AND HEAT LIBERATION

Chemical conversion is a process whereby a chemical in one form is transformed into chemicals in other forms. The many different types of chemical conversion that are possible are dictated by molecular structure, which is determined by chemical bonding.

3.1. ChemicaI Bonding

Molecules are formed by establishment of chemical bonds among atoms. These bonds may be ionic (i.e. involve exchange of electrons) or covalent (i.e. involve sharing of electrons). For combustible materials covalent bonds are by far of greatest importance. Ar- rangements of bonds formed in stable molecules depend on the valences of the atoms involved. A covalent bond, two shared electrons, conventionally is indicated by a bar; for example, H 2 is H--H. Some of the fuels in Table 3 are

H H H I I I

methane, H - - C - - H ethylene, H - - C = C - - H I I [

H H H

methanol,

H I

H - - C - - O - - H I

H

and benzene,

H H I I

C m C / / % H - - C C- -H,

\ / C = C

I H H

which will be represented as H - - © for brevity. Note that ethylene, for example, has a C = C double bond, representing four shared electrons.

Urban and wildland fire phenomenology 321

3.2. Polymers

Many of the solid fuels of concern in fires are polymers. Polymers are large molecules, formed conceptually (in the simplest cases) by breaking a double bond in identical molecules and interconnecting them. 16 Thus, polyethylene is

H H I I

. . . . C m C - i I

H H

H H I H H J l l i i

~-C--C ,--~-C--C • •., I I : 1 I

H H ' , H H

where the broken vertical lines separate the ethylene "monomers". The degree of polymerization is the number of monomers in the polymer chain; the chains may be terminated in various ways, e.g. by placing an H at the end.

Styrene is

H H

C = C , f J

© H

and therefore polystyrene is

H H H H I I I I

. . . . C - - C - - C - - C . . . . I I i i

0 H 0 H

In addition to polyethylene and polystyrene, many other synthetic polymers are experiencing increasingly widespread use. These include polyvinylchloride

(monomer unit

H H I I

- - c - - c - - ), I I

H C1

acrylonitrile

(monomer unit

H H I I

- - c - - c - - ) I I

H C ~ N

and poly(methyl methacrylate), "plexiglas" or "lucite", with monomer unit

I H H - - C - - H H

I I I H - - C C C - - O - - C - - H .

i f fi i H O H

Cellulose, the principal polymeric constituent of natural wood, is built from a glucosan monomer,

H I

H - - C - - O - - H i

C O

C C

-o / \ ? - H C C

H O - - H

Qualitative feelings for burning behaviors of cellulosic materials are common. There is less in- tuition concerning burning behaviors of synthetic polymers. Many, such as polyethylene, polystyrene and poly(methyl methacrylate) soften and form a liquid-like "melt" when they burn. Thus, it becomes unclear as to whether they should fall in fire class A or B. Polyvinylchloride may form corrosive HCI during combustion, while acrylonitrile may produce measureable amounts of highly toxic HCN upon pyrolysis. Thus, the advent of synthetic polymers raises new fire problems.

3.3. Bond Energies

Energetic aspects are of importance for chemical conversions that occur in fires. Energies liberated in chemical processes, such as heats of combustion, need to be known. Energies absorbed, such as heats of pyrolysis of polymers, energies required to convert specified polymers to gases at a given temperature, also must be known. There are many tables 14 of these heats of reaction. However, often it is of interest to calculate energy changes for processes that are difficult to find in tables. Bond-energy methods enable such calculations to be performed, with accuracies that although typically are not high nevertheless are sufficient for many purposes.

The bond-energy approach rests on the idea that a definite amount of energy liberation is associated with the formation of a given chemical bond. The idea is not precisely correct in that energy liberated may depend also on the molecule in which the bond occurs and on the location of the bond within the molecule. In fact there are correction procedures to account for these effects, which may be quite substantial. In a rough first approximation the corrections may be neglected, and the energy liberated in forming a gaseous molecule from its constituent atoms may be calculated simply by adding the energies associated with each bond formed. A list of the bond energies needed for this calculation is given in Table 4, which has been taken from information in Ref. 17 and does not necessarily represent the most up-to-date information, although it is useful for illustrative examples. Accuracies in energy calculations better than 50~ may be anticipated when using Table 4.

3.4. Combustion Reactions

The combustion reaction which occurs in the flames of fires is a chemical combination of fuel with air

322 F.A. WILLIAMS

TABLE 4. Mean bond energies (kcal/mole)

Bond Energy Bond Energy

C--C 85 N=-N 225 C~---C 143 H--H 103 C ~ C 198 O - - H 109 C--H 98 O - - N 150 C- -O 86 N--H 88 C = O 173 S--S 50 C--N 81 C1--C1 57 C ~ N 210 Br--Br 46 C--CI 78 I--I 36 C--Br 67 F - -F 36 C--I 64 H-- C1 103 C--F 102 H--Br 88 C--S 64 H-- I 72 O - - O 33 H--F 135 O = O 117 H--P 76 N- -N 60 H--S 81

to produce CO2, H20 , N2 and heat. Air, in a first approximation, is 0 2 + 4N 2. Thus, for example, the combustion of hydrogen in air is represented as H z + ½ O z + 2 N 2 ~ H z O + 2 N a + Q H 2, where Qn~ is the heat of combustion per mole for hydrogen. Similarly, for carbon monoxide, CO + ½02 + 2N2 CO2 + 2 N z + Q c o . These equations are balanced chemically in that there is no fuel or oxygen left over; such chemical conversions are termed stoichiometric.

Balancing a chemical reaction to achieve stoichiometry may be illustrated by considering the combustion of heptane. Write the reaction as C7 H ~ 6 +

x(O2 + 4N2) ~ 7CO2 + 8 H 2 0 + 4xN2 + QCTH~6, where x is unknown. The coefficients of CO 2 and of H 2 0 have been determined from the chemical formula of the fuel. An oxygen balance then is used to find that x = 11, thereby completing the stoichiometry.

3.5. Calculation of Heat of Combustion

The energies Q in the preceding equations are best calculated from tables of standard heats of formation, the energies liberated when molecules are formed from their constituent elements in their standard states. A somewhat less involved approach is to use the bond energies listed in Table 4. As a simple example consider the combustion of hydrogen. Write the equation for chemical conversion as H - - H + ½ 0 = O ~ H - - O - - H + Q r c From Table 4, this implies 103+½× 117 = 2× 109-QH2, where additivity of energies in reactions has been employed. The negative sign occurs because the heat of combustion is positive if the total bond energies of the products exceed those of the reactants. The result that QH~ = 56.5kcal/mole for combustion of gaseous H 2 with gaseous 02 to form gaseous H 2 0 is within 5 ~o of the correct value. To find QH2 for combustion to liquid H20 , the latent heat of vaporization L must be added to this result. The accuracy obtained here is better than average; it is preferable to use tables for Q if they are available.

3.6. Flame Temperature

Temperatures of flames exceed ambient temperature because the heat released in combustion goes into raising the temperature of the combustion products. The extent to which the temperature is raised depends on the heat capacity Cp of the products. Tables of cp are available. '4 In fact cp varies with temperature, but as a first approximation it may be taken as constant. For gases cp generally lies between 0.2 and 0.5 cal /gK; in a very rough approximation it may be taken as 0.3 cal /gK for all gases. For liquid water % - 1 cal /gK; for most other liquids and for solid combustibles it typically lies between 0.3 and 0,7 cal/gK.

From the molar heat of combustion Q, the heat release per unit mass of products may be calculated as Q/W, where W is the sum of the molecular weights of the species on the right-hand side of the equation for the chemical conversion of one mole of fuel, i.e. the stoichiometric mass of all products per mole of fuel consumed. The flame temperature T I is then found from the adiabatic energy balance Q/W = cp(Ty- Ti), where T~ is the initial temperature, typically room temperature, about 300K. Thus

T s = T , + Q / t W c p ) . (3)

Corrections to this for phase changes may be included by suitably revising Q.

As an example, consider the combustion of propane

in air, C3H s + x(O 2 + 4N 2) --~ 3CO 2 + 4 H 2 0 + 4xN 2 + QC3H~ with x = 5 from the chemical balance. F rom Table 3, since the molecular weight of propane is 44 g/mole, QC3H, = 11,960 × 44 = 526,000 cal/mole, and W = 3 × 4 4 + 4 × 18+20 × 28 = 764 g/mole. Hence, with e e = 0.3g/mole K, eq. (3) gives T s = 300+690/ 0.3 = 2600K, which is about 300K too large. This pro- cedure usually overestimates T I because it neglects effects of dissociation of reaction products, which occurs above about 2000K; dissociation involves, for example, CO 2 ~-CO + ½0 2. There are iterative methods and computer programs for calculating Ty with dissociation included (see, for example, Ref. 18). A short

TABLE 5. Approximate flame temperatures of various stoichiometric mixtures having initial temperature 298K

Pressure Fuel Oxidizer (atm) Tf(K)

Acetylene Air 1 2600* Acetylene Oxygen 1 3410" Carbon monoxide Air 1 2400 Carbon monoxide Oxygen 1 3220 Heptane Air 1 2290 Heptane Oxygen 1 3100 Hydrogen Air 1 2400 Hydrogen Oxygen 1 3080 Methane Air 1 2210 Methane Air 20 2270 Methane Oxygen 1 3030 Methane Oxygen 20 3460

*A maximum temperature that occurs under fuel-rich rather than stoichiometric conditions.


table of adiabatic flame temperatures, taken from Ref. 18, is shown in Table 5. It is seen that temperatures in the range of 2300K are typical for burning in air. These temperatures are of importance in calculating heat transfer in fires. The method for calculating T s that has been outlined here is useful for obtaining quick rough estimates.

4. CHEMICAL KINETICS OF PYROLYSIS

Other chemical conversions, in addition to combustion reactions, can be of significance in fires. These include the formation of smoke and of toxic products in flames as well as conversions of solid fuels to gaseous combustibles. Many of these reactions are pyrolysis processes. For example, smoke may be produced through a sequence of pyrolysis reactions of gas-phase fuels in fuel-rich regions, and carbonaceous residues may arise from liquid-phase pyrolysis of heavier liquid fuels. Attention here is focused on pyrolysis of solid fuels to produce gaseous reactants.

4.1. Chain Reactions in Polymer Pyrolysis

The chemistry of polymer pyrolysis is complex and differs for different polymers. Simplified descriptions are needed to achieve understanding. A useful simplification for many processes of polymer pyrolysis (as well as for the kinetics of combustion reactions them- selves) is the idea of a chain reaction. Chain reactions have active intermediate species, chain carriers, whose presence cause the reaction to proceed more rapidly than it otherwise would. The chain carriers are formed in initiation steps, cause the reaction to proceed in chain-carrying or propagation steps, and are consumed in termination steps.

For polymer pyrolysis, there are many types of initiation steps. In end initiation, the monomer at the end of the polymer chain splits off, leaving a radical (a species with an unsatisfied chemical bond) at the chain end. In random-scission initiation, thermal fluc- tuations break the polymer at random points along its chain, producing radicals on each side of the scission. In weak-links initiation, the polymer is broken in- ternally at preferred high-strain spots, again leaving radical-ended chains.

There are also many types of propagation steps. A relatively easy type to understand is unzipping, in which a single monomer unit is formed and detached at the radical end of the chain. An illustration of unzipping for polystyrene is

H H H H H H H H I [ I I I I [ I

...--C--C--C--C-- , ...--C--C--+ C=C. I 1 I I 1 1 J I 0 H 0 H 0 H 0 H

Propagation steps other than unzipping could be intramolecular transfer steps, namely detachment of higher units of the monomer, e.g. dimers or trimers, from the radical end. Intermolecular transfers, inter- chain propagation processes, also are possible. For example, the radical at the end of an active chain

may break another, stable polymer somewhere in the middle, forming a new stable polymer with half of the attacked chain and leaving the other half active.

Among the possible termination steps is direct combination of the radicals at the ends of two active chains to form a single stable polymer. Another type of termination step is disproportionation, in which two active radicals deactivate each other by an exchange at the end of the chain. For polystyrene, an example of disproportionation may be

H H H H H H H H I I [ I I I J I

. . . - - C - - C - - C - - C - - + - - C - - C - - C - - C - - . . . , I I I I I I I I

O H O H 0 H 0 H

H H H H H H H I I I I I I I

. . . - - C - - C - - C = C + H - - C - - C - - C - - C - - . . . . I I I I I I [ I

© H © H © H © H

Clews concerning pyrolysis mechanisms for specific polymers are obtained from many different experimental observations. 15 One such measurement is the percentage of product volatiles composed of monomer, found when the polymer is heated in a vacuum. Some data of this type are given in Table 6, taken from Ref. 15. If the monomer yield is low then unzipping is unlikely, while high monomer yields are consistent with unzipping.

4.2. Simplified Kinetic Expressions

Rates of polymer pyrolysis may be described by expressions for dM/dt, the time rate of change of the mass M of the condensed phase in a homogeneous system. Such expressions may be complicated for chain reactions. There are conditions under which useful simplified approximations may be obtained. For example, for an unzipping process with a kinetic chain length (or zip length, i.e. the number of propagation steps that occur prior to termination) comparable with the degree of polymerization, each initiation effectively results in unzipping of an entire chain. The rate of mass loss then is controlled by the rate of initiation, and

dM/dt = - k M , (4)

where k is a specific reaction-rate constant for initiation. Often this first-order reaction-rate expression provides a reasonable approximation under more complex circumstances, in which k becomes an effective rate constant that includes influences of many different steps.

The rate constant k depends on temperature T. Often an Arrhenius expression for this dependence provides a good approximation. Thus,

k = Bexp [-Eb/(RT)], (5)

where B and E b are constants, the latter being the overall activation energy for bulk degradation. A table of some measured values of Eb is shown as Table 7, again taken from Ref. 15. There have been a

324 F. A. WILLIAMS

TABLE 6. Yield of monomer in the pyrolysis of some organic polymers in a vacuum

Polymer

Temperature Yield of range monomer, ~o

°C of volatiles

Polymethylene 335-450 0.03 Polyethylene 393-444 0.03 Polypropylene 328-410 0.17 Polymethylacrylate 292 399 0.7 Hydrogenated polystyrene 335 390 1 Poly(propylene oxide), atactic 270-550 2.8 Poly(propylene oxide), isotactic 295-355 3.6 Poly(ethylene oxide) 324-363 3.9 Polyisobutylene 288-425 18.1 Polychlorotrifluoroethylene 347-415 25.8 Poly-fl-deuterostyrene 345 384 39.7 Polystyrene 366-375 40.6 Poly-m-methylstyrene 309 399 44.4 Poly-~-deuterostyrene 334-387 68.4 Poly-~,fl, fl-trifluorostyrene 333-382 72.0 Poly(methyl methacrylate) 246 354 91.4 Polytetrafluoroethylene 504-517 96.6 Poly-ct-methylstyrene 259 349 100 Polyoxymethylene Below 200 100

number of studies in which expressions for k have been derived for more complex mechanisms. ~9 It can often be shown for steady-state pyrolysis that the E~ in eq. (2)of Section 2.4 is Eb/2.

4.3. Competition in Pyrolysis

Certain materials such as wood and paper exhibit two types of combustion, flaming and glowing. The occurrence of these two types may be traceable to the existence of two competing pyrolysis mechanisms for the fuel. Such competit ion may be illustrated most simply by considering pyrolysis of a carbohydrate, the formula for which is (CH20)n, with n = 6 for glucose.

Two conceivable paths are

k~ n C O + n H 2 + n O z

. . . . - / ' / ' ~ nCO2 + n H 2 0 (CH20)n k 2 " ~ n C + n H 2 0 + . ~ n O 2

The rate constant for the initial step is kl in the upper path and k 2 in the lower. The final two arrows represent oxidation, involving combinat ion with 02 to produce combustion products.

Although the final products of combustion are the same, the different intermediaries can cause the burning mechanisms to differ. The species CO and H 2 are

TABLE 7. Activation energies of thermal degradation of some organic polymers in a vacuum

Temperature Activation Molecular range, energy

Polymer weight °C kcal/mole

Phenolic resin - - 332-355 18 Atactic poly (propylene oxide) 16,000 265-285 20 Poly(methyl methacrylate) 150,000 226-256 30 Polymethylacrylate - - 271-286 34 Isotactic poly(propylene oxide) 215,000 285-300 45 Cellulose triacetate - - 283-306 45 Poly(ethylene oxide) 10,000 320 335 46 Polyisobutylene 1,500,000 306 326 49 Hydrogenated polystyrene 82,000 321-336 49 Cellulose - - 261 291 50 Polybenzyl 4,300 386-416 50 Polystyrene 230,000 318 348 55 Poly-~-methylstyrene 350,000 229 275 55 Poly-m-methylstyrene 450,000 319-338 56 Polyisoprene - - 291 306 57 Polychlorotrifluoroethylene 100,000 332 371 57 Polypropylene - - 336-366 58 Polyethylene 20,000 360-392 63 Poly-e-fl-fl-trifluorostyrene 300,000 333-382 64 Polymethylene High 345-396 72 Poly-p-xylyene - - 401-411 73


rc "r

FIG. 1. Illustration of competing rates of pyrolysis.

gaseous fuels and therefore may escape from the solid and support flaming combustion. By contrast, in the lower (dehydration) path H20 is noncombustible while C is a solid. The lower path therefore does not liberate gaseous combustibles but instead forms C which experiences surface burning, a type of glowing combustion process of the solid fuel. While tobacco H2COH burns by a process analogous to the lower path, [ matches burn by processes corresponding to both C / t paths, the flaming resulting from a process like the / H upper path. H \ C

With the two competing processes illustrated, the _ O ~ / \ O1-1 rate of conversion of the fuel is

dM/dt = - (k 1 + k2)M, (6) C J

in which kl and k 2 are given by separate expression of H the type shown in eq. (5). It may be seen that if the activation energies differ, Eb~ ~ Eb2, then different reactions may predominate at different temperatures. This is illustrated in Fig. 1. At sufficiently low T, both rates are negligibly small. Typically k doubles when T increases by an amount on the order of only 10°C. At slightly elevated temperatures, k 2 may be appreciable while kl is negligible. Above T~, k 2 soon becomes negligibly small compared with k 1. For cellulosics, k2 corresponds to dehydration and k I to production of secondary fuels capable of burning in the gas phase.

4.4. Pyrolysis of Cellulosics

Pyrolysis mechanisms of cellulose have been subjected to detailed investigation. Numerous techniques have been employed, and a multitude of facts have H2 C been established. Although the current situation is I complex, a few unifying principles have been de- C vel°ped'2°'21 In particular' there appear t° be tw° / H I principal competing paths, which may be represented H \ as , C

"dehydro- HO / ~ O H t200- L-,~ll, lnse '' +HzO----~har + H 2 + C O 2 +.. . 54~¢~oc'~ t. / ~'~i~,'~i, (exothermic) C

. . . . , ~2/ en~to.i~ermic) I c e l l u l o s e - (280- k ~ (endothermic) H 340°C) \ " t a r ' (primarily

- - l evog lucosan )

The "'tar" is volatile and vaporizes to form a major gaseous fuel to support a gas-phase flame. The gases evolved in the dehydration path are mainly noncom-

bustible, and the char that remains can support only a surface oxidation, glowing combustion. Estimates of rate constants, according to eq. (5), are B2 = 10 ~2 s- and Eb2 = 40 kcal/mole for k2, the char process, and B 1 = 1 0 1 7 s - 1 and Ebl = 53kcal/mole for kl, the tar process.*

A reasonable mechanism has been suggested for the tar-production path. 2° The yield of levoglucosan is so high that probably some sort of an unzipping process is indicated. It has been proposed that the chain may be initiated either by random scission or by end- initiation, through attacks by a hydroxyl group, OH, one of which is attached to the C atom at the end of each chain. After the monomer breaks off, propagation could be sustained by the free oxygen bond. It is the reason for the monomer appearing as levoglucosan which requires explanation. A proposed model for this process is a two-step attack, 2° viz.,

0

/~Ce l lu lose C H/I \H

C I

OH

H2COH f

H \ C H ~ - ' ~ 0 ~ C \ d Cellulose

yf C C I I

H OH

-O

o\ C ~ + H Cellulose , / H

C I

OH

* These values are approximations to those of A. Broido, reported in "Kinetics of Solid-Phase Cellulose Pyrolysis", (see Thermal Uses and Properties of Carbohydrates and Lignins (F. Shafizadeh, K. V. Sarkanen and D. A. Tillman, eds.), Academic Press, New York, 1976).

326 F.A. WILLIAMS

first by the oxygen radical and next by the hydroxyl. The final molecule shown is levoglucosan (fl-glucosan or 1,6 anhydroglucose). The first step is endothermic and the second exothermic, releasing less heat than is required for the first step.

For the dehydration process, it has been reasoned 2° that an out-of-plane, interrnolecular interaction must be the cause. The hydroxyl in an H2COH group of one chain can attack the carbon-oxygen linkage of an adjacent chain, breaking that chain in such a way that half of it is linked to the attacking chain while the other half gives up H20 in forming a stable end- group. Hypotheses for the mechanism of the further decomposition toward char through production of HzO and CO have also been developed. 2° Thus, the dominant features of the pyrolysis of pure cellulose can be understood self-consistently.

Although cellulose is the major constituent of cellulosics such as natural woods, there are other important constituents, notably hemicellulose and, typically in somewhat lower concentration, lignin. 22 These materials have less regular structures than cellulose and show more complex behavior upon pyrolysis. Even cellulose has a macrostructure, exhibiting amor- phous regions and more regular crystalline segments. This macrostructure may affect pyrolysis behavior. Small amounts of inorganic constituents also have measurable influences on pyrolysis. Therefore the overall kinetics of thermal degradation of natural cellulosics vary. Nevertheless, the pyrolysis properties of cellulose always exert an influence on the rates of breakdown of cellulosics subjected to heat, and cellulose provides the best model currently available for these natural substances with respect to their pyrolysis kinetics.

5. CHEMICAL KINETICS OF COMBUSTION

The mechanisms of gas-phase reactions occurring in fires may be discussed by reference to the burning of a candle, illustrated in Fig. 2. The hydrocarbon fuel (wax) vaporizes from the wick under the influence of the heat from the flame. The dark region is fuel rich with insufficient oxygen for appreciable oxidation. The blue is characteristic of the burning zone where gaseous fuel meets oxygen; the blue colour is chemiluminescent, not thermal or equilibrium radiation but rather nonequilibrium radiation from species that have achieved excited states through the chemical reactions of combustion. The yellow is mostly equilibrium radiation from fine, hot soot particles that may be burning with oxygen; the soot has been formed by pyrolysis of fuel gases. Chemical processes that occur in the blue flame have been subjected to detailed investigation.

5.1. Mechanisms and Rates in Methane Flames

Combustion reactions fundamentally are chain reactions involving many elementary steps. Each step proceeds at a rate proportional to the product of the concentrations c (moles/vol.) of the colliding

- - - WICK

W A X

FIG. 2. Schematic illustration of burning candle.

reactant molecules. For example, for A + B ~ p r o - ducts, the rate co (moles of A consumed/vol, s) is co = kCACB, where the rate constant k may be given by an expression like eq. (5). Table 8, taken largely from Ref. 18, lists approximate rate constants for a few elementary steps.

The species CH 3 and H are radicals that serve as chain carriers. The first two reactions in Table 8 are representative initiation steps, with M denoting any stable molecule. In established flames these steps may be relatively unimportant since radicals H, O and OH may reach the fuel molecules by diffusion and consume them more rapidly by propagation steps such as 3, 4 and 5. It is known that formaldehyde, H2CO , plays a role in hydrocarbon oxidation, and step 6 is a potential means for producing it. Steps 7 and 8 describe a path for production of CO through the formyl radical (HCO). Oxidation of CO to CO 2 occurs by step 9, which may proceed more slowly than other steps, leaving unburnt CO if reactions are quenched by rapid cooling. Steps 10 through 13 are part of the chain mechanism for hydrogen oxidation and are quite relevant to hydrocarbon oxidation. The last reaction listed is a representative termination step, involving three-body collisions and having a rate proportional to the product of the concentrations of the three reactants.

5.2. Simplified Rate Expressions

Many steps not shown in Table 8 are known to occur in methane oxidation. Gas-phase oxidations of other fuels involve many additional steps as well. Knowledge of rates of elementary steps and computer

Urban and wildland fire phenomenology

TABLE 8. A few rate constants for reaction steps

Reaction k-Rate constant*

1. C H 4 + M ~ C H a + H + M 2. CH4+O2--*CH3 + HO2 3. CH4+O~CH3 +OH 4. C H 4 + H ~ C H 3 + H 2 5. CH4+OH~CHa +H20 6. CH3 +O~H2 CO +H 7. H2CO + O H ~ H C O + H 2 0 8. H C O + O H ~ C O +H20 9. C O + O H ~ C O 2 + H

10. H + O 2 - , O H + O It. O + H 2 ~ O H + H 12. O + H 2 0 ~ 2 O H 13. H + H 2 0 ~ H 2 +OH 14. H + O H + M ~ H 2 0 + M

1.5 x 1019 exp ( - 100,600/RT) 1.0 x 1014 exp ( - 45,400/RT) 1.7 × l0 la exp ( - 8,760/RT) 6.3 x 10 la exp ( - 12,700/RT) 2.8 x 1013 exp ( - 5,000/RT) 1.3 x 1014exp ( - 2,000/RT) 2.3 x 1013exp(- 1,570/RT) 1.0 x 1014 3.1 x 1011 exp ( - 600/RT) 2.2 × 1014 exp ( - 16,600/RT) 4.0 × 1014 exp ( - 9,460/RT) 8.4 × 10X'~exp ( - 18,240/RT) 1.0 x 10X4exp ( - 20,400/RT) 2.0x 10 ~ T -l**

* Units are cm3/mole s. ** Units are cm6/mole2s for k and K for T.

327

capacities are becoming sufficient to enable computations of histories of chemical conversions to be made with full chemistry for most fuels. However, for many purposes it is helpful to have simplified expressions for overall rates of heat release involving a small number of lumped steps that are not elementary, e.g. expressions corresponding to two overall steps, first combustion of fuel to CO and H 2 0 then oxidation of CO to CO, . Overall rate parameters for such simplified descriptions are becoming available (e.g. Ref. 23).

For many purposes, a one-step approximation to the complex chemistry is sufficient. The molar rate of consumption of fuel F by oxidizer O is represented, for example, as

dcF/dt = - w = - c F c o B e x p [ - E / ( R T ) ] , (7)

in which the overall activation energy E and the overall prefactor B are constants. Over a sufficiently limited range of conditions, a representation of the type shown in eq. (7) often is acceptable.

5.3. Chemical Equilibrium

There are situations in fires under which chemical rates for combustion need not be considered at all because, in a first approximation, chemical equilibrium is attained locally at each point in the gas. These situations may occur only in nonpremixed systems (systems in which the fuel and air are not mixed prior to burning), often termed diffusion flames since burning then involves diffusion of fuel and oxidizer toward each other. They cannot occur every- where in premixed systems (systems in which fuel and oxidizer are mixed at a molecular level) because the equilibrium state involves negligible concentrations of either fuel or oxidizer. The system illustrated in Fig. 2 is nonpremixed and therefore subject to approximation by chemical equilibrium; in fact, most fires involve diffusion flames.

At chemical equilibrium for a reaction step, the forward rate equals the rate of the backward reaction (defined by reversing the arrow, e.g. in Table 8).

Complete chemical equilibrium would involve equilibrium for every step, a condit ion seldom achieved. However, equilibrium often is a good approximation for certain steps involving major species such as H20, CO2 and CO. Equating forward and backward rates results in a relationship between concentrations and temperature for equilibrium (see Ref. 18, for example) that involves an equilibrium constant, K c = kl/kb, where k s and k b are the previously defined rate constants for the forward and backward elementary steps. Combining such equilibrium equations with equations for element conservation (stating that chemical elements are neither created nor destroyed in chemical reactions) and for energy conservation results in expressions for temperature and for concentrations of major species as functions of a local mixture ratio (total local concentrat ion of an element contained in the fuel, divided by total local concentration of the element oxygen) in diffusion flames. These expressions often are obeyed, in a rough approximation, in fires.

5.4. An Example of Diffusion-Flame Structure

These ideas of chemical equilibrium help to explain some major observed characteristics of diffusion flames. The shape of the blue flame in Fig. 2 causes it to be difficult to probe. Measurements are easier to perform in flat diffusion flames, which may be established with the apparatus illustrated in Fig. 3. 24 A liquid fuel is contained in a pool (shaded), and an oxidizing gas stream is directed downward onto the surface of the liquid. When the fuel is ignited, conditions can be adjusted so that a flat flame remains stationary a few millimeters above the surface of the fuel, as illustrated. Quantities vary only in the vertical direction, and the flame structure may be studied by thermocouples and by gas sampling. The liquid fuel may be replaced by a gaseous fuel jet or by a solid fuel.

Representative results for the flame structure in such an apparatus are shown in Fig. 4, for the solid fuel poly(methyl methacrylate). The gas stream had

328 F.A. WILLIAMS

AND

l BAFFLE AI R DUCT I I SANWDVEL ~x~ I~l HE,G.T r m SCREENS CONTROL 1 72~S--S------~-

............ a FUEL

O-RING SEAL ~ ~ j 'r'~ ~/jOVERFLOW DUCT WATER SPRAY ~ " i " / / / / / " / / ~ f~21/[..SJ~ SUCT'OI~I EXHAUST l~J ~____~7~, ~,~L,I SUCTION EXHAUST

RIG C O N T R O L ~ j ~ ! ! ~ I 1 ~ ~

F uWALTERAI, NN ~ ' / / ' / ~ FWLATLERIN IN WATER OUT . i ~ POOL DEPTH CONTROL

lOmm SCALE , ,

FIG. 3. Schematic diagram of diffusion-flame apparatus.

values of the exit velocity U and of the ratio of oxygen mass to total mass in the oxygen-nitrogen stream, Yo2, listed in the figure. There is a two-phase, gas- liquid layer on the order of I mm thick at the surface of the polymer under these burning conditions; the location of the outer edge of this layer is indicated in the figure, as is the location of the center of the luminous blue zone, whose thickness is less than 1 mm.

The monomer, methyl methacrylate, has the chemical formula CsH802 and is the major species liberated in polymer pyrolysis (see Table 6). It is seen from Fig. 4 that this is the major fuel present at the outer edge of the dispersed layer. This material diffuses into the blue zone from below, while oxygen diffuses into the blue zone from above. The heat release is greatest in the center of the blue zone, where these two species meet, as may be seen by the occurrence of the peak in the temperature profile at the center of this zone. The concentrations of the major products CO 2 and H20 also peak near the center of the blue zone, and these diffuse away on each side of this zone. Nitrogen, which does not participate in the reaction, exhibits no distinctive behavior at the blue zone but instead

gently diffuses toward the fuel surface from the oxidizing stream.

This behavior of the main constituents is roughly consistent with the ideas of chemical equilibrium. The mixture ratio, measured on the basis of the ratio of carbon to oxygen or of hydrogen to oxygen, decreases as the distance from the polymer surface increases. If equilibrium calculations are made of temperatures and of concentrations of 02, N2, C O 2 and H20 at each point on the basis of the local mixture ratio, then at least qualitative agreement with measurements is obtained. There are quantitative discrepancies; for example the flame temperature is nearly 500K below the theoretical flame temperature. The magnitudes of these discrepancies are indicative of the extent to which departures from equilibrium occur.

As an extreme idealization, it may be considered that there is essentially no 02 on the fuel side of a sheet of negligible thickness located at zero, the center of the blue zone, and that there are essentially no fuel species (CsHsO2, CO, HE, etc.) present on the oxygen side of this sheet. This "flame-sheet" approximation is useful conceptually as well as for approximate burning- rate calculations, even though the information in Fig.

POLY ( METHYL METHACRYLATE)

N 2 in 0 2

Yo2 =0.178

U = O.315m/s

I-- Z w ,,y w O_

w J O

N O

O

-r"

18

16

14

12 z

r r w 0_

IO t.d ._1 0 ~E

8 1-

d

O 6 ~

4

9 I,-.-

Z w

8 ~ a .

._1 o

7 ~ "r"

m

re T 6 ~ &

I (.) ,i

I

T re 4 ~

T re

U

3 r, - r

t~ U

T

2 a u

T

U

d T

" r


300

500

~00

~.00

)00

;00

O0

O0

O0

I 0 I 2

DISTANCE FROM LUMINOUS FLAME ZONE (mm)

9O

8O I--- Z h i

,n," W

70 o_ W J O

N 60 z

50

FIG. 4. Representative concentration and temperature profiles in a diffusion flame.

4 shows clearly that it is not very accurate in detail. The flame-sheet approximation is a limiting form of the equilibrium approximation.

The many species shown in lesser concentrations in Fig. 4, primarily on the fuel side, are not at all consistent with chemical equilibrium. In addition to the product CO of partial oxidation, these species include the gaseous fuels hydrogen, methane, ethane, propane, ethylene (C2H4) , acetylene (C2H2) , propylene (C3H6) , allene (CH 2 = C = CH2), propyne (CH3C = CH) and formaldehyde (HCHO). These latter species must be produced by finite-rate chemical processes. They are in no way representative of the species expected from combust ion kinetics, such as those discussed in Sec- tion 5.1", and they extend well beyond the blue reaction zone. Instead, they are formed by pyrolysis of the secondary (gaseous) fuel C 5 H802 .

* More sophisticated experimental techniques are needed to measure most of the nonequilibrium species of the combustion kinetics.

The pyrolysis of gaseous fuel proceeds in the dark fuel-rich zone between the fuel surface and the blue zone. Occurrence of the gaseous fuel species observed, rather than other fuel species, can be understood from concepts of kinetic mechanisms of pyrolysis of C 5 H 8 0 2 .24 It is seen that many of the fuel species produced in dark-zone pyrolysis have higher ratios of carbon to hydrogen than the parent fuel.

5.5. Kinetics of Gaseous Fuel Pyrolysis

Numerous chemical reactions occur in the dark pyrolysis zone containing gaseous fuel. These reactions are complex and differ for different fuels; they are not understood thoroughly. 25 If allowed to proceed for a sufficient length of time, they result in production of soot. In the experiment of Fig. 4 there is insufficient residence time in the fuel-rich zone for this to occur. However, in Fig. 2 there is sufficient time, and the soot becomes visible as the yellow zone of the flame. The soot also burns and finally is consumed completely at the upper boundary of the yellow region.

330 F.A. WILLIAMS

It would be helpful to have rough overall rate expressions like eq. (4) or (7) for soot production. Overall rate constants for such expressions are not yet available.

In Fig. 2 some premixing of oxygen with fuel gases may occur at the base of the blue zone. The extent of this premixing is not yet known. The presence of oxygen is known to increase the overall rates of fuel pyrolysis; the term "oxygen-catalyzed pyrolysis" de- scribes this. The percentage of oxygen in the dark zone of Fig. 2 is likely to be greater than that of Fig. 4. Therefore the overall rate of soot production probably is different in the two configurations. Oxygen may aid in soot burnup in the yellow zone; details of this process are poorly understood even though an appreciable amount of information is available on the combustion of individual carbonaceous particles like soot particles.

At the base of the blue zone in Fig. 2 there may well be enough oxygen entrainment for an appreciable amount of premixed combustion to occur. The extent of premixed burning has not been well defined. Oxygen entrainment is enhanced by the buoyant rise of the hot gases. Buoyancy effects are very significant in fires.

6. BUOYANT CONVECTION

Attention thus far has been focused largely on

chemical aspects of fires. Fluid-mechanical aspects, to be considered now, are equally important. Fluid flow influences fires in many ways. An important effect occurs through the buoyancy of the hot gases produced in fires. This buoyancy causes the gases to rise, producing columns of buoyant convection above fires. The rise generates inflow at the base of the fire, thereby bringing fresh oxygen to the flames. Thus, buoyancy may be responsible for both intensified burning and long-range influences in fires.

6.1. Plumes and Thermals

Books, such as Ref. 26, are available on buoyant convection in fluids. It is helpful to identify limiting behaviors of steady-state and transient convection. A fire burning for a sufficiently long time generates a steady convection column, called a buoyant plume. A representative analysis of such plumes for fires may be found in Ref. 27. Alternatively, combustible gases may be released in a very short time and ignited. The transiently rising cloud of hot gases may be called a thermal, in general, or a "fireball" if intense combustion with radiant energy output persists in the cloud. An analysis of such thermals with combustion may be found in Ref. 28.

6.2. Conservation Laws for Plumes

Descriptions of plume structure rest on equations

~z ("'bL"w)1 z +dz

\ (,,'b'e*) Iz [ zooov (z,,-baz) \ / \ /

\ C "°' /

\ \ /

,o\ // jVIRTUAL

ORIGIN

FIG. 5. Schematic diagram of buoyant plume.


for mass, momentum and energy conservation of fluids. These equations may be introduced simply for vertical, axisymmetric plumes by adopting the approximation of a "top-hat" structure, namely treat- ing the density p and vertical velocity w as constant over a plume cross-section of radius b, as illustrated in Fig. 5.

Let z denote the vertical coordinate, p~ the density outside the plume and v the inward velocity in the horizontal direction at the edge of the plume, the so-called "entrainment velocity". Then consideration of mass flows into a slice of the plume of thickness dz, illustrated in Fig. 5, shows that conservation of mass may be written as

d(z~b2pw) = 2~bpo~v. (8)

To write an equation for energy conservation, it is simplest conceptually to measure energies from a zero level defined as that of the gas outside the plume. Then entrainment brings no energy into the plume, and in the absence of energy losses from the plume (such as by radiation) the vertical flow of energy must remain constant, independent of z. Often this energy flow occurs predominantly by convection, and it is given by 7rb2pwcp(T - To~) if the enthalpy with respect to that of the surroundings is cp(T-To~), where c~ denotes the specific heat at constant pressure (assumed constant) for the gas. The pressure p is nearly constant over the cross-section of the plume, and if the average molecular weight W is also constant, then the ideal gas law,

p = p R T / W , (9)

shows that the product p T is constant over a cross- section, i.e. T - T~ = T ~ [ ( p ~ / p ) - 1]. The quantity

F = 27zb2wg(po~ - P)/Po~, (10)

where 9 denotes the acceleration of gravity, is com- monly termed the buoyancy parameter in plume studies. Since the vertical convection of energy is seen from the above reasoning to be given by the product of F with the constant factor poocpToo/(29), the con- stancy of this convection implies that F remains constant. The constant product p®F is the total weight deficiency per second produced by the source of the plume.

In writing an equation for momentum conservation, it may be assumed that in a first approximation, buoyancy and inertial forces maintain a balance. This assumption neglects viscous forces, for example, which are important for small plumes, on the order of one centimeter or less in diameter. Consideration of the inertial and buoyant forces for a slice of the plume of thickness dz results in

~--~ (7cb2 p w2) = Itb2(p oo - P )9 (11)

for momentum conservation. Equations (8), (10) and (11) (with F constant) comprise a suitable set of conservation equations for buoyant plumes. Since plumes

of interest usually are sufficiently large for flows to be turbulent, variables such as w and p in these equations must be interpreted as averages of some sort.

6.3. Turbulent Entrainment

An expression for the entrainment velocity v is needed for use in eq. (8). There is experimental evi- dence 26 that under turbulent conditions with small fractional differences between p and p~, v = ~w, where the entrainment constant ~ typically is of order 0.1. Entrainment involves engulfment of external fluid by turbulent eddies. For large density differences there is some experimental support for the theoretically con-

venient expression ~ = ~ , where c% ~ 0.1 (see Ref. 27, for example). Thus, the fight-hand side of eq.

(8) is approximately 2 n b ~ o ~ w x / ~ . This may be viewed as use of the geometric mean for density in calculation of entrainment rates.

6.4. Plume Structure

Equations (8), (10) and (11) possess solutions in

which ~ b , w and T - T o vary as powers of z, provided that the entrainment approximation just

introduced is employed. Specifically, x / -p /~b =

~ b o ( z / z o ) , where the subscript 0 refers to conditions at the distributed source of buoyancy, usually ground level, and z is measured from a virtual point-source below the ground, as illustrated in Fig. 5. This gives a linear law for plume spread under conditions of small density differences, and it has experimental support. The corresponding variations of velocity and density are given by w = Wo(Z/Zo) -1/3 and (Po~/P)- 1 = [(Po~o/Po)- 1](Z/Zo)-5/3; entrainment and spreading lead to decreases of velocity and of density difference with increasing height.

Use of these results in eq. (8) provides an expression for the distance to the virtual origin, z0 =

(5/6)~fpo/P~o(bo/~) . In terms of the height h above the ground, z = z o + h in the previous formulas. Sub- stitution of the power-law variations into the ratio of eq. (11) to eq. (8) provides, by use ofeq. (10), the relationship F 0 = (16/5) ~ctobo w3 for the buoyancy parameter. In fires, typically P~o/Po "~ 4, ct o ~ 0.05, and w 0 may be estimated as a buoyancy velocity, w o {[(Po~o/Po)- 1]boy} I/2, where 9 ~ 103cm/s2- These formulas then enable plume structures above the flames to be calculated explicitly, as functions of h, for any given ground radius b o of a fire. Structures within the fire core are more complex because of ground influences, partial heat release, radiative energy transfer, etc. 6

6.5. Line Fires and Wind Effects

A sufficiently long line fire in a still atmosphere may have a nearly planar, two-dimensional plume above it. Analogous approaches to the description of such plumes are available. 26

In a wind the plume tends to bend in the direction that the wind is blowing. Estimates of bending may be obtained from a momentum balance in the horizontal

332 F.A. WILLIAMS

direction for a control volume consisting of a horizontal slice of the plume. Information on wind effects and on effects of other phenomena, such as atmospheric stability, may be found in Ref. 26, for example.

6.6. Fire Whir l s

Rotational motion in the atmosphere around a fire may have a profound influence on the fire plume. The inward entrainment velocity transports ambient fluid toward the centerline of the plume. If dissipation is negligible then angular momentum is conserved during this inward transport. The angular momentum per unit mass about the centerline is ur, where u is the rotational component of velocity and r the radial distance from the centerline. With ur constant, as r decreases for a fluid element drawn into the plume, u increases such that u 2 = u~(rl /r2), where subscripts 1 and 2 identify conditions far from and near the centerline. Small rotational velocities in the atmosphere thereby result in large rotational velocities in the plume as the gas spirals inward. This process underlies the occurrence of fire whirls, defined in Section 1.4.

Many natural phenomena fundamentally possess this same mechanism for enhancement of rotational velocities. These include dust devils in deserts, water- spouts over oceans and tornadoes. Intense fire tornadoes, often elements of fire storms, can be strong enough to uproot trees and destroy structures. Weaker whirls in which rotational influences are evident only in smoke in the plume above the fire often are called smoke whirls.

A fundamental investigation of fire whirls is reported in Ref. 29. In laboratory-size whirls, rotation can produce increases in flame heights by a factor of ten and increases in burning rates (rates of consumption of solid or liquid fuels on the ground) by a factor of five. The latter effect arises from increased rates of heat transfer, produced by the increased velocities, and results in an increased buoyancy parameter F for the plume.

Plume analyses can be generalized to include rotational effects. 29 An idea of rotational velocities achieved in fire whirls can be obtained by modeling the rotation as a simple vortex. In the center of the vortex, viscous dissipation damps rotation and causes velocities to behave like those in solid-body rotation, viz., u = fir, where f is the constant angular velocity. The angular velocity peaks at the outer edge of this viscous-core region, and outside the core inviscid conditions are approached, with the angular velocity behaving as predicted by conservation of angular momentum.

To obtain the simplest approximation to the structure of the fire vortex, assume that the viscous core has radius b, the plume radius, that u ~ r for r < b and that u ~ 1/r for r > b. Then continuity of u at r --- b requires that, with u = f r for r < b, the formula u = f b Z / r must hold for r > b. The peak rotational velocity then ocurs at the edge of the plume and is given by fib. The circulation about the plume in the ambient atmosphere is defined as the line integral of

the rotational velocity around a circle centered at the plume axis and is given by F = 2~ru. Use of the inviscid formula for u here gives F = 2~b2f. Hence, in terms of the ambient circulation F, the peak rotational velocity is F/(2~b). This expression may be used to estimate maximum velocities in fire whirls and smoke whirls.

In addition to b, which may be obtained from plume theory and may be estimated to be on the order of the size of the fire (perhaps somewhat less if rotation produces contraction of the plume), it is necessary to know the circulation F to obtain the peak velocity. In very large fires, F might be thought to be attributable to weather patterns. Seldom if ever are fires sufficiently large for the F of the weather to be relevant. Usually F arises from interaction of winds with local topography and may be estimated as the product of a wind-velocity difference with the distance over which this difference occurs. Sometimes the plume of a large fire can replace the topography as the obstacle that interacts with the wind. A separation of flow on the leeward side of the large plume can generate a pair of counter-rotating vortices that are intensified into fire whirls and smoke whirls just downwind from the main fire. In this mechanism, F is on the order of the product of the wind velocity with the diameter of the large fire. Whirls by this and related mechanisms are common especially in large woodland fires.

7. CONVECTIVE HEAT TRANSFER

Increased burning rates in fire whirls are thought to occur through increased rates of convective heat transfer to condensed fuels. The general topic of heat transfer is highly relevant to fire phenomenology, as indicated in Section 2.3. Three modes of heat transfer conventionally are identified as conduction, convection and radiation. The last two are of predominant importance in fires. The first, however, is essential to heat transmission through solids and also is relevant to underlying concepts of convective heat transfer. Therefore it is appropriate to initiate discussion of convective heat transfer by consideration of conduction.

7.1. Conduct ion

The Fourier law of heat conduction states that the energy per unit area per second transmitted across a thin layer of thickness & is q = 2AT~& where 2 denotes the thermal conductivity of the medium and AT is the difference in temperature across the layer. The heat is transmitted in the direction of decreasing temperature. The formula requires 6 to be sufficiently small for the temperature to vary linearly with distance through the layer. Typical values of 2 for gases in fire problems are on the order 10-4cal/cm sK.

Use of the Fourier law to estimate q requires knowledge of the value of & that corresponds to a known temperature difference AT between bound- aries. For steady-state heat conduction in a solid, &


typically is on the order of a characteris t ic body d imens ion E. For heat t ransfer from a gas to a solid, often flow abou t the body causes 6 to be appreciably less than #, since a bounda ry layer may then develop in the gas adjacent to the body, with all of the tempera ture change occurr ing across the bounda ry layer. Use of the Four ier law then entails ob ta in ing the boundary- layer thickness 6.

7.2. Forced Convection

When the gas flow is associated with externally imposed velocities, such as the wind velocity, the m a n n e r in which the gas t r anspor t s the rmal energy is termed forced convection. For forced convection, the rat io 6/¢ depends on the Reynolds number , Re = v(/v, the rat io of dynamic to viscous forces. Here v is a representat ive external flow velocity and v is the kinemat ic viscosity of the gas. Typical values of v are 10-1 cm2/s at room tempera ture and 1 cmZ/s at flame tempera ture ; the value 1 cm2/s often is useful for v in fires.

Heat t ransfer predict ions usually are expressed in terms of a heat- t ransfer coefficient, defined as q/AT (i.e. 2/6), or in terms of a Nussel t n u m b e r Nu--- (q/AT)({/2) = {/6, instead of directly in terms of 6. The heat flux q may be calculated from Nu according to the formula

q = N u 2 A T / ( . (12)

Heat- t ransfer correla t ions for forced convect ion are expressed in formulas for Nu as a funct ion of Re. These formulas essentially provide ~/c5 in terms of Re.

Often an addi t ional nond imens iona l pa ramete r appears in heat- t ransfer correlations, the P rand t l number , Pr = v/ct, where c~ -= 2/(pcv) is the thermal diffusivity of the gas. This pa ramete r measures the relative ease with which m o m e n t u m and heat may be transferred by the fluid. It is of order uni ty for gases,

approximate ly 0.7 for air; reasonable est imates in fire problems are ob ta ined from corre la t ion formulas by simply put t ing Pr = 1.

There are many sources of heat- t ransfer corre la t ion formulas. 17.30 A short , simplified compi la t ion is given in Table 9. Use of Table 9 a long with the formula for q in terms of Nu enables heat fluxes to be calculated.

TABLE 9. Simplified heat-transfer correlations

Forced convection

1. Laminar flow parallel to a fiat plate of length f (20 < Re < 3 x 10 5 ) Nu = 0.66Re 1/2

2. Turbulent flow parallel to a flat plate of length E (Re > 3 × l0 s) Nu = 0.037 Re 4/5

3. Laminar flow normal to a strip of width # (20 < Re < 3 x 105) Nu = 0.57Re 1/2

4. Flow around a sphere of diameter f (17 < Re < 7 × 104) Nu = 0.37 Re 3j5

5. Flow around an infinite cylinder of diameter f (1 < Re < 3 x 105) Nu = 0.35+0.47Re x/2

Free convection

6. Laminar free convection on a vertical fiat plate of length g" (104 < Gr < 10 9)

7. Turbulent free convection on a vertical fiat plate of length (Gr- > 10 9)

8. Laminar free convection on the top surface of a heated horizontal plate of length t (105 < G r < 2 x 1 0 7 )

9. Turbulent free convection on the top surface of a heated horizontal plate of length E (2 x 107 < Gr < 3 x 101°)

10. Laminar free convection on the bottom surface of a heated horizontal plate of length t ~ (3 x 105 < Gr < 3 x 101° )

11. Laminar free convection around a heated horizontal cylinder of diameter ( (103 < Gr < 10 9)

12. Free convection around a sphere of diameter E (3x 108 < Gr < 5x 1011)

13. Laminar free convection around a sphere of diameter (Gr < 200)

Nu = 0.59 Gr TM

Nu = 0.13 Gr 1/3

Nu = 0.54 Gr 1/4

Nu = 0.14Gr 1/3

Nu = 0.27 Gr TM

Nu = 0.52 Gr TM

Nu = 0.098 Gr °'34s

Nu = 2+0.6Gr TM

7.3. Free convection

As discussed in Section 6, gas flow may be produced by buoyancy forces in fires instead of arising from external velocities. T ranspo r t of thermal energy in such flows is termed free convect ion or na tura l convection. In free convect ion, the rat io 6/E depends on the Gra sho f number , Gr, a measure of the rat io of buoyancy forces to viscous forces. The G r a s h o f n u m b e r is G r = g~3flAT/v 2, in which the coefficient of thermal expansion, fl, is simply 1/T for an ideal gas, with T being the absolute t empera tu re (K). Table 9 gives a few corre la t ion formulas for N u as a funct ion of Gr for free convection. Use of these formulas and

eq. (12) for q in terms of Nu enables rates of heat transfer in na tura l convect ion to be estimated.

7.4. Simplified Energy Balances

Heat- t ransfer formulas may be used in energy balances to ob ta in informat ion on various quanti t ies of interest, such as burn ing rates or igni t ion times. Fo r example, for steady gasification of a solid tha t requires an energy L / W per unit mass for vaporizat ion, the gasification rate or burn ing rate m per unit area of the fuel surface may be calculated approximate ly from the energy balance

q = m(L/W), (13)

334 F.A. WILLIAMS

with q given by eq. (12). Equation (13) states that the heat flux into the solid from the gas is that needed for gasification of the solid at the rate m.

In using eq. (12) in (13), AT is the difference between the gas temperature (e.g. a flame temperature) and the surface temperature of the solid, i.e. it is the driving temperature difference for heat transfer. There are many corrections to eq. (13) that may be introduced, e.g. corrections for heat loss from the solid and for reduction in q by the influence of gasification on the convective flow adjacent to the surface. However, eq. (13) without correction often suffices for rough estimates of m.

In transient processes q causes the body temperature to increase with time. If the body is sufficiently small and sufficiently highly conductive to maintain a nearly uniform temperature, then

C V ( T - To) = qAt (14)

expresses an energy balance. Here V and A are the volume and surface area of the body, C is its heat capacity per unit volume, T O is its temperature at time zero and T its temperature at time t. If the body is combustible and is known to ignite at a particular temperature T, then eq. (14) gives the time t required for ignition under the influence of the heat flux q. Like eq. (13), eq. (14) also involves many approximations, such as neglecting the variation of q with time resulting, for example, from the variation of the surface temperature T. Nevertheless, the equation often is useful for estimations.

8. RADIATIVE HEAT TRANSFER

Radiation often is the dominant mode of heat transfer in fires. There are a number of relevant sources of information on radiative heat transfer. These include Refs. 6, 31, 32 and 33. Radiation may be emitted by hot surfaces as well as by flames. These two sources will be considered separately after a rough general formula for the radiant heat flux is given.

8.1. Radiant Heat Flux

The radiant heat flux may be considered to depend on the temperature T of the emitting material and on its emissivity 5 ~< 1. The formula

q = e~T 4 (15)

may be used, where a is the Stefan-Boltzmann constant and has the value 1.36/(1000) 4 cal/cm2s K, such that for 5 = 1 the radiant flux is 1.36cal/cm2s at a temperature of 1000K. Use of eq. (15) to calculate q necessitates knowledge of T and 5 for the emitter.

Equation (15) constitutes a simplification in numerous respects. The flux q is the sum of fluxes of all wavelengths of radiation, and different wavelengths may behave differently; for example, radiation of some wavelengths may be absorbed by materials while that of other wavelengths is transmitted or reflected. Black-body radiation has a particular wavelength distribution with a peak at a wavelength that depends upon T. These wavelength effects can be

important in fires; judicious selection of wavelength ranges is desirable if eq. (15) is employed. In addition to emission, radiation experiences absorption and scattering in propagating through media. Estimates of these effects must be made in calculating radiant fluxes arriving at surfaces.

8.2. Radiation to and from Surfaces

Equation (15) is interpreted most easily for emission by hot surfaces. Surface emissivities e for most materials of interest in fires usually lie in the range 0.5 ~ e ,%< 1. Often 5 = 1 is a good approximation (e.g. for wood), although for certain surfaces (e.g. for some polished metals), 5 may fall into the range 0.01 < 5 <,% 0.1. With information on 5 and on the surface temperature, it is straightforward to use eq. (15) to calculate radiant energy fluxes from surfaces. In comparison with flames, surfaces of burning fuels tend to have lower values of T but larger values of 5, often resulting in comparable values for q.

If radiation is incident on the surface of a material that is sufficiently thick to transmit a negligible amount of radiation, then a fraction of the incident radiation may be considered to be absorbed by the surface and the remainder reflected. With certain idealizations, the fraction absorbed is found to be 5, i.e. the absorbtivity equals the emissivity. In using an energy balance like eq. (13) under conditions such that radiant transfer is important, E times the incident radiant energy flux must be added to the convective q, and the surface emission, eq. (15), subtracted.

8.3. Flame Radiation

Flames in sufficiently small fires often are optically thin, in that if radiation is directed toward the flames, most of the incident radiation emerges from the other side. Flames, like gas (as well as other materials) in general, may be characterized by an absorption length, f,, for radiation, defined as the thickness of the gas, in a planar, uniform configuration, required for producing a specified fractional decrease in the incident radiant energy flux emerging from the opposite side of the layer. A Beer's-law approximation can often be justified, i.e. the ratio of the radiant flux leaving to that incident is e -~/~o, where ~ is the thickness of the layer. The value of E a depends on pressure, temperature, chemical composition and the wavelength distribution of the incident radiation. Typical values of{ a for flames lie in the range 10cm to 50 cm, but much smaller and much larger values also may be encountered.

The emissivity e of a flame is related to its thickness E I and its absorption length t',. The rough approximation

5=~Es/E~ for ~ f < ( ~ (16) for ¢s > {,

may be employed for estimates. With this formula, radiant fluxes emitted by flames can be calculated by use of eq. (15) if T, {s and {o for the flames can be estimated.


Wavelength dependences of flame radiation are complex. For nonsooting fires, the radiation is mainly chemiluminescent, i.e. associated with the chemical reactions occurring, and it is far from black-body radiation. For flames containing fine soot particles, ~a is much smaller (e is larger), and the radiation is approximately of the black-body type, emanating from the hot surfaces of the soot particles. In this latter case, /~ depends on the number density, size and temperature of the soot particles, which in turn depends on chemical kinetics in the flames. Thus, fundamental predictions of radiant fluxes from flames are difficult to obtain. Estimates by use ofeq. (16) with guessed values of / , usually must suffice, unless measurements are available.

Sufficiently large and sooty flames become optically thick, in that a negligible amount of radiation incident upon them is transmitted through. For such flames,

= 1, but this does not simplify eq. (15) appreciably because the temperature T at which emission occurs becomes more difficult to estimate. The radiation is emitted from the edges of the flames, where the smoke temperature is less than the flame temperature. Effec- tive smoke temperatures for emission must be estimated in calculating radiant fluxes from optically thick flames. These temperatures tend to be within a few hundred degrees of 1000K.

8.4. View Factors

In calculating radiant heat transfer, it must be realized that, unlike convective heat flux, radiant flux tends to travel large distances in straight lines. Associated with this travel from a source of finite extent is a decrease in intensity of the radiant flux, according to the inverse-square law. Thus, if radiation is emitted from a sphere of radius rl, the flux at a concentric spherical surface at radius r2 is q(rl/r2) 2, where q is given by eq. (15) for the emitter. If the normal to the receiving surface makes an angle 0 with the vector from the emitter to the receiver, then the additional factor cos 0 appears in the formula for the normal radiant flux incident on the surface of the receiver. These geometrical effects must be taken into account in calculating radiant energy transfer.

The geometrical effects conventionally are included by introducing a view factor F. Tables and graphs of view factors are available, for example, in Ref. 6. For a surface element receiving radiation from a fire, the view factor may be defined by letting 2~F be the solid angle subtended by the fire area, as seen from the receiver. Then F is the fraction of the total view of the receiver occupied by the fire. The normal radiant flux incident on a surface pointed toward the fire is then qF, where q is given by eq. (15) for the fire. Estimations of heat fluxes in fire environments make extensive use of these geometrical considerations.

9. BURNING RATES

Knowledge of rates of heat transfer enables estimates to be made of burning rates of fuels in fires.

For a given fuel object, a burning rate may be defined as the mass of fuel consumed per second, which will be denoted here by -dM/dt. Although it often is possible to relate dM/dt to a heat flux q even for gaseous fuels, attention here will be restricted to liquid and solid fuels, which are more prevalent in fire problems. Such condensed fuels in fires typically have reasonably well defined surface areas over which gasification is taking place. If a characteristic length { of the fuel object is defined in such a way that this surface area of gasification equals ~2, then the formula

dM/dt = - m r 2'' (17)

relates the rate of .mass loss to the burning rate per unit surface area, m. Both m and - dM/dt are variously termed burning rates; it is important to ascertain the units from the context. Methods for calculating these burning rates are addressed here.

9.1. The Feedback Principle

In Section 2.3 it was indicated that usually feedback of energy from flames is responsible for fuel gasification. The manner in which this feedback may be employed to calculate burning rates is indicated by eq. (13), developed in Section 7.4. If the feedback energy flux q is known, then m may be obtained from eq. (13), after which -dM/dt may be found from eq. (17).

In fires, often q itself is determined by m through the influence of the burning mechanism on rates of gas- phase heat transfer. In such situations there is another relationship between q and m, in addition to eq. (13), and the simultaneous satisfaction of these two relationships may be viewed as specifying geparately a burning rate and a feedback flux for the fire. The existence of a feedback q that depends on the response m of the system may lead to distinctive dynamical behavior, such as the existence of more than one steady burning condition. The presence of an additional relationship giving a fire-dependent influence of m upon q may be said to complete a feedback principle for the fire.

The feedback is illustrated schematically in Fig. 6. The line labeled "energy required" is that defined by a formula like eq. (13), according to which it should be a straight line with slope L/W. Curvature has been included in this line in the figure to suggest influences of conductive distribution away of heat at low energy fluxes and of heat losses from the fuel by radiation and convection at high energy fluxes.

The solid lines labeled "energy provided" are representative of dependences of q upon m in fires. These dependences arise through heat-transfer formulas such as those given in eqs. (12) and (15). For example, in eq. (12) Nu may vary with Re according to an entry in Table 9, and the velocity v in Re may increase as m increases if the flow field is influenced by the burning rate, e.g. through plume effects described in Section 6.2.

The tendency for q to decrease with increasing m at large values of m, illustrated by the lower of the two solid lines labeled "energy provided" in Fig. 6,

336 F.A. WILLIAMS

INCREASING RADIATION

STEADY l ,¢,ENERGY FLUX'x

I / / i / "------ENERGY

/ / 'i I / / _ OV,DED I / / / ,,' FOR MULTIPLE ~ / " SI'EAD i STATES

STEADY BURNING - RATE

m (g/cm z s)

FIG. 6. Illustration of feedback energy fluxes in fires.

typically arises from mass-transfer corrections to convective energy fluxes as caused by fuel gasification. These corrections, mentioned in Section 7.4, have not been given here but may be found in the literature. 6 Radiant contributions to q typically are less strongly dependent upon m than are convective contributions; they tend to increase with increasing m. The upper of the two solid lines labeled "energy provided" in Fig. 6 is designed to illustrate a possible influence of adding radiation to convection in calculating q.

Intersection of the curves labeled "energy required" and "energy provided" in Fig. 6 defines potential conditions of steady burning. One such intersection is shown in Fig. 6, with the corresponding values of the steady burning rate and of the steady feedback flux indicated.

9.2. Calculation of Steady Burning Rates

Formulas like eq. (13), as illustrated in Fig. 6, may be used to estimate rates of steady burning in many situations. The simplest cases are those in which q as calculated from heat-transfer principles depends to a negligible extent upon m. Such cases arise if the burning rate of a particular object is to be calculated in a fire environment dominated by other burning objects; the quantities influencing q are first calculated from knowledge of the fire environment, then a steady-state energy balance is employed to find m for the object in question. From the viewpoint of Fig. 6, these cases have an "energy-provided" curve that is approximately a straight, horizontal line at a particular value of q that is independent of m.

A similar approximation often is useful for obtaining a first estimate of the total burning rate of all fuels in a fire. There are fires for which, through competing influences, the "energy-provided" curve is nearly a horizontal line-for all values of m greater than a critical value that corresponds to flame extinction,

and q is essentially zero for values of rn below the extinction value. These characteristics have been discussed in Ref. 34, for example. In such situations, the total burning rate at steady-state conditions in the presence of flames may again be calculated by first finding q from heat-transfer considerations then applying an overall energy balance to all of the fuel to obtain m or - dM/dt.

Computations of steady-state burning rates with the feedback flux q known are best based on energy balances more thorough than eq. (13). First of all, it should be recognized that the energy per unit mass required for vaporization, L/W, is composed of two parts, the energy required to heat the fuel to the temperature at which it gasifies and the energy required for gasification. The latter is a heat of vaporization for a liquid or a heat of pyrolysis for a polymer and may be obtained from tables, as discussed in Sections 2.4 and 3.3; it will be denoted by L,/W here. The former is a rise in thermal enthalpy, denoted here by Cp(T~-Ti), where cp is the heat capacity of the solid or liquid fuel (see Section 3.6), T~ is the surface temperature of the fuel during burning and T~ its initial temperature before the fire. Thus,

L/W = cp(T~- ~)+ Ls/W. (18)

If the fuel experiences additional phase changes, such as melting, prior to gasification, then the energy associated with such processes should be added to cp(T~-T~). For composite fuels, such as cellulosics containing water, the contributions of each component must be added properly in calculating L/W.

Use of eq. (18) in conjunction with eq. (13) to calculate m necessitates estimation of a value for T~. For vaporizing liquid fuels, T~ typically is slightly below the boiling point T b (see Section 2.4), and T~ = T b is a reasonable approximation. For pyrolyzing polymeric fuels, T~ fundamentally depends upon m,


TABLE 10. Approximate temperatures of gasification for some polymers in fires

Gasification temperature, Polymer T= (K)

Poly (methyl methacrylate) 660 Polyoxymethylene 630 Polyethylene 720 Polystyrene 710 Cellulose (flaming) 640 Cellulose (glowing only) 870

according to eq. (2). Substitution of eqs. (2) and (18) into eq. (13) would produce a nonlinear equation for m. Usually it is unnecessary to try to solve this nonlinear equation because eq. (2) predicts so rapid a variation of m with T= for reasonable values of E, that in practice T= varies very little during burning (com- pare Fig. 1). In a first approximation, a constant gasification temperature T~ may be ascribed to each polymer, with temperature changes of a few tens of degrees from this value resulting in either negligible rates of gasification or unattainably high rates. Specification of T= for a polymer is tantamount to specification of a particular combination of m, and E, in eq. (2); it is a less precise approximation to pyrolysis kinetics than is eq. (2), with m zero for T < T=, m arbitrary for T = T= and m infinite for T > T=. Approximate values of T= for a few polymers are listed in Table 10. These values should be viewed as rough estimates sufficient for approximate energy balances; more precisely, according to eq. (2), T= increases as m increases.

Attaining improved energy balances, better than eq. (13), also entails taking into account energy losses from the solid or liquid fuel. These losses may include conduction of heat to the surroundings, convective loss to cool gases flowing about hot fuel surfaces and radiative loss from heated surfaces. The latter often is the largest of these, especially for solid fuels, and it may be estimated from eq. (15) with Tset equal to T=. If the sum of these losses per unit surface area of the

burning fuel is denoted by qe, then the improved energy balance is

q = m [ c p ( T = - T ~ ) + L = / W ] + q t , (19)

where eq. (18) has been employed for L / W . Use of eq. (19) in place of eq. (13) provides better values of burning rates.

Additional complications in energy balances may be introduced by the nature of the fuel. Such complications arise primarily for solid fuels. These fuels often have heterogeneous structures with different parts responding differently in fires. Even homogeneous solids may become heterogeneous under the influence of the fire; for example, wood tends to form a char layer at its surface with virgin fuel gasifying beneath. Time-dependent terms may then become desirable to include in the energy balance, to account for char build-up, for example. Methods for taking such effects into account are not yet well developed. Judicious exclusion of elements of the fuel from the energy balance may be desirable. These complications are discussed in Refs. 6 and 34, for example. In practice, rough estimates almost always can be obtained from eq. (19) without introducing these complications.

Formulas like eq. (19) often have been employed to calculate burning rates. An example concerns a sequence of pool fires with liquid fuels such as gasoline and diesel oil burning in the open in circular pans of different diameters, studied experimentally by Blinov and Khudiakov. 35 The experimental burning rates are shown approximately in Fig. 7. An energy balance for these experiments has been presented in Ref. 36, for example. The decrease in the burning rate per unit surface area with increasing pan diameter at small pan diameters is attributed to a corresponding decrease in q, as given by eq. (12), in the regime of laminar free convection. Note, for example, from eq. {12) and Table 9, q ~ G~I4/[ ~ f314/~ ~ f-114, so m ~ E-1/4 according to eq. (19). If convective heat transfer occurs with the velocity determined by the burning

rate, then q ~ x / ~ / E ~ v / ~ / E ~ x ~ , so that eq.

E

n."

(.9 z n-.

13o

I00

I0

EFFECT OF HEAT LOSS

EFFECT OI g'AT'VE

L A M I N A R T U R B U L E N T

I I I I I 0.1 I I0 I 0 0 I 0 0 0 I 0 0 0 0

PAN D IAMETER, ~ , c m

FIG. 7. Approximate burning rates of pools of liquid hydrocarbon fuels in air as a function of pool diameter.

338 F.A. WILLIAMS

(19) gives m ~ x / ~ or m ~ I/E, a qualitatively similar result. The increase in burning rate and its leveling-off at larger pan diameters may be attributed to turbulent free convection in eq. (12) plus the onset of a significant radiant contribution to q, as given by eq. (15). It may be noted that typical burning rates of fuels in fires lie in the range of 10-3 g/cm 2 s.

9.3. Unsteady Burning and Oscillatory Burning

If there is no intersection of the "energy-provided" and "energy-required" curves in Fig. 6, then conditions of steady burning cannot be established. This might occur, for example, for a particular fuel object in the course of a fire. In this case the entire fuel object may be consumed in a transient gasification process under the influence of a large external energy flux. An estimate of the time required for fuel gasification under these conditions may be obtained as in Section 7.4, viz., t = M(L/W)/(qA), where M is the mass of the fuel object and A its surface area exposed to the ftux q. If steady burning prevails during most of the burning history, then a better approach to calculation of the burning time is to first calculate m by the methods of Section 9.2, then use t = M/(mA), obtained from a mass balance.

It is possible for more than one intersection between the "energy-provided" and "energy-required" curves in Fig. 6 to occur. In such cases, more than one steady state may exist. This situation is illustrated by the dashed line in Fig. 6. For feedback characteristics of this type, static stability reasoning shows that the middle intersection is unstable, while the upper and lower intersections define two different stable steady states, one with a high burning rate and the other low. Abrupt transitions between conditions of high and low burning rates may be induced by gradual variations of the curves in time that result in tangency of the two curves near the upper or lower stable states. Tangency at low m may define a condition of transition from slow to rapid burning, while tangency at high m may predict a sudden transition from rapid to slow burning. In unsteady situations, oscillations may occur in which the fire spends part of its time at a high burning rate and the rest of its time at a low burning rate.

Figure 6 provides a basis for a qualitative discussion of these unsteady phenomena. However, typical fires are too complex for reasonable computations of such behavior to be made from the curves. Burning-rate computations usually are best made as indicated in Section 9.2, with influence of m on q and on L/W included at most in an iterative manner.

Many of the experimental observations of burning- rate oscillations correspond to fires located within compartment and having burning rates that sometimes become controlled by ventilation of air into the compartment. Further considerations are needed to describe burning rates under conditions of ventilation control.

9.4. Ventilation-Controlled Burning Rates

For a sufficiently large fire contained within an enclosure such as a room or a duct, supply of air and exhaust of gaseous products of combustion limit the burning rate. Fires limited in this way are said to be ventilation-controlled. Their total burning rates are determined by rates at which gases flow into and out from the enclosure.

Gas flows to and from enclosures are described by fluid mechanics. Many different situations may occur, depending on geometrical configurations and external conditions. For example, in a duct fire the flow rate of air may be controlled by ventilation equipment such as fans. In a room fire, external winds may determine the rate of air flow, particularly if the room has more than one opening. A simple but instructive situation with relevance to many room fires concerns a room with only one opening through which the flow is controlled by differences of buoyancy inside and outside. This situation is illustrated schematically in Fig. 8.

The hot gases inside the room are less dense than the cooler gases outside. This leads to differences in variations of buoyancy forces with height inside and outside. If flow in the vertical direction is negligible, then conservation of the vertical component of momentum is expressed as a balance between buoyancy forces and pressure forces, instead of the balance between buoyancy and inertial forces introduced for plumes in Section 6.2. The buoyancy-

REsSSuUI~E OUTSIDEINsIDE

p~

A= AREA~..

.....

-3- H

FIO. 8. Schematic illustration of a ventilation-controlled fire within an enclosure having an opening.


pressure balance is expressed as dp/dz = -pg . There- fore the larger exterior density causes the vertical pressure gradient to be steeper outside than inside. These different pressure gradients, illustrated in Fig. 8, require pressure imbalance between the interior and exterior of the enclosure over the height of the opening. Such pressure differences will drive horizontal flows in the direction from the higher to the lower pressure. Since a mass balance will require flows both into and out of the opening to occur, it is seen that the interior and exterior pressures must be equal at some height within the opening, as illustrated in Fig. 8. Below this neutral height the higher external pressure drives exterior air into the enclosure, and above the neutral height the higher interior pressure drives the hot combustion products out.

This physics provides a basis for estimation of ventilation-controlled burning rates for the configuration that has been addressed. If Ap denotes the difference between the exterior and interior gas densities, then the pressure difference Ap for an opening of height H will be on the order of ApgH. A momentum balance in the horizontal direction between pressure and inertial forces may be employed to relate Ap to an average horizontal flow velocity v. From this balance it is seen that Ap is on the order of pv 2. Hence pv2~ ApgH. The total burning rate of fuel, - d M / d t , is a fraction of the total flow rate of air and proportional thereto. The total flow rate of air is proportional to pvA, where A is the area of the opening. Thus it is found that

- dM/dt ~ pvA ~ p[(Ap/p)oH] a/2A,

which shows that the burning rate is proportional to the product of the area of the opening and the square root of its height. A more careful analysis, such as that reported by Thomas in Ref. 6, shows that, approximately,

- d M / d t = 100Axe-g / s , (20)

where A and H are measured in meters. Although there are numerous limitations on the

applicability of eq. (20), 6 it is often useful for estimating ventilation-controlled burning rates in rooms. For a fire in a given enclosure it may be difficult to estimate in advance whether the fire is ventilation- controlled. A suitable approach is to calculate the total burning rate first by the method of Section 9.2, using eq. (19) with ventilation restrictions neglected, and next calculate by eq. (20). The smaller of these two burning rates is the value that would be expected to be observed.

When ventilation controls burning, it modifies gas- phase characteristics, e.g. by reducing flame temperatures through oxygen starvation. The heat-transfer rates are thereby changed, and it becomes difficult to make reasonable direct estimates of the feedback flux q. With - d M / d t known from eq. (20), the value ofm may be calculated from eq. (17) and used in eq. (19) to evaluate q. The resulting value of the feedback flux, needed for consistency to support the ventilation-

controlled burning rate, may aid in estimating conditions of heat transfer within the enclosure.

The history of a fire in a room typically may involve initiation as a small fire needing much less air than that contained in the interior, progression to a larger fire that becomes limited by ventilation, then a return to overventilated conditions as the fire enlarges the openings or depletes the fuel. Such progressions from overventilated to underventilated to overventilated conditions are common. In many respects the underventilated regime is the most hazardous. Conditions within the chamber become dangerous, e.g. there may be buildup of toxic materials such as CO at various locations, increases in amounts of visibility-impairing soot and enhancement of general levels of thermal radiation. Burning gases issuing from the upper part of the opening during ventilation-controlled conditions can produce taller flames that aid in spreading the fire to adjacent combustible structures. Sudden changes in fire characteristics may occur, introducing uncertainties in fire fighting. For these reasons, ventilation-controlled burning is of practical concern in urban fires. It is a topic that can become complicated because of fuel varieties and complexities of configuration--a topic that seems in need of further investigation.

In wildland fires ventilation-controlled burning may occur in the compact organic mantle on the ground. Although the resulting smoldering fires are not spectacular, they may dominate total heat release, site damage and mop-up costs.

I0. FLAME HEIGHTS

Knowledge of radiation intensities in the vicinity of a fire is important for evaluating hazards to personnel and rates of fire spread. These intensities depend on the view factor F, defined in Section 8.4, and F in turn depends on flame heights, often strongly. Therefore flame heights must be known to estimate effects of flame radiation in fires. They also are relevant to convective fire spread through "flame contact" in some upward-propagation configurations. Therefore there is appreciable interest in the calculation of flame heights.

Correlations are available for heights of flames in fires. These correlations are expressed in nondimensional parameters whose selection is motivated by considerations of momentum conservation. Therefore it is appropriate to look more closely into momentum conservation before discussing the correlations.

10.1. Conservation of Momentum

Conservation of momentum has been considered previously. In Section 6.2 it was stated as a balance between buoyancy and inertia, and in Section 9.4 it was expressed as a balance between buoyancy and pressure and also as a balance between pressure and inertia. Each such balance constitutes a simplification that involves neglect of certain forces. A better appreciation of what has been neglected may be

340 F.A. WILLIAMS

obtained from indications of the orders of magnitude of terms in the full equation for momentum conservation.

The general equation for momentum conservation expresses a balance among five different types of forces. These five additive elements are the transient accumulation of momentum, the dynamic or inertial forces, pressure forces, buoyancy and viscous forces. In units of acceleration (force per unit mass), the orders of magnitude of these five elements are, respectively, v/t, I)2/~, Ap/(p(), gAp/p and vv/E z. Here v is a characteristic velocity; all other symbols have been defined previously, with the exception of t, which here denotes a characteristic time for gas-flow transients. The different types of momentum balances may be discussed in terms of the relative magnitudes of these five forces.

Although the transient term may be significant in fireballs and thermals for example, gases respond sufficiently readily that steady flow often is maintained except at large distances. Pressure forces are important in balancing buoyancy to determine atmospheric structure and also for balancing inertial forces in momentum-controlled jets for example. Since these pressure variations occur fairly automatically and will not have to be addressed here specifically, it will be sufficient to focus attention on the dynamic, buoyant and viscous forces. In heat transfer by free convection, all three of these remaining terms may be of the same order of magnitude. A characteristic velocity associated with buoyancy may be found by equating the dynamic and buoyant forces, yielding v =

~ / p . Use of this velocity in the viscous force shows that the ratio of the buoyant force to the viscous force is [9#3Ap/(pv2)] 1/2, which is seen to be

x ~ , the square root of the Grashof number, if the identification IAp/pl = JAT/TI is made. This provides the basis for the appearance of Gr in correlations of heat transfer by free convection (Section 7.3).

In fires large enough to be of practical interest, average viscous forces are relatively small and the relative magnitudes of dynamic and buoyant forces are important. The ratio of these two forces is the Froude number, Fr=v2/(g(Ap/p). In buoyant

plumes Fr is of order unity. However, at the base of a fire Fr may differ from unity. The value of Fr near the fire base is relevant to the low-level structure of the fire and therefore influences flame heights. This controlling Fr may be related to the burning rate through mass conservation. The product pv is identified as the mass of gases produced per unit horizontal area of the fire per second, which is proportional to m. When attention is restricted to classes of fires in which gas densities and density differences remain fairly invariant, it is then found that Fr is proportional to m2/•, i.e. to (dM/dt)2/{ 5. These observations provide the basis for correlation of flame-height data by use of either the dimensional group m2/{ or the dimension- less group rn2/(p2gf ). Both of these groups are related to the relevant Fr.

10.2. Flame-Height Definitions

Imprecision is inherent in definitions of flame heights. Even in steady laminar flows, there are continuous variations in temperature and in soot concentration, and values of flame heights depend, for example, on whether blue or yellow radiant emission is selected as a basis for measurement. In turbulent fires the heights of yellow flames fluctuate rapidly in time. Photographs taken with time exposures may show larger flame heights than averages taken from motion-picture films or visual observations. Flame heights usually are defined as the time-average maximum heights of the yellow flames since such values are most relevant to average radiant energy emissions. Uncertainties in flame heights associated with different methods of measurement typically are on the order of 10 to 20~o, seldom ever as large as a factor of two.

10.3. Flame-Heiyht Correlations

Results of a variety of measurements of flame heights are shown in Fig. 9. The nondimensional ratio of flame height h to a horizontal scale [ of the fire is plotted as a function of a dimensional parameter measuring the Froude number; the units of m and l here are g/cm2s and cm. The sources of the data are

h !

IOOO

IOO

IO

I

o.i

o.oi

REMBERT AND HASLAM CITY GAS.,

GASEOUS TURBULANT

. . . . ~ DIFFUSION G ~ u ~ / FLAMES f . . . . . . . . ~ . ~ ' ~ " " ~ KO SOON ,' W I ~.L, AMS SLOPE= ,20

AND BUMAN "WOOD CRIBS BIRCHWO00 AND CELLULOSE DOWELS

THOMAS-~ ~,,,,,,.,""~ THOMAS ET AL SLOPE=.24 / ~'w~-z-WHISKE Y FONS ET AL

WAREHOUSE SLOPE=,30 / [BROtDO FIRE

/ AND MCMASTERS SLOPE =-35

"~"--- - FO R EST, FAURE

PUTNAM AND SPEICH.~..,~;~

i I I I I I I I - r IO - 5 IO - s 10 -I IO I0 s iO 5 10 7

I0 s ( mZ/9 )

FIG. 9. Representative flame-height correlations.


listed as Refs. 37 to 44. All of the flames are at least partially turbulent.

It is evident from Fig. 9 that there are differences in data of different investigators. These differences are due partially to measurement techniques and flame- height definitions and partially to the use of different fuels; quantities such as Ap/p, which appear in Fr but not in Fig. 9, influence flame heights somewhat. Nevertheless, a reasonable correlation may be obtained by sketching a continuous curve through the average of the correlations shown. It may be noted that the slope of this curve will vary from 0.2 at small scales to 0,33 at large scales. The results shown in Fig. 9 extend over nearly fifteen orders of magnitude in Froude number and four in the ratio of flame height to fuel-bed dimension. They cover sizes ranging from those of laboratory jet-flames to those of mass fires.

To estimate flame heights in practice, it may be recommended that an average curve through the correlations of Fig. 9 be employed rather than a formula based on correlations obtained over a limited range of conditions, unless the situation of interest corresponds to an experiment in which a correlation was measured. Corrections can be introduced for effects of fuel shape (Ref. 42) and fuel type. For example, for reasons of stoichiometry, hydrocarbon fuels tend to have higher flame heights than fuels containing oxygen or inerts, particularly at small scales. However, for purposes of rough estimation such corrections can be neglected for most fuels of practical interest.

10.4. Additional Phenomena lnfluenciny Flame Heights

Data are available to extend Fig. 9 to larger values of Fr. At larger Fr, buoyancy begins to become negligible and the fire becomes momentum-controlled. Under these conditions in the turbulent regime, h//' becomes constant, as might be inferred from an extrapolation of the curves in Fig. 9. Sufficiently small fires in the laminar regime theoretically have h/~ ~ m, so that the parameter on the horizontal axis in Fig. 9 becomes inappropriate for correlation at very small scales.

Flames are affected appreciably by wind (Ref. 42). The flame height typically decreases somewhat as the wind velocity increases, and the flames tilt in the downwind direction. The tangent of the tilt angle might be estimated as the wind velocity divided by the

buoyancy velocity, xfyhAp/p, at the flame height. As indicated in Section 6.6, rotation of the atmos-

phere may have a large effect on flame heights. Estimates of the associated increase in flame height may be found in Ref. 29.

Flames from multiple fires may tend to interact fluid-mechanically and thereby modify individual flame lengths (see Refs. 6, 43 and 45). Also, fires from large, homogeneous fuel beds may lose coherence and break up spontaneously into multiple flames (Refs. 6 and 34). The nature and magnitudes of such effects

vary with conditions and deserve further investigation.

Also in need of additional study is the influence of nearby obstacles on flame heights and flame lengths. The fluid-mechanical interactions produced can have substantial effects on flame heights and thereby modify rates of fire spread. Flames adjacent to walls usually are taller than free flames, and flames encoun- tering ceilings from below may extend appreciable distances in the horizontal direction. Qualitative understandings of such phenomena can be developed on the basis of fluid-mechanical concepts.

!1. FIRE SPREAD

Fire spread concerns the involvement of additional fuel in a fire. Spread rates are rates at which new fuel begins to burn. It is important to distinguish spread rates from burning rates, which concern the rates of consumption of fuel totally involved in the fire. The distinction may be understood by visualizing flames spreading over the upper surface of a fiat combustible material initially ignited along one edge; the local spread rate is the local horizontal velocity at which the boundary of the burning area moves over virgin fuel, while the local burning rate is the local normal velocity (or velocity-density product, mass per unit per second) at which the fuel surface regresses in the vertical direction under the flames. Spread rates are of importance in that they influence the time available for responding to a fire and thereby affect strategies for fire suppression.

11.1. Spread Rates

A simplified approach to the calculation of spread rates of fires may be defined (Ref. 46). The approach involves first identifying a surface of fire inception. This surface represents the boundary between the burning and nonburning combustible. Its location is indicated schematically in Fig. 10 for various types of fire spread. Transport of knowledge of existence of the fire across the surface of fire inception to the unignited fuel is necessary if fire spread is to occur. This knowledge nearly always is transported by means of heat transfer. Therefore the rate of heat transfer across the surface of fire inception determines the spread rate.

In a first approximation it may be assumed that a critical enthalpy change Ah (energy per unit mass) of the fuel is needed for ignition to occur, If p is the fuel density and V is the spread rate then the energy per unit area per second required by the unignited fuel during spread is pVAh. According to an energy balance,

pVAh = q (21)

during steady spread, where q is the energy per unit area per second transferred across the surface of fire inception. The fluxes q are indicated schematically in Fig. 10, Equation (21) is a formula from which the spread rate V may be calculated if q, Ah and p can be estimated.

342 F.A. WILLIAMS

s.EET

FIRE • DENOTES SURFACE

pVAh=q

FIG. 10. Schematic illustrations of fire spread.

Usually p may be obtained from known fuel properties and q from heat-transfer considerations such as those discussed earlier. The value of Ah may be estimated as %(T,- T0, where cp is a heat capacity for the fuel, T~ its initial temperature and T~ a critical fuel temperature required for initiation of burning. This T~ is termed an ignition temperature; for solid or liquid fuels it is very much like the surface temperature discussed in Section 9.2, and its values are essentially the same as the values listed in Table 10 for solid fuels, or the boiling temperature T b for liquid fuels. For wildland fuels, especially fine cellulosic materials, moisture may be a dominant contributor to Ah, and heat of vaporization of the water must be added. Thus, reasonable estimates for Ah usually can be made, if the fuel is properly characterized, so that eq. (21) can be employed to calculate V. Limitations on the use of eq. (21) are discussed in Ref. 46.

11.2. Modes of Transfer Producing Spread

Table 11 lists various transfer modes that may be relevant to the calculation of q for fire spread. The listing is roughly in the order of increasing scale. Although active chemical species such as radicals may initiate burning without significant heat transfer (Section 5.1), the first entry in Table 11 generally is insignificant except possibly at very small scales in

TABLE 11. Transfer modes in fire spread

1. Radical diffusion 2. Heat conduction through gas 3. Heat conduction through condensed materials 4. Convection through gas 5. Liquid convection 6. Fuel deformation 7. Radiation from flames 8. Radiation from burning fuel surfaces 9. Firebrand transport

special circumstances. The conductive and convective heat-transfer correlations of Section 7 are relevant to the second through fifth entries in Table 11. The fifth entry in Table 1t can be especially significant for liquid fuels, whose flame-spread processes are strongly affected by liquid circulation and subsurface heat loss; the circulation sometimes is driven by surface-tension differences. Solid fuels that are heated often change shape; burning materials may be displaced toward unignited fuel elements, or fuels that are not burning may deform through the influence of heat toward the fire periphery. These mechanisms appear as the sixth entry in Table 11. For the seventh and eighth entries, values of q may be estimated from the discussion of radiation given in Section 8.

Firebrands are burning elements of fuel detached from the main fire and carried away convectively by winds. They may spread fire over large distances (~ 1000m) if they fall still burning or sufficiently hot on virgin fuel. Thus, the final entry in Table 11 is quite significant for forest fires and also often for large urban fires. Equation (21) needs reinterpretation or modification for entries 1, 6 and 9 of Table 11 (see Ref. 46).

11.3. Downward or Horizontal Spread along Continuous Surfaces of Solid Fuels

Heat transfer by conduction and radiation con- tributes to q for spread in still atmospheres along continuous surfaces of solid fuels in directions ranging from downward to horizontal with the flames above the fuel. Consider a fuel sheet thermally insulated on one face and of a thickness f that is sufficiently small for temperature variations across the sheet to be negligible. If radiation from the flames dominates q, then in eq. (21) approximately q = etrT4hsinO/~, where h is the flame height and 0 is the angle between the flames (usually the vertical direction) and the exposed surface of the unignited fuel (0 > n/2). Here e and T refer to the flames, use having been made of eq. (15) with view-factor corrections. Roughly speaking, h sin 0/¢ is the ratio of the surface area of the fuel exposed to the flame radiation to the area of the surface of fire inception, taken normal to the direction of fire spread. Equation (21) provides the spread rate,

F = ecrT~hsin O/(EpAh). (22)

Use of eq. (22) necessitates finding the flame height and flame angle, e.g. by the methods of Sections 10.3 and 10.4. The characteristic length { in the flame- height correlation typically is not the fuel thickness that appears in eq. (22) but rather the length of the burning region, which depends on pV/m. For most purposes it is convenient and sufficient to employ a fixed estimate for h in eq. (22) instead of introducing an additional dependence on V. Wind can influence V for horizontal spread by modifying 0. Wind in a direction opposite to the direction of spread reduces sin0 but nevertheless experimentally sometimes increases V because of effects of convective heat transfer not included in eq. (22). This situation arises


in flame spread over thick sheets of poly(methyl methacrylate), for example. Sufficiently large opposed wind velocities nearly always reduce V and eventually cause either cessation of spread or extinction of burning by mechanisms not contained in eq. (22). The magnitudes of the effects of opposed wind on spread rates often are relatively small prior to extinction. Wind blowing in the direction of spread increases V appreciably by increasing rates of radiative and convective heat transfer in q.

If the fuel is too thick then it is not heated throughout its depth prior to flame arrival. In eq. (22) ( must then be replaced by the thickness of the surface layer of the fuel that is heated. This thickness depends on the thermal diffusivity ct of the fuel, defined in Section 7.2. The time that a fuel element is exposed to heating prior to flame arrival is of the order of h/V because the distance parallel to the surface that the fuel is heated is roughly h. Therefore the thickness of

the heated layer is approximately x / ~ V . By substi- tuting this expression for ( into eq. (22) and solving for V we find that

V -- (etrTg)Zh sin20/[otp2(Ah)Z]. (23)

Fuels for which eq. (23) should be used instead of eq.

(22), i.e. fuels of thickness ( > x / ~ , are said to be thermally thick. Equation (22) applies to thermally thin fuels.

Often conduction of heat through the solid or through the gas is more important than flame radiation in contributing to the q of eq. (21). In these situations eq. (22) or (23) must be replaced by a formula based on conductive heat transfer. 46 The condition for conduction through the gas to be greater than flame radiation may be estimated as 2 ( T - T~) > ecrT4h sin 0, where T is the flame temperature and 2 the thermal conductivity of the gas. Comparisons of the relative importance of gas-phase and solid-phase conduction also may be developed. 46

I 1.4. Upward Spread along Continuous Surfaces of Solid Fuels

Upward spread is much more rapid and usually acceleratory. The flames hug the vertical surface of the fuel and tend to be taller mainly because the fuel inhibits the inflow of air needed for completion of combustion. Convective and radiative heat transfer from the flames to the fuel contribute to q. In eq. (21)

q = ~T4h/~/'~h/V for the usual situation of thermally thick fuels if radiant transfer is dominant. Here h is proportional to a power of the burning rate (Fig. 9) which in turn is proportional to pV# since the vertical area of fuel burning is proportional to the spread rate and the thickness g. Also, e increases with increasing flame-layer thickness. Thus, estimation of upward spread rates becomes somewhat complex. Upward spread tends to proceed at velocities on the order of meters per second while horizontal spread velocities often are less than one centimeter per second. More study is needed of upward spread and also of wind-

aided spread, which shares many attributes of upward spread.

11.5. Smoldering

In the spread of smoldering combustion, diffusion of oxygen to the surface of the fuel at which the heat release occurs often controls the rate. In eq. (21) the value of q may be estimated as QoPoDo/~o, where Qo is the heat released in smoldering combustion per unit mass of oxygen consumed, Po is the density of oxygen in the atmosphere (20 ~ of air density for normal air), D O is the diffusion coefficient of oxygen and {o the distance over which the oxygen must diffuse. The value of "(o depends on the size and shape of the fuel. Often energy losses by radiation and conduction from the smoldering region are important, so that q = QoPoDo/Eo- qt, where qt is the flux of heat loss. If qt is too large then smoldering ceases. If ql is negligible then eq. (21) gives

V = QoPoDo/(~opAh), (24)

which typically is on the order of 10-3/~ 0 cm/s, with (o in centimeters. Thus, smoldering spreads very slowly (see citations quoted in Ref. 46). If may be responsible for the eventual development of flaming long after an ignition stimulus has been applied.

11.6. Spread through Porous Beds of Fuels

Porous fuel beds are often encountered, especially in wildland fires. They have wide ranges of properties, from those of dense beds of pine needles to those of thickets of trees. A variety of different heat transfer processes may contribute to q of eq. (21) for spread through porous beds. There are situations in which radiation from surfaces of burning fuel elements provides the dominant mechanism of spread. Under these conditions, from eq. (15), q = ~a(T~-T4), where the emissivity e is that of the burning surface, often nearly unity. When eq. (21) is used for porous beds it must be recognized that p therein is fPs, where p~ is the specific gravity (density) of the fuel elements andf i s the packing fraction, the ratio of the volume of the bed occupied by fuel to the total bed volume. Thus, when radiation from fuel surfaces dominates, eq. (21) gives

V = ea(T~- T~)/(fpsAh) (25)

for the speed rate. Equation (25) rests on the assumption that the fuel

elements are heated throughout prior to fire arrival, i.e. that the fuel is thermally thin. For thicker fuels, the fraction of the fuel that is heated prior to ignition must be taken into account. This fraction is the product of the surface-to-volume ratio s (roughly the reciprocal of the diameter for long sticks) with the thickness of the heated layer of a fuel element. As in Section 11.3,

this thickness may be estimated as xf~/V times the square root of the depth of the bed that is heated ahead of the surface of fire inception. This latter depth is on the order of 1/(fs) for radiant transfer. Hence, for

344 F . A . WILLIAMS

thermally thick fuel p = fPsS[O~/(Vfs)] 1/2 in eq. (21), and solving for V gives

V 2 2 4 4 2 2 2 = e a (T~ - T, ) /[fsaps (Ah) ] (26)

in place ofeq. (25). Additional phenomena that often need to be con-

sidered in estimating rates of spread through porous beds include radiation from flames above the bed, convective cooling of unignited fuel by air entrained from below or from the sides, and convective heating by hot gases in the upper portion of the bed. In wildland fires these effects may depend on the wind and on the slope of the terrain. A variety of different spread characteristics may be calculated (see references quoted in Ref. 46).

11.7. Other Mechanisms of Fire Spread

There are many mechanisms by which fires may spread, beyond those discussed here. Rates of spread through arrays of discrete fuel elements that are not in contact with each other often may be defined as the distance between adjacent fuel elements divided by the time interval between ignition of adjacent elements. 47 These spread rates depend on ignition times. In irregular arrays spread may be unsteady and spatially nonuniform. Mechanisms of spread in arrays are addressed in Ref. 47 and in other references quoted in Refs. 46 and 48, for example.

Fire spread often involves liquid flow in one way or another. For example, a vertically oriented solid fuel ignited at the top may experience downward spread by melting and flow under gravity of liquid fuel carrying flames. 46 Fuels initially in the liquid state usually flow during fire spread even when contained in confining pans. One important mechanism for driving this liquid flow is the variation of surface tension with temperature (see references quoted in Ref. 46). The decrease in the surface temperature of the liquid with increasing distance away from the surface of fire inception causes a corresponding increase in surface tension which in turn provides a force that tends to produce flow of liquid toward the unignited fuel. This mechanism often controls spread rates for liquids that are not sufficiently volatile for flames to spread through the gaseous vapors above them. Another mechanism influencing rates of fire spread over liquid fuels is buoyant convection within the liquid, driven by liquid density differences that are produced by uneven heating of the liquid by the fire. 46

Fire spread through gases occurs for fires in combustible gaseous mixtures as well as for fires above sufficiently volatile liquid fuels. The corresponding rates of spread typically are more rapid than those for other modes; velocities on the order of meters per second or more are common. The mechanism of these spread processes is that of the propagation of premixed flames through gases, a fundamental process in combustion that under uniform, laminar conditions is understood well in comparison with most other modes of spread. 46 The corresponding spread rate is

the laminar flame speed, a tabulated property of combustible mixtures dependent on their thermo- physical and chemical-kinetic properties. Turbulence in the gas can increase the spread rate appreciably. Effects of buoyancy of the gases decrease or increase the spread rate, depending on conditions; the decrease is associated with the buoyant rise of hot combustion products, whereas the increase is poorly understood although it may contribute significantly to fire hazards in large releases of combustible gas.

There are situations in which pulsating or oscillatory spread of fire occurs. An example is spread above liquids that are too volatile to be controlled solely by surface-tension flows but not volatile enough to support purely gas-phase spread. Other examples occur for fires in ducts and corridors and often are associated with onset of ventilation-controlled conditions. Unanticipated pulsations are hazardous to personnel.

In estimating spread rates thought should be given to all mechanisms that may contribute to the flux q across the surface of fire inception. If the mechanism providing the major contribution to q is overlooked then V will be underestimated appreciably. In this sense fires tend to spread as fast as they can.

12. I G N I T I O N

Ignition of fuels is a significant aspect of fire safety. If ignition can be prevented then fire problems are solved. Reduction in the ease of ignition is an important approach to mitigation of fire hazards. Ignition questions also arise in other aspects of fire phenomenology. For example, it was indicated in the preceding section that there are situations in which ignition times influence rates of fire spread. A variety of combustion characteristics are relevant to ignition processes. The characteristics of principal importance to fires are discussed here.

12.1. Ignition Temperature

Consider first the ignition of solids exposed to radiant or convective heating. In a first approximation for many purposes it may be assumed that the surface of the solid must achieve a critical temperature T~ for burning to begin. This temperature is identified as an ignition temperature characteristic of the fuel. Identification of an ignition temperature is useful not only for ignition investigations but also for calculation of flame spread, as indicated in Section 11.1.

The utility of introduction of an ignition temperature lies in the fact that it enables computations of ignition processes to be performed for many different situations on the basis of heat-transfer analyses. For example, the surface of a combustible may be exposed to a radiant energy flux but cooled primarily by free convection as its temperature rises. If the cooling rate is sufficient to prevent the temperature from reaching the ignition temperature then ignition does not occur. Many combinations of radiation, convection and conduction in different configurations may be sub-


jected to heat-transfer computations to estimate whether ignition is achieved on the basis of the ignition-temperature criterion.

In spite of its utility, the ignition temperature represents an approximation dependent on many phenomena. It is a usable approximation mainly because chemical rates are so strongly dependent on temperature. The basis of the existence of an ignition temperature is fundamentally the same as the basis for the existence of a fixed surface temperature during steady burning. This basis has been discussed in Section 9.2. From that discussion it should be evident that values of T~ will differ somewhat not only for steady burning, spread and ignition but also for differing ignition experiments. The variation of the ignition temperature among ignition experiments, however, is small enough for the concept to be useful. Discussions of ignition temperatures may be found in Refs. 49, 50 and 51, for example.

12.2. Spontaneous and Piloted 19nition

The temperatures that the surfaces of many materials (for example, wood) must achieve to undergo ignition depends on whether there is a flame near the surface (exceptions are some exothermic solid or liquid fuels that do not require oxygen to release heat and fuels with very high gasification temperatures that burn through heterogeneous reactions with oxygen at their surfaces). Small flames can serve as ready ignition sources for combustible gases emanating from the heated surface. Ignition in the presence of a small flame is termed piloted ignition, while ignition in the absence of flames is called spontaneous or unpiloted ignition.

Although "spontaneous ignition" is a prevalent terminology here, "unpiloted ignition" is better because "spontaneous ignition" also refers to ignition in the absence of an external heat source. This last process is due to chemical heat generation by the material itself and has occurred during storage of large quantities of fertilizer, of rags soaked with unsaturated "drying" oils, etc. Often also called "spontaneous combustion" or "thermal explosion", s2 the tendency for this process to occur is enhanced by increasing the amount of the combustible material because this decreases the rate of heat loss, which must balance the rate of heat generation (given by the product of eq. (7) with the heat released per mole of fuel consumed) to prevent temperatures from building to ignition points. Conditions for the occurrence of this other type of "spontaneous ignition" may be estimated from heat-transfer formulas if rate parameters for heat generation are known.

In the presence of external heating, temperatures for piloted ignition may be less than those for unpiloted ignition by amounts on the order of hundreds of degrees. For example, ignition temperatures for piloted ignition of cellulose typically are around 640K, while in unpiloted ignition, ignition temperatures of about 870K are observed. Variations in these values on the order of + 50K may occur, depending

on the ignition situation. Qualitatively, piloted ignition begins whenever a combustible mixture is produced in the gas adjacent to the solid surface; unpiloted ignition requires moreover that this combustible mixture ignite by itself rather than just supporting flame propagation.

Unpiloted ignition of solid fuels has much in common with ignition of combustible gas mixtures. If the temperature of a combustible gas is raised uniformly at a specified rate, then exothermic chemistry begins rapidly when the gas reaches a critical ignition temperature. The identification of ignition temperatures for gases is only a rough approximation to the chemical kinetics. An increase in the rate of temperature rise generally produces a higher ignition temperature. For greater precision, ignition temperatures of gases may be associated with particular experiments. For example, an inert solid of specified size and shape may be selected and introduced in a specified manner into a gaseous combustible mixture. The temperature of the solid needed to initiate flame propagation through the gas may be defined as the ignition temperature of the mixture. Although this type of definition improves precision, questions remain concerning the relevance of the resulting ignition temperature to other experiments.

12.3. Iynition Time

Ignition temperatures are useful in estimating critical conditions for ignition to occur at all. Under supercritical conditions there is a time delay between the establishment of the conditions and the occurrence of ignition. This delay is the ignition time. The ignition time approaches infinity at the critical conditions for ignition.

The ignition time depends on the strength of the ignition stimulus. If the externally applied energy flux is increased then the ignition time decreases. This behavior is illustrated schematically in Fig. 11, which

I--

Z

q

~ CRITICAL ENERGY FLUX

FLAMING ----~kk TRANSIENT \

\

LOG ( ENERGY FLUX)

FIG. 11. Schematic illustration of dependence of ignition time on incident energy flux.

346 F.A. WILLIAMS

has been adapted from data on radiant ignition of cellulose.

At energy fluxes below the critical energy flux identified in Fig. 11, ignition does not occur because the fuel fails to reach its ignition temperature. The value of the critical energy flux may be calculated as indicated in Section 12.1 by considering rates of heat loss; this value will thereby be found to depend on many experimental and geometrical factors, such as the size of the fuel sample.

At energy fluxes above the critical value, ignition occurs after a time interval that decreases as the level of the applied energy flux increases. Typically a regime is reached in which the ignition time varies inversely with the energy flux. In this regime, the product of the flux with the ignition time is constant. A constant value of this product implies that a constant amount of energy (per unit surface area) has been delivered to the fuel prior to ignition. This high-flux regime in which a critical energy per unit area is needed for ignition has been indicated in Fig. 11. The critical value may be called the minimum ignition energy per unit area because at lower fluxes the longer ignition times require larger amounts of energy to be delivered.

The transition from critical-flux behavior at low fluxes to critical-energy behavior at high fluxes in Fig. 11 may be understood on the basis of a fixed ignition temperature. As the flux increases, less time is available for heat transfer to occur, and the resulting heat losses become progressively smaller in comparison with the energy input. When the heat losses become negligible, an energy balance shows that a fixed temperature is reached at a fixed product of flux with time. A simplified balance of this type, giving a formula for the ignition time, has been developed in Section 7.4 and is shown in eq. (14). This formula with T = T~ may be used for the entire curve in Fig. 11 if q therein is taken to be the difference between the externally applied energy flux and the total flux of heat loss.

Although these simple concepts explain qualitatively the major features of Fig. 11, closer study reveals many complications. As indicated in Sections 12.1 and 12.2, the ignition temperature increases slowly with increasing energy flux, so that the product of the flux with the ignition time is not exactly constant along the portion of the curve labeled "critical energy"; the critical energy per unit area increases slowly with increasing flux in this regime. In high-flux experiments with cellulose and some other solid fuels, flames have been observed to move transiently over the fuel surface during application of a radiant energy flux and to disappear upon termination of the flux prior to attainment of the critical energy per unit area; this regime of transient flaming is indicated in Fig. 11.49'5 ! Analyses allowing for spatial variations of temperature usually are needed in place of the simplified formula (14) to obtain reasonably accurate comparisons between calculated and measured ignition times. The various heat-transfer, fluid-flow and chemical-kinetic phenomena that may occur often

lead to ignition-time curves more complicated than Fig. 11. Detailed ignition-time studies, such as those reported in Ref. '53, help to clarify ignition mechanisms.

12.4. Ignition of Gaseous Fuels

Combustible gases may be ignited by hot surfaces, sparks, flames or pressuFe pulses (as from explosions). There is an analogy to Fig. 11 in that hot surfaces often provide low fluxes characterizable by an ignition temperature (Section 12.2) while sparks provide high rates of energy deposition, characterizable by minimum ignition energies, which are tabulated. Discussion of ignition of gases may be found in Ref. 54, for example. Ignition times in gases may be studied from the viewpoint of detailed chemistry on the basis of buildup of chain carriers (Section 5.1) or through overall-rate approximations (Section 5.2) with emphasis on thermal effects.

12.5. Flammability Limits

If there is too little or too much fuel in a fuel-air mixture then the gas cannot be ignited by a flame or by any other ignition source. For example, methane-air mixtures with less than about 5 % or more than about 15 ~o methane on a volume or molar basis cannot burn. The maximum and minimum concentrations of fuel that permit flame propagation are called the rich and lean limits of flammability, respectively. Much information is available on flammability of gases, and tabulations of flammability limits may be found, s5'56 Flamma- bility limits depend on pressure and temperature and may be narrowed by addition of inert gases. In addition to contributing to understanding of ignition of condensed fuels, knowledge of flammability and of flame quenching of gases is important for studying fire-safety problems involving combustible gases. For example, in fuel tanks of automobiles the concentrations of gasoline vapors are above the rich flammability limit; the fact that these vapor-air mixtures are too rich to burn is significant for safety.

12.6. Flash Points of Liquid Fuels

Ignition of liquid fuels generally occurs in the gaseous vapors of the liquid. The equilibrium concentration of these vapors depends on the temperature of the liquid, according to eq. (1). If the liquid temperature T is low enough then the vapor mole fraction X~, given by eq. (1), will be less than the mole fraction X t at the lower flammability limit. Under these conditions the vapor-air mixture is too lean to burn, and a small flame applied to the gas above the liquid will not result in propagation of a flame over the liquid surface. If the liquid temperature T is increased to a value Te, at which Xe = Xe, then this flame propagation will occur. The minimum liquid temperature Te permitting flame propagation through the vapor is called the flash point of the liquid.

When T reaches Te, the presence of a pilot flame causes a flame to flash rapidly over the liquid surface. A sustained fire may develop that continues after the


pilot is removed, or additional heating may be needed to sustain the combustion, depending on the situation. Standard experimental arrangements and procedures have been defined for measuring flash points of liquid fuels because values of Tt are relevant to fire hazards. Results of these tests agree only approximately with calculations based on the use of measured flammability limits of vapors in eq. (1) because additional processes such as gas-phase diffusion influence the results of the tests somewhat.

If the liquid is heated without a pilot flame nearby then higher temperatures can be reached prior to the occurrence of ignition. Often even the boiling temperature T b is too low to cause unpiloted ignition, and only by applying even higher temperatures in the gas, e.g. by convective heating, can unpiloted ignition be produced. Nevertheless, the vapors are hazardous at liquid temperatures above Tl because they can be ignited by weak stimuli such as sparks with energies on the order of a millijoule. One strategy for keeping combustible liquids safe is to keep their temperatures below their flash points.

13. EXTINCTION

Methods for suppressing or extinguishing unwanted fires have been studied for many years. Specialized equipment is available, such as shovels, bulldozers and aircraft for forest fires and sprinklers, extinguishers and high-pressure hoses for urban fires. Proper exercise of suppressive measures in the field requires good judgment. Knowledge of mechanisms of flame extinction, more thorough than provided by the fire triangle, aids in development of this judgment.

13.1. The Damk6h l e r Ex t inc t ion Cri ter ion

There are a variety of viewpoints that can contribute to understanding of flame extinction in fires. Once concerns the feedback-flux competition (Fig. 6) discussed in the Section 9. T M Another is based on a balance between a chemical time z~ and a residence

time r, in the gas-phase flames. The latter is discussed here because it provides a framework within which many different suppression measures can be viewed.

Damk6hler 57 defined a nondimensional parameter as the ratio of a flow or diffusion time to a time for chemical heat release in the gas. This Damk6hler number may be written here as

O = z , / z c. (27)

A formula for it may be obtained by introducing expressions for zr and re. In terms of a characteristic length E, velocity v and diffusivity ct, the relationships z, = Etv and % = f2/a may be employed. The chemical time may be estimated from a formula like eq. (7) as zc = Co/O9, where c o is an initial concentration of reactant. The resulting formula for the Damk6hler number is

This formula indicates how D varies with conditions. It can be reasoned s8 that if the parameters in eq.

(28) are based on conditions at the boundary of the fire, then an important burning property, such as the maximum temperature, depends on D in a manner described by an S-shaped curve, as illustrated in Fig. 12. Continuous variation of boundary properties causes D to vary continuously. Evidently if D is increased beyond the point marked "ignition", then a discontinuous change to a much higher temperature must occur. Similarly, if for conditions along the upper branch D is decreased continuously to the point marked "extinction", then a discontinuous change to a lower temperature occurs. On the lowest branch in Fig. 12 temperatures are near ambient and there is no burning; the upper branch represents burning conditions, and the middle branch usually is unstable and irrelevant.

These characteristics of the S curve indicate that there exists a critical value of D below which

Trnox

I !

EXTINCTION ) ~IGNITION D FIG. 12. Schematic illustration of dependence of the maximum temperature on the Damk6hler number.

348 F.A. WILLIAMS

extinction occurs. Below the critical D, say, for D < DE, where De denotes the critical value, the residence times in the gas are too short in comparison with the chemical conversion time for significant heat release to occur. For D > De, the opposite is true, and flames can exist. The extinction is sharp, involving small changes of D to move from vigorous burning to no combustion for the situation illustrated in Fig. 12. Although this is typical, there are situations in which the S curve is stretched out and extinction occurs more gradually.

The Damkthler criterion for extinction is an idealization but nevertheless has proven useful in obtaining overall chemical kinetic parameters for gas-phase combustion, near extinction, from experimental measurements. 59

13.2. Extinguishment Methods

Seven different strategies for extinguishing flames of fires are listed in Table 12. All of the strategies may be understood qualitatively on the basis of the Damkthler criterion for extinction. Isolating the fuel is usually achieved by applying a fire suppressant to the fuel; it tends to reduce D by reducing the fuel concentration eF in the gas (see eq. (28)). Some extinguishing agents, such as CO2, are believed to at least partially isolate the oxidizer and thereby reduce D by reducing the oxygen concentration c o in the gas. Cooling condensed fuels tends to reduce cF as well as the flame temperature T in eq. (28), thereby reducing D; cooling the gas also reduces T, which has a strong influence on D. Chemical inhibition of the combustion reaction, whether homogeneous (in the gas) or heterogeneous (on surfaces), modifies the overall kinetic parameters B and E in eq. (28) in a manner that leads to a reduction of D; these chemical changes are produced by use of chemical suppressants. The last entry in Table 12 is useful mainly for small fires, such as extinguishment of matches, and the consequent decrease in D arises from an increase in v in eq. (28).

13.3. Calculation of Conditions for Extinguishment

It is not yet possible to use the criterion D < De, along with eq. (28), to calculate conditions needed for achieving extinction of the flames of real-world fires because values of the parameters that appear (E, B, De) are not generally available. A small amount of the information needed has been obtained. 59 For practical calculations, rougher approximations must be employed.

A useful simplification of eq. (28) is based on the observation that since E/RT is large, the value of D

TABLE 12. Approaches to extinguishment

1. Isolate the fuel 2. Isolate the oxidizer 3. Cool condensed fuels 4. Cool the gas 5. Inhibit the chemical reaction homogeneously 6. Inhibit the chemical reaction heterogeneously 7. Blow the flame away

depends strongly on the flame temperature. In a rough approximation, it may therefore be assumed that D is zero (i.e. the gas-phase chemical reaction ceases) if the flame temperature falls below a critical flame temperature TE for extinction. The extinction criterion D < De is thus replaced by the simpler criterion

Tj. < TE, (29)

where T I is the flame temperature, discussed in Section 3.6.

Calculation of extinction on the basis of the criterion in eq. (29) may be performed by use of energy balances. Equation (3) is modified by inclusion of the heat losses associated with suppressive measures. For example, if water is added to the fire, then the energy that the fire expends in heating the water is subtracted from Q/W in eq. (3). Methods for writing more detailed energy balances for fires subjected to suppressive action are available (e.g. Refs. 34 and 60).

Use of the extinction criterion T I < TE requires knowledge of the value of the extinction temperature TE. This value is different for different fuels, and for any given fuel it depends somewhat on fire conditions. 59 Values of TE tend to be larger for smaller fires with shorter residence times. Nevertheless, a universal value that has been found to be useful for fires involving cellulosic fuels, such as wood, is TE = 1580K. 6° Although reducing the residence time may increase this TE by amounts up to 200 or 300K, the value quoted is conservative in that it does not give an underestimate of the amount of suppressant needed for extinguishment. The variation of TE from one fuel to another also is on the order of 300K for most fuels. The value 1580K may be used for practically all carbon-containing fuels composed of the elements C, H, O, N; for hydrogen burning in air TE is significantly smaller, about 1000K, and therefore hydrogen fires are much more difficult to extinguish.

There are even less fundamentally based rules of thumb for achievement of extinction. For example, in hydrocarbon fires in many situations extracting 2500cal/g of fuel gasified extinguishes the flames. 6° Use of such simplified criteria may circumvent the need for looking closely at energy balances.

It must be emphasized that these numerical values for extinction conditions refer to effects of cooling on fires, mainly water addition. The extinction mechanism is said to be thermal, not chemical. Extin- guishing agents that modify the chemical kinetics certainly will affect the critical value TE, increasing it appreciably if they are good chemical suppressants. Values of TE for chemical suppression are unavailable today.

13.4. Complicating Factors in Fire Suppression

There is more involved in fire extinguishment than merely extinction of flames. For example, cellulosic fuels experience glowing combustion as well as flaming, and glowing may persist subsequent to flame extinguishment. The glowing may rekindle flames under suitable conditions. Therefore subsidiary


measures for extinguishment of glowing are necessary if complete extinction is required (as it often is in urban fires).

In large fires, especially wildland fires, the fire often may be considered suppressed completely prior to extinguishment of all flaming combustion. Forest fires conventionally are said to be contained when enclosed by natural or constructed barriers and controlled when it is judged that there is no longer any possibility of escape. Although mop-up generally continues until the fire is out, it is often impractical to attempt to extinguish all flaming rapidly in forest fires, and suppression might be defined as achievement of a condition under which the fire is no longer a threat to surrounding areas of fuel. Thus, suppression may be tied to a criterion of elimination of fire spread. These complications lead to the existence of a variety of possible criteria for fire suppression (see Ref. 34, for example).

14. FIRE SUPPRESSANTS

Of the various materials used to suppress fires, some work only in the gas phase, while others act on the fuels in their solid and liquid states as well. Inert gases, such as CO2, N2 and steam operate only in the gaseous phase, cooling and diluting the flames. Their influences are thermal. The influence of water as a suppressant also is thermal, whether it be applied by a hose, a fog nozzle, sprinklers or in release from containers (e.g. aircraft-borne). The thermal effects of water are large because of the addition of its heat of vaporization, and these effects sometimes can be enhanced by applying the water to a solid fuel where it increases the feedback flux requirement (Fig. 6) as well as cooling the flames.

Water undoubtedly is the most common and most useful fire suppressant. Applied to the wood in wood fires, it promotes flame extinction and helps to prevent reignition. There are water-additive mixtures, such as protein foam, light water or aqueous film-forming foam that enhance the effectiveness of water in liquid- fuel fires, enabling the water to be applied directly to the fuel without spreading the fire excessively. These water additives produce thick films on top of the fuel that inhibit its transport to the flames, thereby again effectively increasing the feedback flux required to generate gaseous fuel. There are continual innovations in the technology of the use of water as a fire suppressant. Investigation of the establishment of foams with desired properties is a challenging subject. However, the suppression phenomenology in many ways is less complex than that of suppressants that involve chemical effects. The discussion here will be directed toward chemical suppression mechanisms.

14.1. Influences of Additives on Solid-Fuel Pyrolysis

Many different materials when added to polymers modify their behaviors during pyrolysis. These modifications can influence the manner in which the polymer burns and may thereby aid in fire sup-

pression. Cellulose affords a good illustration; addition of sodium bicarbonate (NaHCO3) , for example, reduces its tendency to experience flaming, as measured, for example, by an increase in the ignition temperature for sustained flaming combustion. There is support for a mechanism involving catalysis of the dehydration route defined in Section 4.4.

The major gaseous combustible from cellulose is its volatile tar. Reduction in tar production reduces chances of having flames. Methods for directly inhibiting the depolymerization (i.e. for reducing the rate constants of this path) are unknown. However, the total tar yield can be decreased by increasing the rate of the competitive dehydration. Sodium bicarbonate is believed to catalyze dehydration (a catalyst in this sense is a material that speeds a process--e.g, by providing an alternative, more rapid mechanism for it to occur--without itself appearing in the products of the process). Although the catalysis does not preserve the original cellulose, it reduces significantly the amount of combustible gases generated.

As a general chemical principle for both condensed- phase and gaseous reactions, it may be stated that the effective way to inhibit, retard or suppress a process is to introduce or accelerate a competing process. Negative catalysis involving direct reduction in rate constants is possible in principle but rarely in practice. Searches for inhibitors in reality are searches for catalysts of alternative paths.

By catalyzing dehydration, sodium bicarbonate causes cellulose to lose its structural integrity more rapidly, at lower temperature, when exposed to the heat of a fire. It also increases the weight of the residue that remains after pyrolysis. It promotes char formation and glowing combustion at the expense of flaming combustion. For example, it catalyzes the glowing combustion of sugar cubes, which without addition of a material like sodium bicarbonate are very difficult to burn. 6x

14.2. Fire Retardants

Fire retardants are materials added to combustibles to reduce their tendencies to burn. Often they are incorporated into manufactured items. Their selection is based not only on considerations of their influences on pyrolysis of the solid but also on investigation of their effects on gas-phase flames and combustion products.

Many different synthetic polymers are in widespread use. Effective fire retardants can be quite specific, differing from polymer to polymer. Therefore much activity must be devoted to retardant selection and testing. Sometimes specialized approaches are introduced, such as the use of intumescent paints that swell under the heat of a fire, providing a layer on the surface of the fuel that hinders heat and mass transfer.

Cellulosic materials remain a major component of materials found in urban environments. They also comprise virtually all of the fuels in wildland environ-

350 F.A. WILLIAMS

TABLE 13. Some possible fire retardants for cellulosic materials

Name Formula

Sodium bicarbonate NaHCOa Potassium bicarbonate KHCO 3 Potassium carbonate K2CO 3 Sodium carbonate Na2COa Monosodium sulfate NaHSO4 Monosodium phosphate NaH2PO 4 Trisodium phosphate NaaPO4 Monoammonium phosphate NH,H2PO , Diammonium phosphate (NH,)2HPO , Ammonium sulfate (NH,)2SO 4 Ammonium sulfamate NH4SO3NH2 Ammonium chloride NH,C1 Sodium chloride NaCI Sodium tetraborate Na2B,O 7

ments. Fire retardation of cellulosic materials has been studied because of concerns about urban fire hazards. The retardants developed clearly also can be considered for use as fire suppressants in attempting to control wildland fires.

Some of the materials that have been studied as possible fire retardants for cellulosics are listed in Table 13. The overall effectiveness of these retardants varies appreciably (see, for example, references quoted in Ref. 6). Most of them, mainly the earlier entries, catalyze dehydration and thereby enhance rates of weight loss at low elevated temperatures. Some, such as sodium tetraborate, do not influence these weight losses and therefore may not interact in the dehydration route. They help to suppress flaming by other means that are less well understood. Effects include possible chemical modifications of gas-phase combustion processes, discussed later. Some of the retardants in Table 13 (phosphates, such as diammonium phosphate) also apparently interact with glowing combustion and tend to reduce smoke production, for reasons poorly understood. Many basic questions remain concerning mechanisms of fire retardation.

Practical concerns in selection of fire retardants and suppressants extend well beyond questions of their effectiveness as retardants. For example, toxicity of the material itself is an important consideration in its handling. Methods for applying retardants and permanence of the application are relevant. Influence of suppressants on the environment must be considered; for example, use of the attractive material sodium tetraborate in suppressing forest fires was abandoned because of problems with sterilization of soil and corrosion of tankers, although boron compounds are effectively used as fire retardants for cellulosics in urban applications. For suppressants that are to be applied during burning thought must be given to questions such as their solubility in water, their influences on penetration of water droplets to the fuel, their adhesion to the surface of the fuel requiring protection, weights required for preventing flaming, etc.

14.3. Flammability

Tests of flammability of materials are needed for evaluating fire hazards and also for ranking the effectiveness of fire retardants. Many standardized tests have been developed, 62 often by the American Society for Testing and Materials (ASTM). These tests provide specific numbers that are used to assign flammability ratings. However, as stated in Section 2.1, it is best to exercise judgment in drawing conclusions from the test results.

For example, 6~ there is a test for fabric flammability that involves applying a flame to a fabric for 12 s in a particular manner, removing the flame and observing whether flaming of the fabric continues. For a particular fabric with a particular retardant, the flames died, so the fabric would be rated non- flammable. However, if the exposure had been for a shorter time, say 3s, then flaming would have persisted and consumed the fabric. The explanation lies in catalysis of dehydration by the retardant; the amount of tar-forming cellulose remaining after the 12 s exposure was insufficient to support flaming even though the fabric retained its structural integrity. Thus, drawing conclusions concerning flammability properties on the basis of only the 12 s exposure is unreasonable in this example.

In general, rational ratings of flammability require thought. The ASTM tests can be improved, but it is likely always to be possible to find situations in which blind acceptance of test results give false impressions of flammability. All types of fire testing can pose challenging questions; for example, for certain fires, wood, even though it burns, may constitute a more fire-resistant wall than steel for some construction purposes because the latter may soften and lose its structural integrity sooner. Careful and objective evaluation is essential in designs for fire safety.

14.4. Halogen-Containing Fire Suppressants

There are fire suppressants that are known to operate largely by modifying the chemical kinetics of the gas-phase combustion processes. The main examples are halogen-containing compounds, compounds involving C1 or especially Br. The compound CFaBr is readily available in commercial fire extinguishers. Partial knowledge has been acquired of the chemical kinetic mechanisms by which these suppressants aid in flame extinction. 63

In Section 5.1 it was indicated that entries 10 and 11 in Table 8 are significant chain-carrying steps in hydrocarbon combustion. Removal of radicals such as H atoms slows the overall rates significantly. A material such as CF a Br decomposes prior to reaching the flame, giving HBr, among other products. The step H + HBr--, H 2 + Br removes the very active H atom, replacing it by the less active Br atom. In fact, the chemical inhibition mechanism is much more complex than this and is not thoroughly understood. However, the step shown is one of the important steps and is illustrative. The result is a decrease in the overall rate


of heat release in the flame. The mechanism provides a contribution to extinguishment that falls under entry 5 in Table 12.

Practical considerations that arise in the use of extinguishants like CF 3 Br include the ease with which they can be stored and delivered to flames. Some fires, notably those in classes B and C, can be dangerous to attack by water-based suppressants. Gases like CF3Br have an important place in protection against such fires. It is found, however, that they do not suppress glowing combustion effectively. This is to be expected on the basis of the chemical mechanism by which they operate; glowing does not involve free radicals like H atoms in important ways. These suppressants in fact enhance problems of smoke and possibly of toxic products if applied to glowing fires. Therefore in wood fires they are best used only on the flames; another suppressant should be available for extinguishing glowing.

Halogens are contained in many fire retardants incorporated in polymers. There may be multiple mechanisms through which they are beneficial in reducing polymer flammability. The mechanisms are not understood well. The mechanism for inhibition of combustion, indicated above, may play a role.

14.5. Dry-Powder Extinguishants

Powders are also available in commercial fire extinguishers. Typical powders are some of those listed in Table 13, e.g. KHCO 3. In addition to the effects discussed in Sections 14.1 and 14.2, such powders may also modify the chemical kinetics of combustion in the flames (e.g. Ref. 64). Both heterogeneous (on the surfaces of powder particles) and homogeneous (after powder vaporization) chemical processes have been suggested as modes of this chemical effect. Thus, they may fall under item 5 or item 6 in Table 12. It is difficult to distinguish between their chemical effects and their thermal effects associated with energy absorbed by heating and vaporization. Their mechanisms of extinguishing flames are poorly understood.

14.6. Prospects for New Suppressants

New fire retardants may be expected to appear as new plastics come into use. However, in the field of fire suppression, development of new suppressants that can be applied directly to fires is likely to be slow. Most of the effective materials seem to be known already through empirical observations, even though their mechanisms of action may not be understood. Some super-effective materials that inhibit gas-phase combustion appreciably when present in very small quantities are known, such as iron pentacarbonyl, but they are very toxic and not suitable for practical extinguishers. Most of the progress is likely to come from better-reasoned application of available suppressants.

15. M O D E L I N G URBAN FIRES

There is a significant amount of activity in fire

modeling directed toward calculating histories of urban fires. 65'66 The objective usually is not to calculate specific fires that occur in the real world but rather to provide information relevant to the design of buildings, to improvement of test methods and to possible establishment of regulations that may be incorporated in improved building codes. Most of the work employs electronic computers for calculations, statistical or deterministic, based on algebraic or differential equations. It is of interest to look at some of the aspects that are modeled.

15.1. Fire Growth in Rooms

Among the numerous ways in which urban fires begin are cooking accidents in kitchens and cigarettes falling on upholstery. The latter, for example, may start as smoldering, develop to flaming then involve propagation to furniture and to other items in the room. The fundamentals that have been presented here enable estimates to be made of rates and times for each of these processes if enough information on materials and geometry is available. The modeling approaches 6s calculate these histories on the basis of principles like those that have been presented. The calculations may demonstrate that certain materials or material arrangements are unusually hazardous. These predictions may be tested by experiment. Special hazards associated with newer materials such as polyurethane foams may be clarified in this manner.

15.2. Flashover

Experimentally and also in modeling it is found that often during fire growth in rooms a time is reached at which the fire suddenly spreads very rapidly to encompass the entire room. This phenomenon is termed ftashover; the time interval between ignition and flashover is the flashover time. Flashover times are important in determining the time available for fire suppression.

A variety of mechanisms may be responsible for the existence of flashover. One mechanism is radiative, involving approximately uniform heating of the materials in the room to the ignition temperature. Another mechanism may involve fire growth until flames reach the ceiling, at which time they often move rapidly to other parts of the room because of consequent modifications in convection. An objective of modeling often has been to predict the flashover time on the basis of such mechanisms.

15.3. Accuracies of Modeling Predictions

Often it is difficult to predict fire histories and flashover accurately. In one test a bed was ignited adjacent to a plastic curtain that covered a window. After about ten minutes, the curtain melted and fell to the floor without igniting. The bed burned for some time more, but the fire finally died without developing flashover. It was estimated that if the bed had been 5 cm closer to the curtain then the curtain would have ignited before falling, thereby causing flashover in the

352 F.A. WmLIAMS

room. Thus, the occurrence of flashover may depend critically on particular dimensions.

Modeling cannot be sufficiently precise to predict important phenomena such as flashover with accuracy in all situations. It is best to seek only qualitative indications of fire hazards by exercising models.

15.4. Motion of Smoke and Flames in Corridors

After flashover fires often become ventilation-controlled and spread smoke and flames to portions of the structure outside the room. Calculation of smoke motion through corridors and chambers is a problem in fluid mechanics subject to modeling. This motion can be influenced appreciably by small changes in conditions such as the degree of opening of a door or a window. The modeling can be performed with reasonable accuracy for simple geometries, but for a building with many ducts, corridors and rooms the predictions usually have large uncertainties. The motion of smoke and of other combustion products is important because it strongly affects the ability of people to escape from the burning structure. Many fire deaths may be traced to entrapment by smoke and inhalation of toxic products of combustion. Modeling smoke motion may contribute to defining improved procedures for escape from fires in buildings and to developing associated protective measures.

15.5. Burning Structures and Mass Fires

Modeling the burning of a building totally afire in some ways is simpler than modeling the progression of the fire through the building. Interior details may be neglected and the building treated as a fuel element. Calculations then may be made of necessary conditions for adjacent structures to ignite. These fire- spread calculations also are subject to inaccuracies, as indicated in Section 11. It is important to be able to estimate suppressive measures needed to prevent spread. This is a challenging and undeveloped area of fire modeling. In practice actions are based on judgment conditioned by past experience.

If a fire spreads to a number of structures a mass fire may develop (Section 1.4). Modeling mass fires is a difficult task. 67 A few concepts are available, but routines for implementation on computers are not. The occurrence of fire storms and of conflagrations in urban environments has much in common with the occurrence in wildland fires.

16. MODELING WILDLAND FIRES

Modeling approaches have been developed for predicting the histories of wildland fires. 68'69 Uses include improvements in systems for fire-danger ratings, establishment of better strategies for fire suppression and improvement in planning of prescribed burning of forests to reduce fire hazards.

Although urban and wildland fires have many attributes in common, there are significant differences. For example, the fuels differ, living fuels being more prevalent in wildland scenarios; the fuel type

influences the burning behavior and the efficiency of suppressants. The effects of atmospheric conditions on fires may produce different results in urban and wildland settings. For example, low humidity with abundant fuel enhances fire danger, and this condition most often is encountered in late summer in wildlands but in winter (especially indoors) in urban environments. Fuel, atmosphere and topography have been cited as major factors controlling the behavior of wildland fires; topography is less relevant for most urban fires which are influenced instead by relative arrangements of fuels. Firebreak concepts are important for management of wildland fires; they also play a role in fire-safe design of urban areas but are applied there less often.

16.1. Types of Wildland Fires

Wildland fires are divided into three classes, ground fires, surface fires and crown fires.

Ground fires burn on the ground and consume organic material beneath the surface litter. There are many examples of ground fires, e.g., fires in peat bogs; they are difficult to detect and also to extinguish.

Surface fires burn surface litter, logs, grass and other material on the surface. They are the most prevalent type of fire and dominate the cost of wildland firefighting.

Crown fires burn in the tops of trees and shrubs. They move rapidly, occasionally independently of surface or ground fires, and they possess the greatest destructive capacity.

All three types of wildland fires are subject to analyses concerning their behavior and their effects.

16.2. Effects of Wildland Fires

Wildland fires away from inhabited areas are not necessarily detrimental in an overall sense. There are regions in which natural fires are an integral part of the life cycle of the vegetation (Ref. 70). The fires consume excess cellulosic material and produce heat, smoke and ash that can be beneficial ecologically. There exists, for example, a flowering plant in Southern California that grows and blooms only after the ground above its seed has been heated by passage of a fire. Effects of fires become undesirable for example when they are sufficiently severe to destroy large trees (e.g. crown fires), when they threaten inhabited areas or when they detrimentally upset ecological balances (e.g. increasing flood propensity). Large forests fires can spread to urban areas, producing extensive damage and requiring cooperative suppressive measures by urban and wildland firefighters. Controlled, prescribed burning in well- planned wildland management can aid in the wildland's need for natural fires and help to reduce the danger of occurrence of undesirably destructive fires.

16.3. Factors Influencing the Behavior of Wildland Fires

Many of the factors that influence the behavior of wildland fires also affect histories of urban fires.


Relevant factors include the composit ion of the fuel, its inorganic and moisture content, fuel loading (mass per unit ground area), fuel-element size, fuel-element shape (surface to volume ratio), the packing fraction (porosity or packing ratio) of the fuel bed, unevenness in fuel distribution, continuity of fuel elements, humidity of the atmosphere, wind velocity, wind direction, local circulation in the atmosphere, the slope of the terrain, elevation, irregularities in the topography and the overall configuration of the wildland. Most of these factors have been taken into account in the models of wildland fires. For example, it has been found that the response of fine natural fuels to changes in temperature and humidity (i.e. absorbed moisture) explains much of the variation in wildland fire behavior, and the intrinsic free moisture of living vegetation often inhibits fire spread.

16.4. Character o f Wildland Fire Modelin 9

Since spread is the principal practical concern in wildland fires, the modeling is mainly that of fire spread (see Section 11). As with urban fires, it is often impossible to predict the history of a specific fire accurately. Phenomena of concern to firefighters, such as the occurrence of a blowup (a sudden, unexpected increase in fire intensity or spread rate, often accompanied by firewhirls or by long-distance transport of firebrands resulting in spot fires ahead of the fire front) cannot be predicted well by modeling. However, computat ions may be performed to provide a general idea of the fire behavior and also of the influences of changes in parameters on that behavior. Analyses of the type introduced in Section 11 underlie the modeling; rough estimates for some of the characteristics computed, such as spread rates, may be obtained by the methods of Section 11. Modeling of urban and wildland fires is a method for achieving a better understanding of their phenomenology.

Acknowledgement--I am indebted to a number of people who participated in the course on which this article is based and who contributed information used in preparing the article. These include Frank Albini, Robert Levine, Daniel Olfe, Brady Williamson and Carl Wilson. Deserving of special recognition is Abe Broido, who not only helped immeasurably with the concept and material for the course but also contributed numerous incisively stimulating observations on a broad range of fire topics over many years.

Continued support of research efforts in the fire area by the National Bureau of Standards, the Naval Research Laboratory and the National Science Foundation during the preparation of this manuscript is gratefully acknowledged.

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