Mechanisms of fire spread

MECHANISMS OF FIRE SPREAD F. A. WILLIAMS

Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, Cahfomia 92093 Mechanisms involved in many types of fire spread are described in a manner that sacrifices accuracy for the purpose of emphasizing general aspects of the underlying heat-transfer, fluid-flow and chemical-kinetic phenomena. Consideration is given to conditions for transition from one mode of propagation to another. Research on fire spread has been pursued intensively in recent years, and in the present contribution an attempt is made to provide a framework within which various studies can be placed. Entries to current literature are provided. Areas of apparent importance that do not seem to have been emphasized are suggested.

1. Introduction From smoldering upholstery to woodland wildfires, fire spread is of concern to those interested in fire safety. Improved understanding of mechanisms by which fires spread can aid in prevention and control of nonstationary fires. Therefore fire spread has been an active area of research. One objective of this paper is to identify recent publications on the subject, thereby providing sources of reference for those wishing to delve more deeply into the intricacies of the field. Other reviews provide additional entries to the literature. ~-~ In an effort to develop some understanding and as a method for systematizing the presentation, a simplified description of spread processes is defined here and is employed to categorize different modes of spread. A focal point of the discussion will be the spread rate since this is the principal quantity that studies wish to ascertain. The discussion will be mainly theoretical, directed toward identifying mechanisms; experimental methods will not be reviewed even though significant advances in experimental techniques have beeta made recently. 6 Theoreticians often attempt to include all potentially important phenomena in their models of fire spread, believing that they cannot properly describe the process if something that contributes is neglected. Nevertheless, there is merit to the opposite view which holds that the best avenue for developing

understanding is to neglect all but essential phenomena and to study thoroughly limiting cases in which different phenomena are controlling. With the objective of facilitating the latter approach, an effort is made herein to define boundaries of regions having different modes of fire spread. Although this can be done conveniently only on the basis of a highly simplified description, identification of the main phenomena may be followed by accurate analysis of the resulting simplified problem and then, if desired, inclusion of additional phenomena as perturbations. It will be seen that such an approach can, for example, explain why downward flame spread along sheets occurs by different mechanisms for such similar materials as cellulose and polymeth~methacrylate.

2. Fundamental Concept of Fire Spread Clearly, fire spread is a meaningful concept only under situations in which both burning and nonburning combustibles can be identified. Here "burning" is used in its most general context to include processes ranging from glowing combustion to vigorous flaming. Fire spread can occur only if there is some type of communication between the burning region and the nonburning fuel. The critical problem in ascertaining mechanisms of fire spread is to establish the character of this communication.

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For purposes of analysis it is necessary to define the boundary between the burning and nonburning combustible, herein termed the surface of fire inception. The communication responsible for fire spread consists of transport of something across the surface of fire inception. Although in principle for small, slow fires that "something" could be active molecular species, in all real cases known to the author it involves heat in one form or another, conduction, convection, radiation, burning liquid or hot firebrands. Therefore the first problem in describing fire spread lies in identifying the major mode of heat transfer across the surface of fire inception.

densed fuels that pyrolyze the value of T~ is controlled by chemical kinetics of pyrolysis and increases slightly as V increases. Except in spread of nonflaming combustion, piloted rather than spontaneous T[s usually are applicable to fire spread. When phase changes of the fuel prior to attainment of the ignition temperature are negligible energetically, 2Lh = c(T i - To) , where T Ois the initial temperature of the fuel and c its average heat capacity (per unit mass) between temperatures T O and T~. In fact there are a number of situations in which the nature of the fuel may change significantly (and to an extent dependent on the type of fire) prior to reaching T~, so that formulas more complex for Ah are appropriate if known.

2.2 Fundamental Equation of Fire Spread

2.4 Discussion

Complete knowledge of the mode of heat transfer is equivalent to specification of q, the net energy per unit area per second transported across the surface of fire inception. The spread rate V is expressible in terms of q through application of an energy-conservation principle if the additional concept of an ignition temperature is introduced. With p the fuel density and Ah the difference in thermal enthalpy (per unit mass) between the fuel at its ignition temperature and the virgin fuel, energy conservation is written as

Although theories of fire spread often employ Eq. (1) directly, there are many analyses in which it does not appear explicitly. Main examples of the latter for small-scale spread are studies in which differential forms of conservation equations are solved. Even if Eq. (1) goes unmentioned, the results can be interpreted a posteriori in terms of the fundamental equation. At larger scales, geometrical aspects of analyses must differ, depending for example on whether the fuel is continuous or discrete. For discrete fuels fire often moves in jumps and Eq. (1) is applicable only in an averaged sense, V being the raio of the jump distance to the time interval between jumps. Very large fires that propagate principally through spotting produced by firebrands fit rather unnaturally into Eq. (1) since V is related to the distance and time over which transport of active firebrands occurs; it becomes necessary to associate with firebrands an artifical energy of pAh. In what follows use will be made of Eq. (1) to discuss fire spread at successively larger scales within each of three geometrical categories. But first, application of Eq. (1) for identifying dominant spread mechanisms will be considered.

2.1 The Surface o f Fire Inception

oVah = q.

(1)

The importance of this relationship, which may be termed the fundamental equation of fire spread, has been discussed clearly by Emmons.V 2.3 Ignition Temperature Although the concept of an ignition temperature T~ perpetually has been shrouded in controversy, there is nothing wrong with this useful idea when it is properly applied. The important point to realize is that the value of this quantity, herein defined as the maximum fuel temperature on the surface of fire inception, depends on the dynamical situation. Sufficiently thorough analyses that include chemical processes result in predictions of T d in many applications less thorough analyses suffice, empirical estimates being employed for T~. Values of T i vary widely, ranging from nearly adiabatic flame temperature for flame spread through premixed gaseous combustibles to less than the boiling point for flame spread over volatile liquid fuels. For con-

3. Definition of Dominant Spread Mechanism For any given spreading fire, different modes of heat transfer across the surface of fire inception will have different values of q associated with them. It seems logical to define the dominant spread mechanism as the one having the transfer mode that produces the largest contribution to q. Except for borderline

MECHANISMS OF FIRE SPREAD cases in which two or more modes contribute equally, it should be a good first approximation to neglect all communication other than the dominant mode. From Eq. (1) it is seen that the dominant mechanism alternatively may be defined as the one producing the largest value of V; fires will spread as fast as they can. Through comparison of q or V for different spread mechanisms, the dominant mechanism can be ascertained. Sometimes this can be done experimentally, by measuring q's associated with different modes of heat transfer in the propagating fire. Usually such measurements are very difficult to perform with sufficient accuracy, and therefore mechanisms are inferred either indirectly or theoretically. An example of an indirect inference would be to measure the dependence of the spread rate on pressure, temperature, oxygen content of the atmosphere, material properties of the fuel or geometrical configuration and to compare the results with theoretical predictions based on different spread mechanisms. Purely theoretical determination of dominant mechanisms is aided by calculating q's or V's for different mechanisms individually with all other transfer modes neglected, then evaluating conditions under which these quantities are equal for various pairs of mechanisms. Some such calculations will be given in subsequent sections.

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flame propagation ignites the material its surface continues to regress normally under the influence of a burning process which is distinct from that of spread. Although Eq. (1) is sufficiently general to be applied approximately to normal burning as well, this application is not considered herein, except in cases termed one-dimensional, for which the spread process is in fact normal burning.

4. One-Dimensional Aspects of Fire Spread 4,1 The Role of Chemical Kinetics The simplest example of the approximate application of Eq. (1) comes from laminar flame propagation in a premixed gaseous fuel (e.g., Ref. 8). This example also affords the opportunity of introducing another basic principle sometimes needed to describe fire spread properly, viz. equating the chemical time Tc with the residence time % in a critical zone of the gas phase. The role of chemical kinetics in fire spread can be two-fold. For condensed fuels there are cases in which chemistry controls gasification rates, the pyrolysis rate of the fuel being an important rate process. In addition, the rate at which the gaseous fuel ignites is controlled by chemical kinetics and sometimes influences spread rates. This latter effect is relevant here.

3.1 Classes of Spread Equation (1) is applicable most directly for one-dimensional spread in which conditions are fairly uniform on each side of the surface separating burning and nonburning fuel. Examples approximating this case in some respects are considered in the following section. More often, two-dimensional or three-dimensional characteristics of the spread are essential. In such cases use of Eq. (1) often entails application of physical principles to identify the portion of the virgin fuel that participates in fire spread. For example, for fast-moving crown-fires the bulk of the trees are unaffected untii after spread has occurred. In subsequent sections examples of these more complex types of spread are considered, classified according to whether the fuel is treated as being continuous or discrete. 3.2 Spread and Normal Burning For two-dimensional spread it is important to distinguish between the spread process and normal burning. For example, in flame spread over the surface of a thick polymer, after the

4.2 Conductive Transfer The mode of forward heat transfer for the laminar flame is heat conduction. Hence

q = Xg(T~ - To)/l,

(2)

where h is the thermal conductivity of the gas andgl is a length characteristic ,of the temperature gradient. To eliminate l, the approximation % -= l / V may be introduced, whence % = ~c produces l = xcV.

(3)

Expressions for % are well known (e.g., Ref. 9). For the gaseous flame, the temperature T~ appearing in the Arrhenius factor of Tc is approximately the adiabatic flame temperature. Combination of Eqs. (1) through (3) produces the well-known formula V = (k~/pc%) 1/~ for the flame speed. It may be noted that in this example the gas-phase chemistry enters in finding the characteristic length for heat conduction and contributes an inverse dependence of q upon V.

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4.3 Smoldering

4.5 Configurations of Smoldering

Smoldering combustion of mattresses, upholstery, sawdust, wood shavings, paper, cardboard, fabrics, cigarettes and incense affords a second example which often may be viewed as being approximately one dimensional. A certain amount of experimental data, 5,1~ as well as some theoretical analysis (e,g., Ref. 13) can be found on the subject. It may be remarked that rather detailed theory having exothermic smoldering as a surface reaction has been completed, 14,15 the latter 15 including a diffusion flame downstream from the smoldering region, but these are twodimensional analyses more properly viewed as first steps toward treatment of flame spread; they ignore surface regression in the smoldering region, and results are not compared with experiments on smoldering.

Two distinct configurations can be considered for smoldering combustion. In one the fuel is porous and oriented so that the product gases percolate through it. Examples would be a cigarette during a puff or a sawdust pile ignited from below. If heat losses (radiative and conductive) from the glowing surface can be ignored, then in this configuration Q = q by energy conservation, and Eqs. (1) and (4) yield V = H o P o D o/lgpAh for the spread rate, which typically is of the order of 10-3/l~cm/s, with l~ in cm. This result relates the spread rate to the~diffusion length, usually on the order of a lateral dimension of the fuel bed or of a fuel element. Since this configuration has the maximum q, it corresponds to the maximum rate of spread, and the result illustrates how slowly smoldering propagates. In the second configuration the product gases do not flow through the unignited fuel, and the mode of fuel heating is conduction or possibly radiation. Examples include downward propagation of glowing combustion on paper sheets, burning incense and the cigarette between puffs. Cases having convective augmentation may be identified, such as upward propagation of smoldering along a cotton thread, in which convection may return a significant fraction of Q to the unignited fuel. Heat losses can influence V appreciably in the second configuration, since q = Q qL, where qL includes losses from the burning surface and also from the unignited fuel. Differing dependences of V on geometrical and environmental parameters can arise through different modes of qL" That qL can depend explicitly on V may be seen, for example, by observing that when conduction is the main means of heating the fuel, the analog of Eq. (2) for the solid, in conjunction with Eq. (1), provides a V-dependent heated-length l needed for estimating the lateral area over which qL occurs. Smoldering often is influenced by processes, such as this heat loss from the side, which are not one-dimensional.

4.4 Diffusion-Controlled Glowing and

Extinction In principle, smoldering reactions can occur in two different regimes, diffusion-controlled or kinetic-controlled. In the former case the oxygen concentration at the burning surface is small compared with the ambient oxygen concentration, and the rate of heat release is controlled by the rate of diffusion of oxygen to the surface, a non one-dimensional aspect of the phenomenon. If H o is the heat released in smoldering per unit mass of oxygen consumed (beginning with fuel and oxygen at ambient temperature) and D O the diffusion coefficient for oxygen, then the energy liberated per unit area of the burning surface per second in the diffusion-controlled regime is

Q= HoPoDo/Ig,

(4)

where Po is ambient oxygen density and lg a characteristic length for diffusion in the gas. With kinetic control, the mass fraction of oxidizer in the gas at the burning surface is approximately the same as that in the ambient gas, Yo, and the rate of heat release depends strongly on the burning temperature (equal to the ignition temperature T,) which appears in an Arrhenius expression for the rate of heat release, Q = HoY"oA e x p ( - E / R T i ) . In both regimes, an energy balance for the burning surface determines T,. Often combustion in the kinetically controlled regime is unstable, '6 causing Y~A exp(-E/RT~) < poDo/l, to define an approximate criterion for extinguishment of smoldering. Therefore only Eq. (4) is employed in the following discussion.

4.6 Radiation-Controlled Spread through

Porous Beds Modeling has been developed for flame spread through porous beds of fuel composed of fairly uniform elements, ranging from excelsior, grass and pine needles to wood cribs. 4,17-~4 There are many one-dimensional aspects to these spread processes, especially for fine fuels having either close packing or a large depth of the bed, and excluding wind-

MECHANISMS OF FIRE SPREAD driven or up-slope propagation. Spread is seldom slow enough for transfer to be controlled by heat conduction (Eq. 2); often radiation from burning embers or from flames within the bed is the dominant mode. In such cases, if both burning and unignited fuel are sufficiently extensive to be optically thick for thermal radiation, then q = ebcrb(T~ -- To4),

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radiation from flames above the bed, convective cooling of unignited fuel by air entrained from below, from ahead or from the sides of the bed, and convective heating of unignited fuel by hot gases in the upper portion of the bed. These effects are wind-dependent and slope-dependent and are likely to be of significance in many real-world situations, such as during propagation of crown or grassland fires.

(5)

where ~ bis an emissivity of the burning region, crb the Stefan-Boltzmann constant and Tb a suitable average temperature of the burning zone. When Eq. (5) is used in Eq. (1) to obtain a simple expression for the spread rate, it must be recognized that p = Kp~, where p~ is the specific gravity of the fuel elements and K denotes the packing fraction, the ratio of the fuel volume to the total volume of the bed. In the resulting formual for Vthe least certain quantity typically is Tb, the value of which often appears to be around 1000~ 4.7 Thermal Thickness of Fuel During

Spread If the fuel elements are thick or have a distribution of size, then their interiors may not be heated appreciably prior to arrival of the flame, and a modification to the spread-rate formula arises. In terms of the characteristic length 1to which heat penetrates the unignited bed, the time of exposure of a fuel element prior to flame arrival is t = l~ V, and transient, one-dimensional conduction of heat causes the element to be heated to a depth of order = N/-~, where a = h~/p~c is the thermal diffusivity of the fuel. If characteristic dimensions of all elements exceed 8, then in Eq. (1) p = cr~p~, where cr is the surface-to-volume ratio for the fuel elements. There results V = q~ / (Ah2p ~~ 2K~a l), which is applicable only when the value of V so obtained exceeds that found by use of the simpler formula p = Kp~. The value of l must be known in order to apply the last result; when transfer is radiative, the product of the number density of fuel elements (roughly Kcr2/a for sticks of length a) with a representative absorption cross-section for a fuel element (e.g., a/cr) approximately equals the reciprocal of l, and typically l - 1/(K~). 4.8 Complications in Spread through Porous

Beds Non one-dimensional aspects of spread through porous beds include influences of

4.9 Ventilation-Controlled Fires There are one-dimensional aspects to certain fires in enclosures, particularly to fires in ducts or in long corridors. 25,~6 Such fires can be ventilation-controlled in the sense that the total rate of heat release is proportional to the total flow-rate of oxygen. With quantities referred to a unit cross-sectional area of the duct, under ventilation-controlled conditions Eq. (4) is replaced by Q = HoYom A, where m a is the mass" flux of air into one end of the tunnel. Often losses are mainly attributable to conduction through the walls, and with q = Q - qL, Eq. (1) again gives the spread rate, where p is now the mass of fuel per unit volume in the duct. The fuel typically consists of a relatively thin lining on the wall, so that P is much less than the specific gravity of the fuel, and spread rates can be very high, comparable with the air velocity at the inlet. Just as in the case of the porous bed, thick fuels may be heated incompletely during spread; transient conduction may determine an effective value of p. It may be remarked that corridor fires often do not achieve the idealized conditions considered here, there being non onedimensional influences from backflow and from starting transientsY

5. Fire Spread along Continuous Surfaces

Upward-spreading and downward-spreading fires constitute two distinct classes of fire propagation along continous fuels. The former class, defined more precisely as fires for which the unit vector lying in the fuel surface and pointing in the spread direction has a positive vertical component, is characterized by rapid and acceleratory spread. The latter class, for which this vector has a negative vertical component, typically is slow and steady. Horizontal spread, except for cases of the ceilingfire type in which flames are below a combustible sheet, 9 closely resembles downward spread and is discussed here under that category, which will be considered first.

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5.1 Downward Spread over Solid Fuels Downward and horizontal spread over solid fuels has been a topic of intensive research in recent years, a,6,9,~4,~s,2s-s~ Four principal modes of transfer producing propagation have been identified--heat conduction through the gas, heat conduction through the solid, radiation from the flames and motion of burning fuel. These will be considered in order. Heat conduction through the gas is the dominant mode for flames spreading down sheets of paper, for example. 35 Theories have been developed for predicting rates of spread by this mechanism (e.g., Ref. 29). For spread over a thermally thin sheet of thickness L and width w, whose back face is insulated, Eq. (1) may be multiplied by L w to obtain the total energy per second required for ignition. It may be assumed that this energy is transmitted by conduction from a flame located a normal distance d from the surface of the sheet, typically 1 ram, over an average distance l in the tangential direction, upstream from the surface of flame inception. Therefore L w q = lwh g(Tf - Ti) / d is a reasonable approximation, where h is the thermal conductivity of the gas and ~Tr the flame temperature. It may be noted that-since the existence fo this forward conduction in the gas phase violates the boundary-layer approximation for gas flow parallel to the surface, gas velocities near flame inception must be sufficiently small if this mechanism is to be operative. Under these conditions l and d will be of the same order of magnitude, and it is therefore found that

q = X.(T~- T,)/L,

(6)

which may be substituted into Eq. (1) to obtain the spread rate. Just as with spread through porous beds, if the fuel is too thick then it is not all heated before flame arrival, and in Eq. (6) L is replaced by ~ = [ ( l / V ) (k/pc)] 1/2 whenever 3 < L. In this last expression, the length I which appears in the heating time l~ V is approximately l = d; in an Oseen approximation 29 it is related to the gas velocity V~ with respect to the flame by l = h~/p~cgV~ arid results in the formula V = V~p~c X IT, 2 ~ g g f - T~)2/pc~ks(Ti - To) for the spread rate. Heat conduction through the solid is the dominant mode for flames spreading down sufficiently thick sheets of polymethylmethacrylate, for example. 9 A detailed theory has been developed for predicting the rate of spread according to this mechanism. 49 Equation (2) becomes applicable for q with kg replaced by the conductivity k, of the solid;

however, the result of combining the modification of Eq. (2) with Eq. (1) is not a closedform expression for Vbecause I remains undetermined. Unlike spread by forward conduction of heat through the gas, but rather like propagation of a premixed flame in the gas, Eq. (3) with V being the gas velocity Vg must be applied to the gas adjacent to the pyrolyzing fuel in order to obtain 1. This introduces chemical kinetics of the gas-phase reactions into the spread rate, an effect not present in control by gas-phase conduction. The analysis becomes somewhat involved because the rate of the gas-phase reaction depends on the gasphase concentration of fuel which in turn depend strongly on Ti, a surface temperature of the fuel determined by diffusion-flame. combustion downstream. A simplified treatment, applicable if (a) Ti is high enough for the gas-phase reaction to occur in an ignition mode at Ti rather than in a premixed-flame mode at Tf and if (b) the ratio of a solid-phase conduction-length 44 to a gas-phase length for buoyancy or forced convection is comparable with or larger than the square of a nondimensional activation energy for gasification, has been given earlier. 9 The modifications needed if either of these two conditions are violated, also have been indicated. 49 If the first condition (a) is violated, it is merely necessary to evaluate r c at TI rather than at T~, keeping in mind that the fuel concentration still is established through T~. If the second condition (b) is violated, then buildup of fuel and ignition occur in a thin sublayer of the gas-phase boundary-layer adjacent to the fuel surface, and predicted dependences of the spread rate on experimentally adjustable parameters thereby are modified. The resulting modifications have been summarized. 49 Radiation from the flames is the dominant mode of flame spread for horizontal propagation over carpeting materials in the presence of a certain amount of externally applied radiation, for example. 46 There are a number of theories for including effects of flame radiation on spread rates (e.g., Refs. 29, 46, 51 and 52). Formulas like Eq. (5) but with suitably defined overall shape-factors are applicable. A rough equation is q = ~f~bT~hf sin O f ~ L ,

(7)

where the flame height h# its emissivity ~f and its average angle with respect to the fuel surface Of all may be subject to significant uncertainties in estimation. Here again, L must be replaced by g if g < L; the formula for following Eq. (6) remains applicable with

MECHANISMS OF FIRE SPREAD l now a forward length of radiant heating, typically of order h i . The value of h i depends on the total rate at which mass of the ignited portion of the fuel is consumed. 53's4 Use of Eq. (7) in Eq. (1) provides the spread rate directly; in this respect spread by radiant transfer is similar to spread by gas-phase conduction, as described by Eqs. (1) and (6). A sample result for thermally thick fuel is V = (~fabT))2hf sinZOf/ P~Gh~(Ti - To) 2. Motion of the burning fuel is a dominant mode of spread for downward propagation of flames over vertical sheets of polyformaldehyde, for example, s Such processes have not been analyzed theoretically. It is possible to construct simple models if hypotheses are made for the physical processes involved. Many thermoplastics melt and drip while burning. If a horizontally oriented cylinder of melt having diameter 8 is assumed to flow down the surface of the solid fuel under the influence of gravitational acceleration g (in the direction of motion), then a balance of the weight with the viscous force acting on the cylinder gives plSeg = txl V, where Pl and P~I are the density and viscosity of the melt, and where the velocity gradient has been approximated as V/8. Use of the previous formula for 8 (with l = 5), viz. 8 = kl/ptctV, then gives

V= [(hl/p~tct)2 Oxt/pt)g] 1/3

(8)

as an estimate of the rate of flame spread. Spread by dripping often is irregular, and therefore Eq. (8) can provide the average rate, at best. In addition, other physical effects, such as shrinkage under surface tension, also can produce significant motion of burning fuel under certain conditions. 37 Thus, it is difficult to obtain quantitative descriptions of some of the modes of spread that are observed. 5.2 Limits for Different Mechanisms of

Spread Boundaries between regimes of spread by the four mechanisms just discussed can be defined from the results obtained. To compare heat' conduction through the gas with that through the solid, use is made of Eq. (6) and of Eq. (2) with h~ replaced by h~. For thermally thick materials L is replaced by the thickness of the heated layer of the solid in Eq. (6), and it is easily shown 44 that for control by solid-phase conduction this thickness is approximately 8 = h~ / p~G V = l, for the l which appears in Eq. (2). It therefore follows that

G(T~ - T,) > ~ ( T , - To)

1287

(,~)

is the criterion for gas conduction to dominate solid conduction in thermally thick materials. The appropriate value for T~ here is that near the point of flame inception, whieh will be close to the extinction temperature for the diffusion flame. For cellulosic-type fuels burning in air, extinction temperatures tie in the vicinity of 1300~ and thermocouple measurements of burning paper 3'~,4'5 show flame temperatures near flame inception to be in the vicinity of 1400~ Use of this last value, along with T i = 370~ and X~ = 2 x 10 4 cal/em sOK for cellulose 31'a5,52 and T i = 395~ and h = 4.5 x 10 -4 cal/cm s~ for polymethylmethacrylate, 9 gives as the ratio of the right-hand side to the left-hand side of Eq. (9), the values 0.67 for cellulose and 1.66 for polymethylmethaerylate. Thus, it becomes reasonhble that these two materials exhibit different modes of flame spread; the higher thermal conductivity of polymethylmethacrylate is the main cause. When the sample becomes thinner than a, the length in the denominator of Eq. (6) decreases, but that in the denominator of Eq. (2) does not; therefore there is a tendency for sufficiently thin samples to have spread rates controlled by heat conduction through the gas. For polymethylmethaerylate, estimates indicate that this change-over occurs at LV = 6.7 • I0 -4 em2/s, which corresponds to a halfthickness L = 0.04 em for downward spread. 9 From Eqs. (6) and (7) it may be seen that the criterion for conduction through the gas to dominate flame radiation is

h~(Tf - T~) > e w b T ~ hf sin O f.

(10)

This is mainly a condition on flame height, which in turn is a condition on total rate of fuel consumption. Typically radiation can be dominant in a heated environment but seldom for an isolated fuel-element initially at room temperature, even if it is very massive. Similar comparisons easily are made between solid-phase conduction and flame radiation, between gas-phase conduction and fuel motion, etc. In the latter case the spreadrate V, rather than q, is employed in the comparison, because the mechanism of spread by fuel motion is interpretable only artificially in terms of q (through Eqs. 1 and 8). 5.3 Effects of Curvature Geometrical factors can modify mechanisms

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of downward or horizontal spread over continuous but nonplanar surfaces from those operative over flat faces of fuel sheets. It is well known that flames often spread preferentially along edges of combustible materials. 34 There are two reasons for this, one gas-phase and one solid-phase. In the gas, convex curvature facilitates accesses of oxygen to the flame and increases the ratio of the flame area to the surface area of the fuel, thereby enhancing heat input from the flame per unit surface area of the fuel. The consequent increase in the rate of heat transfer per unit area of the fuel surface tends to augment heat input from the flame as well as heat loss from warmed but unignited fuel not blanketed by flame. In the solid, convex curvature reduces the area available for heat conduction in the direction of flame propagation, thereby reducing the amount of solid-phase conduction. Due to this latter effect, a thermally thick material may become thermally thin at its edges; if the material is thermally thin away from its edges, then of course the solid-phase effect is absent. For polymethylmethacrylate it has been observed 41 that rates of flame spread down rods 1.27 cm in diameter are greater than rates of spread down thick sheets, a result attributable to an approach toward thermally thin behavior and also manifest in a greater tendency toward control of spread by gas-phase conduction instead of solid-phase conduction. There are other situations in which concave rather than convex curvature can enhance spread rates, for example by reducing rates of radiative and conductive heat-loss to the gases or by allowing hot gases to penetrate into otherwise protected pockets. 6,55 There are also special effects of a fluid-mechanical nature, e.g. recirculations, that can be induced by the geometrical configuration and can affect downward or horizontal rates of spread. 5.4 Horizontal S p r e a d over L i q u i d Fuels For flame spread over horizontal surfaces of liquid fuels, in addition to the mechanisms applicable to solids, spread may occur through liquid flow driven by surface tension. 56-62 Unlike solids which usually gasify kinetically, liquids typically maintain evaporative equilibrium at their surfaces. If the equilibrium vapor pressure at the inital temperature of the liquid is sufficiently high for the gas mixture at the surface to possess a fuel concentration above the lower flammability limit, then the liquid temperature is said to be above the flash point, and flames spread rapidly in a premixed fashion in the gas, as described by Eqs. (2)

and (3).61 If the liquid is cold and highly viscous, then transfer mechanisms may resemble those for solids. Between these two extremes, spread driven by the gradient in surface tension usually predominates. Consider a liquid fuel of depth h I subject to a gradient ~' = ( c k r / d T ) ( d T / d x ) in surface tension cr. Balancing the surface force with the viscous force gives as a characteristic liquid velocity,s9 interpreted as a spread rate, V = cr'h t/~xt, where ixt is the liquid viscosity. This result shows that V increases linearly with ht, but it is good only for quite shallow pools, typically breaking down at a value of h t between lmm and 1 cm. For deeper pools a boundary layer will develop at the surface, and an effective value for h z must be used in the formula for V. To estimate this effective value, assume that the specified or' extends only over the horizontal distance l and that ~r' = 0 beyond this distance. Then, since there is no surface force to be balanced beyond l, in a first approximation the velocity will be zero there. This necessitates there being a horizontal velocity gradient of order V~ l. A balance of inertial and viscous forces in the horizontal direction then provides an estimate for the effective value of h z, through pzVZ/l = ~ l V / h ~ . Use of this result in the previous expression for V gives

V = (~r'l~ z) ( ~ llplo") ~/3,

(11)

in which now only the horizontal scale l appears (directly and also through the estimate ~r' = ( & r / d T ) ( T i - To)/l). Consideration of heat transfer is needed for obtaining l. In principle, the liquid can provide the heat transfer determining l, but the Peclet number typically is so high that the resulting l, given by l = k l / p t c t V , is very small, less than 10 -~ cm. This means that the same types of transfer processes considered previously for solids, especially radiation and gas-phase conduction, are relevant. Often radiation from large flames over the burning pool is dominant, and l is of the order of the flame height hr. Complications may arise, e.g. if the liquid is transparent to radiation or if conduction through metallic sides of the liquid container is significant. An additional complication, sometimes of importance, is buoyant convection within the liquid. Since the hot liquid under the flame is less dense than that ahead, buoyant convection alone can produce flame spread, in the absence of gradients in surface tension. The pressure difference for the hot and cold liquid is of order g h t ( d p l / d T ) ( T i - To), where hI is the depth of the heated layer. Balancing

MECHANISMS OF FIRE SPREAD the resulting pressure gradient against viscous forces gives V = ( g h ~ / ~ z l ) ( d p t / d T ) ( T ~ - To),

(12)

which may be compared with the V in Eq. (11) by using the cube-root factor therein for h z. If the V from Eq. (12) is the larger, then buoyant convection predominates. 5.5 Upward Spread Buoyant convection in the gas, similar to that just discussed, is responsible for the rapid and irregular spread of flame along the lower surface of a horizontal sheet of a combustible solid. 6,9 Upward spread along continuous fuels is also unsteady and extremely rapid. 4,a2,6a-66 During upward spread flames bathe the unignited surface, grow in size and soon become turbulent. 636~ Studies of heat transfer from turbulent flames on surfaces ~7 become relevant. Theory 6a'~5'~6 shows that for infinitely thick materials upward spread is perpetually acceleratory, while for materials of finite thickness a constant but large rate of spread can be approached. A simplified model serves to illustrate these points. During upward spread, heat transfer will occur by radiation and conduction from the flame. The transfer rates on surfaces adjacent to the luminous flame are so great that transfer elsewhere can be neglected in comparison. If for simplicity only radiation is considered, then because of the different geometry Eq. (7) becomes q = er with ~ given by the formula after Eq. (6). The rough approximation h f / V m a y be employed in g for the time of exposure prior to ignition, whence by use of Eq. (1) it is found that V = [ef~rbT~/(T i - To)]2hf/(psCshs).

(13)

The time-dependent quantity in Eq. (13) is the flame height hr. Correlations for flame heights are available for various configurations 53"54 but are sparse for flames rising up surfaces. By analogy with theearlier results, it is reasonable to assume that he = a M n, where c~ and n are constants and where M is the total rate of mass loss of fuel per unit width of the sheet. Typically ~3,54 0.20 -< n -< 0.33 in the absence of upward surfaces, and although higher values (e.g., 0.8) occur for flames adjacent to surfaces,65 the value of n apparently never reaches unity. For fully developed spread up sheets of finite half-thickness L, the length of the burning portion of the sheet is approximately x

1289

= L ( V / r ) , where r is the velocity of normal regression of the burning surface. Since the total rate of mass loss per unit width is x times the rate of mass loss per unit area (p~r), the formula M = psVL may be used in the expression for hf, the substitution of which into Eq. (13) then determines the steady rate of spread V. For developing spread or for a sheet of infinite thickness, x is the distance from the bottom of the sheet to the burning front, and V = d x / d t . Use of M = p rx in the formula for hf, with r approximately constant, then transforms Eq. (13) to d x / d t = 13xn, where 13 is a constant. The acceleratory character of the spread is then evident from the solution x = [(1 - n) 13t] 1/~1-~1, for which V = 13 [(1 - n) 13t] ,,/~1-,~. As n approaches unity the rate of spread becomes exponential in time. More involved analysis produces more thorough results. 65,66 6. Fire Spread among Discrete Elements If the fuel elements are discrete, then the number of options open to the fire for selection of spread mechanisms is reduced. In particular, heat conduction through the fuel seldom can be the cause of item-to-item propagation, although occasionally it can be a contributing factor. The principal mechanisms usually are radiation and convection of hot gases and of burning particles. Comparisons of the type expressed by Eq. (10) typically are most relevant. The complexities associated with Tc and with liquid flow tend not to arise. 6.1 Small-Scale Spread through Discrete Fuels There have been a few studies of fire spread at small scales involving discrete fue*ls. For a linear array of vertically oriented matchsticks, radiant effects are small and convective transfer from the flame on a burning stick to an adjacent, unignited stick is the main mode of propagation. 6s There is a critical relationship between stick height and stick spacing that must be satisfied for spread to occur. The overall rate of spread can depend on rates of propagation along continuous surfaces if, for example, spread along the continuous surface is needed to bring the flame sufficiently close to an unignited, discrete element to cause its ignition. The rate of spread along the continuous surface can be enhanced by small amounts of energy input from adjacent elements that are burning. Materials tend to deform because of outgassing as they burn, and this can en-

1290

FIRE AND EXPLOSION RESEARCH

hance or retard the rate of item-to item spread by m o v i n g the flames w i t h respect to u n i g n i t e d elements. A heated u n i g n i t e d element also can deform and b e n d closer to the flames, t h e r e b y igniting sooner. F o r spread along arrays of vertically oriented sheets of paper, flames t e n d to be larger a n d radiation can become more important. 69 6.2 Large-Scale Spread through Discrete

Fuels Within enclosures 4,27,7~ configurations Often are such that in principle fires can spread along continuous surfaces of combustibles, b u t yet observed histories are such that discrete steps usually a p p e a r to occur, with additional items b e c o m i n g i n v o l v e d in fire at each step. The process may r e s e m b l e item-to-item spread, complex radiative a n d convective patterns being responsible for the propagation. It is a challenging p r o b l e m to predict in advance the sequence of item involvement for a given initiation of fire in a given enclosure. For fires in structures having a n u m b e r of enclosures, 72 it can also be challenging to attempt to predict the sequence and rate of spread between enclosures. Effects of w i n d , fuel type, h u m i d i t y (as it influences the fuel), fuel loading and structural configuration are significant. 4 These same effects, with terrain i n c l u d e d in structural configuration, are significant for large-scale fires in wildlands. 74,75 Radiation from flames can be the major transfer m e c h a n i s m in spread of w i l d l a n d fires; methods are available for estimating spread rates b y this mechanism, effects of firebreaks, etc. 7 Convective transfer can enhance spread rates of w i n d - d r i v e n or up-slope fires. Large fires with well d e v e l o p e d convection-columns can spread much more r a p i d l y through a mechanism of f i r e b r a n d transport. Statistical approaches may be n e e d e d to describe influences of convective transport of firebrands on spread rates. 4,76 Large-scale spread among discrete elements of fuel is not understood well today. 7. C o n c l u d i n g Remarks The intensive research on fire spread in recent years has been focused mostly on steady spread over continous fuels. M t h o u g h m u c h work remains to be done in this area, 49,61 it w o u l d seem that u n k n o w n s in other areas are greater and now deserve more attention. Areas most worthy of e m p h a s i s appear to be configurational e n h a n c e m e n t of spread rates, such as edge effects; u n s t e a d y spread, especially

acceleratory, over continuous surfaces; spread among discrete elements at small and large scales; and spread d u r i n g fire suppression. Many small-scale, laboratory studies can be devised to investigate spread among discrete fuels or spread u n d e r well-controlled conditions involving retardent-containing atmospheres or fuels. Such studies can be subject to detailed, scientific analysis while at the same time improving u n d e r s t a n d i n g of practical problems in fire spread. The broad topic of fire spread needs c o n t i n u e d emphasis because of its impact on fire suppression.

Acknowledgment This work was supported by the National Science Foundation through the program on Research Applied to National Needs, Grant GI-34775. REFERENCES 1. FRmDMAN,R.: Fire Res. Abst. Rev. 10, 1 (1968). 2, WILLIAMS,F. A., BAaREaE, M. AND HUANC, N. C,: Fundamental Aspects of Solid Propellant Rockets, pp. 522-527, AGARDograph 116, 1969. 3. MACES, R. S. AND MCALEW III, R. F.: J. Fire Flare. 2, 271 (1971). 4. THOMAS,P. H.: Fou-brand 1, 16 (1975). 5. FRIEDMAN,R.: "'Ignition and Burning of Solids," Symposium on Fire Standards and Safety, National Bureau of Standards, April 1976. 6. FEaNX~DEz-PELLO,A. AND WILLIAMS, F. A.: "Experimental Techniques in the Study of Laminar Flame Spread over Solid Combustibles," Combustion Sei. Tech., to appear. 7. EMMONS,H. W.: Fire Res. Abstracts Rev. 5, 163

(1963). 8. WILLIAMS,F. A.: Combustion TheoN, pp. 98-100,

Addison-Wesley, 1965. 9. FER~XNDEz-PELLO,A. Ar~OWILLIAMS,F. A.: Fif-

teenth S~tmposium (International) on Combustion, p. 217, The Combustion Institute, 1975. 10. PALMER,K. N.: Combustion and Flame 1, 129 (1957). 11. EGERTON,A. C., GUGAN,K. AND WEINBERG, F. J.: Combustion and Flame 7, 63 (1963). 12. GVGAN, K.: Combustion and Flame 10, 161 (1966). 13. K1NBARA,T., ENOO, H. AND SETSUKO,S.: Eleventh

Symposium (International) on Combustion, p. 14. 15. 16. 17.

525, The Combustion Institute, 1967. SmlCnANO,W. A.: Aeta Astronaut. 1, 1285 (1974). OHK1,Y. AnD TSUGg, 8.: Combustion Sci. Tech, 9, 1 (1974). UBHAYAKER,S. K.: Combustion and Flame 26, 23 (1976). Fons, W. L., CLEMENTS, H. B. AND GEORCE, P.

MECHANISMS O F FIRE SPREAD

18.

19.

20. 21.

22. 23.

24.

25. 26. 27.

28.

29.

30.

31. 32. 33. 34. 35. 36.

37. 38. 39. 40.

M.: Ninth Symposium (International) on Combustion, p. 860, The Combustion Institute, 1963. HOTTEL, H. C., WILLIAMS, G. C. AND STEWARD, F. R.: Tenth Symposium (International) on Combustion, p. 997, The Combustion Institute, 1965. ANDEBSON,H. E. AND ROTHERMEL,R. C.: Tenth Stlmposium (International) on Combustion, p. 1009, The Combustion Institute, 1965. FANG, J. B. AND STEWABO, F. R.: Combustion and Flame 13, 392 (1969). BERLAD,A. L., ROTHERMEL,R. C. AND FRANSDE~, W. H.: Thirteenth Symposium (International) on Combustion, p. 927, The Combustion Institute, 1971. FRANSDEN,W. H.: Combustion and Flame 16, 1 (1971). ROTHERMEL, R. C.: "A Mathematical Model for Predicting Fire Spread in Wildland Fuels," USDA Forest Serivee, Research Paper INT-115, 1972. PAGN1, P. J. AND PETERSON, T. G.: Fourteenth Symposium (International) on Combustion, p. 1099, The Combustion Institute, 1973. ROBERTS,A. F. ANDCLOUGH, G.: Combustion and Flame 11, 365 (1967). DE RIS, J.: Combustion Sci. Tech. 2, 239 (1970). QUINTIERE, J.: Fifteenth Symposium (International) on Combustion, p. 163, The Combustion Institute, 1975. TARIFA, C. S., DEL NOTARIO, P. P. AND TORRALBO, A. M.: Twelfth Symposium (International) on Combustion, p. 229, The Combustion Institute, 1969. DE RIS, J.: Twelfth Symposium (International) on Combustion, p. 241, The Combustion Institute, 1969. LASTRIr~A,F. A., MAGEE, R. S. AND MCALEVYIII, R. F.: Thirteenth Symposium (International) on Combustion, p. 935, The Combustion Institute, 1971. CAMPBELL,A. S.: Combustion Sci. Tech. 3, 103 (1971). MILLER, B. ANDMEISER, JR., C. H.: Textile Chem. Color. 3, 118 (1971). MILLEB, B. AND GOSWA~I, B. C.: Textile Res. J. 41, 949 (1971). LEE, B. T. AND WILTSHIRE, L. W.: J. Fire Flare. 3, 164 (1972). PARKER,W. J.: J. Fire Flam. 3, 254 (1972). SIRIGNANO,W. A.: Combustion Sci. Tech. 6, 95 (1972). MoussA, N. A., TOONG, T. Y. AND BACKER, S.: Combustion Sci. Tech. 8, 165 (1973). HIRANO, T., NOREIKIS, S. E. AND WATERMAN, T. E.: Combustion and Flame 22, 353 (1974). HIRANO, T., NOREIKIS, S. E. AND WATERMAN,T. E.: Combustion and Flame 23, 83 (1974). SIBULK1N, M., KETELHUT, W. AND FELDMAN, S.:

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Combustion Sei. Tech. 9, 75 (1974). 41. SIBULKIN, M. AND LEE, C. K.: Combustion Sci. Tech. 9, 137 (1974). 42. PERRINS, L. E. AND PETTETT, K.: J. Fire Flam. 5, 85 (1974). 43. CAMPBELL,A. S.: J. Fire Flare. 5, 167 (1974). 44. FERN~,NDEZ-PELLO, A., KINDELAN, M. AND WILUAMS, F. A.: Ing. Aeron. Aston. 135, 41 (1974). 45. HIRANO, T. AND SATO, K,: Fifteenth Symposium (International) on Combustion, p. 233, The Combustion Institute, 1975. 46. KASHIWAGI, Z.: Fifteenth Symposium (International) on Combustion, p. 255, The Combustion Institute, 1975. 47. SmULKIN, M. AND HANSEN, A. G.: Combustion Sci. Tech. 10, 85 (1975). 48. SIBULKIN, M., KiM, J. AND CREEDEN JR., J. V.: Combustion Sci. Tech., 14, 43 (1976). 49. FERNANDEZ-PELLO, A. AND WILLIAMS, F. A.: "A theory of Laminar Flame Spread over Flat Surfaces of Solid Combustibles," Combustion and Flame, to appear. 50. PAGNI, P. J.: unpublished. 51. TAR1EA, C. S. AND TORRALBO,A. M.: Eleventh Symposium (International) on Combustion, p.. 533, The Combustion Institute, 1967. 52. ALmNI, F. A.: Eleventh Symposium (International) on Combustion, p. 553, The Combustion Institute, 1967. 53. THOMAS,P. H.: Ninth Symposium (Intemational) on Combustion, p. 844, Academic Press, 1963. 54. KOSDON, F. J., WILLIAMS, F. A. AND BUMAN, C.: Twelfth Symposium (International) on Combustion, p. 253, The Combustion Institute, 1969. 55. THOMAS,P. H. ANDLAW, M.: "Fire Spread along Paper," Fire Research Note No. 584, Joint Fire Research Organization, 1965. 56. BURGOYNE, J, H., ROBERTS, A. F. AND QUINTON, P. G.: Proc. Roy. Soc. (London) A308, 39 (1968). 57. GLASSMAN, I., HANSEL, J. G. AND EKLUND, T.: Combustion and Flame 13, 99 (1969) i 58. MACKINVEN,R., HANSEL,J. G. AND GLASSMAN,I.: Combustion Sci. Tech. 1, 293 (1970). 59. SIRIGNANO,W. A. AND GLASSMAN,I.: Combustion Sci. Tech. 1, 307 (1970). 60. TORRANCE,K. E.: Combustion Sci. Tech. 3, 133 (1971). 61. AKITA,K.: Fourteenth Symposium(International) on Combustion, p. 1075, The Combustion Institute, 1973. 62. TORRANCE,K. E. AND MAHAJAN,R. L.: Fifteenth Symposium (International) on Combustion, p. 281, The Combustion Institute, 1975. 63. MARKSTEIN, G. H. AND DE RIS, J.: Fourteenth Symposium (International) on Combustion, p. 1085, The Combustion Institute, 1973. 64. HANSEN, A. AND SIBULKIN, M.: Combustion Sci. Tech. 9, 173 (1974). 65. OBLOFF, L., DE RIS, J. AND MARKSTEIN, G. H.:

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F I R E AND EXPLOSION RESEARCH

Fifteenth Symposium (International) on Combustion, p. 183, The Combustion Institute, 1975. 66. SISULKIN,M. AnD KaM, J.: "The Dependence of Flame Propagation on Surface Heat Transfer II. Upward Burning," unpublished. 67. DE RIs, J. ant) ORLOFF, L.: Fifteenth Symposium (International) on Combustion, p. 175, The Combustion Institute, 1975. 68. VOGEL,M. ANDWmLIAMS,F. A.: Combustion Sci. Tech. 1,429 (1970). 69. EMMONS,H. W. AND SHEN,T.: Thirteenth Symposium (International) on Combustion, p. 917, The Combustion Institute, 1971. 70. THOMAS,P. H.: Fourteenth Symposium (International) on Combustion, p. 1007, The Combustion Institute, 1973. 7t. DE RIs, J., M~RTV KANUBV,A. AND YUEN, M. C.:

Fourteenth Symposium (International) on Combustion, p. 1033, The Combustion Institute, 1973. 72. BUTLER,C. P., MARTIN,S. B. AND WIERSMA,S. J.: Fourteenth Symposium (International) on Combustion, p. 1053, The Combustion Institute, 1973. 73. CaocE, P. A. AND EMMOnS, H. W.: "The LargeScale Bedroom Fire Test, July 11, 1973," FMRC

Technical Report RC74-T-31, Serial No. 21011.4, 1974; see also Technical Report RC75T-31 and FMRC Serial No. 21011.6, 1975. 74. ADAMS, J. S., WmL1AMS, D. W. AND TRECELLASWILLIAMS, J.: Fourteenth Symposium (International) on Combustion, p. 1045, The Combustion Institute, 1973. 75. STEVENSON, A. E., SCHERMEaHORN, D. A. AND MILLER, S. C.: Fifteenth Symposium (International) on Combustion, p. 147, The Combustion Institute, 1975. 76. ALBINI, F. A. ANDRAND,S.: "Statistical Considerations on the Spread of Fire," Institute for Defense Analyses, Research and Engineering Support Division, Washington, 1964; McMAsTERS, A. W.: "A Statistical Fire Spread Model for Forest Fires," Western States Section, The Combustion Institute, Paper No. 73-15 (1973); O'RECAN, W. G., NOZAKI, S. AND KOURTZ,P.: "A Method for Using Directional Rates of Spread to Predict Forest Fire Configurations," Western States Section, The Combustion Institute, Paper No. 73-17 (1973); DAvis, J. R.: "Fire Spread in Arizona's Oak Chaparral," Western States Section, The Combustion Institute, Paper No. 73-20 (1973).

COMMENTS I. Glassman, Princeton Univ., USA. It may be worthwhile to point out that in the figure depicting the depth and extent of heating in the fuel, the diagram for the liquid fuel should be modified to show that the heated liquid extends in front of the flame, particularly at the surface. Our work at Princeton has shown that the surface tension reduced flow carries liquid ahead faster than the flame moves, as one would expect. I think it is important to stress that the heating ahead phenomenon that Prof. Williams portrays for all flame spread situations simply must cause sufficient vaporization (or gasification) of the fuel so that one is just within the lean flammability limit. Indeed I have tried to stress conceptually for many years that the flame spread was a lean flammability phenomenon. With this type of consideration it is then possible to explain the role of chemical suppressants added to fuel. Phosphorus alters the degree of vaporization so that one must heat to higher surface temperatures to move within the flammability. The halogen type suppressants are known to narrow flammability limits so once again one must heat to high enough temperatures so that the degree of richness is great enough so that there are sufficient radicals, over and above those suppressed by the

halogen, to propagate the flame. I do not believe Prof. Williams mentioned that in some cases such as certain types of flashover that two spread mechanisms probably exist. For fuel which have a very large latent heat of gasification the rate of flame spread is slow. However if a large flame exists so that there can be extensive radiative heating ahead due to the high heat of gasification the vapor pressure of the fuel components can rise rather suddenly ahead of the flame and the propagation jumps to the situation similar to very low flash point fuels and speeds of hundreds of centimeters per second can be obtained. This type of qualitative explanation of a time dependent phenomena could appear to explain the dangerous situation which sometimes arises with certain types of celular plastics such as polyurethane.

Author's Reply. Although I did not have time to emphasize the surface-tension-driven flow in my oral presentation, it is discussed in the written paper, where reference to the Princeton work may be found. The figure that Prof. Glassman refers to is a rough sketch of the surface of fire inception for the liquidfuel case and does not appear in the written paper. Merely as a convention, I drew the surface to just

MECHANISMS O F FIRE SPREAD touch the forward-most flamelet, but I certainly agree that there is heated liquid moving ahead of that flame in the surface-tension-driven mode, as well as in the buoyancy mode, of spread. I agree that for spread along continuous surfaces, with certain exceptions such as smoldering, it is both possible and interesting to view the process at least qualitatively as involving attainment of a fuel-lean flammability limit in the gas at the surface of fire inception. In Ref. 49 of my paper we have considered using this idea of Prof. Glassman in a detailed mathematical theory for one type of flame spread, and we indicated the spread-rate predictions that would be derived therefrom. If it is assumed that the limit concentration remains constant, then the predicted functional dependences of the spread rate, for spread along surfaces of polymethylmethacrylate, does not agree as well with experiments as do predictions of theories that treat the gas-phase oxidation in somewhat greater detail, specifically as a one-step Arrhenius process. This is perhaps understandable in view of known dependences of flammability limits on geometrical configurations, for example the differences between flammability limits for upward and downward propagation in tubes. I have always felt that there exist a number of mechanisms for flashover, some purely fluid dynamical and others strongly chemical. The mechanism that Prof. Glassman describes in his final paragraph certainly is one of the logical mechanisms. I cannot comment on whether that mechanism is applicable to polyurethanes because my knowledge of these materials is insufficient.

Howard W. Emmons, Harvard University, USA. Professor Williams has presented an excellent review of flame spread in a form which brings out the effective physical mechanisms. These formulas however contain some illy defined quantity--usually a conduction length, l, 8, etc. If we look at the early days of heat transfer similar quantities were used---the heat transfer coefficient being h = h/8. The further development of this field has replaced these "'unknown" lengths by the heat transfer coefficient itself in an appropriate dimensionless form such ~s the Nusselt Number Nu = hL/h where L is the size of the equipment (pipe diameter, height of well). If we now return to flame spread, the future would be expected to use Williams' equations as hints of what dimensionless flame spread variables are appropriate for various fire spread situations and that these dimensionless spread variables will be used to produce dimensionless correlations of a wide

1293

range of fire spread experimental and analytical results.

Author's Reply. I agree completely, and I wish I had written that in the paper.

R. F. Chaiken, Bureau of Mines, USA. While a thermal transport approach is very useful in understanding fire spread phenomena, it should be recognized that the definition of ignition may be difficult both experimentally and theoretically. For example, nonsteady-state fire spread in ventilated ducts generally has gaseous flames which precede the solid fuel pyrolysis front. This makes it difficult to measure the propagation rate and sometimes leads to "skip" or noncontinuous spread. In connection with the concept of ignition temperature it should be remembered that a good many mechanisms, both chemical and physical, are embedded in that concept. This would affect how one would choose a surface of fire inception as well as a value for T~. Author's Reply. In my presentation I have not emphasized transient spread, which often can occupy a large part of the history of ventilation-driven duct fires, since the steadystate spread rates of these fires typically are very high. I agree that in situations of this type, e.g., in corridor fires, skip or jump-like spread often occurs, but I believe that these phenomena are not inherent in a uniform, straight duct but rather reflect irregularities of fuel loading, duct shape, air flow, or something else. In the paper I have tried to emphasize that the ignition temperature may indeed depend on many physical and chemical mechanisms which will cause its numerical value to vary with conditions in different ways. Nevertheless, it appears to be a useful concept for enhancing understanding of fire spread.

IV. A. Moussa, Massachusetts Institute of Technology, USA. In obtaining a simplified equation for the smoldering speed in cellulosic materials, you have assumed the heat flux generated in the smoldering zone to be governed by oxygen diffusion. Because of the low temperature in this smoldering zone, one immediately wonders whether oxygen diffusion or surface reaction kinetics is the governing rate. (In fact the study we presented in this symposium dealt specifically with this problem and determined that oxygen diffusion is indeed governing.) On what basis have you assumed oxygen diffusion to be governing? What would you assume for the smoldering of other materials, say polyurethane foams where the reaction temperature (as measured in our laboratory) is much lower than for the case of cellulosic materials?

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FIRE AND E X P L O S I O N RESEARCH

Author's Reply. The existence of phenomena of well-defined extinction of smoldering led me to assume that in most cases smoldering will be diffusion-controlled. Extinction of surface combustion can be observed for carbon b u r n i n g and for cellulose glowing, for example, W h e n there exists a welldefined extinction, b u r n i n g usually is diffusioncontrolled until near-extinction conditions are approached, and w h e n conditions of kinetic control are established, a small change in conditions typically produces extinction, by the mechanism elucidated by Frank-Kamenetskii and others. The fact

that overall activation energies for kinetically controlled heat-release typically are large compared with thermal energies ultimately is responsible for the very narrow range of kinetieally controlled conditions. If the material is capable of smoldering at low reaction temperatures, then the possibility of kinetic control is greater. For example, if a material can smolder at a reaction temperature equal to ambient temperature, then there is no extinction phenomenon, and kinetic control may occur over awide range of conditions. I don't know much about polyurethane foams.