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Reference publications

NBSIR 84-2895

Modeling of Smoldering

Combustion Propagation

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

June 1984

U S. DEPARTMENT OF COMMERCE

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84-2895

1984

NATIONAL BUREAU OF STANDARDS

NBSIR 84-2895

NATIONAL BBRESCT cf ST/ "r &f:DS

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MODELING OF SMOLDERING

COMBUSTION PROPAGATION

T. J. Ohlemiller

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

June 1984

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director
\
TABLE OF CONTENTS

Page

Introduction 1

Fire safety hazards involving smoldering 2 Some experimental characteristics of smolder propagation 3 Plan of this paper 5

Chemistry of Smoldering Combustion 6

Gas phase oxidation 7 Oxidative polymer degradation 9

Char oxidation 13

Substantial heat sinks: pyrolysis and water vaporization 15

Simplified kinetic model 16

A General Model of Smolder Propagation in a Fuel Bed 21

Thermophysical considerations 21

Single particle equations 24

Bulk solid equations 31

Bulk gas equations 36

Non-dimensionalization 40

Magnitude of dimensionless groups and simplifications 47

Approaches used with some related problems 55

Forward and reverse smolder propagation 57

Smolder Propagation Models in the Literature 62

Phenomenological models 62

Numerical models 65

Models in related problem areas 76

Concluding Remarks 80

References 83

Nomenclature 89

iii
LIST OF TABLES

Page

Table 1. Experimental Characteristics of Smolder Propagation 95

Table 2. Dimensionless Groups with Approximate Magnitudes 96

iv
LIST OF FIGURES

Page

Figure 1. Example of smolder wave structure in a permeable

horizontal fuel layer; wood fibers with bulk density of

0.04 g/cm . From ref. 24 102

Figure 2. General structure of a smolder wave in a bed of fuel

particles showing gradients on wave scale and on particle scale 103

Figure 3(a). Profiles of temperature, oxygen concentration and solid

density for typical case of forward smolder

(from ref. 18) 104

Figure 3(b). Profiles of temperature, oxygen concentration and solid

density for typical case of reverse smolder

(from ref. 19) 103

v

MODELING OF SMOLDERING COMBUSTION PROPAGATION

T. J. Ohlemiller

Introduction

Smoldering combustion is defined here as a self-sustaining, propagating

exothermic reaction wave deriving its principal heat from heterogeneous oxidation of the fuel (direct attack of oxygen on the fuel surface). The primary

context for considering this type of process here is fire safety. Such

processes also occur in other contexts such as cigarette smolder or underground coal gasification; results from these areas will be discussed to a

limited extent where pertinent.

Smoldering, like flaming, is a combustion process which spreads through a

fuel when heat released by oxidation is transferred to adjacent elements of

the fuel. While spread of smoldering and flaming both occur through coupled

heat release and heat transfer mechanisms, smoldering typically yields less

complete oxidation of the fuel, lower temperatures and much slower propagation

rates. However, all of these smolder characteristics can vary widely with

oxygen supply. Stable smoldering is possible in some circumstances at air/

fuel ratios only a few percent of stoichiometric . At the opposite extreme, a

strong oxygen supply can raise the intensity and temperature of smolder to

such a degree that gas phase reactions become dominant and flaming propagation

takes over.

Air/ fuel ratio as used here refers to the ratio of the fluxes of air and fuel

entering the reaction zone (as seen by an observer moving with the reaction

zone)

.

1
Fire safety hazards Involving smoldering . Only rather recently has

smoldering been recognized as a major fire safety hazard In the United States.

Clarke and Ottoson (1) found cigarette ignition of bedding and upholstery

materials to be the single largest cause of residential fire deaths. The

cigarette, itself a smoldering cellulosic fuel, is nearly ideal as a smolder

initiator in susceptible fabric and filling materials. In such materials, a

self-sustaining smolder process can often be established well before the

cigarette is consumed. This particular problem is coupled to both fabric and

filling response to heating (2, 3, 4, 5). Once established in this manner,

smoldering becomes a steadily growing generator of carbon monoxide and other

toxic gases as the size of the reacting region enlarges. This smoldering

process may spread stably for an hour or so then abruptly transition to

flaming combustion. The hazard in this context (and others) is thus two

fold - toxic gases during smoldering and rapid destruction following flaming

transition. Which of these is most responsible for the observed death toll is

not fully clear; there has been some analysis of this question which implicates both smoldering and subsequent flaming (6).

The increasing interest in residential energy conservation has resulted

in a strong demand for effective insulating materials, particularly for attics

where heat losses are greatest. Cellulosic loose-fill insulation is quite

effective and comparatively inexpensive. It is essentially ground wood, being

made mainly from recycled newsprint, reground to a fibrous, fluffy form that

is blown into place. The oxidation chemistry of wood clearly renders this

material combustible. A variety of additives, typically in fine powder form,

are included in an effort to control this combustiblity . They are rather

effective in suppresing flaming but are less so for smoldering. When the

2
insulation is improperly installed, heat sources such a recessed light

fixtures can cause smolder initiation (7, 8, 9, 10). Once started, this

smolder becomes self-sustaining, spreading through an attic space and posing

the same toxicity and flaming-potential hazards as in the above case.

The two problem areas above, upholstery/bedding and cellulosic insulation

smoldering, have received the most attention; they are also the most extensively studied in some respects. (It should be noted that the former problem

is by far the more common, probably because of much more frequent encounters

with the ignition source, i.e., a cigarette.) There are numerous other

problems, however. For example, wood and certain other low permeability

building materials (particleboard sheathing and some types of rigid foam

insulation) can smolder, especially in configurations where surface heat

losses are suppressed (11, 12, 13). Spontaneous heating and ignition, usually

to smoldering combustion, have long been a problem with a variety of natural

products, usually cellulosic in nature (14, 15). In industries which process

oxidizable materials in finely divided form there is a danger not only from

dust explosions (a flaming problem) but also from smoldering ignition of dust

layers accumulating within or on top of the hot processing equipment (16). A

significant fraction of grain elevator fires, for example, begins with

smoldering (17).

Some experimental characteristics of smolder propagation . Table I is

brief summary of the smolder behavior observed for a variety of fuels in

several configurations. The list of results is by no means complete but it

gives a representative picture of smolder in organic materials. The study bv

Palmer (li) presents the most complete parametric examination of smolder

3
behavior for configurations that are rather complex but realistic. The study

by Ohlemiller and Lucca (18) is the first systematic comparison of configurations simple enough to be modeled in detail. (The forward/reverse terminology

is explained below.)

Cellulosic materials show up most frequently in Table I because they are

so common and nearly all of them smolder. There are, of course, numerous

other materials with the potential to smolder but both the material and the

way it is used must be favorable to smoldering in order for it to emerge as a

fire safety hazard. Note that all the materials listed are fibrous or particulate and thus tend to have a rather large surface to volume ratio; they also

comprise a fuel mass permeable to gas flow and diffusion. The current discussion will ultimately focus on fuels in such a physical form.

The various results in Table I generally support the description of

smoldering as a slow, low temperature combustion phenomenon that responds with

increased vigor when the oxygen supply is enhanced. The slowness is apparent

in the magnitude of the smolder velocities. It is interesting to note that

these do not vary greatly (except for a cigarette during a draw) in spite of

wide variations in fuel type and configuration. This probably reflects an

oxygen-supply-rate-limited character of the process; a process limited by the

fuel oxidation kinetics would be expected to vary more with various materials.

The low temperatures generally reflect the fact that smoldering causes quite

incomplete oxidation of the carbon and hydrogen in the fuel. Part of this

incompleteness is probably a consequence of the detailed nature of the surface

reactions but part is also a result of how various configurations minimize or

preclude gas phase oxidation of surface-derived molecules (pyrolytic or

oxidat ive)

.

4
Figure 1 gives a detailed view of the structure of a smolder wave in a

fairly typical configuration, i.e., a horizontal fuel layer smoldering by

natural convection/diffusion in ambient air. The fuel is a permeable, low

density layer of wood fibers. The details of these results are discussed more

fully in Ref. 24 but several features are noteworthy. The thickness of the

fuel layer decreases quite substantially as a result of smoldering. The wave

structure is very much multi-dimensional. Heat losses (e.g., from the top

surface) have a substantial impact on the thermal structure. It is inferred

in Ref. 24 that the overall shape is probably largely determined by oxygen

diffusion and that two successive overall stages of heat release are coupled

together to drive the propagation process. It is apparent that realistic

smolder problems can be quite complex. The work of Baker on the structure of

the reaction wave in a smoldering cigarette elaborates upon this considerable

complexity (25-28); this body of work provides more details on the structure

of this rather unique smolder problem than can be found for any other problem

of this nature.

Plan of this paper . The smolder initiation problem has been quite

extensively studied at least from a thermophysical point of view (29, 30).

Propagation subsequent to ignition is considerably more complex and much less

studied. In the present work, questions pertaining to ignition are not

examined. Transition from smoldering to flaming is not examined in any detail

due to the current, almost total lack of information on the nature of this

process. The focus here is on the coupled processes of chemical heat generation and heat transfer involved in sustained smolder propagation. This review

begins with an overview of the types of chemical processes involved. The

nature of both the heat sources and the heat sinks is examined; it will be

5
apparent that even for cellulose, the most extensively studied fuel, detailed

chemical mechanisms are lacking. Next a rather general model of the propagation process is posed in order to provide a complete picture of all the interacting phenomena, both chemical and physical, that may play a role. This

model is more comprehensive than any previously posed in the literature though

it is limited to situations described by a continuum solid/gas hypothesis.

The model is not solved, rather it serves as a benchmark for assessing the

simplifications present in the models thus far solved in the literature. The

general model is re-cast in dimensionless form to permit inferences about the

sizes of parameter groupings which justify simplications. Finally, the

smolder models in the literature are examined in light of these considerations; the existing models will be seen to fall well short of adequately

describing realistic smolder propagation problems such as that in Figure l.

Chemistry of Smoldering Combustion

In very general terms, smoldering involves the exothermic attack of heat

and oxygen on condensed phase polymeric materials at a rate sufficient to

overcome heat losses and thus be self-sustaining. In the absence of oxygen,

the typical polymeric fuel will be endothermically degraded by heat to smaller

volatile molecules, sometimes also leaving a solid residue of a variable

aromatic nature referred to as a char; this char pyrolyzes much more slowly

than the initial polymer*. The participation of sufficient oxygen in this

process results in its having a net exothermicity . The heat could arise from

one or more of three sources:

*There is some confusion in the literature over the use of the term"char".

Here the term refers to the black solid residue which typically remains after

the end of the initial rapid weight loss upon heating of a polymer.

Pyrolytic gasification of this char is usually much slower than that of the

initial polymer.

6
(1)

oxygen participation in the polymer degradation process

(2) oxygen attack on the volatilized molecules produced by

the degradation, or

(3) oxygen attack on the char residue.

The extent of the contribution to smoldering from each of these potential

sources (with the exception of some aspects of (3)) is not very well

quantified for any fuel (and, of course, it will vary with the chemical nature

of the fuel and, possibly with the conditions of heating of a given fuel). To

model smolder propagation, one needs rate expressions for all significant heat

release processes and competing heat sink processes.

In examining these potential sources in greater detail, frequent

reference to results for cellulose will be made; it is the most-studied

smolder-prone polymer.

Gas Phase Oxidation . It is assumed, from indirect evidence, that gas

phase oxidation of volatilized molecules does not provide the majority of the

exothermicity which drives smolder propagation. Flames, per se , are usually

not visible during smoldering. Most frequently used retardant chemicals that

act as flame suppressors are not of much use in stopping smoldering (3, 31).

(There are some chemicals that suppress both flaming and smoldering, at least

in some tests (32)). Finally, smoldering is most frequently encountered in

fuels with a large surface to volume ratio which would encourage solid surface

7
attack of oxygen and tend to suppress gas phase radical chain reactions

through surface quenching of the free radicals (33, 34). While this argues

against a dominant role for gas phase oxidation, it does not preclude a significant supplementary role.

Any supplemental contribution from gas phase oxidation reactions can be

expected to increase as the peak smolder temperature increases (and as the

surface to volume ratio of the fuel decreases). Baker presents evidence for

significant gas phase oxidation (on a time scale of ~ 10s) of H

2 and CO

produced from tobacco in the temperature range 500-900°C (35). Extrapolation

of Dryer's (36) kinetics for homogeneous oxidation of "wet" CO indicates that

CO can be oxidized on a time scale of seconds at 600°C. Propane, fairly

representative of many hydrocarbon molecules, can be oxidized even below 400°C

on a time scale of ten seconds (34). However, all of these cases except the

cigarette involve experimental conditions where the radical chain reactions in

the gas would not be suppressed by the proximity of a large amount of solid

surface

.

Gas phase heat release contributions from the oxidation of other more

complex molecules arising from the polymer degradation process will be highly

dependent on their chemical nature (and thus, in turn on the nature of the

original polymer). This has received little attention in the context of

smoldering and more work is needed. Open tube flow reactor studies of the

overall oxidation rate of specific polymer gasification products would

presumably yield only the upper limit on the potential reaction rate during

smoldering; the suppressing effect of a large amount of solid surface areas

would be missing. Packed tube studies would be more pertinent but the

8
surf ace/volume ratio and the nature of the surface would become additional

variables. For example, the oxidation of propane, mentioned above, is

suppressed up to 550°C in the presence of an extended surface of glass (34).

It is interesting to note that, in contrast to the inferred secondary

role of gas reactions in smoldering, in one configuration of in situ coal

combustion (reverse combustion, defined below), the sole heat source is taken

to be gas phase oxidation of vaporized fuel molecules (37, 38).

It appears likely that the many smoldering processes whose peak temperature is well below 600°C receive only supplemental heat input from gas phase

reactions, if any. On the other hand, the usual view that smoldering is

driven mainly by analogs of the classic graphite oxidation reactions when

oxygen attacks the char may be too simple. It ignores the possible exotherm

from oxygen attack on the original polymer or partially degraded versions of

it; it also glosses over the complexity of the char itself which is not pure

carbon

.

Oxidative Polymer Degradation . Look next at the question of

exothermicity from oxidative degradation of the polymer. Nearly all polymers,

thermosetting or thermoplastic, are subject to exothermic oxygen attack if the

temperature is sufficiently, high; such processes have been extensively

studied, though mainly at temperatures and heating rates lower than in

smoldering (39). Thermoplastic polymers will normally contract under surface

tension forces when heated, minimizing their surface area and causing endothermic pyrolysis to dominate; smoldering is thus precluded. However, if the

thermoplastic polymer is coated on a rigid support with extended surface ar*-

i

9
that does not decrease with temperature (e.g., pipe lagging), a considerable

exotherm can result, sometimes causing smoldering ignition (40). Char-forming

polymers that initially have a large surf ace-to-volume ratio tend to retain it

during degradation; exothermic oxidative degradation, if favored chemically,

thus can continue in parallel with char-forming reactions (the two may be

coupled in some polymers). In all types of polymer these oxidative reactions

Lend to compete with purely pyrolytic reactions which are usually endothermic.

Given this competition between oxidative and pyrolytic degradation, the net

outcome in terms of exo- or endothermicity depends on the chemical nature of

the polymer and the circumstances of the heating (heating rate, ambient oxygen

concentration, surf ace-to-volume ratio). Thus, in attempting to assess the

importance of oxidative degradation of a polymer as a heat source in smolder

propagation, one must focus on a specific situation. Unfortunately, it is

very difficult to learn many details of the smolder chemistry by studying the

propagation process itself.

Thermoanalytical techniques (thermogravimetry (TG) and differential

scanning calorimetry (DSC)) provide a controlled environment (fixed oxygen

level, linearly programmed heating at l-lO°C/min) that can yield at least an

engineering characterization of smolder chemistry. These techniques have been

applied to various materials in a high specific surface area form including

cellulose fibers (41, 42), lignin powder (42), wood fibers (44), polyurethane,

polyisocyanurate and phenolic foams (13) and tobacco powder (45, 46). One

finds a qualitatively similar result, for heating in air, in all cases. The

DSC thermogram, which provides a measure of heat production or absorption of

the sample during heating, typically shows two major exothermic peaks. The

first peak begins just as the sample starts to degrade rapidly and lose

10
weight. The second corresponds to oxidation of the char residue left by the

first stage of degradation. Given these facts alone, it would be reasonable

to identify the first peak with oxidative pyrolysis of the original polymer or

its immediate condensed phase degradation products (the first heat source

listed above).

Shafizadeh and Bradbury (47) investigated this question of oxidative

pyrolysis in more detail for pure cellulose. It was shown that during

prolonged isothermal heating in air at 190°C, oxygen attack on cellulose is

indeed substantial, leading to a build-up and ultimately a steady-state

concentration of hydroperoxide groups and greatly accelerated production of CO

and C0 ?

. However, it was found that for isothermal heating at temperatures

above 300°C, weight loss in air is no faster than in pure nitrogen implying

that, by this temperatue, the oxidative pyrolytic reactions are overwhelmed by

the purely thermal pyrolytic reactions. On the other hand, Shafizadeh, et al.

(41) found, upon comparing linearly programmed heating (15°C/min) of cellulose

in nitrogen and in air, that the presence of oxygen substantially enhances the

rate of cellulose gasification up to 350°C where DSC results show the first

exotherm to be peaking. In ref. 41 the first exotherm is attributed to

chemisorption of oxygen on the condensed phase material (presumably radicals)

formed during pyrolysis, not to oxidative attack on the original cellulose

molecules. The chemisorption hypothesis appears to have derived from studies

of the considerable exothermicity evolved when O

2 is adsorbed on cellulosic

chars; however, these chars were formed at substantially higher temperatures,

400-500°C (45). Most recently, Shafizadeh and Sekiguchi (43) presented

further evidence derived from nuclear magnetic resonance and Infrared studies

of the condensed phase. The solids examined were residues left from

11
isothermal (325-600°C) and programmed heating (lO°K/min) of cellulose. An

apparent qualitative correlation between the amount of aliphatic carbons in

the residue and the size of the first exotherm measured from this residue by

DSC in air, lead to the conclusion that the source of this first exotherm is

oxidative attack on these aliphatic carbons. For residues created by first

heating cellulose in nitrogen at 400°C, the proposed heat source seems quite

convincing. For cellulose exposed directly to air during heating, the

evidence is less convincing; it is this latter situation that is most

pertinent to cellulose smoldering. The overall work of Shafizadeh and coworkers is strongly suggestive of a pyrolytically-initiated, oxygen-altered

exothermic degradation process contributing to cellulose smolder but its exact

nature is not clear at present.

A significant secondary result of the study by Shafizadeh and Sekiguchi

(43) was a demonstration that the heat measured during DSC experiments with

cellulose residues in air is proportional to the quantity of non-pyrolyzable

solid and not to the amount of pyrolysis vapors generated. This supports the

widely-used assumption that the DSC is measuring the heat of heterogeneous

oxidation and not gas phase heat from oxygen attack on evolved fuel-like

molecules

.

The impetus for defining the nature of the first exotherm seen in DSC

thermograms of cellulosic materials lies in its dominant role during smolder

initiation (9, 10) and its substantial contribution to smolder propagation

(18, 24). It is likely that the first exotherm seen with the other materials

mentioned above can play a similarly significant role in their smolder

characteristics. Thus the available evidence points to the likelihood of a

12
significant role for oxidative pyrolysis as an appreciable heat source in many

smoldering processes.

Char Oxidation . The solid residue left at the end of the first DSC

exotherm (defined here as char) is typically a predominantly carbonaceous

material with a substantially enhanced surface area due to pore formation (23,

48). The important role of this char in many smoldering processes is not in

dispute; its oxidation is highly exothermic (heat source number (3) listed

above). The heat release from this source can be dominant in some configurations; however, it can be less critical or even absent in others. The crucial

factor is oxygen access to the various stages in the fuel degradation process,

as will be discussed below. Polymer chars are not unique chemical entitles

but rather depend on the thermal history of formation; with continued heating

they change further (49, 50). The mechanism of emergence of a pore system,

which for some starting materials is totally absent, has not received any

scrutiny in the context of smoldering in spite of its obvious pertinence to

surface oxidation. Much more is known about this subject for coals where the

existing pore system evolves during gasification (51, 52); models of the coal

pore system and its evolution have been developed (53, 54).

Low temperature cellulosic chars (typical of smoldering) are much more

readily attacked by oxygen than is pure carbon. Shafizadeh and Bradbury

hypothesize, however, that the oxidation of the char proceeds by a mechanism

similar to that proposed elsewhere for carbon (55, 56). The proposed oxidation mechanism of carbon begins with oxygen chemisorption at a surface free

radical site. The surface complex falls into one of two broad classes. The

first class of complexes is quite stable and dissociates only at very high

13
temperatures (up to 1000°C). The second class is mobile and reactive, quickly

forming both CO and CO

2 which then desorb from the surface. These mechanistic

proposals have not yet been fully confirmed or quantified.

The formation, reactivity and oxidation products of cellulosic chars are

greatly influenced by the presence of inorganic additives or impurities.

Alkali metals such as sodium, calcium and magnesium have a strong influence on

the ability of cellulosic materials to smolder (12, 57). This complicates

modeling efforts in as much as kinetic rate constants found in the literature

for the oxidation of a cellulosic char are unlikely to hold for cellulose

obtained from some other source. Shafizadeh has shown that small amounts of

inorganic materials such as sodium chloride or boric acid can alter the rate

of heat release from a cellulose char by at least a factor of two up or down

(relative to a char from pure cellulose) depending on the additive (58); the

effect is achieved in large measure through changes in the product ratio

CO/CO2 from the surface oxidation of the char but significant kinetic effects

may also exist (Al).

Other reactions can gasify a carbonaceous char. Thus CO

2

, 1^0 and H2 (which may arise from pyrolytic or oxidative reactions) can remove carbon from

the solid in the form of CO or CH^ . These reactions are believed to be less

important in many smolder problems. Attack by the first two of these gases is

endothermic; the third is exothermic. Available data indicate that they will

proceed at a significant rate only above 650-700°C (52) even in the presence

of catalytic impurities (59). Many smolder processes proceed stably at

substantially lower temperatures. The peak temperature is largely dependent

on the rate of oxygen supply, however, so, as with gas phase oxidation

14
reactions, the relevance of these is dependent on the smolder conditions of

interest. Cigarette smolder Is one problem in which both the diffusive oxygen

supply and the forced flow oxygen supply are normally sufficient to keep the

peak temperature above 850°C; Baker (35) has shown that CC>2 reduction on

tobacco char is significant in this case. In the most common mode of in situ

coal gasification (forward mode, defined below), these non-oxidative gasification reactions are of central importance since they account for the bulk of

the coal char conversion to gases (60).

Substantial Heat Sinks : Pyrolysis and Water Vaporization . It was noted

above that endothermic pyrolysis of the polymeric fuel competes with

exothermic oxidative degradation. The endothermicity of the pyrolytic reactions is comparatively small on a unit mass basis but it can be an important

heat sink during smoldering, nevertheless, since a large fraction of the fuel

may undergo this process. These reactions are again specific to the polymer

of interest. Pyrolytic degradation of polymers has been extensively examined

(39, 61). Such degradation typically proceeds by free radical chain reactions

in the condensed phase involving a complex interplay of chain initiation,

propagation, transfer and termination reactions. For complex polymers such as

cellulose, the monomeric unit may come apart in further parallel reactions

resulting in a large variety of volatile products. If the polymer forms a

pyrolytic char, aromatic condensation reactions building a cross-linked char

must be progressing in parallel with the other reactions. Despite this

complexity, full details of which are rarely available, the net process of

weight loss can frequently be described by relatively simple expressions.

Bradbury, et al (62), showed, for example, that pyrolytic weight loss from

cellulose can be empirically described by a three reaction model.

15
Water constitutes a significant fraction of some organic fuels,

particularly the smolder-prone cellulosic materials. It is also a major

product of oxidation. Movement of this water in and out of the condensed

phase by condensation and evaporation can provide a substantial local heat

effect and can also alter the overall rate of smolder propagation. The

specific effects can depend significantly on whether the water, once

vaporized, is carried into higher temperature regions thus remaining a vapor,

or is carried into a region where it will recondense. The rate of evaporation

or condensation is frequently taken to be mass transfer limited (by the

boundary layer external to the fuel particle) with the particle phase water

vapor pressure assumed to be in equilibrium with the particle temperature.

This last assumption may well fail when the particle water content is low

(63); empirical correlations may prove necessary (64).

Simplified Kinetic Model. The degree of detailed description of the

above chemical processes necessary in modeling smolder propagation depends on

the purpose of the model. In fire research, one is seeking to understand

first the controlling factors in the propagation process with the ultimate

goal of learning how to prevent it. In a first cut at this, it is usually

sufficient to insert into the model some reasonably accurate description of

the major chemical heat effects; this calls for "global kinetics" in which

real chemical species other than oxygen are not described. The drawback of

this approach is that it offers no clues about how changes in chemical

mechanisms might be exploited to suppress smoldering. If the goal were to

model the toxicity of smoldering, one would have to incorporate the

generation/consumption reactions of the major toxicants (usually CO); such a

goal can rapidly lead to increased model complexity. Some goals mandate

16
seemingly intractable complexity. Modeling of cigarette smolder with the goal

of learning to control the numerous flavor-related or health-related species

is such a problem (65). However, if the species of ultimate interest are

present in minor amounts, so as not to have a signficant impact on the smolder

wave structure when they are created or destroyed, the problem can be

uncoupled. The wave structure can be solved using "global kinetics" only; the

history of the minor species can then be tracked as they ride through this

pre-determined wave structure (45).

In any event, it should be apparent from the preceding discussion, that

even for cellulose, the full mechanistic details of the smolder chemistry,

much less the quantified rate expressions, are not available. Inevitably

then, one must resort to substantial simplifications in modeling smolder

chemistry.

It was noted above that thermal analysis techniques are of some use in

characterizing the "global kinetics" of gasification and the heat effects

accompanying it. The results must always be utilized with caution since the

heating rates during smolder (0(10 to 10 °C/min)) can sometimes greatly

exceed the useable heating rates in thermal analysis (_<0(10^ °C/min)). Such

differences in heating rate could potentially alter the controlling chemical

and/or physical processes. Higher heating rates cause the peak rate of any

elemental reaction to shift to higher temperatures where the reaction then

proceeds faster. The limiting step in a given heterogeneous oxidation

reaction could shift at higher temperatures from chemical control (adsorption,

surface reaction or desorption) to physical control (oxygen diffusion).

Furthermore, competing parallel reactions with differing activation energies

17
will be shifted upward in temperature differing amounts (less with increasing

activation energy); this could shift the controlling mechanisms in a complex

elemental reaction sequence. In spite of this caveat, these techniques are

frequently used; available higher heating rate techniques suffer from

considerable experimental inconvenience (66, 67). It is somewhat comforting

that the extrapolation in heating rates from thermal analysis to smolder is

less than with nearly any other combustion process; nevertheless it poses

unanswered questions.

There are available in the literature a wide variety of techniques for

fitting kinetic expressions to thermogravimetry data (68, 69). These can be

used to obtain the "global kinetics" of gasification and heat generation/

consumption; typically, techniques employing multiple heating rates give

kinetic parameters most suited to extrapolation. This has been done for a

polyurethane (70), tobacco (45) and wood fibers (44). Since, as noted above,

the thermal analysis results for these materials yield only two global

reaction peaks in the presence of oxygen and one in the absence of oxygen, one

can obtain a reasonably adequate picture of the "global kinetics" of the fuel

gasification by fitting the following scheme to the thermal analysis behavior

(44).

18
Char* + Gases*

This scheme has the principal qualitative features of smolder chemistry

discussed above. An endothermic pyrolytic path competes for the original fuel

with an exothermic oxidative degradation path. Both paths form a char which

is subject to exothermic oxidation. Gas phase oxidation of the product gases

is not included in the thermal analysis scheme. Thermogravimetry will not see

such processes, of course. Furthermore, one can infer from the short gas

residence time in a DSC sample holder (< Is) and available data on oxidation

of such species as CO or H

2 (likely products as a typical char is gasified

around 400-500°C) that the DSC will measure only the heterogenous heat effects

(36, 71). (This may not be the case with inhibited chars that gasify at

higher temperatures but recall that Shafizadeh and Sekiguchi (43), as noted

earlier, provided experimental evidence that the DSC is measuring only

heterogeneous reaction heat for cellulose.) If gas phase chemical heat

effects are to be included in a smolder model, they must be quantified by some

other means

.

This reaction scheme leaves much to be desired as a description of

smoldering combustion chemistry. Even as a pragmatic device for approximate

description of chemical heat effects during smoldering, its use must be

19
examined case by case. Recall, for example, that Bradbury, et al (62)

required a three step reaction scheme just to empirically describe the

pyrolytic gasification of cellulose whereas Eqn. (1) would assign only one

step to this process. In this case, Bradbury, et al required the extra

complexity in part to describe the temperature-dependent variability in

quantity of char formed during cellulose degradation. Failure to include this

effect could yield misleading results in a model of cellulose smolder applied

over a broad range of conditions. Note also that Eqn. (1) does not directly

fit the most recent mechanistic ideas proposed by Shafizadeh and Sekiguchi

(43) for the first exotherm from cellulose, i.e., that is caused not by

immediate cellulose oxidation but rather by oxidation of a condensed phase

cellulose degradation product. On the other hand, Eqn. (1) has been used to

give a reasonable description of a related material, wood fibers (44).

In summary, modeling of smolder propagation requires empirical rate

expressions for the major heat sources and heat sinks. If the peak smolder

temperatures do not exceed 600°C, the major heat effects are probably attributable to oxidative polymer degradation, char oxidation, polymer pyrolysis

and water movement. Thermal analysis techniques can provide empirical rate

and heat effect data on the first three of these but these data must be

utilized with caution to model the higher heating rate processes during

smoldering. A considerable amount of information on the details of the

chemistry involved in smoldering combustion of even such a common material as

cellulose is still lacking; this precludes construction of smolder models with

anything more than a global representation of the fuel chemistry.

20
A General Model of Smolder Propagation in a Fuel Bed

Thermophysical Considerations . Although the degradation and oxidation

chemistry of the fuel is quite complex, it is only one side of the smolder

propagation problem. The rate of oxidation of the char, for example, can

depend not only on the intrinsic chemical processes occurring at active sites

on the char surface but also on the specific surface area available for

reaction (m /g of char), the local temperature and the local oxygen concentration. The local oxygen concentration depends in turn on the rate at which it

can reach the reaction neighborhood. Typically, in fire-safety related

smolder problems, the oxygen originates in the ambient atmosphere and must

penetrate the permeable bed of fuel by buoyant flow and diffusion. As seen

from Table 1, the fuel bed frequently consists of a large array of fuel

particles ; their flow permeability is typically much less than that of the

fuel bed itself. Pores may exist or develop in the fuel particles during

degradation allowing oxygen to diffuse inward against a net outward movement

of gasification products. If the pore system is evolving as a consequence of

degradation and oxidation reactions, the surface area available for the oxidation reactions is also changing; a pore system can quickly yield an internal

area for oxidative reactions that is much greater than the external geometric

surface of the fuel particle. The familiar zone concepts for single particle

gasification are then pertinent (72, 75, 52). Zone I is the limiting case in

The term particle is used in a very broad sense here. An open cell polymer

foam, for example, actually consists of a continuous solid phase permeated by

a continuous gas phase. We identify the particles in such a case as the

intersecting segments of polymer (generally cylindrical) that frame the

contiguous gas bubbles, mentally isolating a typical intersection as the

center of a typical particle. Solid wood presents a similar but more complex

(and anisotropic) structure; a particle in this structure would be the long,

thin, flat double cell wall that subdivides the gas space.

21
which oxidation reactions (or other gasification reactions) on the internal

surface of the particle are very slow compared to the rate of oxygen diffusion

inward through the pore system; the oxygen concentration throughout the

particle interior is thus virtually uniform. Zone III is the opposite

limiting case in which the rate of oxidation in the particle interior is so

fast that oxygen cannot penetrate and oxidative reactions are confined to the

external surface. Zone II is the more general case in which the oxygen supply

and consumption rates in the particle interior are comparable, leading to a

non-uniform oxygen concentration. In certain circumstances there may be

temperature gradients within the particles as well. In the present smolder

problem, there is the further complication that the oxygen concentration and

gas temperature around the exterior of individual fuel particles is varying as

the overall smolder wave moves through the array of fuel particles (see

Fig. 2). Transport of heat and oxygen to the outer surface of fuel particles

and through the particle array become additional rate processes that must be

considered in determining the overall movement of the smolder reaction front.

The model must account for all of these processes which interact with the

smolder chemistry.

Similar types of interactions between physical and chemical processes

occur in other combustion problems involving coal beds and incinerators as

well as in a variety of industrial processes. A systematic development of

solutions to simpler and more tractable versions of such problems is the

subject of Ref. (73). Analytical solutions to a wide variety of single

particle and multiple particle problems are presented.

22
In describing the following model equations, a major simplification is

imposed. The smolder wave is assumed to extend over many fuel particles so

that it can be treated as a continuum. Without this assumption, it is necessary to treat each fuel particle and the gas around it separately as it inter

acts with the gas and all other individual fuel particles; such a model

becomes highly specialized to the particular geometric arrangement of

particles and their shapes. By making this continuum wave assumption, it is

no longer possible to treat such problems as the smoldering of a stack of

several logs ir. a fireplace or wood stove, for example. However, all of the

problems noted in Table I are amenable to the continuum assumption. Even the

spread of smolder over a single piece of solid wood is treatable if it is

viewed as a very low permeability fuel bed of small particles as defined in

the preceding footnote (the anisotropy of the wood would have to be carefully

accounted for, however).

In formulating a continuum model of smolder wave propagation through a

large array of fuel particles, conservation equations are required for the ga

phase and the particle phase. Since the fuel bed interacts with its

surroundings, a set oi boundary conditions is also required to define that

interaction. In some cases, such as that in Fig. 1, it is probable that a

complete description of the full propagation problem requires a model of the

behavior of the surrounding gas coupled to that of the fuel bed. This

complete problem has never been addressed. Here it is assumed that some

approximate boundary conditions (described below) suffice to isolate the fuel

bed alone as the system to be studied.

23
Recall that the goal here is a rather complete formulation of a model not

for the purposes of solution but as an explicit exposition of all the interacting elements in the general problem of smolder propagation. The relative

simplicity of existing models in the literature will then be apparent.

Single Particle Equations . Consider first a situation in which gradients

in species concentration and temperature exist within the individual fuel

particles (Fig. 2) . As will be seen, species gradients are much more likely

than temperature gradients in view of the continuum assumption. In contrast

to the gradients in the particle phase, the gas phase around each particle is

assumed to be locally homogeneous due to rapid molecular and/or turbulent

mixing on the scale of a particle; this is a usual assumption in packed bed

problems. There still may exist concentration or temperature differences

between the locally homogeneous gas phase and the fuel particle surface across

a boundary layer. Transport across this boundary layer is treated by the

usual packed bed heat and mass transfer correlations (70).

Because of the gradients within the particles, it becomes necessary to

formulate conservation equations for the interior of a typical particle.

Then, at each locus along the continuum smolder wave it is necessary to

account for the interaction between this typical particle and the surrounding

gas at that locus within the fuel bed. Note that since the particles may be

porous, both gas and solid exist within the particle boundary as well. At any

locus within the particle it is assumed that gas and solid are in thermal

equilibrium because of the typically very small dimensions of the pores (small

fraction of particle radius). The general problem of heat and mass transfer

within a single particle is three-dimensional. It is reduced here to one

24
space dimension with the assumption that the particles approximate a sphere, a

cylinder or a thin flat plate. Furthermore, it is assumed that the fuel

particles are all initially the same size; a few comments will be added below

on the effects of a broad distribution of sizes. The particles are also

assumed to be rigid structures. They may lose mass by attrition from the

outer surface or from the Internal pore surface but they do not go through a

plastic state that would allow various forces to deform them; this is a

reasonable approximation for most fuels of interest but it could be an oversimplification for some flexible polyurethane foams, for example.

With these assumptions, the following conservation equations apply to the

interior of a typical fuel particle. See the Nomenclature table for an explanation of the symbols used.

Conservation of gas mass within particle pores:

~dt ^*P pGP^

+ ^a "9r PGP

VGP^ =

^ (\ P i

V

G Ai) (2)

The first term accounts for transient accumulation of gas within the particle

pores, the second term for flow of gas out through the pores (three possible

particle geometries depending on value of a); the third term is a summation

over the reactions which gasify the particle. As noted above, these comprise

both surface oxidation reactions and volumetric pyrolysis reactions. If the

particles are completely non-porous , however, the surface reaction term

belongs only in the boundary condition at the particle periphery. (Note that

pyrolysis will immediately generate pores in a non-porous, non-fluid particle;

they will persist unless the particle passes through a subsequent fluid

25
state.) The pyrolytic reactions implicit in the third term do not have a

surface area dependence so Ayp^ drops out of those rate expressions.

Conservation of solid mass within particle:

d_

3t U 1 *P

> PP 1 ^(\pa V

G£ ( 3 )

The first term is the transient loss of solid as a net result of the chemical

reactions described by the second term; there is no convective term since the

particles are rigid.

Momentum of gas in particle pores:

Here the first two terms account for acceleration of the gas in the pores, the

third for the pressure gradient driving the gas flow, the fourth for the drag

due to the pore walls (assumed laminar). The fifth term accounts for the

acceleration of the gas generated by the various reactions; it appears here

because the gas continuity equation has been substituted into the equation.

Conservation of typical gas species in particle pores:

( 4 )

( 5 )

26
The gas continuity equation (Eqn. (2)) has been substituted here as well. The

first term is due to accumulation of species j in the pores, the second term

to convection of the species in the pores, the third to diffusion in the

pores. The first part of the reaction term accounts for creation or destruction of species j by any or all of the £ reactions in the particle. The

second part of the reaction term accounts for dilution by gasification

products and comes from substitution of the gas continuity equation. Note

that the dlffusivity D

£ p is taken to be the same for all species and it is the

effective value in the pores; this can be much less than the free gas value

when the pores are comparable to the mean free path of the gas molecules (500-

1000 A pores at 1 atm.) (76). It has been assumed here that there are no gas

phase reactions in the pores affecting the species since residence times are

generally short and temperatures low (recall the earlier discussion of gas

phase reactions on the smolder wave scale); this could require modification in

some cases. Oxygen is the gas species of foremost interest.

Conservation of typical condensed phase species in particle:

Here Eqn. (3) has been substituted generating the second part of the reaction

rate term. Again there is no convective term because the solid is assumed to

be rigid.

( 6 )

27
Conservation of gas and solid energy in particle:

9h

! O Vpp at

ah + 4>dP, GP

ah

p gp at + 4> d p

GP

P

KGP GP 3r

j_ _a_ a 3r

r

GP

D

ep i , STP

+

h

GP

) V

Gi (7)

Here Eqns . (2), (3), (4), (5) and (6) have been substituted in the original

energy balance to obtain the form shown. The first and second terras are due

to transient accumulation of sensible enthalpy in the solid and gas, respectively. The third term is due to the enthalpy of the gas convecting in the

pores. The fourth term is due to composite gas/solid conduction of heat. The

fifth term describes the net diffusion of sensible enthalpy with gases in the

pores. The first part of the reaction term is the chemical heat effect (at

standard conditions); the second part is a dilution term arising from sensible

enthalpy differences between the solid and gas phases. Note that all kinetic

energy, pressure work and drag work terms are neglected. Radiation transfer

within the particle is also neglected.

Several auxiliary relationships are needed to complement the preceding

equations. Each reaction rate (or global reaction) must be explicitly

described in terms of temperature and species dependencies. The temperature

and species dependence of the transport and thermal properties is required.

The perfect gas law is typically assumed to relate gas density, pressure and

temperature. The pore system in the particles can be expected to evolve

considerably as the various reactions proceed. The mass loss dependence

28
of 4> p and A^pi could be obtained empirically although little work has been

done; most measurements pertain to coal chars (52). There have been a number

of attempts to model the evolution of pore systems during gasification of

single coal char particles; these are reviewed in Ref. 52. The pore size

distribution also affects the effective pore diffusivity D

ep

and the flow drag

constant ap^,; a model or empirical data are again necessary.

The two most pertinent boundary conditions on the external particle

surface are those for a typical species and for energy transport.

Species :

p v Y T

mGP GP jGP

— pM <t> D

GP T ep ( 3r )r

K t YjGP

” YjG^

+ “

*P

) ] ^gAJ ( 8 )

Here the first term describes convection of the species from the pore system

below the particle surface. The second term describes net diffusion of the

species to or from the pore system. The third term describes net diffusive

plus bulk flow (from particle interior) mass transfer through the boundary

layer around the particle. The lumping together of flow and diffusion in one

term is conventional if not strictly correct; one must be careful that mass

conservation and stoichiometric requirements are satisfied. The fourth term

is the source or sink of the particular species due to all reactions on unit

area of the exterior surface of the particle.

Energy

+ H t

g

) ( l ^p) J Q £,t)

+

^P PGP

V

GP

h

GP (9)

29
The first term accounts for composite gas/solid heat conduction in the

particle at its exterior surface. The second term describes convective heat

transfer across the boundary layer around the particle; it includes conducted

heat and sensible enthalpy in products from the particle Interior. The third

term gives the net source or sink of heat due to reactions on the exterior

surface of the particle; note that the reaction heat here is the value at the

actual surface temperature. The fourth term is the sensible enthalpy carried

by gases emerging from particle pores.

Inspection of Fig. 2 leads one to conclude that this approach to

describing the particles with internal gradients cannot adequately describe

two modes of energy transfer on the scale of the smolder reaction wave. These

are solid-solid conduction and solid-solid radiative transfer. The gradients

in the particles are assumed here to be one-dimensional (normal to the

exterior surface) whereas the existence of these two transport processes

implies an asymmetric temperature distribution in each particle (threedimensional) .

As will be seen, the gas phase around the particles is always treated

here as a continuum on the smolder reaction wave scale. In this case, one

could approximate the wave-scale heat conduction process as a gas/solid

composite process appearing in the gas phase energy equation only; this could

even include a radiation-corrected conductivity. The radiation might be

better treated by taking the wave-scale distribution of particle surface

temperatures (e.g., bottom of Fig. 2) and using it in a radiative transfer

model such as the four-flux approximation for a two-dimensional smolder wave

(77).

30
Bulk Solid Equations . If the internal temperature and species gradients

(normal to the particle exterior surface) are sufficiently small so that

reaction rates within the particle are everywhere the same (Zone I behavior),

then it is unnecessary to consider these gradients. The necessary conditions

for this will be examined below. If these gradients can be neglected, there

is no need to consider individual particles; the condensed phase can be

treated as a continuum on the smolder reaction wave scale with no particle

scale gradients of interest.

In the equations below, one new physical phenomenon is introduced, that

of shrinkage of the fuel bed as it is gasified. It was noted in connection

with Fig. 1 that this can be a quite significant effect with some fuels. It

is a result of shrinkage of individual particles and possibly deformation of

the particles, especially in the case of fibers. A complete description of

this phenomenon would require a set of mechanical force balance equations for

the fuel bed, coupled together with the conservation equations given below.

Rather than do this, an approximation is considered which could be adequate in

some cases. Only the possiblility of coherent motions of the bed particles is

considered; situations in which individual particles break loose from their

neighbors and move several particle diameters under the influence of a force

such as gravity are not considered.

The bulk fuel phase equations presented here were derived with a twodimensional smolder wave in mind; it appears that a third space dimension

would not add any new complexity. The coordinate system is fixed in space.

31
Since the fuel bed can shrink, a differential equation describing the

void volume in the bed is needed.

ii - V .[Vp (1 - <t>)] = n

p

Ap (10)

This indicates that the transient change in void volume is due to any net

flow of particle volume in/out of the control volume and to loss of particle

volume upon fuel gasification. Here Ap is the flow displacement volume of

each particle. It is assumed here that this volume can be empirically

measured as a function of mass loss from a particle (e.g., by displacement of

a non-wetting liquid such as mercury). Then Ap is proportional to the rate of

mass loss from the particle.

The particle velocity vector Vp in Eqn. (10) is a result of fuel bed

shrinkage upon gasification. In a situation such as that shown in Fig. 1,

shrinkage downward is probably the most important effect but there is horizontal shrinkage as well with a net particle movement toward the direction of

smolder propagation. At least for some fuels, bulk shrinkage of the fuel can

be empirically characterized to a first approximation by simply measuring the

mass loss dependency of the bulk volume of a fuel bed small enough to be

heated uniformly. Given this function, one can estimate the components of Vp

during smolder propagation with the following:

v Py ' H ft [ym/n,

o

)]

1/3

dy' ; vpx

= j‘ [VjOn/n,) ] 1/3

dx' (11)

Here Vg (m/m

c

) is the empirical, mass-loss dependent, fractional bulk volume

function, assumed isotropic. Its rate of change integrated over a path in the

32
horizontal (x) or vertical (y) direction from the point of interest to an

"anchor" point gives the velocity components of the movement of that point.

The "anchor" point in the y-direction is the bottom of the layer; in the xdlrection, it is the unburned end of the fuel bed. This approach to

describing bed shrinkage is clearly approximate; it ignores mechanical aspects

of the fuel bed that could hinder the computed movements. At some point

during shrinkage, real fuel beds frequently develop cracks because the

particles are not free to move. If the cracks are large on the scale of the

smolder reaction wave, they may preclude the continuum approach used here.

Conservation of gas in the bulk condensed phase:

This is the gas within the pores of the particles. It convects through the

fuel bed control volume only as a result of the shrinkage-induced movement of

the particles (second term). The gasification reactions in the particles may

produce this gas (third term); this is the same set of reactions as in

Eqns. (2)-(7). The net transport of gas out of the particles and into the

bulk free volume of the fuel bed has both diffusive and bulkflow components.

Here it is described in a single lumped mass transfer term (fourth term) as is

usual practice for packed bed mass transfer.

- K ( 12 )

33
Conservation of solid in the bulk condensed phase:

^ [(!-)( l-+p

)p

p

] + V • [v

p

(l-+)(l-*

p

)p

p

] = - < 1-4>)I (\P1VGt*pt) <>3)

A

This is basically similar to Eq. (12) except that no solid moves across the

particle boundary.

Conservation of typical gas species in bulk condensed phase:

This is a typical gaseous species within the pores of the particles; oxygen is

again the species of foremost interest. The gas continuity equation has been

substituted in this equation producing the dilution terms that appear as the

second part of the reaction and mass transfer terms. Note that any smolderwave-scale diffusion of these gas species appears not here but in the bulk gas

phase species equations.

Conservation of typical solid species in the bulk condensed phase:

The bulk solid continuity equation (Eq. (13)) has been substituted here

producing a result similar to Eq. (1A) except for the mass transfer term.

YjGP (1A)

3A
Conservation of gas/solid energy in bulk condensed phase:

\ 3h

CP

dh

P

- - _ _ )

( l - 4) ' D + (l — 4 ]p + & p v • Vh +fl-4)pv »Vh >

] 9P

pGP 3t L WP

J P 3t

9P

MGP P GP ^ 9P

JM

P P Pj

-V • [(1 - *)yrr

p

] + V . S = HA^tTg - T

p

)

+ " h

G

' bp- (1 " ^ l

( A

VPl V

Gt

R

T£^ + K \b J-

( Y

KGP

" Y

KG^ ( h

GP

" h

GK'

jL

+ (1 - 4) I £

R

p l , STP ^ (16)

Eqns . (13)-(15) have been substituted in the original energy conservation

statement for the fuel bed control volume to obtain the form above. As in the

energy equation for a single particle, kinetic energy, pressure and drag work

terms have been neglected. The first four terms (brackets) describe enthalpy

accumulation and convection for both the gas and solid in the bulk condensed

phase. The fuel bed now has a well-defined thermal conductivity although the

condensed phase contribution to it (fifth term) may not be so easily separated

from the gas phase contribution as is done here. Radiative transfer (sixth

term) requires a model such as the four flux approximation mentioned above.

Scattering could appreciably alter radiative transfer in some fuel beds of

interest and, of course, the changing chemical nature of the solid during

smoldering will substantially alter its radiative properties (generally making

the solid more absorbing and less scattering as it is converted to a char).

The seventh term describes convective heat transfer between the bulk solid an i

bulk gas phases. The next two terms describe dilution of enthalpy upon s i i

<-

gasificaton due to a difference in the specific enthalpy of gas and solid

phases. The last term gives the net rate of chemical heat release due to the

various reactions in the bulk condensed phase.
As with the single particle equations, various auxiliary relationships

are needed to complement the above conservation relations. Among those not

mentioned in that previous context or in the preceding discussion is the

specific area of the bulk fuel bed (exterior area of particles), Ayg, as a

function of mass remaining; this is calculable if the particle size and shape

are well characterized but it can pose a problem with some fuels. The solid

to gas transport coefficients K and H should be calculable from various packed

bed correlations; the mass loss rate for the particles will probably be, in

most cases of smoldering, too small to alter these values appreciably from

those of inert particles.

Bulk Gas Equations . Regardless of whether the condensed phase requires

single particle treatment or bulk treatment, the gas phase around the

particles is treated here as a continuum without transverse gradients on the

particle scale. The following set of bulk gas conservation equations is

coupled to either of the two sets of solid equations given above in order to

obtain a complete description of the smolder wave propagation process.

Conservation of gas mass in bulk gas phase:

It

+ V ’ (VG

PG^ = K \b l

(YKGP

" Y

KG )

Here the gas is convecting (at velocity Vq) as a result of buoyancy or

external pressure forces. The very slow gas movement that would be caused by

drag from the shrinkage of the particle bed is neglected in this development.

The source term on the right is the same as the last term (sink) in Eq. (12).

36
The mass fraction of component K in the particle in last term) is

evaluated at the particle/bulk gas phase interface.

Momentum of gas in bulk gas phase:

*°G IT * *P

G

(’G

• *) V

G + *VP = - *P(i - CjUgVg - c2 (y

V

G *VB ^ Y

KGP " \g^'

J\.b

( 18 )

Note that this is a vector equation obtained by summing x and y components.

The gas continuity equation (Eq. (17)) has been substituted here introducing

the acceleration term on the right hand side (last term). The second and

particles. This a two-dimensional differential form of the Ergun equation

(78) applicable to laminar and turbulent flow. The parameters c^ and C

2 are

dependent on bed porosity and particle size; they are best measured

empirically (by measurements of pressure drop -versus flow rate on uniform beds

of particles). In cases where a well-defined hydrostatic pressure gradient

exists in the fuel bed, a perturbation pressure can be substituted for the

total pressure P and its gradient used to convert the gravitational force term

(first on right hand side) to a more familiar buoyancy term (79).

Conservation of typical gaseous species in bulk gas phase:

third terms on the right hand side describe the flow drag caused by the fuel

37
Again the gas continuity equation has been substituted introducing the dilution term on the right. The reaction term (last term) includes only homogeneous gas phase reactions. The diffusion term (third term) contains the

effective diffusivity of the various species in the bed, D

eB’ assumed the same

for all species. In a packed bed, as the Reynolds number (based on particle

diameter) Increases, the flow streamlines became increasingly complex due to

repeated splittings and diversions by the particles. The result is

effectively a mass transport process, dispersion (analogous somewhat to turbulent eddies), that ultimately dominates over molecular diffusion but is

similarly described. Thus D

g g changes smoothly from an effective molecular

diffusivity to a dispersion coefficient with increasing Reynolds number (80).

An added complication, however, is that dispersion typically differs in directions parallel to the flow and transverse to the flow. The possible impact of

this should be considered in the context of a specific problem.

Conservation of gas energy in bulk gas phase:

Here Eqns. ( 17)— ( 19) have been substituted to obtain the form shown. Note

that, as in Eq . (16), bulk heat conduction (third term) is proportioned to

this phase in accord with the local fractional free volume (or area), <J> .

This is a reasonable approximation for the general situation where local bulk

4">G IT + • *>G

- ’ • (VT

C

) - -havb(t

g

- T

p)

+ K \b l

^ hKG

h

G^ ( Y

KGP

Y

K.g)

^PG

D

eB

V

^KG ^ ^ + ^ ( R

mGQm,STp)

m

‘mG vm, STP ( 20 )

38
gas and bulk solid temperatures are unequal. Note that radiation in the gas

is neglected on the assumption of short pathlengths and small emlssivities

.

The other terms in Eq. (20) are analogous to similar terms in Eq. (7) or

Eq. (16), explained previously. Again, kinetic energy, pressure work, drag

work and gravity work are neglected. As before, various supplementary relations giving temperature and species dependence of thermal properties, etc.,

are needed to complete these equations.

A complete problem description requires the bulk gas phase equations

(Eq. (17)— (20) plus supplementary relations), coupled to either the single

particle equations (Eq. ( 2 )— ( 7 ) plus supplementary relations plus interphase

boundary conditions such as Eq. (8) and (9)) or the bulk solid phase equations

(Eq. ( 10)— ( 16) plus supplementary relations). In addition one needs boundary

and initial conditions on the bulk gas and bulk solid (if treated as such) for

the particular fuel bed configuration of interest. It was noted previously

that the gas phase around the fuel bed may, in some cases, be a closely

coupled part of the total propagation problem. This could be particularly

true when the smolder process induces a buoyant flow in the surrounding gas

and the details of that flow have a strong influence on the rate of oxygen

supply from the ambient air to the fuel bed. Since such buoyant flows themselves can pose a difficult problem, the fully coupled, ambient gas-plus-fuel

bed situation has not been addressed. Instead, as a first approximation, the

ambient gas is assumed to be homogeneous just as was the case with the single

particle equations. Heat and mass transfer from the ambient air to the fuel

bed are assumed to occur across a thin boundary layer with known heat and mass

transfer coefficients. Appropriate boundary conditions on the bulk gas and

the bulk solid will be analogous to Eq. (8) and (9), minus the surface

39
reaction terms; the bulk solid boundary condition would typically contain

radiative exchange with the surroundings as well. Note that the use of the

single particle equations (Eq. (2)— (7)) to describe the condensed phase does

not permit application of fuel bed boundary conditions on the solid phase.

The boundaries would then be felt by the solid only through its interaction

with the bulk gas; this could yield a poor description of the solid thermal

profile near the bed boundaries. This is the second area related to heat

transfer on the smolder wave scale in which the use of the single particle

equations poses problems. However, these problems can probably be avoided in

most smolder problems of interest. As will be seen below, the condensed phase

particle scale gradients of most frequent significance are in species and not

temperature. If this is the case, the bulk condensed energy equation can be

used to obtain the bulk solid temperature (with some further minor

assumptions), assumed uniform throughout the typical particle, while the

remaining single particle equations are used to correctly describe particle

scale species gradients and reaction rates. Then the heat transfer problems

associated with the single particle equations do not appear.

Non-Dimens ionalizat ion . The preceding sets of equations can be

substantially simplified for specific problems once various parameter values

have been defined. To facilitate the identification of situations where

various terms can be neglected, the equations have been non-dimensionalized

.

The non-dimensionalization via the reference quantities in Table II is

designed to render all important terms of order unity. The choice of

reference quantities imposes some interrelationships among them. The expression in Table II for the reference bulk gas flow velocity v^R comes from the

assumption that flow drag and buoyancy are balanced in the bed of fuel

40
particles. The reference particle flow velocity (due to particle shrinkage),

v pK , is taken to be comparable to the smolder propagation velocity and estimated from the average equivalence ratio of the smolder wave assuming all

oxygen is transported in at the flow velocity, VqR

. The expression for the

reference length on the scale of the smolder wave, £ , comes from the assumption that heat conduction and convection are comparable in the particle bed.

The reference time, t R , is thus essentially of the order of the smolder wave

passage time. The actual ranges of numbers used for these reference values

(as listed in Table II) cover the spectrum seen experimentally in a wide

variety of configurations (Table I plus some other cases as well).

The large set of dimensionless parameters in Table II is not necessarily

the minimum number nor the optimum set for a specific problem. This set

suffices to assess the relative importance of various terms in Eqns . (2)-(20)

for a wide spectrum of smolder propagation problems.

The dimensionless set of conservation equations is as follows.

Conservation of gas mass in bulk gas phase (from Eq. (17))

3( Pr

<j>)

it. —£— + v

3t

• (

V

*) - ^(> (\GP * \G))

( 21 )

Conservation of x-momentum in bulk gas phase (x component, from Eqn. (18)):

~ ~ ^v

x A A A ^ vx

A ~ A ^ V x\ ~ 9p ' ~

i T

A

”

3

* PG —+ A

* V

^+ PG ^

= ~*Z + V

y v - It. V

x 6 x Kl - Vx ^(’7k(\gP - \G ))

( 22 )

41
Here the ambient hydrostatic pressure gradient has been separated from the

total pressure gradient term resulting in converting the gravity term to a

buoyancy term (second term on right hand side), as discussed above. The y

component of momentum is substantially the same as the x component in this

general analysis so it is not considered separately.

Conservation of typical gaseous species in bulk gas phase (from Eq. (19)):

(23)

Conservation of gas energy in bulk gas phase (from Eq. (20)):

7 • ( \

7 t

g

) -

"l2 'H'B ( TG

Tp) + l

”2K ( hGK

hG^Y

KGP

' Y

KG^ KG

+ I(’9K "kC7 • ( CG

Deb

7 \)) + ^ (’13. RJ KG in (24)

Conservation of void volume in the particle bed (from Eq. (10)):

*17 A

17 p

(25)

42
Conservation of gas in bulk condensed phase (from Eq. (12)):

19

3t

(

l - *15 *K Pcp] + *20 ’

• [

V

P

(' - *15 ) *P PGp]

(i - ,

l5 i )£1 + (i - ) J

(»2U £,)

(*22K (

Y

KGP

Y Kg)) (26)

Here, and in other equations below, the first chemical reaction term in the

condensed phase (subscript l) has been assumed to be among the largest and its

reference rate (RRj) is used to normalize the other terms in the equation;

thus it stands out of the summations over condensed phase reaction rates.

Conservation of solid in the bulk condensed phase (from Eq. (13)):

( *U / 'l5 ) ^

(l - »

15

*)(l - *p )pp + V • v

p

(l »

15

*)(l - n,

6 *p)p

- -

(l - «

15 )

R

t)

(27)

Conservation of typical gas species in bulk condensed phase (from Eq. (14)):

3Y

" **

*23j (

l - *15 ) *P PGP

+

*24j 0

- *15 *K PGP *P

* ’ Y

j GP

(‘ - *15 p)(*25jl

-

*26

j )

*1 +

(> - *15 ) J^StJ - *26

j) *21, K)

- TT 22 j (

Y

iGP - V) * YjGP ^(*26K (

Y

KGP

- YKg)) ( 28 )

43
Conservation of typical solid species in bulk condensed phase (from Eq. (15)):

3Y

A A

(lI lA/lT l5

) PP + V

P

* ^ Y

jP "

(

l/

(

l “

*15 *)) | (^25^ + * 263 )

R

i)

(29)

Conservation of gas/solid energy in bulk condensed phase (from Eq. (16)):

(' -

*15 ^['V*

A A 3h

GP

15'

3t

+ *27 *P PGPV

P

* V h

GP +

- V

(

l - *16 *p) *P

"

V T

1

+

(

h

G

h

Gp)(

l *15 *) j[(*l8£R

i) +

l

+

(l - tt 15 <J>) I^3U * lBlR

£ )

* N A 3h

p

• + (*- *16 *r>KW pp

r

A 3t

16 *p)

A0 V • Pp p TMp]

•

<

l>

CN 00

A

s - *

29

A / A A \b \

t

g " Tp )

\+ y

TT _ _ ( Y Vi

A - h

' KG

30K\ KGP KG / \ GP <

(30)

Alternatively, if the particle phase has species and temperature gradients on

the scale of a single particle, Eq. ( 2 )— ( 9 ) apply instead of Eq. ( 10)— ( 16) and

the former equations, when non-dimensionalized via the reference quantities in

Table II, become as follows.

Conservation of gas mass within particle pores (from Eq. (2)):

’32 ^ (*P pGp)

+ *33 k ^

(*“ PGP *P

V

Gp)

‘ R

1 + J,(*2U

R*) (31)

44
Conservation of solid mass within particle (from Eq. (3)):

34

9t

(

l “ *16 ^p) Pp] ~R

1 " ifilU R

i)

Conservation of momentum of gas in particle pores (from Eq. (4)):

A /S ^ Q p

A A * A "35 *P PGP ~ +

*36 ^P

PGP

V

GP ~ = “^P 7^ "GP'GP "GP L

0 \"37l "l

dt dr dr *

- - v„„ I (» 37 t

R

t)

Conservation of typical gas species in particle pores (from Eq. (5))

3Y

j GP

"on x iPon *Lr- + . ^D PPOV

1 3 (;a: ; !!igp PTM D. 39 j YP

KGP * ' "40j TPKGP GP * 41 j *a * \* ''GP "ep

dt dr r dr 3r

= ( ’24jl - W R4

+ l

[

( OIL( v - v_) a

. -- ir oc . Y

F C'Zj 25j jGPJ 21i 1 ]

"21:

Conservation of typical condensed phase species in particle (from Eq. (6))

(l - »

16 ip) *38j Pp

-4- R

1 Jj("2U

Ri)

Conservation of gas and solid energy in particle (from Eq. (7)):

v(‘ - "i6*p)»

3hp a a ^hgp A a a <*l>Qp

P 3~ + *45 ^P

PGP

+

*46 ^P PGP

V

GP ~ dt dt dr

3T

i a ra :

w ‘p\ 1 a ^47 ''a *

( r ^P * r 3r \ 3r / r 3r |G y48K\GP PGP

D

ep

r

a ayKGP

3r

1 +

*49 (

h

P

h

Gp) + ^ ho, (

l + *5U ( h

P hGp))

( 32 )

(33)

(34)

(35)

(36)

45
The two boundary conditions of principal interest at the exterior particle

surface become (from Eq. (8) and (9)):

3Y

1T 54 <f’p PGP

V

GP

Y jGP

_1I 42*P PGP

D

ep

3r

’52\> 7^ ( T‘p- ‘ (

* -

3r

—P= (Y -Y 1 + fl- 1 jGP G

} 1

*16 ^P^ l ^53*

*16^ I (

1T 43 £j

R

$)

A A A A + n 55

<*

) P

PGP

V

GP

h

GP

(37)

(38)

It should be noted that in the preceding non-dimensionalization process, all

reaction terms, whether in the bulk gas phase or on the particle surface, were

assumed to have power law species dependencies and an Arrhenius temperature

dependence; this form is mirrored in the definition of RR (see ir in

Table II). Many char gasification reactions are more aptly described by a

Langmuir-Hinshelwood kinetic rate law which is somewhat more complex but

behaves like a power law over limited species concentration ranges (52, 81).

In Eqns. (21) to (38) all wave-scale gradients are of order unity as are

A the time derivatives. Transverse gradients (3/9r) in the single particle

equations are to be assessed as are their analogs in the bulk gas and solid

equations, i.e., terms Involving a difference between bulk gas and bulk solid

A species concentration or temperature. The dimensionless reaction rates R^ in

A the bulk gas phase and R^ on the particle surface are of order unity for the

principal reactions of interest when T approaches unity. The net result is

that all symbols with a caret above them denoting a non-dimensionalized value

are of order unity; the relative importance of each term In a given equation

then is indicated by the magnitude of the tt group it includes. For the transverse gradient terms first mentioned, the situation is somewhat more complex,

as will be discussed.

46
Magnitude of Dimensionless Groups and Simplifications * First, note that

in all of the dimensionless equations involving a time derivative of a gas

phase variable (exclude Eq. (25), $ is not really a gas phase variable) the

it’s are small, typically 10 or less (see , tt^ , tt^., tt^, > *27* *32’

^35 ’ ^ 393 * an<^ *45 This implies that, to a good approximation,

these transient gas terms can be neglected; the bulk gas as well as the gas

within the condensed phase can be treated as quasi-steady. This is a consequence of the large disparity in densities between the gas and condensed

phases as well as of the chosen time scale here, i.e., the wave passage time.

If rapid ignition or other fast transient was of interest, the storage

capacity of the gas phase could become significant.

The solid/gas density discrepancy also renders negligible the gas

convecting with the bulk condensed phase (in its pores; second term in

Eq. (26) and (28); third term in Eq. (30)). Note that the corresponding tt*s

(

Tr 2 Q » ^04

i and ^27^ are a H °f order 10 so these terms can be neglected with

the same degree of accuracy as that in the quasi-steady gas phase assumption.

(This does not apply to the gas convection in the single particle equations

since it can be occurring at a much higher velocity.)

The preceding approximations leave Eq. (26) and (28) with only two types

of terms, mass (or species) generation by reaction and mass (or species)

transfer across the external boundary layer around the condensed phase

particles. These terms must balance each other. Eq. (26) reduces to the

algebraic statement that the net mass transfer rate equals the net particle

gasification rate. Eq . (28) with the aid of Eq. (26), reduces to the algebraic statement that the rate of mass transfer of a typical species to or fror

47
the condensed phase equals its net rate of consumption or production by the

surface reactions in the particle phase. If this algebraic equality is

solvable for the concentration of the species of interest in the condensed

phase (e.g., that of oxygen, if the reaction is first order in oxygen mass

fraction), this relation can be used to eliminate that unknown concentration

elsewhere in the model equations (82).

The gas momentum equations (Eq. (22) and (33)) can usually be simplified.

Table II shows that the it's multiplying the acceleration terms ( , tt^ in

Eq. (22); ir

, ^

2,11 Eq * (33)) can be small or of order unity. Most

realistic cases are expected to make these terms small. Only a fuel bed with

a very open structure, producing a low flow drag, while still sustaining a

compact smolder wave would yield significant acceleration effects in Eq. (22).

The conditions for significant acceleration effects in single particles are

analogous—rapid gasification out of small particles with very open, low flow

drag pores. Typically only the drag, pressure gradient and buoyancy (bulk gas

only) terms need be retained (see and tt^). Even the turbulent part of the

flow drag is typically small unless the Reynolds number of the particles

exceeds 10 or so (83) and it is not even included for flow in the particle

pores (Eq. (33)).

The preceding approximations can be expected to apply to most smolder

propagation models. Their validity in specific applications should always be

checked, however.

Note that the tt’s multiplying the interphase transport terms (^k’

^12' *22 j

* *29* ^OK ^ have a very large potential range of values as shown in

48
Table 11. Tne mass and heat transfer coefficients used to evaluate these tt's

are based on standard packed bed correlations (73). These transport processes

must always be occurring at a significant rate in this type of problem, thus,

the terms in the equations which describe them must be of order unity. This

implies that when' the tt value in the term is large, the temperature or mass

fraction difference in the term must be small and conversely. Thus, if tt

and are large, the temperature difference between the bulk gas and the

bulk solid must be small. If it is sufficiently small as to have no significant effect on the reaction rates, there is no real need to keep track of this

difference. The bulk gas and bulk solid energy equations can be combined

yielding a single local temperature to be tracked. Because of the close

analogy between heat and mass transfer, one might expect the temperature and

mass concentration differences between the two phases to be small or large in

similar conditions. In the reaction zone, however, if the condensed phase

reactions are the major cause of the temperature difference between phases,

the temperature difference will tend to be larger than the concentration

difference (111). In any event, when tt^, 7r 2 ?

j an<^ ^OK are ^ ar §e compared to

unity, the concentration differences are also small and it becomes unnecessary

to separately track a bulk gas phase and bulk condensed phase oxygen mass

fraction, for example.

For cases with internal gradients in the particles, the interphase

transport coefficients show up in tt^ and in Eqns. (37) and (38); variations in their values have somewhat different implications. Recall that the

boundary layer heat and mass transfer resistances are in series with those in

the particle. A large value of tt means that the heat conduction in the

particle is much faster than heat transfer across the boundary layer around

49
the particle. Then the temperature within the particle will be nearly uniform

but it will differ appreciably from the surrounding bulk gas. If so, Eq. (30)

can take the place of Eq. (36). The converse case, small affords no

simplification; it merely means the bulk gas and particle surface temperatures

are nearly equal. Note that the range of expected values for is biased

toward values greater than unity; typical fuel particles smaller than about

1 cm tend to satisfy this criterion for uniform internal temperature. Similar

reasoning applies to so that large values would again imply uniformity,

this time of gas species concentrations, within the particle pores. Table II

shows, however, that the range of expected values for is biased strongly

toward values much less than unity. This is a consequence of the small

dif f usivities likely to be encountered in particle pores. Thus temperature

uniformity is much more likely than species uniformity in the particles as a

consequence of slow external transport. Even when external transport

considerations such as these do not yield diminished gradients in the

particles, the gradients may still be small, as shown below.

The single particle equations show that gas species within the pores can

move by both convection and diffusion. The convection is due to the selfgenerated internal pressure gradient in the particles as a result of their

gasification. The species of interest are those that affect the reaction

rates within the particle; oxygen is of primary interest but CO 2

, CO, H

2 and

H

2 O could be pertinent in high temperature cases, as discussed previously.

Oxygen must diffuse in against the net outward flow. Internally generated

species both diffuse and flow outward. The absolute and relative magnitudes

of tt

, - . and tt. , . in Eq. (34) indicate the importance of convection and diffu40j 41 j

sion of the typical species. Note that was evaluated using the maximum

50
value of VGpR (i.e., assuming that flow alone carries all of the gasified mass

from the particle) so that * s less than or, at most, equal to unity. On

the other hand may be very small (large particles, small pores) or very

large (small particles, large pores). Suppose that is much greater than

unity. Since the total diffusion term will be of order unity at most (to

balance the reaction rate), then the species gradient which causes diffusion

must be proportionately much less than unity. This implies the species

gradient can be neglected and Eq. (28) can be used in place of Eq. (34).

Conversely, suppose that is much less than unity (and )• Then

diffusion is restricted and convective flow may play an important role in

species movement. The diffusion term does not drop out of the problem since

the species gradient for an incoming gas can become greater than unity. Thus

oxygen, diffusing in against the convective outflow penetrates only a fraction

of the particle diameter. In the extreme limit, the oxygen cannot penetrate

the pores significantly and oxidation reactions are confined to the outer

surface of the particle. This is analogous to Zone III behavior but we have

the added complication of the outward flow of gases contributing to it. If

these gases originate in an oxygen-free region of the particle, they must come

from the pyrolytic reactions. This behavior is most likely to be encountered

in the pyrolysis portion of a fast moving smolder wave when the initial fuel

particles are large and have no pores or only very small pores. Note, then,

that these physical effects can shift the competition between pure pyrolysis

and oxidative pyrolysis; if oxygen cannot penetrate the pore system, oxidative

pyrolysis is confined to the particle exterior while pure pyrolysis is

uninhibited. In sections of the smolder wave where such pyrolysis is minimal

or negligible (i.e., where char oxidation is occurring), the gasification

process will essentially cease in portions of the particle where oxygen does

51
not penetrate and the net convective outward flow will be due only to the

oxidation reactions. Zone III behavior in this case involves no flow from

within the particles; it erodes from the exterior (shrinking core) and all of

the internal species and flow equations can be dropped.

Similar considerations regarding convective and conductive heat transfer

within a particle are influenced by the values of x., and tt._. A value of tt._

that is much greater than unity implies that the temperature gradient in the

particle is much less than unity and, thus, Eq. (36) can be replaced by

Eq. (30). Conversely, much less than unity implies large temperature

gradients so that conduction still may be important. Convection is important

only if diffusion is unable to remove the products rapidly (small as

above). If convection is not important, the flow equation for the particle

Interior can be dropped.

Note that ir^ has a greater potential range of values below unity than

does . This is again due to the possibility of restricted diffusion in

pores; the pores typically have no great effect on the thermal conductivity or

thermal diffusivity of the solid particles. One can infer from this that

smolder problems with significant internal species gradients in the particles

but negligible temperature gradients will be encountered more frequently than

problems with both types of gradient. Recall that a similar conclusion was

reached when examining the Influence of external transport limitations. Of

course, if the species and temperature gradients are both small, the single

particle equations are not needed.

52
Laurendeau (52) presents a further argument to the effect that

temperature gradients can frequently be neglected. Originally derived for

catalyst particles, it pertains here to fuel particles in which only char

oxidation is occurring. It also pertains here only to cases in which ir^ is

at the low end of its range in Table II since the assumption is made that

transient thermal processes in the particle are negligible. Given these

restrictions, one can estimate the temperature gradient within a particle by

balancing heat conducted out of the particle against char oxidation heat

generation; the latter is limited by the rate of oxygen inward diffusion in

the regime of interest. The final result becomes

eT ^ ((t

pc - Tpc )/Tpc ) = -

D

ep

pGP

Y

OXS

(W

PS ' PS X

P PS

(39)

Here 6^ is the fractional variation in particle temperature (difference

between center and surface temperatures (Tp^. and Tpg, respectively), divided

by the surface temperature), Y is the oxygen mass fraction at the particle

UX o surface and Q , the heat of the char oxidation process. Even for fairly

large pores (500A) the value of 3^ computed above is likely to be less than

0.05. Laurendeau quotes studies showing that < 0.05 typically yields

results quite similar to isothermal particles.

A few other simplifications in the single particle equations are

possible. Inspection of Table II shows that and Eq. (33) can be

much less than unity (and almost always are); these acceleration terras can

then be dropped from the momentum equations leaving a balance between the

self-generated pressure gradient and laminar flow out through the particle

pores. The last term on the left hand side of Eq. (36) describes the net

53
enthalpy transported by diffusion. While Table II shows that may he

small or large, it is probable that the algebraic sum of diffusive enthalpy

fluxes represented here will be small unless one species has an exceptional

specific enthalpy. Neglecting this term should give the same degree of

approximation as that already included in the assumption of equal

dif f usivities for all species. The surface reaction terms in Eq. (37) and

(38) are typically either dominant (non-porous particles) or negligible

(typical porous particles); they should be retained or dropped accordingly.

All of the preceding assumed that the fuel particles are of a single

size. In the absence of internal gradients, it does not really matter if this

is so or not (unless the oxidation reactions are confined to the exterior

surface of the particles). However, if there are internal gradients, a

distribution of particle sizes can add substantial complexity in behavior. If

the largest particles are not small compared to the smolder wave length in

regions of average or smaller particles, the smolder front becomes inherently

three-dimensional and untreatable by the continuum assumption used here.

Qualitatively, one can envision the behavior of such a system. Large

particles take longer to be gasified than small particles when the internal

gradients are significant (Zone II or Zone III behavior). The smolder front

would tend to move ahead and to surround large particles ultimately leaving

very large particles behind to be slowly consumed or quenched. Similar qualitative behavior can be exhibited by a fuel bed of small particles which are

very unevenly packed, i.e., regions of higher packing density (and lower

permeability) can mimic the behavior of very large particles.

54
If there is a distribution of particle sizes, at least some with internal

gradients, and they are all small enough to satisfy the continuum wave assumption, the approach here can be used, in principle. Recall, however, that the

boundary conditions (Eq. (8) and (9) or (37) and (38)) apply at the particle

radius which is different for each segment of the size distribution. This

implies a need to solve the single particle equations for each particle size

increment of interest at each station along the smolder wave at each time

step—a virtually hopeless task. In fact the numerical solution task is

daunting even for a single particle size and, as will be seen, it has not yet

been attempted in the context of smolder propagation. Qualitatively we can

again envision the behavior. The smallest particles will meet the criteria

for Zone I behavior; there will be no significant temperature or species

gradients within these particles and they will be consumed first at a rate

dictated by the reaction chemistry. Somewhat larger particles will most

probably have significant species gradients but a uniform internal temperature. Still larger particles will have species and temperature gradients; the

largest of these will be consumed last at a rate heavily dependent on heat and

mass transfer processes (possibly including external transport processes).

Approaches Used With Some Related Problems . The formidable nature of a

packed bed reactor problem with internal gradients in the particles has long

been recognized. This problem appears in catalyst beds, blast furnaces,

various ore treatment processes, incinerators and in coal and oil shale

retorting. A widely used approach is to develop an analytical description of

the behavior of a single particle and then to place this model of a typical

particle into the context of the spatially varying surroundings (temperature,

reactant gases) produced by the reaction wave progressing through the particle

array.

55
Single particle reaction models are discussed extensively in Refs. 52 and

73. Virtually all analytical models are based on isothermal particles and the

pseudo-steady-state approximation (as was used to obtain Eq. (39)).

Laurendeau (52) summarizes the essence of various classes of models. External

diffusive transport can be controlling (Zone III behavior); for a spherical

particle this leads to a particle consumption time proportional to the square

of the initial diameter. If external transport is not rate limiting, attention turns to processes at the exterior particle surface or within the

particle pores. Laurendeau separates these models into macroscopic and microscopic types. The simplest macroscopic model ignores pores in the fuel and

considers only reaction on the progressively shrinking external surface of the

particle; reactant diffusion through any ash residue may also need to be

considered. The more general macroscopic models consider the distribution of

reaction throughout the particle as a result of reactant diffusion in through

pores. Laurendeau indicates that the major practical difficulty with such

models is inadequate information on the evolution of the effective reactant

diffusivity in the pore system as it changes through the course of particle

consumption. Microscopic models focus typically on a single "representative"

pore and model its evolution with extent of reaction; more sophisticated

models consider a distribution of pore sizes. The chemistry in all of these

analytical models is typically restricted to a single nth order gasification

reaction. Szekely et al (73) present a more extensive development of such

models and also discuss the use of such models in the context of a bed of fuel

particles.

In the context of smoldering combustion, it appears that such analytical

modeling of events in a fuel particle interior could be applied to problems

56
(or segments of problems) where the only process in the particle Interior is

char oxidation. Such application has not yet been made but this approach

should find application to some practical smolder problems in the future. The

most general smolder case in which the fuel particle may undergo volumetric

pyrolysis that generates a pore system coupled with oxygen diffusion leading

to oxidative pyrolysis is probably not well enough defined yet to facilitate

any truly detailed modeling.

Forward and Reverse Smolder Propagation . The behavior elicited from the

preceding model equations is strongly dependent on the boundary conditions

imposed. Most real world smolder propagation problems involve fuel beds

immersed in air with oxygen penetrating the bed by diffusion and flow, as in

Fig. I, producing a two- (or three-) dimensional smolder reaction wave.

Clearly it is preferable to begin exploring the behavior of these equations

* with simpler configurations, preferably one-dimensional (1-D) on the smolder

wave scale. This requires that the smolder front and gas movement are

parallel and that the smolder wave thickness is small compared to any general

curvature of the smolder front.

There are two somewhat simpler limiting cases implicit in the preceding

model equations that have received most attention, since they can be treated

as one dimensional. The smolder reaction wave can move parallel to and in the

same direction as the flux of oxygen (one-demensional forward smolder) or it

can move in the opposite direction (one-dimensional reverse smolder). An

observer moving with the reaction front sees a counter-current reactor in the

forward smolder case (fuel and air come from opposite directions); reverse

smolder propagation yields a co-current reactor (fuel and air come from the

57
same direction). The impact of this difference in relative directions of

movement can be considerable. Figure 3 shows a qualitative comparison of

wave-scale oxygen and temperature profiles for these two cases. This is drawn

from the experimental work of Ref. (18) on smolder in beds of wood fibers but

a similar result could be inferred from the in situ coal gasification

literature.

The reverse smolder profiles are distinctly more simple. If one moves

with the smolder front, the air and fuel enter the reaction zone intermingled

with each other (though moving at different rates) and they can begin to react

together as soon as the local temperature is elevated by heat transferred from

the hotter regions. The wave resembles a laminar pre-mixed flame in this

sense. As with a laminar flame, steady (or virtually steady) propagation is

possible. Oxidation and pyrolysis reactions co-exist, competing for the

condensed phase on the basis of their relative kinetic rates (assuming that

transport limitations on the fuel particle scale do not enter in).

The forward smolder wave, on the other hand, seems to consist of several

distinct zones. At the front of the wave (right side in Fig. 3) there is a

plateau between a leading water condensation region and a trailing water

evaporation region. All the water driven out of previously smoldered fuel, or

formed from it, is trapped in this region which grows in length as the smolder

wave progresses. (Other reversibly condensible materials can similarly

accumulate but are usually much less prevalent.) DeRis (84) has pointed out,

in a different context, that the net heat sink across this plateau is simply

equal to the rate of sensible enthalpy accumulation in the material within the

growing plateau region. The pyrolysis zone immediately following the plateau

58
is a major heat sink; note the lack of oxygen in this region for forward

smolder. This pyrolysis front moves forward into virgin fuel as rapidly as

heat transfer processes allow though it is ultimately limited by the fact that

the heat being absorbed here is generated in the char oxidation region at the

left. All oxygen consumption and heat generation occur at the left as the

entering oxygen attacks the pre-heated residue (char) from the pyrolysis

region. For the usual modest oxygen flow rates, this oxidation zone can only

move forward after it has consumed all the fuel (char) locally so that oxygen

can reach the next element of fuel. This links the rate of movement of the

char oxidation zone directly to the oxygen supply rate and the char oxidation

stoichiometry. Thus the heat generation zone and the heat sink zone

(pyrolysis + water plateau) move forward at what are, in general, differing

rates. Typically the pyrolysis front moves faster and a hot region builds

between them (unsteady behavior). In the absence of heat losses, this is a

major sink for the net heat generated; the remainder must exit with whatever

solid residue (e.g., ash) is left by the oxidation zone. A substantially

simplified solution for forward propagation of an exothermic reaction in the

absence of pyrolysis as a heat sink (85) shows that the peak temperature tends

to increase indefinitely; this is particularly rapid when the heat generation

zone moves forward at the same rate as the heat flow (convection dominated

case). In the more general case of Fig. 3, the pyrolysis front cannot consume

heat faster than it is generated in the oxidation region and the relative

rates of movement should tend toward the ratio of the net specific reaction

heats for each zone (assuming no heat losses and a diminishing heat content In

the oxidation zone residue).

59
It was found in Ref. (18) that the differing wave structures of Fig. 3

lead to differing oxidation chemistry dominating the propagation process, at

least for the wood fibers studied there. In forward smolder, oxidation of the

char residue from the pyrolysis zone must dominate, as noted above. In

reverse smolder, this char is present in the reaction zone along with oxygen

and the degrading fuel. The oxidation kinetics of this wood fiber char were

found to be much slower than the oxidative pryolysis reactions which form the

char (first DSC exotherm as discussed in section on Smolder Chemistry). The

char was thus left behind intact by passage of the reverse smolder wave. Thus

the forward/reverse differences can include not only alterations in temperature and oxygen profiles but also a complete shift in the nature of the

dominant oxidation chemistry.

Purely forward and purely reverse smolder propagation are quite rare in

the fire safety context. These pure combustion modes are found in other areas

such as in situ coal gasification and oil shale retorting. Forward and

reverse smolder are of interest here because their one-dimensional versions

provide the simplest smolder propagation configurations for detailed modeling.

Also, portions of realistic fire safety problems may approximate these

simplest propagation modes. The smolder configuration in Fig. I is more

typical of real world fire safety problems. This can be viewed as coupled

forward and reverse smolder zones since oxygen enters the reaction zone from

ahead of the wave permitting an exothermic oxidative pyrolysis process

(reverse smolder zone); oxygen also enters from above and behind permitting

exothermic char oxidation (forward smolder zone) (24).

60
In view of the preceding discussion one can suggest a logical progression

of increasingly more complex and complete modeling efforts that would build

ultimately to quantitative solutions to the general model equations above.

The sequence is as follows: (l) One-dimensional reverse smolder with no

Internal gradients on the particle scale. Since this problem resembles a

familiar deflagration wave, it is probably the simplest starting point.

Comparisons with experiment are fairly easy; one-dimensional behavior can be

approached if the smolder wave thickness is small compared to its radius of

curvature. (2) One-dimensional forward smolder with no internal, particlescale gradients. This case exhibits a more complex wave structure. It is

also more difficult to produce experimentally; the transient growth in the

total wave thickness demands an increasingly large fuel bed diameter to permit

a one-dimensional wave front (18). Both of these experimental simulations are

most readily done with forced air flow but the case of buoyant flow plus

diffusive oxygen supply is also of interest as a prelude to the next stage in

the modeling sequence. (3) Two-dimensional smolder propagation with no

particle-scale gradients (cases such as that in Fig. 1). Such cases with a

buoyant/diffusive oxygen supply are a mix of forward and reverse smolder

behavior; they are found in many real-world situations. Various modes of

simulating the coupling to the surrounding gas phase would require study at

this point. (4) One-dimensional reverse smolder with particle scale gradients

(single size particles); species-only gradients would cover many realistic

fuels. (5) One-dimensional forward smolder with particle scale gradients;

again, gradients in species alone would cover many real world fuels. (6) Twodimensional smolder with particle scale gradients; this would require solutions to Eq. C 2 ) — ( 9 ) and Eq. (l7)-(20) with varying simplifications, as

discussed above, depending on the area of application.

61
The numerical task becomes increasingly forbidding as one follows this

sequence. Bearing in mind that these models describe the coupling of thermophysics and chemistry, one cannot justify proceeding too far along this path

without better descriptions of the chemical rate processes than are presently

available. As will be seen, at present only steps (1) and (2) of this

suggested sequence have been partially explored.

Smolder Propagation Models in the Literature

Phenomenological Models . There have been only a few efforts to take

various sub-sets of the preceding equations and solve them fully as descriptions of particular types of smolder problems. Before describing these it is

of interest to briefly mention some phenomenological models of smolder propagation that have appeared in the literature. In most of these, an attempt is

made to derive a simple algebraic relation typically showing the dependence of

smolder velocity on other parameters; the basis is usually a heat balance,

greatly simplified.

Cohen and Luft (19) measured smolder propagation velocities through a

variety of materials in a horizontal layer configuration like Fig. 1. They

attempted to correlate these results with a qualitative, empirical model. The

model is based on a macroscopic one-dimensional balance of conductive and

convective heat transfer in the bed and the assumption that the air supply is

determined by a buoyancy/drag balance in the bed. No consideration is given

to a diffusive oxygen supply even though it is probably totally dominant for

the thin layers examined (_<_ 2 cm). A variety of parametric dependencies are

predicted but not adequately tested; some of the inferred and some of the

62
measured numbers are Implausibly low (2-10% total solids gasification

inferred; measured peak smolder zone temperatures as low as 165°C).

Kinbara, Endo and Sega (20) did a series of experiments on downward

smoldering in vertical rods of various materials, mainly densely packed

cellulosics. Their initial model was later refined by Sega, who also did

additional similar experiments (86, 87). The emphasis in the modeling is on

explaining the observed dependence of downward propagation velocity on sample

width and on ambient temperature. The basis of the model is the onedimensional steady state heat conduction equation for the bulk condensed

phase; the principal approximations (besides the l-D description of a 2-D

problem) concern the rate of heat generation and heat loss along the reacting

solid surface (roughly cone-shaped); both are linearized, the former on the

assumption that heat generation is limited by oxygen transport to the reacting

surface. In the original paper an unclear argument is used to infer, from an

analytical solution of the heat equation, a smolder velocity expression with

similar sample width and ambient temperature dependencies to those seen

experimentally. Sega (87) later offered a more plausible derivation of a

similar expression; similar approximate descriptions of heat generation and

heat losses were used, however.

Williams (88) briefly discussed smolder propagation in a review of fire

spread mechanisms. One-dimensional arguments are applied in a very broad

manner noL really addressed to any specific problem. For forward smolder a

hypothesized balance between an oxygen-diffusion-limited rate of heat release

and convective heat-up of the fuel (no heat losses) leads to a linear relation

between smolder velocity and ambient oxygen level; it also leads to the

63
inference that, smolder velocities tend to be low because of the typically

large diffusion length for the oxygen (i.e., because of the slow rate of

oxygen supply). Furthermore, on the hypothesis that kinetically-controlled

smolder is unstable (and thus will extinguish spontaneously), Williams poses

an approximate criterion for smolder extinction: smolder will cease as soon

as the kinetic rate of oxygen consumption falls below the diffusive supply

rate, i.e., as soon as kinetics become the rate-limiting step in the heat

release process. Both hypotheses are qualitatively reasonable but their

application to any specific problem will modify and complicate them

considerably.

Gugan (89) devised a simple model to explain one feature observed with

rod-like fuels smoldering in ambient air, i.e., the roughly conical shape of

the steady-state reaction zone as seen, for example, in a quiescently

smoldering cigarette. He hypothesized that at the reaction surface, oxygen

consumption by reaction is balanced by radial diffusion of oxygen from the

surrounding air. Longitudinal diffusion is ignored even though the aspect

ratio of the cone is typically near unity. Insertion of a measured smolder

velocity into the resulting expression for the reaction surface shape yields a

result that is qualitatively like the measured profile. Palmer (ll) used a

similar idea to explain his observation of depth-dependent upward smolder

velocity in deep dust piles. Ohlemiller (2A) has used this idea to suggest

that the general shape of the reaction zone in Fig. 1 is dominated by oxygen

diffusion

.

It was noted previously that reverse smolder yields a deflagration wave

analogous to a pre-mixed laminar flame. This suggests the application of an

64
approximate steady-state laminar flame speed theory to this type of problem.

Corlett and Brandenburg (38) have applied the Zeldovich/Frank-Kamenetskii/

Semenov (90) theory to reverse combustion In the context of coal gasification. Their approach ignores any considerations about events in the fuel

particle interiors, implicitly assuming Zone I behavior (total kinetic

control). There appear to be some inconsistencies in the development,

incorporation of water and fuel volatilization heats in one part of the

analysis while ignoring them in others; the only reaction that is really

included is a one-stage exothermic oxidation. The results are qualitatively

plausible. The wave speed depends directly on the thermal conductivity of the

fuel bed and the rate of heat release. The peak temperature is shown to

increase slowly with air supply. The inferred wave thickness is of the order

a few centimeters. All of these inferences are in accord with either more

detailed smolder models (91) or smolder experiments (18). Corlett and

Brandenburg also applied a similar analysis to forward combustion in coal;

this is a much more dubious exercise in view of the generally unsteady

character of forward smolder.

Numerical Models . Moussa, Toong and Garris (92) did a series of measurements on low density fibrous cellulose cylinders smoldering horizontally in

stagnant atmospheres of varying oxygen content. By varying the oxygen

percentage and total pressure, they defined a domain of steady, sustained

smolder bordered by extinguishment or flaming. The material was basically

similar to rope or string. This is a small scale, radially symmetric analog

of the smolder situation shown in Fig. 1. As such, it will have a mixed

forward/reverse character. Baker's experimental results for cigarette smolder

confirm this (25, 26, 27). The smolder configuration is similar to that of

65
Kinbara, el al, discussed above; it is also quite analogous to the quiescently

smoldering cigarette examined by Gugan. The smolder behavior is twodimensional in character but is treated by a one-dimensional model. The

smolder reaction wave is split into two parts for subsequent analysis; a

planar interface is assumed to separate an endothermic pyrolysis zone from a

char oxidation zone. In view of the later studies of cellulosic materials in

Refs. 24 and 28, this concept of a purely endothermic pyrolysis zone is

probably incorrect; the high permeability of the cellulose cylinders would

readily permit sufficient oxygen to make the region of pyrolysis exothermic

(competing oxidative and pyrolytic reactions as discussed in the Smolder

Chemistry section). The cellulose pyrolysis is treated in much the same way

as is done by Bradbury et al (62), i.e., two competing reactions form either

volatiles or a char. The amount of char formed to serve as the sole fuel in

the model is then variable, depending on the balance of these two reactions

(function of heating rate) but this point is not explored. Since this

pyrolysis process is taken to be endothermic, it requires a balancing heat

input from the char oxidation zone to permit steady smolder propagation. The

char oxidation zone and the pyrolysis zone are coupled together at the

hypothetical interface by the heat balancing (and temperature matching)

requirement.

The pyrolysis zone is analyzed In some detail; somewhat simplified

steady-state analogs of Eq. (13), (15) and the sum of Eq. (16) plus Eq. (20)

were solved numerically to infer the requisite input heat flux for a given

value of steady smolder velocity. The energy equation is the sum of the two

given here because gas and solid are assumed to be in local thermal equilibrium (quite plausible for this problem). The implicit assumption of

66
negligible temperature gradients within the fibrous fuel particles is probably

quite good here; Internal species gradients do not enter this part of the

problem. Exterior surface heat losses are ignored in the pyrolysis region.

The treatment of the char oxidation region is much less detailed and much

more approximate. The char oxidation zone is assumed to be an isothermal

cylinder of length equal to the cellulose cylinder diameter; oxidation is

taken to occur on the outside of this cylinder only. Baker’s results (26, 27)

for a similar problem, i.e., cigarette smolder, imply that these assumptions

are qualitatively plausible but not semi-quantitatively accurate. The char

oxidation rate can be limited kinetically (single step reaction) or by mass

transfer through a natural convection boundary layer on the exterior of the

cellulose cylinder. Heat is lost from the char both by convection and by

radiation. The net flux available is dependent on the assumed char temperature (taken equal to the interface temperature); a steady-state model solution

occurs where the heat flux (and temperature) matches that demanded by the

endothermic pyrolysis zone.

The treatment of the char oxidation zone and the use of a char oxidation

expression from the literature (rather than one measured for the particular

material) make this a qualitative model. The assumption that the pyrolysis

zone of cellulose is endothermic rather than exothermic in this configuration

makes the model an even more approximate description of the experimental

situation though this assumption may be appropriate for other fuels. The

sometimes quantitative agreement between model and experiment shown in the

paper must be viewed as fortuitous. As a qualitative description of this type

of smolder situation the model appears useful. It predicts that extinguish67
merit occurs with decreasing oxygen availability (ambient oxygen mole fraction)

at about the point where the oxidation kinetics are becoming limiting (in

rough agreement with William's criterion). It also predicts an increasing

smolder velocity and peak smolder temperature with increasing oxygen availability. The model cannot predict the transition from smolder to flaming

since it makes no provision for flaming reactions.

Ortiz-Molina (93) and Ortiz-Molina, Toong, Moussa and Tesoro (4) also

applied the above model to subsequent studies of smolder propagation in blocks

of polyurethane and in composites consisting of polyurethane blocks wrapped

with cellulosic materials. The smolder configuration was qualitatively

similar to that used above though the samples were bigger in lateral dimensions. The model is as qualitatively applicable to a single foam block as it

is to the cellulose cylinders discussed above and again it correlates with

some of the experimental trends. Its qualitative applicability to the

composite situation is more dubious. Such composites are frequently at the

heart of upholstery and bedding smolder but the controlling factors in their

smolder have not been adequately studied. The cellulosic outer material can

usually smolder on its own; the polyurethane core may or may not smolder on

its own. The smoldering behavior of the composite would appear to involve

some potentially complex interactions rather difficult to model in any detail.

Ortiz-Molina, et al., suggest that the cellulosic fabric merely provides extra

heat to sustain the foam core smolder process. While it undoubtedly does do

this, further questions such as how the smolder speed and extinguishment limit

of the composite are determined remain to be answered.

68
Muramatsu, Umernura and Okada have, in effect, taken the pyrolysis zone of

the same smolder problem as above and treated it in somewhat greater detail;

the context is quiescent smolder of a cigarette (64). The model is onedimensional; Baker’s data show that this is a good approximation for this

portion of a cigarette (28). The pyrolysis zone is again taken to be nonoxidative; Baker’s data (28) make this questionable. Ultimately the tobacco

pyrolysis process is taken to be thermoneutral but the heat effect at 0 to 4%

oxygen appears most pertinent (28); this has not been determined. Since

oxidation is noL considered, species gradients in the tobacco particles are

not considered. Temperature gradients in the particles are also not

considered; for quiescent smolder these are very likely negligible. Steadystate analogs of Eq. (13), (15), (19) and the sum of Eq. (16) plus Eq. (20)

are solved numerically. The summation of energy equations is again used

because of the assumed local thermal equilibrium between gas and solid.

Unlike the analysis of Moussa, et al. above, the energy equation here includes

lateral exterior surface losses by convection and conduction; it also Includes

radiative transfer via a temperature-dependent thermal conductivity.

Radiative transfer can become important in low bulk density fuel beds if the

radiation path length is a significant fraction (> few %) of the thermal wave

thickness. Changes in void fraction (Eq. (10)) are ignored; this is reasonable for this portion of this smolder problem. Water is one of the species

leaving the condensed phase. Eq. (19) is applied to water vapor only; a loss

term through the permeable paper around the cigarette is included. The rate

of water evaporation is described by an empirical expression with an Arrhenius

temperature dependence and a power law dependence on the departure of

condensed phase water content from a temperature-dependent equilibrium value.

69
The tobacco pyrolysis is taken to occur as a result of four parallel, independent reactions gasifying four separate, pre-existing components

(unidentified); the kinetic parameters are measured experimentally. A

measured value of smolder velocity is imposed. The temperature (up to 500°C

and total density profiles show good agreement with measured values. Beyond

500°C the experimental temperature profile bends downward unlike the predicted

profile; this may be due to neglected exothermicity . A somewhat startling

feature of the predicted evaporation of water is that continues all the way up

to a solid phase temperature of 200°C; this has not been checked directly by

experiment. Unfortunately, the limited parametric studies of this model shed

little light on the general problem of factors controlling smolder

propagation.

Quite recently, Lelsch (94) modeled the two-dimensional problem shown in

Fig. 1, i.e., smolder propagation in a permeable horizontal fuel layer. The

model, however, is one-dimensional. It considers only steady-state propagation. The materials of application are grain particle layers (corn/soybean)

and wood sawdust, for which experimental results are also presented. The

model is more general than the previous two above in that pyrolysis and char

oxidation reactions are both allowed everywhere in accord with the Arrhenius

kinetics each follows. Reactions within the grain or wood particles are not

considered, rather they are explicitly assumed to occur only on the outside of

the particles; no justification is given for this. The small particles used

in the experiments are more likely to exhibit Zone I rather than the assumed

Zone III behavior. The reaction kinetic constants for the wood case are

inferred from the literature rather than being measured; for the grain case,

they are backed out of the model predictions after insertion of a measured

smolder velocity.

70
The model consists of steady-state analogs of Eqns. (13), (15) and the

sum of Eq. (16) plus Eq. (20) (without radiative transfer). Again here the

use of a single local temperature for gas plus solid is quite justifiable.

The one-dimensional energy equation also contains a term describing radiative

losses from the top of the fuel layer. A serious deficiency is the failure to

Include an analog of Eq. (19) for oxygen in spite of the fact that oxygen

supply was recognized to be the major controlling factor in the experiments on

smolder propagation. Instead, an assumed ’’average" level of oxygen Is used in

the char oxidation kinetic expressions. As a result, the model becomes

completely kinetically-controlled in its heat release rate rather than oxygen

supply rate-controlled. The few computed values of smolder velocity are in

reasonably good agreement with experiment but this is not a particularly

meaningful test since the average oxygen level is, in effect, a floating

parameter; the value used, 0.16 atm, is seen to be quite high when reference

is made to Fig. 1.

Cigarette smolder during the in-draw of air is a forward propagation

problem, albeit a specialized one. Summerf ield, Ohlemiller and Sandusky (45)

developed a one-dimensional unsteady model of this process during a constant

draw (steady flow) from ignition. The real problem is very much twodimensional on the smolder wave scale and may even have significant species

gradients in the tobacco particles (it appears to be a borderline case). The

two-dimensionality on the smolder wave scale is due to the tendency of the

peripheral cigarette paper to bum back allowing air to preferentially enter

the cigarette at the base of the coal (21, 95); this causes the peripheral

region to bum much faster than that near the tobacco rod centerline (21).

The question of species gradients in the particles does not seem to have b< <

71
explored in the context of this problem; the present author is not aware of

any pore size distribution data for tobacco. The results of Baker (35) imply

that this question is pertinent to oxidation only below 400 °C; above this

temperature, external mass transfer is found to control the rate of oxidation

of the solid. In the model of Summerfleld, et al., the 2-D effects due to

paper burn-back are approximated in a 1-D manner; the air inflow resistance of

the paper is assumed to drop to a very small value when the temperature of the

adjacent tobacco is greater than a fixed paper burn temperature. Gradients

within the particles are assumed negligible throughout the entire smolder

zone. Only two reactions were included: non-oxidative pyrolysis and char

oxidation; kinetic expressions were obtained at the much lower heating rates

found in thermal analysis. Water behavior was not included; for this fuel,

water could play a significant role in modifying temperature profiles.

Endothermic char gasification by CO

2 was not included though Baker shows it to

be significant in real cigarettes (35). Unsteady versions of Eq. (13), (15),

(16), ( 17)— (20) were solved for the special conditions of high gas flow rate

such that smolder wave-scale species diffusion and heat conduction are small

compared to convection (small and tt^). Radiative transfer in the solid

is included by means of the forward/reverse approximation. Note that the

local gas and solid temperatures are not assumed to be equal; Baker’s data for

cigarettes show differences of more than 100°C in some regions even at a low

flow rate (2 cm^/s) (25).

The model was not extensively tested although it does correctly predict

the upward trend of smolder velocity with oxidizing gas flow rate and oxygen

content. In agreement with Baker, the char oxidation process is found to be

limited by external mass transfer. The good quantitative agreement with

72
experiment in the paper is mostly a fortuitous result of the choice of paper

burn-back temperature. The predicted smolder velocity is very sensitive to

this parameter because it controls the amount of inflowing air that bypasses

the hottest part of the char oxidation zone. This same phenomenon makes these

model results a rather poor basis for interpreting other forward smolder

situations. For example, because of this air bypass route, this forward

smolder problem does not have the oxygen profile structure shown in Fig. 3 for

a l-D reaction wave and its qualitative behavior is not constrained in the

same ways as were discussed in conjunction with that figure. In particular,

there is no longer a very simple relation between total rate of oxygen inflow

and the rate of movement of the char oxidation zone. Some of the noted parameter effects are probably qualitatively applicable to other, more typical

forward smolder problems in some circumstances. An increase in the convective

heat transfer coefficient between bulk gas and bulk solid caused a nearly

proportional increase in smolder wave speed; this should apply only to the

pyrolysis portion of a wave such as that in Fig. 3 and only when convective

heat transfer dominates over conduction/radiation. Suppression of radiative

transfer in a few test cases of this problem caused a large increase in peak

reaction zone temperature; evidently it constitutes a substantial mechanism

for heat dissipation from this zone. Order of magnitude changes in the preexponential kinetic factors for the two overall reactions had minimal effect

on the problem; this is indicative of oxygen-supply-rate-limited behavior.

The most detailed model of reverse smolder propagation presently in the

literature is that of Ohlemiller, Bellan and Rogers (91). This is a l-D,

time-dependent model based on analogs of Eq. (13), (14), (16), (17) to (20).

Gas and solid energy equations are combined on the assumption of local therm i!

73
equilibrium; local species equilibrium between gas and solid is also assumed

(large tt 2k » ^2

’

7, 22K* ^29 ’ *301^* Wave-scale diffusion is included.

Radiative transfer is included by means of the forward/ reverse approximation.

The model is applied to smolder propagation in a polyurethane foam where the

initial characteristic particle dimension was 0.1 to 1 mm. Particle scale

gradients were not explicitly considered in this paper. Subsequently it was

reported that the pores in the char are mainly in the 20-80 A range. The char

dimensions and time scales studied were such that temperature and oxygen

gradients in the charred particles were thus quite small. However the initial

foam material had no pore structure so that oxidative attack on the initial

fuel particles was undoubtedly diffusion limited; gas dif fusivities in the

solid phase tend to be several orders of magnitude less than those in even the

smallest pores. This was not explicitly modeled. Instead the kinetic parameters for the only two reactions included in the model were obtained by

thermal analysis using samples with the same foam structure. Because of the

differences in heating rates (5X or more) this is bound to be somewhat

inaccurate, but it is difficult to estimate how much so. Water behavior is

not included; polyurethanes contain minimal moisture but may form it during

smolder.

The model solutions correctly predict the qualitative trends of behavior

seen with the polyurethane foam, i.e., increasing smolder velocity and peak

temperature with increased O

2 supply and the differing rates of increase

depending on how O

2 supply is increased (higher velocity with fixed O

2 level

or fixed velocity with increasing O

2 level). Quantitatively the model is less

successful, predicting temperatures and smolder velocities that are somewhat

low. It is inferred that, like virtually every other smolder propagation

74
process, this reverse smolder process is oxygen-supply-rate-limited (as

opposed to kinetically limited). The rate of propagation is set by a balance

between the rate of heat generation (proportional to the rate of oxygen

supply) and the rate of heat transfer (conduction plus radiation) to the next

element of fuel. Reaction kinetics in this model serve only to dictate the

distribution of the oxygen between the two reaction pathways included.

Cutting the pre-exponential factor of the dominant char oxidation reaction by

10X had no effect on the total rate of heat generation (all C>2 continued to be

consumed in a thicker wave) and it produced only a small decrease in

propagation rate.

Reverse smolder tends to leave part of the char behind as a residue since

the wave typically engulfs (heats and "ignites") the next element of fuel

before the first element is all consumed. This residue is an insulator

against the only path for heat losses in such one-dimensional propagation,

i.e., losses from the originally ignited end of the fuel bed. This model

shows that this insulating residue plays a significant role in helping the

fuel bed achieve a self-sustaining smolder situation. If the reaction

kinetics are slowed thus making ignition more difficult, sustained heating

will eventually build up a residue layer that will insulate the slowed

reaction and allow it to sustain and propagate itself. It is unclear whether

the model thus implies ignition can always be achieved no matter how slow the

reaction kinetics (the limit would seem to be adiabatic ignition). Such a

prediction is overly pessimistic with regard to the real world where 2-D heal

losses would impose a lower limit on kinetic rates able to support selfsustained smolder propagation. Nevertheless there is a strong Implication

that large decreases in kinetic rates must be achieved by smolder retardant75
to suppress reverse smolder. Because of Its very limited chemical description, this model misses another pathway to smolder suppression, i.e., fuel

pyrolysis. This process is endothermic, typically, and it removes material

that could be oxidized heterogeneously to sustain smolder. Exploration of

this would require only a slight extension of the model. This model was not

capable of predictions about transition to flaming since it included no gas

phase reactions.

Models in Related Problem Areas . It was noted previously that there are

several other combustion problems that qualitatively resemble smolder propagation. The problem area that has received the most analysis is gasification of

coal and oil shale both in situ and in packed bed reactors. Both forward and

reverse propagation are employed; reverse propagation may be used to establish

a high permeability channel for subsequent use in a forward propagation mode.

Numerous models, of varying degree of approximation, have been developed to

explain the parametric dependence of propagation velocity and to predict the

evolved gas composition (37, 96-108). The chemistry involved is rather

different from smoldering as are various parameter values affecting propagation; in addition, the fuel bed permeabilities may be much lower Implying

strong pressure gradients. Nevertheless, the gross qualitative behavior

should pertain to smoldering under some conditions. Here we briefly examine

the nature and results of a few of the most complete models.

The forward propagation models in Refs. 104, 105 and 106 are all

basically the same. These are one-dimensional models based on analogs of

Eq. (10) (with Vp = 0), (13), (15), (16) and Eq. (l7)-(20). Diffusion in the

bulk gas phase is neglected with minimal justification; it appears to be

76
potentially significant for at least some cases considered. It is recognized

that there probably are particle-scale gradients in part (or all) of the range

of coal or oil shale particles encountered in practice. However, the role of

these gradients is essentially ignored at this modeling stage. Heterogeneous

reactions are either in Zone I or Zone III. The number of reactions

considered is large, e.g., eleven in Ref. 106. The reason for this is the

attempt to predict the major effluent species and hence the heating value of

the gas. Both gas phase and heterogeneous oxidation are included but the

former is treated in a manner that obscures its real significance. These

models are reasonably successful in predicting the rate of movement of the

reaction and condensation fronts as well as the major gas constituent levels

for the few cases compared with experiment. They appear to comprise the best

qualitative description of 1-D forward "smolder" propagation currently in the

literature. The temperature profile is qualitatively the same as that in

Fig. 3; the predicted dependence of propagation velocity on oxygen input level

is qualitatively the same as that found experimentally in Ref. 18 for smolder

propagation through a bed of wood fibers. Unfortunately these models have not

been exercised in a way which sheds much further light on the forward smolder

propagation problem.

Reverse combustion of coal and oil shale appears to have received rather

less modeling effort. The model of Kotowski and Gunn (37) is fairly detailed

though substantially less so than the forward models just discussed. This

a steady-state, 1-D model based on analogs of Eq. (17), (19) and (20); the

energy equation is for the combined gas plus solid since the two temperature

are assumed locally equal. Diffusion is again neglected even though it n.iv b>-

important for some cases studied. Particle-scale gradients are not

77
considered. A conservation equation for the condensed phase fuel is avoided

by a heuristic description of the temperature-dependent fraction of volatilized material. (This sort of empiricism misses the heating-rate-dependent

temperature shift in the gasification process.) Only one reaction appears in

the energy equation, gas phase oxidation of fuel vapors. Water vaporization

is considered as a heat sink but not as a rate process. The resulting temperature and oxygen profiles resemble those in Fig. 3. Propagation velocity and

peak reaction wave temperature increase with oxygen supply in agreement with

experiment. Increased fuel bed conductivity is predicted to increase the

propagation velocity as expected. An unexpected (and unexplained) prediction

is that an increased pre-exponential rate factor will decrease the propagation

velocity; this is contrary to the expectation based on the simple deflagration

model of Corlett and Brandenburg (38) mentioned above. The model of Amr (108)

is time-dependent and slightly more detailed than that of Kotowski and Gunn

but basically similar; the limited parametric studies do not add further

insights pertinent to smolder propagation.

There have been some linear stability analyses of forward and reverse

propagation in the context of in situ gasification. In this gasification

process, one attempts to propagate a reaction wave between drill holes into

the fuel seam below ground. It Is of interest to determine what parameters

determine whether the reaction front will be broad and flat or will contract

into channels. The reverse combustion stability analysis of Gunn and Krantz

(109) is based on perturbations of a very rough steady-state propagation model

(gas/solid energy equation with conduction and convection only; final temperature an imposed function of gas velocity). The solutions predict that reverse

propagation is unstable at low gas velocities, particularly toward large

78
wavelength disturbances. The results are used by the authors to explain

several anomalous experimental aspects of reverse combustion in coal seam

tests as being due to channeling (diameter about l meter). Krantz, Keyashian,

Zollars and Gunn (l 10) also did a linear stability analysis based on the same

type of steady-state propagation model for wet forward (and reverse) propagation. Wet in this case implies sufficient water for the steam vaporization

and the combustion fronts to coincide. In view of Fig. 3, this type of model

seems to be an even rougher approximation for forward propagation. The solutions predict that forward wet combustion is unstable for many conditions of

Interest to in situ gasification. Experimentally, less water-saturated

forward combustion in coal seams is found to be stable.

Most recently Britten and Krantz (112) did a linear stability analysis of

a more complex reverse propagation model. Analogs of the sum of Eqns . (13)

plus (17), the sum of Eq. (16) plus Eq. (20) and Eq. (19) for gas phase oxygen

were solved analytically by asymptotic expansion. Radiative transfer was not

included. Neglect of wave-scale diffusion in the gas phase oxygen equation

was justified by restriction of the solution domain to very large Lewis

numbers pertinent to very low porosity coal seams. Only one reaction was

considered, gas phase oxidation of fuel vapors arising from the condensed

phase; the gasification of these vapors was not explicitly included in the

model equations. (A related model by Britten and Krantz (113), analyzed

similarly, considers only direct heterogeneous oxidation of the solid.)

The zeroth order solutions (steady-state) are basically similar to the

phenomenological model of Corlett and Brandenburg (38). The predicted variations of peak temperature and propagation velocity with flow rate of the

79
oxidizing gas are quite reasonable. The zeroth and first order solutions were

spatially perturbed about a planar front to determine which wave numbers of

spatial disturbance grow. The model predicts a stable planar front only in

the absence of a change in permeability of the solid across the reaction

zone. For all other cases, there is a disturbance wavelength (whose value

depends on the various parameters) which is amplified more rapidly than any

other. The results are shown to be plausibly in accord with the limited data

on the behavior of in situ coal burns.

In contrast to these stability predictions for in situ gasification, the

present author has found, in small scale laboratory tests (15-25 cm diameter

fuel beds), forward smolder propagation readily goes unstable even at very low

flow velocities (few mm/s) while reverse smolder is stable at least to l cm/s

gas flow velocity. Egerton, Gugan and Weinberg (21) reported that continued

forward smolder in cigarettes is highly unstable. These results do not necessarily contradict each other given the small scale of the smolder experiments,

the 2-D influence of a peripheral boundary not accounted for in the preceding

analyses and the large disparity in parameter values in the two types of

experiments, particulary fuel bed permeability.

Concluding Remarks

The general problem of smoldering combustion propagation in particulate

fuel beds is very complex both chemically and thermophysically . Since the

starting material is a polymer or mix of polymers, the number of elementary

reactions involved becomes indefinitely large posing a problem much more

complex than the well-studied, related problem of carbon oxidation. It is not

80
feasible to model all of these reactions even if they are someday elucidated

for even one common smolder-prone substance such as cellulose. The

engineering approach around this problem has been the use of a few overall

lumped "reactions" with apparent kinetic parameters determined typically by

thermal analysis. The potential pitfalls here due to disparities in heating

rale have not been adequately explored. It must be borne in mind that this

engineering approach to the chemistry can only help quantify the relative role

of chemistry in the total problem. It cannot provide any real insights about

chemical solutions to the problem of smolder propagation.

The thermophysical problem can be more explicitly described but the

complexity of the resulting equations precludes general solutions. The

general case is made intractable by the existence of species and temperature

gradients simultaneously on the fuel particle scale and on the reaction wave

scale. Given this, one must start with simplified cases in an effort to build

a body of knowledge about the quantitative interactions among the numerous

model parameters. The current state of development is quite inadequate. Even

the l-D limiting cases of forward and reverse smolder with no particle-scale

gradients are not fully characterized. The qualitative nature of their

behavior is reasonably well understood. The dominant role of oxygen supply

rate is recognized.

The seemingly secondary role of oxidation kinetics during self-sustained

propagation is deceptive. Any realistic solution to smolder propagation in

smolder-prone materials such as cellulosics will almost certainly hx rhnn. i<

in nature. Most product uses preclude physical approaches such as inhibit in,

oxygen supply rate and, in any event, smolder has a propensity to adapt

81
tenaciously to even very low oxygen supply rates. The role of kinetics enters

in setting the level of the ignition temperature (or heat flux) and the conditions for extinguishment. Some qualitative models such as that of Moussa,

Toong and Garris nicely demonstrate the principles involved but require

considerable elaboration in order to be fully realistic. The limiting cases

of l-D forward and reverse smolder are not particularly useful in this regard;

they are unusually resistant to extinguishment because of their l-D character

and they approximate relatively few real world smolder problems. A fully

developed model of the problem in Fig. I would provide the first basis for

quantitative exploration of all the chemical and physical factors involved in

sustained smolder of a realistic configuration.

Finally, it should be pointed out that the opposite boundary on smolder

behavior, transition to flaming, has only been superficially explored experimentally and not modeled at all. The process is implicit in the general model

equations above given the right boundary conditions, but the nature of the

phenomenon is only dimly understood. Since transition from smolder to flaming

poses a sudden increase in life hazard, much more effort is needed toward

understanding and controlling this process.

82
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9_, 5 (1978).

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83
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84
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Gasification: Phenomena, Conceptual Models and Research Status.

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Compounds and Its Catalysis by Inorganic Materials" paper presented at

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41. Shafizadeh, F., Bradbury, A., DeGroot , W. and Aanerud, T., I & EC Prod

.

Res. Devel . 21 , 97 (1982).

42. Ohlemiller, T. J., results to be published.

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44. Rogers, F. and Ohlemiller, T., Combust. Scl . Technol. , 24 , 129 (1980).

45. Summerf ield, M., Ohlemiller, T. and Sandusky, H., Combust .Flame 33 , 263

(1979).

46. Muramatsu, M. and Umemura , S., Beitrage Zur Tabakf or s chung 11, 79

(1981).

47. Shafizadeh, F. and Bradbury, A., J. Appl . Poly. Sci . 23 , 143 (1979).

48. Bradbury, A. and Shafizadeh, F., Carbon 18 , 109 (1980).

49. Shafizadeh, F. and Sekiguchi, Y., Carbon 21 , 511 (1983).

50. Sekiguchi, Y., Frye, J. and Shafizadeh, F., J. Appl. Poly. Sci. 28, 3513

(1983).

51. Gan, H., Nandi, S. and Walker, P., Fuel 51 , 272 (1972).

52. Laurendeau, N. , Prog. Energy Combust. Scl. 4, 221 (1978).

53. Simons, G. and Finson, M. , Combust. Sci. Technol. 19 , 217 (1979).

54. Simons, G. , Combust. Scl. Technol. 19, 227 (1979).

55. Shafizadeh, F. and Bradbury, A., J. Thermal Insul. 2, 141 (1979).

56. Bradbury, A. and Shafizadeh, F., Combust. Flame 37 , 85 (1980).

57. McCarter, R. , J. Consumer Prod. Flamm. 4, 346 (1977).

58. Shafizadeh, F. Bradbury, A. and De Groot, W., "Formation and React ivit

of Cellulosic Chars" paper presented at Western States Section/

Combustion Institute (1981).

85
59. McKee, D. , "The Catalyzed Gasification Reactions of Carbon" in The

Chemistry and Physics of Carbon, (P. Walker and P. Thrower, eds.) Marcel

Dekker, New York (1981).

60. Gregg, D. and Olness , D. "Basic Principles of Underground Coal

Gasification", Lawrence Livermore Laboratory, Preprint UCRL-52107,

August 1976.

61. Jellinek, H. (ed.), Aspects of Degradation and Stabilization of

Polymers

,

Elsevier, New York (1978).

62. Bradbury, A., Sakai, Y. and Shafizadeh, F., J. Appl. Poly. Sci . 23 , 3271

(1979).

63. Perry, J. (ed.), Chemical Engineers Handbook , 3rd ed. McGraw-Hill, New

York (1950), p. 800-808.

64. Muramatsu, M. Umemura , S. and Okada, T., Combust. Flame 36 , 245 (1979).

65. Baker, R. , Prog. Energy Combust. Sci. 7, 135 (1981).

66. Hedges, J. "Ignition and Pyrolysis of Polymer Films", Ph. D. Thesis,

Univ. of Utah, Dept, of Chem. Eng., June 1978.

67. Anthony, D. Howard, J., Hottel, H. and Meissner, H. Fifteenth Symposium

(International) on Combustion , The Combustion Institute, Pittsburgh, Pa.

(1974), p. 1303.

68. Flynn, J. and Wall, L. , J. Res. Nat'l Bur. Stds. Sect. A 70 , 487 (1966).

69. Jellinek, H. (ed.) ibid , chap. 12.

70. Rogers, F. and Ohlemiller, T. , J. Macromol. Sci-Chem. A15(l), 169

(1981).

71. Lewis, B. and Von Elbe, G. , ibid , Chap III.

72. Walker, P., Rusinko, F. and Austin, L., "Gas Reactions of Carbon” in

Advances in Catalysis XI , 133 (1959).

73. Szekely J., Evans, J. and Sohn, H. , Gas-Solid Reactions , Academic Press,

New York ( 1976)

.

74. Szekely, J., Evans, J. and Sohn, H. , ibid , p. 109-111.

75. Szekely, J. Evans, J. and Sohn, H. ibid , chap. 7.

76. Szekely, J. Evans, J. and Sohn, H. , ibid , p. 30.

77. Hottel, H. and Sarofim, A., Radiative Transfer , McGraw Hill, New York

(1967) pp. 430-433.

78. Stanek, V. and Szekely, J., Canad . J . Chem Eng . 50 , 9 (1972).

86
79. Heliums, J. and Churchill, S., Chem. Eng. Prog. Sympos. Series, No. 32

Vol 57 , 75 (1961).

80. Szekely, J., Evans, J. and Sohn , H., ibid , p. 263.

81. Szekely, J. Evans, J. and Sohn, H. , ibid , p. 152.

82. Smith, J., Chemical Engineering Kinetics 2nd ed, McGraw-Hill, New York

(1970), p. 446.

83. Bird, R. , Stewart, W. and Lightfoot, E, Transport Phenomena , Wiley, New

York ( 1960), p. 198.

84. DeRis, J., Combust. Scl . Technol

.

2\ 239 (1970).

85. Johnson, B., Froraent, G. and Watson, C., Chem. Eng. Sci. 17, 835 (1962)

86. Sega, S., Bull. Jap. Assoc. Fire Sci. Eng. 25 , 17 (1975).

87. Sega, S., Bull. Jap. Assoc. Fire Sci. Eng. 25 , 25 (1975).

88. Williams, F. , Sixteenth Symposium (International) on Combustion , The

Combustion Institute, Pittsburgh, Pa. (1976) p. 1281.

89. Gugan, K. , Combust. Flame 10 , 161 (1966).

90. Glassman, J., Combustion, Academic Press, New York (1977) p. 68-75.

91. Ohlemiller, T., Bellan, J. and Rogers, F. , Combust. Flame , 36

,

197

( 1979).

92. Moussa, N., Toong , T.-Y. , and Garris, C. , Sixteenth Symposium

(International) on Combustion, The Combustion Institute, Pittsburgh

(1976) p. 1447.

93. Ortiz-Molina , M. , Ph.D Thesis, Dept, of Mechanical

Massachusetts Institute of Technology, June 1980.

Engineering J

94. Leisch, S., "Smoldering Combustion in Horizontal Dust Layers",

Thesis, University of Michigan Aerospace Engineering Dept.,

November 1983.

Ph.D

95. Baker, R. , Nature, 264, No. 5582, 167 (1976).

96. Mayers, M. , Trans. ASME 59, No. 4, 279 (1937).

97. Silver, R. , Fuel, XXXII, 121 (1953).

98. Koizumi, M. , Sixth Symposium (International) on Combustion , Reinhold,

New York (1956) p. 577.

99. Gottfried, B. , Soc. Petrol-Eng. J. , Sept. 1965, p. 196.

100.

Gottfried, B., Combust. Flame, Feb. 1968, p. 5.

87
101. Gunn, R. and Whitman, D. , "An In Situ Coal Gasification Model (Forward

Mode) For Feasibility Studies and Design", DOE Laramie Energy Research

Center Rept. LERC/RI-76/2 , Laramie, Wyoming (1976).

102. George, J. and Harris, H., Siam J. Numer. Anal. 14 , 137 (1977).

103. Dockter, L. and Harris, H. , "A Mathematical Model of Forward Combustion

Retorting of Oil Shale", DOE Laramie Energy Technology Center Rept.

LETC/TPR-78/ 1 , Laramie, Wyoming, 1978.

104. Winslow, A., Sixteenth Symposium (International) on Combustion , The

Combustion Institute, Pittsburgh, Pa., (1976) p. 503.

105. Thorsness, C. and Rozsa, R., "Lawrence Livermore Laboratory In Situ Coal

Gasification Program: Model Calculations and Laboratory Experiments",

Lawrence Livermore Laboratory Preprint UCRL-78302, Livermore, Ca.,

(1976).

106. Thorsness, C. , Grens , E. and Sherwood, A., "A One-Dimensional Model For

in Situ Coal Gasification Lawrence Livermore Laboratory Rept.

UCRL-52523, (1978).

107. Branch, M. , Prog. Energy Combust. Sci. 5 , 193 (1979).

108. Amr, A., Combust. Flame 41 , 301 (1981).

109. Gunn, R. and Krantz, W. , Society of Petroleum Engineers Paper SPE-6735,

Oct. 1977.

110. Krantz, W. , Keyashian , M. , Zollars, R. and Gunn, R. , Society of

Petroleum Engineers Paper SPE 7523, Oct. 1978.

111. Smith, J., ibid , p. 365.

112. Britten, J. and Krantz, W. , "Linear Stability of Planar Reverse

Combustion in Porous Media", presented at 1984 Spring Meeting, Western

States Section, Combustion Institute, Boulder, Colorado.

113. Britten, J. and Krantz, W., Proc. NATO Workshop on Chemical

Instabilities

,

Austin, Texas, March 1983.

88
Nomenclature

a^, - empirical constant relating drag on gas flowing through fuel

particle pores to gas flow velocity.

AVPi - surface area/unit volume of particle available for reaction; this

factor drops out of rate expressions for volumetric reactions such

as pyrolysis.

C - specific heat of initial (unburned) fuel particles.

PO

Cj, C2 ~ coefficients for laminar and turbulent drag terms, respectively in

differential Ergun equation.

Ay

g - external fuel particle area/ unit volume of fuel particle bed.

g x

- component of gravitational acceleration in x direction.

eb

- effective diffusivity or dispersion coefficient for gas species in

bulk gas of fuel particle bed.

D

£ p

- effective diffusivity of gases in particle pores.

- activation energy.

H - heat transfer coefficient for flow around exterior of fuel

particle.

89
GP

- sensible enthalpy per unit mass of gas in fuel particle

P° reS E (

YjGPhjGP.

N

) V sensible enthalphy per unit mass of gaseous species j in fuel

particle pores.

- sensible enthalpy per unit mass of solid in fuel

particle = I. [Y._ h \

J jP JP-

- mass transfer coefficient for flow around exterior of fuel

particle.

- reference length in non-dimensionalization.

ox overall stoichiometry of smolder wave, mass of oxygen/unit mass of

fuel consumed.

n - number of particles/unit volume of fuel particle bed.

- local pressure of gas in bulk gas around fuel particle,

- pressure of gas in particle pores.

p' - local pressure deviation from hydrostatic value,

EXO

- net exothermicity of smolder wave per unit mass of fuel consumed,

90
,STP

- heat of reaction l per unit mass of reactant, measured at standard

temperature and pressure.

0^ j - heat of reaction Jt per unit mass of reactant, measured at local

reaction temperature.

Qm , STP

heat of reaction m per unit mass of reactant, measured at standard

temperature and pressure.

R - universal gas constant.

R - rate of reaction of bulk gas species j/unit volume of fuel bed.

hi - rate of reaction i in fuel particle; (mass/unit volume) or

(mass/unit area).

r - dimension perpendicular to particle surface (radius, halfthickness); a = 0 for slab-like particles; a = 1 for cylindrical

particles; a = 2 for spherical particles.

t - time.

S - net radiation flux vector due to solid fuel particle

emission/absorption/ scattering.

1 - effective ambient temperature for buoyancy force.

A

91
local temperature of bulk gas around fuel particles,

local temperature of solid in fuel particle,

velocity of gas in fuel particle pores.

components of bulk gas velocity in X and Y directions; also

denoted v v in Eq . (22).

x, y

bulk volume of fuel particle bed (gas + solid) assumed dependent

only on fraction of initial solid mass that remains, (m/m

c

).

local mass fraction of gas species j in bulk gas around fuel

particles

.

local mass fraction of gaseous species j in fuel particle pores,

local mass fraction of solid species j in fuel particle,

pre-exponential factor in chemical reaction rates,

rate of change of flow displacement volume of a fuel particle,

thermal conductivity of bulk fuel particle bed (gas + solid).

thermal conductivity of fuel particle.
>o

r^.^1

F£j

- stoichiometric coefficient (mass basis) for formation of gaseous

species j from reaction l in fuel particle.

. - as above but for consumption of j .

v - mass of gas produced per unit mass of solid reactant consumed,

G*.

p - viscosity of gas in particle pores,

G i

$ - porosity (fractional free volume) of fuel particle bed,

- porosity (fractional free volume) within a fuel particle.

BO

- initial bulk density of fuel particle bed.

GF

- density of gas within a fuel particle.

KP

- density of solid part of fuel particle.

- summation over all gaseous species in pores of fuel particles

- summation over i reactions occurring in fuel particle.

summation over m reactions involving gas species in bulk gas

phase.

93
- summation over all the gaseous species present in bulk gas (and in

KG

particle pores)

Subscripts

- arbient

- initial value

- reference value

94
EXPERIMENTAL

CHAHA

r

Tr.RISTICS

OF

SMOLDER

PROPAGATION

o

o!

c

ft •3

X ^ V A a d

ft o

•3 ^X

P 3

ft TJ A • ^ a.e

® 3 «

> co

• u

o • C C —

<

E C 9 9 e >

c

ft a* a > • o

•o X

C 9

• U —< ft ft ft > a

> vp S

• ^ u —« 15 o o

B «*-. o

9 0-9

9 e

** o

§ s A X

G

9Pu uo

5K -<

*-( > x a

DO X

•h a> x x

ft 9 U

e

• o

ft c > ft

o a

6 ft A *3

> r, O A

-4 ft «m k

U

a c

® «*

00 X• 4J

TJ

ft •o 9 C

9 9 9

P • U ~4 C 9

>

i S

• 6 3

• C

*3 —

« i im > 9

9 P

§ £

o

E )O

*3 —

<

a> v*-« rc p

p « —< p «t

E e

Cl, Eii

c , u

E 1 ft B | c. U. «J|

K

u

ft>

c u

O • /*>• O ft © 3

c o 9 O HN U m - •-* TJ 0C u V|w

ft

*3-

I o

e:

w»p I ft

8 x

co

I o

u u 3

<-> ft -3

I > O C >s —' O ft u m ai

• —< X

, a m . ft 3

01 ft a p w -o

r-> pni

I l O O

a Lii

o.c o

LJ ©n*. e- I c ©

> m

c ^

u

5 a C >s

v— v

OE| 9

u| w a

o a 4-' C

00

ft | ft m

>

C

ft 3 ft- C >sa

• AE 9

ft> t>| 0! 9 • «3 C A O O

3 m ft U IM e «-< 3 x

ft ^ 9 o

4J ft i_) *-» ^ *3

•H . X

E ft -c 0C

ft c a 3

ft > p 0

4J c ft u c o c x ^ o W AJ

o

>

o c 3u| o o^ c

:* • o o

I w- m

H

E • u c

o ^ W. E

E C ^ -c c &

3 X

E l 0

X’j| n

DC

6 & IT E ^ C ^ c

a. O

k

a'>>e

P *3 O 9 X O'p

E o

9

>>9

P *3 O C X 9

P ^ &. U A

E ft.

•3 O

9 *3 P O — PC X C. E

4j a r. E ^ *r! papP ec ^ O ^

.

vC i-> U.

• 9 C

OX - P

*3 9

-* A *3 p O ^

moo SC >

• A O ^ W-4 U

9oX *3 U 9

u a*

*-t p x a

E *3 U P

A ^ 3

• a

E *3

CC P u

•3

t N l

>s «H O

Q P IM

—i oX

^ ^

P P A X «*- 9 A x 3 O E *3 P A X ft. 3 ni U

O • a

•3 0®

ft —« "3 P —4 A O

ft E C A

3 i d 1 3§

a

p u

o ^ x oo

CM

ft • ^ o

X ^

A

Pi T5

ft P

oo a 22

A —

.

o u —• c

3 Z

r~* —< P

ft m

U «f

ft O

X w

k e

ft 9

O

ft -c r c

c o

5 22O C O 3 u c

0 A —

o - - IT X

ft C — ft U •< p U

£!

95
Table II

Dimensionless Groups with Approximate Magnitudes

Definitions of reference quantities and estimated ranges for key values:

T

R ^EXO/C

po-^

625 - 1500 K

GR

Wa\ /

T

R T

A) C

1 MR

10~4 - 10 1 m/s

PR

V GR P G0^oxo) n p _ ^ ox BO

10" 7 - 10

-3 m/s

*r

= t

r

x bo

P B0

V

PR

C

P0^

T

R

-T

o^

10 3 - 1 m

t = £ /V

R R PR

1 - 10" s

V

GPR '^ 1 ^ P

p V

PR^ PGPR

A

VBR

A r)

P R ^ C

1 P K

V GR

£

R

/<}? R^

P PR ^ r FR

C lR UGPR

V GPR

/({> PR^

hGR , hp

R

, etc. (all enthalpy ref. values) = C

p0 (T

R

- T

o

)

Yrgr ^KGPR » ^KPR = or niaximum values

4> , * R

, * BR , * pR , PGR » P pR , P GpR = initial or maximum values

V S R’ AVBR ’ \p£R» DepR’ DeBR

= maximuIT1 values

96
Dimensionless groups and magnitudes employing above quantities :

77 1 " (*r^rv gr^

0(10" 3

)

11 2K

lKA

V’BR

£ R

,1 KGPR

/V

GRP GR

<,i R^

0(10~ 2 - 10 3

)

71 3

" ^RP GR

/t: RC

l U R^

0(10" 2 - 10" 9

)

17 4 " ^RP GR

V GR^RC

1 W R^

0(10" 3 - 1)

17 5 = ^RS x

P A

/ C

1U R

V GR^ 0(1)

77 6 =

( C

2VGR

/C

1 P R^

0(10" 6 - 10)

11 7K ^VER^KGPR^l1^ 0(10

-4

- 1)

"Bj

' ^ £

R

Y

jC-R

/t: RV

GR^

0(10" 3

)

77 9j =

^ D

eBR

Y jGR/£

R

V

GR^ 0(.l - 1)

a b

17 1 Omj

=

^ V^ V

F"

V

C^nij Zm^ P GR

Y OXR^ ^ P GR

Y GFMR^

6Xp r-VRI

GR

) /t> GR

V

GR

) 0(0

77 1 1 =

t T

R

X £f/

£ R

P GR

V GR

h

GR^ 0(.l - 1)

17 12

=

f ^‘VB

R

T

if l/ P GR^ ^ Gif GR? 0(10~ 2 - 10A

)

97
0 ( 1 ) ”l3mj "l0mj ''• ym,STP

/hGR^

V

F

V

C^mj^ 0(1)

V - (Wv pr

1

r) 0(1)

”15

=

*R 0(.l - 1)

”l6

= 5?R 0(.l - 1)

”l7 =

( n

P

APRVV

PR

) 1 0(1)

*18£ ^ VG At P£R

Z P£^ PGPR

Y

OXPR^ ^ PPR

Y

PZR^

exp ( E

P£

/RT

R'* £R

/V

PR P PR^ 0(l)

= PR PPR^

*19

=

( PGPrVL R

RR

i)

0( 10" 3

)

*20

= IVfrWW) 0(10" 3

)

*21 Si = (^A^l ^ 1 0(1)

*22K

=

f KA

\-BR

Y KGPR'

/RR

l ^

/-N

<r O

<r 1 o o

*23

j = ^PR PGPR

Y jGPR^ L RRR

l ^ 0( 10" 3

)

*24 j = ^PR PGPR

V

PR

Y jGPR /£

R

RR

l ^ 0(10~ 3

)

*25 £j = ^ V

F"

V

c^£j /VG£^ 0(1)

98
71 0 ( 1 ) 26j = Y jGPR 0(1)

*27 ^PRP GPR

h

GPR/p PR

h

PR^ 0(10

-3

)

*28

=

^ S R

/p

PR^ PR

h

PR^ 1 0(1)

T 29

=

^ H

^\'BR

T

R'

t R

/,p PR

V

PR^PR

/l o(io" 2 - io4

)

*30K

=

^ KA

\

, BR

Y

KGPR

h

GPR

£

R

/p

PR^ PR

h

PR ; 0(10“4 - 104

)

^3U = lQ Jl,STP

/V

G^

h

PR^ 0(1)

*32

=

^ P GPR

/C

R

RR

1

; 0(10“ 3

)

*33 = lP GPR

V GPR

/r

PR

RR

l^ 0(1)

*34

= 'v P PR

/t

R

RR

l

0(1)

*35 = 1

P

GPR

V GPR* PK

/P PR

t R

; o(io" 15 - io

-3

)

*36 ^ P GPR^ GPR

/P

PR ; 0(10’ 1C - 1)

*37£

=

' r pR^ GPR

RR

£ /PPR^

0(10“ 10 - 1)

*38j

= "°PRYjPR/t: R

RR

l^ 0(1)

*39j

= ^PRP GPR

Y jGPR/t

R

RR

l

> odo"3

)

99
11 40j =

^ P GPR<J> PR

V

GPR

Y jGPR/r

PR

RR

l^ 1 0(1)

*

41 j = GPR^ ePR

Y

jGPR^

r PI^

R P

0(10"6 - 104

)

*42 '• P GPR<f> PR

D

ePR

/r

PR

K '^ o(io" 6 - i)

71 435. j = ' *PRRR

£

/A VP£lP jPGR^ 0(10~4 - 1)

*44

=

^ P PR

h

PR

V

Gl

/t: R

Q

1 ,STI^

R

1^ 0(0.1 - 1)

*45

= ^PRP GPR

h

GPR

V

Gl

/t

R

Q

l ,STP

RR 1-^ 0(10~4 - 10” 3

)

*46 ^PRP GPR

V

GPR

h

GPR

V

Gl

/v

PRQ

l ,STP

RR 1^ £0 (0.1 - 1.0)

*47 ^ ^PR

T

R

V

Gl^

r PR^ 1 ,STP

RR

1^ 0(10~ 3 - 10 4

)

2

* 48K

=

^ h

KGPRP GPR

D

ePR

Y

KGPR

V

Gl

/r

PRQ

1 ,STI^

R

1^ 0(10~6 - 104

)

*49

=

^ h

PR

V

Gl /Q

l ,STP^ 0.(0.1 - 1.0)

*50£

= ^ £ ,STP

RR

i!

1 ,STP

RR

1^ <.0(1)

*51£

=

^ h

PR

V

G£/Q £,STP^ 0(0.1 - 1)

*52

=

( X PR

/r

PR

H ^ 0(0.1 - 100)

100
104

V ) 53l ^ Q

Jt,STP <J> PR

RR

£

/V

GJlAVPAR

HT

R^

71 54 ^PRP GPR

V

GPR

/K ^

" 55 l<; PR P GPR

V

GPR

h

GPR /HTR^

0 ( 10“ 4 -

£ 0 ( 1 )

< 0 ( 1 )

101
HEIGHT

ABOVE

BOTTOM

(cm)

HEIGHT

ABOVE

BOTTOM

(cm)

HEIGHT

ABOVE

BOTTOM

(cm)

102
TYPICAL /

POROUS

PARTICLESCALE

GRADIENTS

PRODUCT

GASES

SMOLDER WAVE-SCALE GRADIENTS

Fig. 2) General structure of a smolder wave in a bed of fuel particles

showing gradients on wave scale and on particle scale.
STRUCTURE

OF

FORWARD

SMOLDER

WAVE

104

of

forward

smolder

(from

ref.

18

).
STRUCTURE

OF

REVERSE

SMOLDER

WAVE

105

DISTANCE

Mr.

3(b).

Profiles

of

temperature,

oxygen

concentration

and

solid

density

for

typical

case

of

reverse

smolder

(from

ref.

18).
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See/ nstruction s)

1. PUBLICATION OR

REPORT NO.

NBSIR 84-2895

2. Performing Organ. Report No, 3. Publication Date

June 1984

4. TITLE AND SUBTITLE

Modeling of Smoldering Combustion Voo'^ac^cAi\i\'\

5. AUTHOR(S)

T.J. Ohlemiller

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7 . Contract/Grant No. NATIONAL BUREAU OF STANDARDS ___________________

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

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

10.

SUPPLEMENTARY NOTES

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

11.

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

bibliography or literature survey, mention it here)

Smoldering combustion, which can pose a serious life safety hazard, is encountered

most frequently in various cellulosic materials and in open-cell polyurethane foams.

It is probable that the principal heat source driving this process is heterogeneous

oxidation but gas phase reactions may also contribute at higher temperatures. The

chemistry involved is best-defined for the case of pure cellulose but even here

the details are limited and actual mechanisms poorly understood; simplified kinetic

descriptions, typically derived from isothermal or theromoanalytical experiments,

currently provide the only tractable inputs for smoldering combustion models. The

general problem of smolder wave propagation through a permeable bed of fuel particles

is posed; coupled to the chemistry, one must also consider the physical processes

of heat and mass transfer on both the particle scale and on the smolder wave scale.

The general equations can be somewhat simpified, after non-dimensionalization, for cases where certain dimensionless parameters are very large or very small compared

to unity. Existing smolder propagation models are all greatly simplified compared

to this general model, neglecting gradients on the particle scale and considering

only one-dimensional gradients on the wave scale. These models are reviewed; their

contributions and deficiencies are noted.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) cellulosics; combustion; plastics; polyurethane foams; smoldering combustion

13.

AVAILABILITY

[X] Uni i mi ted

| | For Official Distribution. Do Not Release to NTIS

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

20402.

[^] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

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15. Price$13.00

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