C

Combustion

Forman A. WilliamsDepartment of Mechanical and AerospaceEngineering, University of California San Diego,La Jolla, CA, USA

Definition and Introduction

Since combustion is an essential part of allfires, including wildfires and fires at thewildland-urban interface, thorough knowledgeof combustion is a significant underlying elementin addressing the topic of this encyclopedia. Achemical process that liberates heat, combustiontypically involves finite-rate chemistry in fluidflow with heat and mass transfer. The scienceof combustion is focused on obtaining basicdescriptions of combustion phenomena byexperimental and mathematical methods. Theprinciples are sufficiently well developed that thesubject qualifies as an applied science.

Unwanted fires involve specifically the com-bustion of available fuels in air. Combustion stud-ies contribute to the development of methods forfire prevention, fire detection, fire hazard evalu-ation, fire damage assessment, and fire suppres-sion. For example, investigations of mechanismsfor extinction of combustion suggest elements ofoperation for fire extinguishers employed in firesuppression. Identification of fire retardant mate-rials is aided by combustion knowledge. Strate-

gies for controlling large fires employ estimatesof combustion behavior. In general, much of thefield of fire control research concerns combustionresearch.

Fundamentals of Combustion

Concepts of thermodynamics are of fundamen-tal importance in combustion. Thermodynamicproperties of fuels that pertain specifically tocombustion include heats of combustion and adi-abatic flame temperatures. One definition of theheat of combustion of a fuel is the heat releasedwhen the fuel reacts isothermally in air at a givenpressure and temperature to form gaseous carbondioxide and liquid water as reaction products.This is often termed the higher heating valueof the fuel, the lower heating value being thatreached if the final products contain steam insteadof liquid water, so that the heat of vaporizationof water is not recovered. The standard heat ofcombustion, the heat of combustion at normalatmospheric pressure and at standard room tem-perature, is given in Table 1 for a number of fuels.

Also listed in that table are the adiabatic flametemperatures of the fuels, defined as the tem-perature reached in a fuel-air mixture containingthe right amount of air required for burning tothe specified products at constant pressure in asystem initially at standard room temperature.The right amount of air is termed the stoichiomet-rically required amount, and if there is more airpresent, as in fuel-lean systems, or less, as in fuel-

© Springer International Publishing AG 2018S. L. Manzello (ed.), Encyclopedia of Wildfires and Wildland-Urban Interface (WUI) Fires,https://doi.org/10.1007/978-3-319-51727-8_60-1

2 Combustion

Combustion, Table 1 Selected heats of combustion and flame temperatures for various fuels (From Williams 2002)

Fuel State Formula Standard heat ofcombustion at298.15 K (kJ/g)

Oxidizer Pressure (atm) Adiabatic flametemperature (K)

Carbon Solid (graphite) C 32.8 – – –

Hydrogen Gas H2 141.8 Air 1 2400

O2 3080

Carbon monoxide Gas CO 10.1 Air 1 2400

Methane Gas CH4 55.0 Air 1 2220

Air 20 2270

O2 1 3030

O2 20 3460

Ethane Gas C2H6 51.9 Air 1 2240

Ethylene Gas C2H4 50.3 Air 1 2370

Acetylene Gas C2H2 48.2 Air 1 2600

O2 1 3410

Propane Gas C3H8 50.4 Air 1 2260

Heptane Liquid C7H16 48.4 Air 1 2290

Dodecane Liquid C12H26 47.7 Air 1 2300

Benzene Liquid C6H6 41.8 Air 1 2370

Fuel oil Liquid – 42–47 Air 1 2300

Coal Solid – 20–36 Air 1 2200

Wood Solid – 19–23 Air 1 2100

rich systems, then in most cases the final flametemperature achieved under adiabatic conditionslies below the value in the table. For acetylene,the values listed are maxima, which occur un-der fuel-rich conditions. For hydrogen, methane,and acetylene, adiabatic flame temperatures arealso listed for combustion in pure oxygen ratherthan in air, and for methane values are givenfor an elevated pressure; these different valuesare seen to be higher. Fuel oils, coal, and woodvary in composition, and so the entries in thelast three rows of Table 1 are average estimates.All of the listed flame temperatures here exceed2000 K, which is why rates of heat transfer fromfires are so large, although, because of dilutionand measurement difficulties in turbulent flames;maximum observed temperatures generally areless than this, but still above 1000 K, sufficientlyhigh to generate rapid fire spread.

The preceding information pertains to flamingcombustion, in which the maximum temperaturesoccur in the gas phase. Smoldering is another

mode of combustion, experienced in particular bycellulosic fuels such as wood and paper, in whichoxidation occurs largely heterogeneously afterdiffusion of oxygen to the surface and interior ofthe material. Often called glowing combustion,the maximum temperatures for this pathway arelower, roughly in the range of 500–900 K. Thetwo different possible modes arise from two dif-ferent paths of pyrolysis (thermal decomposition)of the material, illustrated schematically for acarbohydrate (e.g., nD 6 for glucose) as:

(CH2O)n

nCO + nH2

k2

k1

nC + nH2O

nCO2 + nH2O

+nO2

+nO2

where k1 marks the flaming path and k2 theglowing path (dehydration). For cellulose, a poly-mer of C6H10O5, the two competing initial stepsmay be described as the processes:

Combustion 3

C

(200–280°C) k2

“dehydro-cellulose”(slightlyendothermic)

(exothermic)char + H2 + CO2 + ...H2O+

(endothermic)“tar” (primarilylevoglucosan)

cellulose(280–340°C)

k1

in which the “tar” is volatile and vaporizes toform a major gaseous fuel to support a gas-phaseflame, while the gases evolved in the dehydrationpath are mainly noncombustible, and the charthat remains can support only a surface oxidation,glowing combustion.

In combustion processes, the chemistry pro-ceeds at finite rates and involves a number ofelementary reaction steps. Two principal aspectsof the chemical kinetics of combustion are thereaction mechanism (the sequence of elementarysteps involved) and the rate of each elementarystep. The relative rates of the steps vary withconditions, because they depend on the temper-ature, pressure, and composition of the system.Thus, steps that are important for some condi-tions become unimportant for others, which leadsto changes in mechanisms as conditions change.Reaction mechanisms are understood well formany but not all combustion processes.

Reaction mechanisms in combustion usuallyinvolve chain reactions, in which a reactive inter-mediate species (such as hydroxyl, OH), called achain carrier, which is created in some steps anddestroyed in others, accelerates the overall rateof conversion of fuel to products. During muchof the combustion process, these intermediariescan achieve steady-state concentrations (in whichtheir total rate or creation equals their total rate ofdestruction), or they can be involved in reactionsteps that attain partial equilibrium (in whichthe forward and backward rates of the steps areequal). If steady-state or partial equilibrium ap-proximations can be verified for a particular com-bustion process, then improved understanding ofits chemical kinetics is obtained, and simplifieddescriptions of the overall rate of conversion offuel to products may be developed. The resultingdescriptions are called reduced chemical-kineticmechanisms, and methods for identifying and

validating the required approximations in com-bustion are continually evolving.

When reaction mechanisms are not fullyknown, but measurements of overall rates of fueldestruction, oxygen consumption, or heat releasecan be made, then empirical one-step overallreaction rate expression can be obtained. In termsof the fuel concentration f (g/cm3), the oxidizerconcentration g (g/cm), pressure p (atm), andtemperature T (K), the empirical rate w (g/cm3

sec) often can be fitted to the formula:

w D Aflgmpne�E=RT

where R (8.31 J/mol K) is the universal gas con-stant and the overall activation energy E, reactionorders l and m, pressure exponent lCmCn, andprefactor A are constants. Typically 40 kJ/mol� E �200 kJ/mol and 0 � l, m, lCmCn � 2,although values outside these ranges can occur.The exponential factor here is called the Arrhe-nius factor, and the rate formula is said to be anArrhenius expression, honoring hisfighere Fig1early work (1899).

A distinguishing attribute of many combustionprocesses that gives them a qualitative differencefrom other chemical rate processes is the ten-dency for the combustion rate to increase as thecombustion proceeds. In chemistry this propertyoften is called autocatalysis. There are two causesfor the progressive increase in the rate of com-bustion. One is chain branching, exemplified bythe elementary step H C O2 ! OH C O (ina sense, the most important elementary step inall gas-phase combustion in air), where H, O,and OH are chain carriers, so that one carrierthereby generates two; since the combustion rateincreases as the concentration of chain carriersincreases, branching accelerates the process. Theother cause is the increase of most elementaryrates with increasing temperature, seen as theexponential factor in the preceding rate formula;since combustion is exothermic (releases heat),the temperature tends to increase as the processproceeds, and therefore the rate increases. Theresulting self-acceleratory character of the reac-tion introduces features such as explosions, char-acterized as branched-chain explosions if chain

4 Combustion

branching dominates, or as thermal explosions ifthe increase in the rate with temperature domi-nates.

Combustion Waves

When equations are written for mass, momen-tum, and energy conservation for planar, steady,adiabatic combustion waves through which reac-tants are converted to products, it is found thata one-parameter family of waves may exist. Thepossibilities are conveniently illustrated in thepressure-volume diagram shown in Fig. 1. Here pdenotes the ratio of the pressure in the burned gasto that of the initial combustible and v the ratio ofthe gas volume per unit mass of those combustionproducts to that of the initial reactants. The locusof the burnt-gas state in this diagram is called theHugoniot curve, and the negative of the slope ofthe straight line connecting the burnt state to theinitial state (called the Rayleigh line) is found tobe proportional to the square of the propagationvelocity of the wave. Therefore, the slope mustbe negative; this divides the Hugoniot curve intotwo branches, the upper one corresponding todetonations and the lower one to deflagrations. Itis seen from the figure that there is a minimumpropagation velocity for detonations, exhibiting

tangency at the indicated upper Chapman-Jouguet (C-J) point, and a maximum propagationvelocity for deflagrations, with tangency at thelower C-J point.

The two dashed lines in the figure illustraterepresentative intermediate conditions, and eachhas two intersections, corresponding to weak andstrong waves, as indicated in the figure. Thewaves encountered in combustion are weak (infact, nearly isobaric) deflagrations and strong, orquite often C-J, detonations. An increase in theheat released in combustion increases the sepa-ration between the initial state and the Hugoniot.Additional properties of these waves are summa-rized in Table 2, where the subscripts C and –refer to upper and lower C-J condition, respec-tively. In detonations a strong leading shock waveheats the mixture to initiate combustion, while indeflagrations initiation occurs by conduction ofheat from the hot products to the cold reactants.

These combustion waves can occur whenthe reactants, fuel, and oxidizer are well mixed,termed premixed systems. In fires, the fuel andoxidizer (typically air) usually are not well mixedinitially; they are non-premixed systems, alsocalled diffusion flames because the fuel andoxidizer must then mix by diffusion beforecombustion can begin. The evolution of fires,however, is complex, and the mixing needed

Combustion, Fig. 1 Aschematic diagram of thelocus of burnt-gas states forcombustion wave, fromWilliams (1985)

Typical Strong detonation solution

Hugoniot curve

Upper C-J point

Typical weak detonation solution

100

1

Typical weak deflagration solution

Lower C-J point

Typical strong deflagrationsolution

pA

B

C

D

EF u

Combustion 5

C

Combustion, Table 2 A summary of types of combustion waves and their properties (From Williams (1985))

Section inFig. 2–5

Pressure ratio p�(p1/p0)

Velocity anddensity ratios v�(v1/v0)D(p0/p1

PropagationMach numberM0 � (v0/af,0)

DownstreamMach numberM1 �(v1/ae,1)

Remarks

Strongdetonations

Line A–B pC < p <1 vmin < v < vC(vmin > 0)

M0C < M0 <1 M1 < 1 Seldomobserved;requires specialexperimentalarrangement

UpperChapman-Jouguetpoint

Point B pD pC (pC > 1) vD vC (vC < 1) M0 DM0C

(M0C > 1)M1 D 1 Usually

observed forwavespropagating intubes

Weakdetonations

Line B–C p1 < p < pC (p1 >1)

vC < v < 1 M0C < M0 <1 M1 > 1 Seldomobserved;requires veryspecial gasmixtures

Weakdeflagrations

Line D–E p� < p < 1 v1 < v < v� (v1 >1)

0 < M0 < M0� M1 < 1 Often observed;p� 1 in mostexperiments

LowerChapman-Jouguetpoint

Point E pD p� (p� < 1) vD v� (v� > 1) M0 DM0�

(M0� < 1)M1 D 1 Not observed

Strongdeflagrations

Line E–F 0 < p < p� v� < v < vmax

(vmax <1)M0min < M0 <M0� (M0min > 0)

M1 > 1 Not observed;forbidden byconsiderationsof wavestructure

for forming combustion waves may well occurduring a fire or prior to its initiation, makingknowledge of combustion waves essential inevaluation fire hazards. Of the two types ofwaves, detonations generally are the mostdestructive because of their associated damaginghigh pressures. Assessments of possibilitiesof detonation development therefore becomeimportant. If a spark discharge, frictionalheating, or a small flame, for example, initiatesa deflagration in a confined space such as along tube, then as the deflagration proceeds,it often undergoes a time-dependent transitionto a detonation. Studies of deflagration-to-detonation transitions, which are continuingtoday because of their complexity, thus becomedirectly relevant.

Ignition, Extinction,and Flammability

Combustible mixtures at low temperatures inprinciple are reacting, but their rates are soslow that they are negligible. Because of thestrong temperature dependence of the rate of heatrelease, heat losses thus can prevent combustionfrom the beginning. Rates of heat loss fromsystems typically increase approximately linearlywith increasing temperature, as indicated in Fig.2. If the thermal conductivity or the surface-to-volume ratio of the system increases, thenthe slope of the heat loss curve increases. Sincethere also is a maximum rate of heat release incombustion, there can be three intersections ofthe heat release and heat loss curves, as illustratedin the figure. The intermediate intersection is,

6 Combustion

HEAT RELEASE

INCREASINGTRANSFER

COEFFICIENT

HEATLOSS

CRITICAL IGNITION CONDITION

00

500 15001000

TEMPERATURE (K)

RA

TE

(w

/g)

2000

Combustion, Fig. 2 A schematic illustration of the dependences of the rates of heat release and of heat loss ontemperature, illustrating criticality (From Williams 2002)

however, statically unstable, so that the twostable intersections are the intersection withthe negligible rate of heat release at the bottomand the vigorously burning intersection at thetop. Small systems with large rates of heat lossexhibit only the slow reaction intersection, whichis the reason that size limits are imposed forstorage of exothermic materials. As the size isincreased, a tangency condition is approached,which would correspond to extinction if thesystem were burning rapidly. If, on the otherhand, the system had been slowly reacting, thenbeyond that tangency condition, the unstableintersection would return the system to its stateof a negligible rate of reaction. At a still smallerslope, another tangency condition is approached,beyond which there is no longer any intersectionfor a steady heat balance that is not a rapidlyburning intersection. That second tangency thusdefines the critical condition for ignition, therebeing no possibility of slow reactions beyond thatcondition.

Extinction is the process in which a com-bustible system reacting at an appreciable rate is

brought to a condition in which it is reacting ata negligible rate, illustrated by approaching theupper tangency condition in the figure. Strate-gies for achieving extinction of combustion canbe classified as isolating the fuel, isolating theoxidizer, cooling the fuel or the gas, inhibitingthe chemical reaction, or blowing the flame away.Examples in the first of these categories includemechanical methods, such as the use of shovels toencase fuel in noncombustible material, while acomponent of the effectiveness of carbon dioxidefire extinguishers falls in the second. Applicationof water to fires is an example in the third cate-gory, and gaseous or powder chemical fire extin-guishers operate in the fourth. The developmentof efficient and effective means to extinguishcombustion continues to evolve.

Ignition is the process by which combustiblemixtures, reacting at negligible rates, are causedto begin to react rapidly. Ignition can be achievedby external stimuli whenever the high-rate in-tersection exists. It occurs spontaneously, with-out external stimuli, (spontaneous combustion)beyond the indicated critical ignition condition,

Combustion 7

C

when the only balance is the high-rate balance.The spontaneous process occurs when the systemis big enough for the loss rate to depend suffi-ciently weakly on the temperature of the system.

Conditions for achieving ignition by appli-cation of thermal stimuli can be expressed asignition energies if the rates of application arerapid or as ignition temperatures (variously calledspontaneous ignition temperatures or autoigni-tion temperatures) if the rates of application areslow. In the former case, all of the energy appliedmay contribute to ignition, while in the lattercase, some of it is lost, and steady conditions areapproached, in which the applied rate equals theloss rate. There are several methods of ignition,for example, exposure to a sufficiently hot sur-face, to a hot inert gas, to a small flame (pilotedignition), to an electrically heated wire, to aradiant energy source, to an explosive charge, orto an electrical spark discharge. Ignition criteriafor the last two of these can be expressed bestin terms of ignition energy and for the first twoin terms of ignition temperature; the others fallin between. The energy that must be supplied toa system to achieve ignition usually exceeds theignition energy because of losses; for example, inspark ignition, the electrical energy that must besupplied to the spark (the spark ignition energy)exceeds the ignition energy because of heat con-duction to the electrodes and energy carries awayin the gas by shock waves.

Ignition energies and ignition temperatures fora few fuels in air are given in Table 3. Values of

these quantities can vary appreciably with chem-ical composition and experimental conditions.Among practical fuels, ignition temperatures tendto be around 700 K for coal and 600 K fornewspaper, dry wood, and gasoline.

In gas mixtures of fuel and oxidizer, it hasbeen found experimentally that ignition cannotbe achieved if the fuel or oxidizer concentrationis too low. The critical concentration of eitherfuel or oxidizer, below which ignition is impos-sible, is the flammability limit of the mixture.These limits can be expressed as minimum andmaximum fuel percentages, between which thefuel percentage must lie for the mixture to becombustible. The minimum percentage is calledthe lower flammability limit (LFL), and the max-imum is the upper flammability limit (UFL).Flammability limits for a few fuels in air atatmospheric pressure and room temperature arelisted in Table 3. If the intent is to burn themixture, then it is necessary to keep the fuelpercentage between the LFL and the UFL. Tohandle mixtures safely, without the possibility ofcombustion occurring, the fuel percentage shouldbe kept below the LFL or above the UFL. Inpartially filled gasoline tanks of automobiles, forexample, the fuel percentage in the air above theliquid usually is above the UFL (which is about6%).

The LFL and the UFL vary with pressureand temperature and normally tend to approacheach other as either of these quantities is de-creased. Thus, as the pressure is reduced, the

Combustion, Table 3 Ignition and flammability properties of selected fuels in air (From Williams 2002)

Fuel Minimumignitionenergy (mJ)

Spontaneousignitiontemperature (K)

Lowerflammability limit(% by volume ofgaseous mixture)

Upper flammabilitylimit (% by volumeof gaseous mixture)

Minimumquenchingdistance (cm)

Maximumburning velocity(cm/s)

H2 0.02 850 4 75 0.06 300

CH4 0.29 810 5 15 0.21 45

C2H6 0.24 790 3 12 0.18 47

C2H4 0.09 760 2.7 36 0.12 78

C2H2 0.03 580 2.5 100 0.07 160

C3H8 0.24 730 2.1 9.5 0.18 45

C7H16 0.24 500 1.1 6.7 0.18 42

C6H6 0.21 840 1.3 8 0.18 47

CH3OH 0.14 660 6.7 36 0.15 54

8 Combustion

LFL and UFL converge and meet each other ata critical pressure, the minimum pressure limitof flammability. At pressures below this criticalvalue, the mixture does not burn at any fuelpercentage. The pressure limit of flammabilityfor any given mixture decreases with increasingtemperature. At pressures well above this limit,if the temperature is high enough, the mixturewill experience autoignition, without any externalignition stimulus; the minimum critical value forthis higher pressure is the explosion limit. Thepressure at the explosion limit usually decreaseswith increasing temperature, although for manyfuels, such as hydrogen and hydrocarbons, theopposite trend occurs over an intermediate rangeof conditions in many experimental situations, forchemical-kinetic reasons. Between the flamma-bility limit and the explosion limit, the mixturecan be ignited by an external stimulus but doesnot ignite spontaneously.

A standard apparatus in which the LFL andUFL are measured is a vertical tube 5 cm indiameter and 100 cm in length, with its endseither closed or open to the atmosphere. The tubeis filled with the gas mixture, and a strong sparkis discharged either at the top or at the bottom ofthe tube. The result of the experiment depends onwhether the tube is open or closed and on whetherthe spark is at the top (downward propagation) orat the bottom (upward propagation). The limitsusually are widest for upward propagation in aclosed tube. The limits usually reported are thewidest because the results are used most often inconnection with safety. Since the limits dependon the experiment, they are properties not only ofthe gas mixture but also of the configuration ofthe system. Limits generally widen as the size ofthe gas container increases. Tabulations of limitsare useful only if the dependence on the config-uration is not too great. This condition usuallyis satisfied because differences in the UFL forupward or downward propagation in closed oropen tubes seldom exceed a few percent; cor-responding changes in the LFL seldom reach afactor of two. In addition, when tubes greater than5 cm are employed, the widening of the limitsis generally found to be small. One reason forselecting 5 cm is that, when smaller diameters

are employed, the flammability range begins tonarrow appreciably. If the tube diameter is toosmall, then combustion cannot be achieved atany fuel percentage. The critical diameter belowwhich combustion is impossible is the quenchingdiameter of the mixture.

If a deflagration propagating through a gasmeets a tubular restriction with a diameter lessthan the quenching diameter, then combustionfails to penetrate into the tube. Similarly, if thedeflagration encounters parallel plates whose sep-aration distance is less than a critical value (thequenching distance), then combustion fails toproceed between the plates. Experimentally, thequenching diameter is about 20–50% greater thanthe quenching distance. Quenching distances fora few fuels in air at atmospheric pressure androom temperature are given in Table 3. The valueslisted there are those for the fuel percentage thatgives the minimum value; the quenching distanceincreases significantly as the fuel percentage de-parts from the optimum. There is a correspondingdependence of the ignition energy on the fuelpercentage. Values of quenching diameters areemployed in the design of flame arrestors, inwhich fine grids are placed in gas lines to preventflames from propagating through them.

Flammability limits and quenching distancesare manifestations of the same general phenom-ena and are associated with heat loss and withthe strong dependence of heat release rates ontemperature. Because of this strong dependence,rates of heat loss that are only roughly 10% of therate of heat release can cause extinction of a de-flagration. Qualitatively, it is as if a system at anupper intersection in Fig. 2 were to be subject toheat loss lines with slopes that generally increaseuntil the upper tangency condition is reached,beyond which only a slow reaction intersectionis possible.

Since it is the ratio of the rate of heat loss to therate of heat release that is of significance in ex-tinction, reduction of the rate of heat release is analternative way to extinguish deflagrations. Thiscan be achieved by adding flame inhibitors to thecombustible to reduce the rate, by reducing thetemperature (through dilution, e.g., by additionof water or an inert gas to extinguish the com-

Combustion 9

C

bustion), or by slowing the chemical kinetics ata constant flame temperature (through chemicalinhibition that interferes with the branched-chainreaction, e.g., by introducing bromine-containingspecies that combine with chain carriers to reducetheir concentrations in the reaction zone). Anextreme version of slowing the chemical kineticscan arise if a change in conditions producesan abrupt change in the kinetic mechanism toone that releases heat at a much slower rate.An example of this may be found in hydrogen-oxygen combustion, in which the influence ofthe less reactive radical HO2 acts as a sink forchain carriers by combining with the much morereactive radical H, removing it to form the stablemolecules H2 and O2, thereby killing the mainchain-branching step, as becomes dominant atflame temperatures below about 900 K.

Approximate correlations between quenchingdistances and minimum ignition energies havebeen developed. It can be stated that, to ignitea combustible, an amount of energy must bedeposited locally that is roughly sufficient to raisethe temperature to the adiabatic flame tempera-ture in a disk-shaped volume of diameter equalto the quenching diameter and of a thicknessequal to the deflagration thickness, defined below.The resulting ignition criterion can be expressedsolely in terms of either the quenching diam-eter or the deflagration thickness by using theresult that the quenching diameter generally liesbetween about 20 and 60 times the deflagrationthickness (essentially because heat loss rates thatare roughly 10% of the heat release rates causeextinction, as indicated earlier). The basis of thisignition energy criterion is that if it is not satis-fied, then the heat loss to the cool combustiblesurrounding the ignition point causes extinctionafter the energy is deposited.

Premixed Flames

Deflagrations exhibit thicknesses and propaga-tion velocities that depend on their structures,as can be reasoned by dimensional analysis, interms of a thermal diffusivity (dimensions lengthsquared divided by time) and an overall chem-

ical reaction time; resulting typical deflagrationthicknesses generally are around a fraction ofa millimeter at normal atmospheric conditions.The corresponding propagation velocity, for anygiven fuel, at given atmospheric conditions, istermed the laminar burning velocity. It can beobtained from measurements the dependence ofthe cone angle of a flame above a Bunsen burneron the flow rate into the burner, for example, orfrom measurements of the flow rate for liftoffof a flat horizontal flame formed on a flat-flameburner. The laminar burning velocity depends onthe equivalence ratio, the ratio of the fuel-to-oxidizer mass (or mole) ratio to the value of thatratio for stoichiometric conditions (conditionsunder which no fuel or oxidizer is left over aftercombustion). Burning velocities exhibit maximanear equivalence ratios of unity (actually, usuallysomewhat greater than unity because of variationin the chemical kinetics and transport properties).Table 3 lists in the last column these maximumburning velocities for the fuels included there.

Gaseous planar premixed flames are subject toinstabilities. There is a hydrodynamic instabilityresulting from the fact that the density of theburnt gas is less than that of the reactants, butdiffusion effects usually overcome this instabil-ity. Exceptions are found in fuel-lean hydrogenflames and in fuel-rich flames of propane andhigher hydrocarbons, for which high molecu-lar diffusion coefficients of the reactant that ispresent in deficient quantities lead locally to themixtures becoming more nearly stoichiometric asthat reactant diffused preferentially toward thehot reaction zone; this phenomenon producescellular flame structures in these mixtures. Indetonations, on the other hand, diffusion effectsare very small because of the high flow veloci-ties with respect to the wave, but the finite-ratechemistry that occurs behind the leading shockwave leads to a convective-acoustic instabilitythat results in most detonations exhibiting cellularstructures; diamond-like patterns result from this,there being narrow lines of very high pressuresthat can leave traces on smoked foils, as well asoften on shreds of debris, which are helpful inforensic analyses to determine whether a detona-tion occurred in an accident scenario.

10 Combustion

Diffusion Flames

In fires, the fuel and oxidizer usually are not ini-tially premixed, and hence they involve diffusionflames. Diffusion flames may spread through ar-rays of solid or liquid fuels. The process of flamespread usually involves heating of the nonburningfuel to a temperature at which it begins to giveoff combustible gas that can participate in thecombustion process. A surface of fire inceptioncan be defined as a conceptual surface separatingburning and nonburning fuel. A spread velocitycan then be defined as the velocity at which thissurface moves through the fuel array and canbe estimated from the heat flux q imparted bythe combustion and the energy required to bringthe virgin fuel into participation. Many differentphysical phenomena can be involved in flamespread (radiation, conduction, convection, flamecontact, firebrand transport, buoyancy, surfacetension, etc.), some of which are illustrated inFig. 3. Although there have been many studies offlame spread, understanding of the most impor-tant phenomena that arise is still progressing.

Energy balances also can be useful for es-timating steady burning rates of condensed fu-els in fires that burn with diffusion flames. Thefundamental balance involved there is that thefeedback energy flow rate from the flames to thefuel must equal the energy required to generatethe combustible gases from the fuel, there be-ing possible radiative, convective, and conductivecomponents in the feedback. The flow processesoccurring may be laminar or turbulent, dependingon the size of the fire. An idea of whether theflow will be laminar or turbulent can be formedby considering the simpler configuration of alaminar gaseous jet flame in an open oxidizingatmosphere.

There are correlations (based on experimentand on theory) between energy emission ratesand flame heights of diffusion flames of gaseousjets. Flame heights and shapes usually are definedfrom visual observations, but in more scientificstudies, they often are related to the positionat which the mixture is stoichiometric, sinceit varies from being very fuel rich at the jetexit to very fuel lean in the atmosphere. Flame

Combustion, Fig. 3 Aschematic illustration ofdifferent possiblemechanisms of flamespread (From Williams1982)

POROUSBED

LIQUIDPOOL

FORESTFIRE DENOTES SURFACE

OF FIRE INCEPTION

FUELROD

VERTICALSHEETq

qq

q

q

Combustion 11

C

40

LAMINAR TURBULENT

BLOWOFF

LIFTOFF

EXIT VELOCITY (m/sec)

HE

IGH

T (

cm)

00 40 80

Combustion, Fig. 4 A schematic illustration of the flame height for open-jet diffusion flames, showing laminar andturbulent regimes, liftoff heights, and blowoff velocity (From Williams 2002)

heights defined in different ways typically differby amounts on the order of 10%, although dif-ferences between average and maximum heightsin turbulent situations can exceed a factor oftwo. Figure 4 is a schematic illustration of thedependence of the average flame height on theexit velocity of the fuel jet; the numbers in thefigure would apply roughly to a methane jetissuing from a tube 0.5 cm in diameter into airat atmospheric pressure and room temperature.

At low exit velocities, the diffusion flame islaminar, and its height increases in proportion tothe exit velocity u. Theory and experiment agreein showing the laminar height to be proportionalto ud2/D, where d is the exit diameter and D is thethermal diffusivity of the gas. Buoyancy can playan important role, and it causes departures fromthis type of dependence if the exit is not round.Transition to turbulence begins at exit velocitiesabove a critical value, and the portion of the flamethat is turbulent then rapidly increases with u.The dashed line in the figure marks the lowerboundary of the turbulent portion. The heightof the turbulent flame is independent of u andproportional to d, as predicted if the moleculardiffusivity D is replaced by a turbulent diffusivityproportional to ud.

The base of the flame is observed to be liftedabruptly to an appreciable distance above the jetexit when u exceeds another critical value. Thisvalue depends somewhat on the exit geometry

of the jet, and depending on the fuel and theoxidizer, it may occur in the laminar or turbulentregime. The liftoff is caused by extinction of thediffusion flame in the region of high rates of strainat the base of the flame. At greater heights, therate of strain is less because the jet has spread,and the flame again can begin. The height atwhich the lifted flame begins, called the liftoffheight, increases approximately linearly with u.At a large enough exit velocity, the liftoff heightis nearly equal to the flame height, and blowoffof a roughly spherical flame occurs at a thirdcritical exit velocity, the blowoff velocity. If uexceeds the blowoff velocity, then a diffusionflame cannot be stabilized by the jet. Even inthis relatively simple experiment, then, diffusionflames behave in complex ways that currently arelargely but not completely understood.

Cross-References

� Ignition� Physics-Based Modeling�Rate of Spread� Smoldering Combustion

References

Borman GL, Ragland (1998) Combustion engineering.McGraw Hill, New York

12 Combustion

Clavin P, Searby G (2016) Combustion waves and frontsin flows. Cambridge

Fristrom RM (1995) Flame structure and processes. NewYork, Oxford

Glassman I, Yetter RA (2008) Combustion, 4th edn. Aca-demic, Burlington

Kanury AM (1975) Introduction to combustion phenom-ena. Gordon Breach, New York

Law CK (2006) Combustion physics. CambridgeLewis B, von Elbe G (1987) Combustion flames and

explosions of gases, 3rd edn. Academic, OrlandoLiñán A Williams FA (1993) Fundamental aspects of

combustion. OxfordOppenheim AK (2006) Dynamics od combustion systems.

Springer, BerlinPenner SS (1957) Chemistry problems in jet propulsion.

Pergamon, New York

Prud’homme R (2010) Flows of reactive fluids. Springer,Heidelberg

Quintiere JG (2006) Fundamentals of fire phenomena.Wiley, Hoboken

Strehlow RA (1984) Combustion fundamentals. McGrawHill, New York

Williams FA (1982) Urban and wildland fire phenomenol-ogy. Prog Energy Combust Sci 8:317–354

Williams FA (1985) Combustion theory, 2nd edn. AddisonWesley, Redwood City

Williams FA (2002) Combustion. Encyclopedia of phys-ical science and technology, 3rd edn. Academic, SanDiego

Zel’dovich YB et al (1985) The mathematical theory ofcombustion and explosions. Consultants Bureau, NewYork