7.1 Homogeneous and heterogeneous combustion - … Homogeneous combustion ... Homogeneous and...

413VOLUME V / INSTRUMENTS

7.1.1 Homogeneous combustion

OverviewOur understanding of the phenomenon of combustion

dates from the second half of the Eighteenth century, whenAntoine-Laurent Lavoisier demonstrated that the processwas not due to the release of phlogiston, then considered tobe one of the constituent parts of matter, but instead to thecombining of one component of the air, oxygen, withparticular gaseous, liquid or solid combustible materials. Itsspecial characteristic is the manifest emission of light andheat which often takes on the typical aspect of flames.

In reality, combustion processes have been the mainsource of energy for mankind ever since prehistoric times, andeven today they play a central role in our economy, supplyingabout 90% of the energy we consume. In spite of the relentlessand, in certain respects, challenging search for alternativesources of energy, this pre-eminent position is bound toremain unchallenged for a long time to come. The fuelscommonly used as sources of energy are natural gas, which isprincipally made up of methane; petroleum products, whichare composed of mixtures of hydrocarbons; and coal. Whenthese combine with oxygen (the comburent substance) carbondioxide and water are formed through a series of chemicalreactions that, as a rule, are very complex because they takeplace in various stages and involve, as will be explainedfurther on, many intermediate species. Leaving aside thesedetails, it is nevertheless easy to formulate the global reactionsthrough which combustion products are formed from the fuelsmentioned. These can be expressed as follows:

[1] CH42O22H2O CO2 for methane

m m[2] CnHmn 1O2

nCO21 H2O4 2for a generic hydrocarbon

[3] C O2CO2 for coal

These reactions release significant quantities of heatenergy whose values can be calculated from the enthalpies offormation of the various species involved (see below).Alongside the preceding combustion reactions, there is also,for example, the combination of oxygen with hydrogen toform water, and the combination of hydrogen with chlorine

and bromine to form the corresponding hydrochloric andhydrobromic acids. In short, the category of combustionprocesses includes all reactions that are very rapid andstrongly exothermic.

The peculiar nature of combustion processes stems fromthe interaction of an ensemble of physical and chemicalphenomena which give rise to particular and diversifiedsituations. The simplest is that of a gaseous mixturecontaining a fuel, methane, for example, oxygen and anyinert gases such as nitrogen which may be present, in whicha very rapid exothermic reaction is triggered that produces astrong heating effect with consequent sharp variations of thetemperature and concentrations of the reagents in time andspace. The geometrical configuration and the physicalconditions of the mixture, in particular the temperature, ofthe mixture create the right conditions for its ignition to takeplace.

If the fluids are in motion, changes in components andoperating conditions cause different identifiable situations toarise, each of which is characterized by specific problemsfor the description and handling of combustion processes.The case of a gaseous mixture formed by a fuel and acomburent (combustion-support) that moves with laminarflow in a given direction at constant rate will be consideredfirst. If a combustion reaction is triggered at a particularposition in the direction of the flow, a flame is generatedwhich causes a sharp increase in the temperature of themixture. The reactive event takes place in a thin layer inwhich, besides the temperature increase, an abrupt decreasein the concentration of the reagents also occurs, while theconcentration of the reaction products increases at anequivalent speed and attains its corresponding value in thefinal mixture. In substance, one can identify a reaction frontthat moves at a given rate of direct propagation in thedirection opposite to that of the gaseous flow. If thenumerical values of the two velocities are identical, theposition of the flame front remains unchanged in time andthe flame is said to be stationary. This description identifiesthe behaviour of what are known as premixedone-dimensional flames. A different situation occurs in thecase of diffusion flames, in which the two gaseous reagentsflow into two separate zones and then converge in a thirdzone where the combustion takes place.

7.1

Homogeneous and heterogeneouscombustion

In both of the cases described above, however, twofurther situations stand out, which result from the fact thatthe flow of the currents are, respectively, either of thelaminar or of the turbulent type. When the flow is laminar,mixing processes occur only at the molecular level, insteadwhen the flow is turbulent, they are also the result of theaction of turbulent eddies. These characteristics have asignificant influence on the behaviour of combustionprocesses and point out the importance of the fluid-dynamicbehaviour of the systems concerned.

The difference between premixed and diffusion flamesprovides us with a criterion for classifying the characteristicsof a system subject to a combustion process. For example, inthe carburettor of an internal combustion engine, petrol andair are mixed and then, after undergoing compression andignition, they burn as a premixed flame, with a front thatpropagates inside the combustion chamber. In the case of alighted candle, on the other hand, the heat released by theflame causes the candle’s wax to vaporize and mix with theair in the area that is close to the wick, where the combustionprocess takes place and assumes the typical characteristics ofa diffusion flame. Other examples of diffusion flames arepresent in burner jets or in the reaction engines of jetaeroplanes. If, however, the burner is very large, it is betterto premix the fuel with the air.

In cases where the chemical reaction is preceded by adiffusion process, typical concentration-profiles of thevarious reagents are formed. For example, if the reagents,which are made up of gaseous fuel and air, flow respectivelyinto an internal cylinder and into the ring of a co-axialexternal cylinder, the combustion process that takes placewhen they come together occurs on a flame frontcharacterized by cylindrical symmetry, with the consequentformation of the component concentration-profiles havingthe same symmetry.

Heterogeneous combustion processes are entirelydifferent. In these processes, particles of a solid, for example

coal, or tiny droplets of liquid hydrocarbons, act as fuels. Inthe first case the reaction occurs through the interaction ofoxygen molecules with the carbon atoms that constitute thesolid material, and the diffusive processes that take place inthe gaseous film adjacent to the surface of the solidsignificantly influence the kinetics of the combustionprocess. The combustion of liquid droplets is even morecomplex, as the heat resulting from the reaction causes thepartial evaporation of the volatile liquid components whichdisperse in a gaseous state and react with the oxygen.

In conclusion, combustion processes show thearticulated variety of situations summed up in the diagramreproduced in Fig. 1. After examining the thermodynamicand kinetic aspects of combustion processes, the mostsignificant typologies listed in the diagram will be analyzed,taking into consideration the determination of ignitionconditions, premixed laminar and turbulent flames, anddiffusion flames. Certain aspects that concern the influenceof combustion processes on air pollution, taking into accountespecially the chemical reactions involved in the formationof nitrogen oxides and solid carbonaceous particles, will alsobe discussed.

Thermodynamic aspects. Flame temperatureA generic chemical reaction is used as a reference,

formally expressed as follows:

[4]

where vi is the stoichiometric coefficient of the ith

component. At constant pressure, the heat released by thereaction, QR, is equivalent to the variation in enthalpyassociated with the reaction itself (DH), but with theopposite sign; if the pressure is 1 bar, this can be computedon the basis of the standard molar formation enthalpiesDf H° of the various species, linearly combined according tostoichiometric coefficients (assumed to be positive forreaction products and negative for reagents):

ν ν ν νA B M NA B M N+ +⋅⋅⋅ + + ⋅⋅⋅

COMBUSTION AND DETONATION

414 ENCYCLOPAEDIA OF HYDROCARBONS

PREMIXED REAGENTS

COMBUSTION PROCESSES(rapid, strongly exothermic reactions)

NON-STATIONARYSTATES

(time-dependentprocesses)

phenomena of ignitionand extinction

STATIONARYSTATES

laminar andturbulent flames;

their velocityis determined by

the physico-chemicalcharacteristicsof the mixture

HOMOGENEOUStheir characteristics

are determinedby the flowof gases and

by interdiffusivephenomena

DIFFUSION FLAMES

HETEROGENEOUStheir characteristics

are determinedby the exchange

of heatand material

with the surfaceof the solid

Fig. 1. Classificatory scheme of the various different types of combustion.

[5]

Table 1 reports the values of the formation enthalpies ofvarious species involved in combustion processes. They referto standard conditions, which correspond to a pressure of 1bar and a temperature of 298 K.

If one knows the molar heat capacities cp,i of the varioussubstances, one can calculate the degree to which variationsin enthalpy depend on temperature using the equation:

[6]

Heat capacities, in turn, are usually expressed as afunction of temperature by means of a polynomial functionof the following type:

[7]

whose parameters ai, bi, ci, ... are known as calorimetriccoefficients. Their values for some of the compoundscommonly involved in combustion processes are set out inTable 2.

One of the important parameters that characterizes aflame is the temperature that is reached after combustion(Tf), starting from the value To. If the flame is adiabatic, i.e.if all the heat released by the reaction contributes only toheating the reacting mixture, a simple balancing of energiespermits the calculation of the final temperature, also calledin this case adiabatic flame temperature. This value isobtained by equating the heat that is absorbed by the mixtureof the reaction products to the heat released by the reaction(at tempercature To), calculated by means of equations [5-7]:

[8]

where Ainiai, Binibi, Cinici …, in which thesummations extend to the products of the reaction. Theadiabatic flame temperature is calculated by solvingequation [8] for the unknown quantity Tf . The degree ofextension of the polynomial depends on the degree ofaccuracy one wishes to achieve.

In practice, no chemical reaction ever takes place to afull extent, but tends to reach an equilibrium compositionthat depends on the pressure and the temperature reached bythe system in accordance with the equation:

[9]

where pi is the partial pressure of the component i atequilibrium, which is equal to its mole fraction times thetotal pressure P, where P, DrG° is the variation in thestandard free energy of the reaction and R is the gasconstant. The quantity DrG°, in turn, is calculated on thebasis of the standard free energies of formation of thevarious components, Df Gi° (see Table 1), by means of anadditive equation similar to [5]. Finally,KgfM

nMfNnN…fA

nAfBnB…, where fM,… are the coefficients

of fugacity of the various components which measure thedeviation from ideal behaviour; plainly, if the componentsand the mixture behave ideally, then Kg1.

Actually, apart from global reactions such as thoseindicated above, in a combustion process there are also otherequilibrium reactions involving the reaction products thatmust be taken into consideration, such as:

[10] CO2CO 12 O2

[11] H2OH21/2 O2

amongst others. Since the composition of the system atequilibrium depends on the temperature that is reached, [8]must be solved together with equations of type [9] whichallow the evaluation of a composition as a function of thetemperature. By and large, a system of non-linear algebraicequations is obtained, which can be easily solved by meansof adequate computational programmes.

Chemical and kinetic aspects of the combustion of hydrocarbons

The rate ri of a generic chemical reaction expressesthe number of moles of a species of reference i that aretransformed per unit of time and per unit of volume. Thisdepends on the volume and composition of the system:for a combustion reaction, in its simplest form it isusually expressed as a function of the primary reagents,as follows:

p pp p

K K TM N

A B

G RTM N

A B

r

ν ν

ν ν γ

⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅

= ( ) = − °e ∆

A T T Bf= −( ) +0 22 3

2

0

2 3

0

3T T C T Tf f−( ) + −( ) + ⋅⋅ ⋅Q H A BT CT dTR r T

T

Tf

− = + + + ⋅⋅⋅( ) =∫∆0

° 2

0

c a bT cTp i i i i, = + + +⋅⋅⋅2

∆ ∆r T r i p ii

T

H H c dT= + ∑∫298

298

ν ,

Q H HR r i f ii

= − = −∑∆ ∆ ν

HOMOGENEOUS AND HETEROGENEOUS COMBUSTION


Table 1. Standard formation enthalpies and standardformation free energies at 298 K for various gaseous

compounds involved in combustion processes

Compoundf H °kJ/mol

fG°kJ/mol

CH4 74.86 50.85

C2H6 84.78 33.00

C6H14 154.74 0.25

C8H18 208.59 16.41

C6 H6 82.98 129.75

CH3OH 202.05 163.45

C2H5OH 234.75 168.18

Table 2. Values of the calorimetric coefficientsfor the calculation of molar heat capacities (J/moleK)of some gaseous substances involved in combustion

Compound a b103 c106 Temperatureinterval (K)

CH4 22.36 48.15 – 273-1,200

CO 27.63 5.02 – 273-2,500

CO2 26.67 42.29 –14.25 300-1,500

H2 28.81 0.28 1.17 273-2,500

H2O 34.42 0.63 5.61 300-2,500

N2 26.38 7.62 –1.44 273-2,500

O2 26.21 11.50 3.22 273-5,000

[12]

The square brackets indicate the molar concentration ormolarities of the components, a and b are two empiricalparameters known as orders of reaction, while k(T) is thereaction rate constant that depends on the temperaturethrough the Arrhenius equation k(T )A exp(ERT ), whereE is the activation energy. The values of these kineticparameters for some hydrocarbons are reported in Table 3.

The preceding equation [12] expresses the rate of thecombustion process taken as a whole, in other words,without taking into consideration the fact that in reality thetransformation in question takes place through various stageswhich involve several intermediate species, consisting forthe most part of molecular fragments. This lumped approach,while ‘ignoring’ the details of the transformations examined,nevertheless provides an effective starting point forexamining some typical aspects of combustion processessuch as ignition and the formation of premixed flames.

Very accurate investigations have been conductedwhich, through the use of sophisticated diagnostictechniques, have supplied interesting information on thedetails of the combustion reactions of hydrocarbons. Apartfrom the traditional analytical techniques such aschromatography and mass spectrometry, new techniquesbased on the use of laser devices have been developed. Forexample, by means of the diagnostic procedure known asLIF (Laser Induced Fluorescence), molecular species areidentified by exciting them energetically with a laser rayand then analysing the radiation that they emit byfluorescence. In this way, species can be identified that arepresent only in very small concentrations as they have avery short average life. Moreover, accurate experimentalmethodologies have been finalized to determine the rate ofcombustion reactions. Lastly, it is worth mentioning thatthe study of homogeneous combustion processes is beingconducted also using methodologies that belong toquantum chemistry which, thanks to the use of supercomputers, have now made it possible to calculate thereaction rate constant of a number of the elementaryprocesses involved in said combustion processes with areasonable degree of accuracy. The expressions of thevelocities of these reactions are based on their molecularnature, as they reflect the characteristics of the elementaryact through which the reaction takes place. For example, if

the process occurs following the collision between twospecies A and B, the rate of reaction is expressed asfollows:

[13]

where n is a number in the order of unity.Thanks to the ensemble of methods of investigation

mentioned, a considerable range of values of kineticparameters is now available for many of the elementaryreactions involved in homogeneous combustion processes.One of the most thoroughly investigated cases has been thatof the oxidation of methane at a high temperature, which hasled to the general scheme illustrated in Fig. 2. The schemeshows the most important reactions involving compoundsthat contain one or more carbon atoms. It can be observedthat the methane molecule is attacked by the radical speciesH, O, OH, highly reactive because of the unpaired electronpresent, indicated with a point. CH3 radicals are thusformed, which react with the oxygen atoms to produceformaldehyde, CH2O. By removing a hydrogen atom fromthis species the formic radical is obtained, which via thermaldecomposition forms carbon monoxide and atomichydrogen. Furthermore, two CH3 groups can combinetogether to form ethane which in turn can be attacked by H,O and OH to form the ethyl radical which will then beoxidized. Hydrocarbons with a molecular weight greaterthan methane and ethane are also attacked by these radicals,forming unstable species that are subject to rapid demolitionwith the formation of smaller molecular fragments. Thus theproblem of the oxidation of the higher hydrocarbons can beexplained by the same oxidation mechanism involvingfragments of small size, whose characteristics are known asa result of the investigations carried out on the oxidation ofmethane.

It is interesting to observe that in the complex sequenceof elementary reactions involved in such processes, the onesthat play an important kinetic role are:

[14] HO2OHO

[15] COOH CO2H

which, curiously enough, do not depend on the nature of thefuel used.

As combustion reactions are normally conducted withair, and therefore in the presence of nitrogen, nitrogenouscompounds are also formed, and in particular nitrogenoxides. The reactions that correspond to their formation areintegrated with those relating to combustion as they involvesome of the above-mentioned radical species. Whenoperating at a high temperatures and for extended contacttimes, the formation of nitrogen monoxide NO may takeplace via the Zeldovich mechanism:

[16] ON2NO N

[17] NO2NO O

In fuel-rich mixtures, the following reaction also comesinto play, which tends to replace equation [17]:

[18] NOHNO H

At relatively low temperatures, NO can be formed alsofrom N2O through the following sequence of reactions:

[19] ON2MN2O M

[20] ON2O2NO

k AT n E RT= −e

r k Ta b

= ( ) fuel oxygen



Table 3. Kinetic parameters of certain fuels

Fuel A(one-step)E

kJ/mola b

CH4 1.3109 202.64 0.3 1.3

C2H6 1.11012 125.60 0.1 1.65

C6H14 5.71011 125.60 0.25 1.5

C8H18 7.21011 167.47 0.25 1.5

C10H22 3.81011 125.60 0.25 1.5

C2H5OH 1.51012 125.60 0.15 1.6

C6H6 2.01011 125.60 0.1 1.85

[21] HN2ONH NO

where M indicates an inert molecule that facilitates thereaction by subtracting energy from the reagents by means ofcollisions. A second type of mechanism that leads to theformation of nitrogen oxides, called prompt NO, is morecommon at low temperatures and with short contact times. Itis assumed that hydrocyanic acid HCN, formed by theinteraction of carbon radicals with nitrogen, plays the role ofintermediary, which is formed by the interaction of carbon asfollows:

[22] CH N2HCN N

[23] CH2N2HCN NH

Subsequently, the hydrocyanic acid is converted, througha complex pattern of reactions, into nitrogen oxide. Attentionto the reactions that lead to the formation of nitrogen oxidesis justified by the fact that these species are among the mosthighly polluting agents in the atmosphere.

Ignition point and temperatureIf a mixture of gaseous fuel and oxygen is contained in a

closed chamber, the components tend to combine with oneanother and to release heat. If the rate at which the heat isoutwardly dispersed is high, the temperature will remain low,and the reaction will proceed slowly under a particularcondition called slow combustion.

If, however, the temperature exceeds a certain criticallimit which, as we shall see, depends on the specificcharacteristics of the mixture and the container, the rate atwhich heat is released can give rise to a great increase in therate of the reaction, to the point of causing an explosion. Thecondition corresponding to the start of such a process is

known as the ignition point and underlies all processes ofcombustion. The phenomenon arises from the exponentialprogression of the Arrhenius equation, which establisheshow the reaction rate depends on the temperature. A methodfor approximating the value of the critical ignitiontemperature was devised by Semenov. It is based essentiallyon two premises: that the mixture remain well mixed andthat the rate at which heat is exchanged between the zonewhere the combustion takes place and the externalenvironment be proportional to the difference between theirtemperatures. Thus, the quantity of heat dispersed per unit oftime can be expressed by the following equation:

[24]

where S is the surface that delimits the system, T0 theexternal temperature and h the coefficient of heat exchange.Q. depends upon the physical characteristics of the mixture

being used, in particular on its thermal conductivity anddegree of homogenization. The heat that is released per unitof time can instead be expressed as follows:

[25]

where V is the system’s volume and r the rate of the reaction.In the treatment that follows, the latter shall be taken to be akinetic expression of the order of zero, assuming that theexponents a and b in equation [12] are also equal to naught.This approach, although only approximate since it does nottake into account the effect of the reagents’ consumption atthe start of the ignition process, nevertheless yields resultsthat are satisfactory.

The energy-balance is expressed by equating the rate ofincrease of the reactive mass to the difference between theheat that is released Q and that which is dispersed Q:

Q VQ rR+ =

Q hS T T− = −( )0



CH4 CH3 CHO COCH2O

C2H5 C2H4 C2H3

CH2O, CHO CH3

C2H2 CH2

CH3,CH2,CHOCH3CHO CH2CO CH CO

CH3CH

CH3

H, O, OH H, O, OH M, O2, H

H, O, OH H O

HCH3

C2H6

CH3CH3 H

M, O2

O

H

O OHO, OH O H O, O2

O, O2

OH O

M

H, O, OH

M, H

H

Fig. 2. Schematic representation of the process of oxidation of methane that shows the most important reactions involving radical species and molecular fragments.

[26]

where t is time, cp the specific heat and r the density of thereactive mixture. If the following adimensional values areintroduced:

[27]

[28]

[29]

and bearing in mind that, in general, RT0 E1, by means ofa series of algebraic transformations it can be demonstratedthat the energy-balance equation [26] will acquire thefollowing form:

[30]

The preceding differential equation appears simple atfirst glance, but, actually, it cannot be analytically integrated.Its particular characteristics can, nonetheless, be note byrecording in a diagram the variations of the two terms to thesecond member, which refer to the amounts of heat that havebeen either released or dispersed, respectively. In this mannerthe trends reported in Fig. 3 are obtained, where the threestraight lines refer to the three different values of coefficienta. In the graph, the stationary state QQ corresponds tothe intersection between the curve that represents the amountof heat that has been released and the straight line thatrepresents the amount of heat that has been subtracted. Asshown in Fig. 3, intersection occurs in case I. On thecontrary, no intersection occurs in case III, which impliesthat the amount of released heat is always greater than that ofdispersed heat. Under these conditions, the system is

intrinsically unstable and ignition takes place. The thresholdbetween the two effects is illustrated in case II, in which thestraight line representing dispersed heat is tangential to thecurve representing released heat. At this point, the followingconditions apply:

[31]

conditions that are called critical, and which lead to thefollowing result for the critical parameters:

[32]

Thus, if a is less than e (base of natural logarithms) there isignition; if it is greater there is a stationary state. Inaccordance with [27], the following equation relating criticalignition temperature Tc to temperature T0 can be derived:

[33]

If, for example, we assume T0500 K and thereasonable value of 10,000 cal/mol for the energy ofactivation, we obtain a value of Tc equal to 550 K, whichshows that the amount of preheating that would precedeexplosive self-ignition is relatively small.

This theory can be improved if the consumption of reagent is taken into account and expresses thecombustion reaction rate with a kinetics of the order a0with respect to the reagent fuel. Leaving aside the detailsof the calculation, it can be demonstrated that the criticalvalue of the coefficient of heat exchange is expressed bythe following equation:

[34]

where BC0 QR Er CpRT02, in which C0 is the reagent

concentration at time zero. When a0 or when values of Bare very high, the preceding equation is the same as [32], afact that shows how Semenov’s approach allows the value ofac to be calculated with a reasonable margin of certainty.

Ramified chain reactionsIn addition to thermal factors, chemical factors may also,

in certain cases, play such an important role in oxidationreactions as to give rise to explosive evolutions. The radicalspecies that form in the course of certain stages of theprocess, by virtue of their high degree of reactivity due tothe presence of unpaired electrons, can generate, byinteraction with stable molecules, new molecular fragmentsthat provoke an avalanche of reactions typical of chain-reaction processes. In these processes, four characteristicstages can be distinguished:• Initiation, during which molecular fragments are

generated via thermal decomposition:

[35] A2X

where X stands for a generic radical species.• Propagation, during which the number of radicals

remains unaltered:

[36] XBYC

where Y stands for another radical. An example of thisis the reaction between the radical OH and methane,that yields water and a methyl radical:

a e a Bc = − ( )

1 2 703

2

3.

T TRTEc

c− =0

2

a ec c= = , ϑ 1

Q QdQd

dQd+ −

+ −= = ϑ ϑ

dd

e aϑτ

ϑϑ= −

aRT hSEAVQ

E RT

R

= 0

2 0e

τ =EAQ

c RTtR

pE RTr

0

2 0e

ϑ = −( )ERT

T T0

2 0

c V dTdt

hS T T A Q VpE RT

Rr = −( ) − −0

e



0

˙˙

III

II

I

Fig. 3. Diagram showing the curve corresponding to the function eÿ and the straight lines aÿ as a function of ÿ for different values of a. Line I is subcritical, line II is critical and line III is supercritical.

[37] OH CH4H2O CH3

which occurs in the course of the oxidation of methane.• Ramification, a special type of propagation in which the

number of radicals produced is greater than the numberof radicals participating in the reaction:

[38] XA2Y

An example is provided by the reaction:

[39] HO2OH O

• Termination, a reaction during which radicals areeliminated, for example as a result of their pairing:

[40] XYA

These reactions are often facilitated by the action of athird body, a stable inert species that subtracts energy in thecourse of the exothermic process during which the tworadicals combine with one another.

The possible explosive degeneration of a chain-reactionprocess is the result of the ramification reactions which, ifnot controlled by the termination reactions, cause acatastrophic increase in the number of radicals that thesystem generates. One can carry out an approximate analysisof the phenomenon by assuming that the rate at whichradicals are generated is constant and that the velocities ofpropagation and termination are both proportional to theconcentration of the radicals present, which will begenerically indicated by the symbol X. In a system closed toexchanges of matter, the accumulation of radicals over timeis described by the following equation:

[41]

where I0 stands for the rate at which radicals are producedduring the initiation reaction, and f and g, respectively, standfor two functions, one of temperature and the other ofpressure, that reflect the efficiency of the ramification andtermination stages. The term ffg is known as theramification factor. Bearing in mind that for t0 theconcentration of radicals is nil, the integration of thepreceding equation yields the following expression for theconcentration of radicals present as a function of time:

[42]

Its development depends on whether f is positive ornegative: if it is negative, i.e. if fg, the function tendsasymptotically towards the value:

[43]

which is the stationary solution that can be derived directlyfrom [41] by assuming the first member to be equal to zero.If instead f is positive, the concentration of radicalsincreases exponentially. In this case the system is subject to acatastrophic explosive evolution because the production ofradicals via the ramification process prevails over theirdestruction via termination.

Ignition diagramsThe mechanism that has been described can be employed

to interpret the combustion of various substances, including

hydrocarbons. If the reaction takes place in a closedcontainer and changes in pressure are measured over time,very different trends will be observed depending on the kindof fuel that is used and the conditions in which the reactiontakes place. Fig. 4 shows the increase over time in thepressure of a mixture in relation to different values of itspressure at the start of a reaction. In curves a and b, whichcorrespond to the lowest pressures, there is a period ofinduction during which a reaction has yet to take place. After

X· = −I

g f0

X· = −( )Ie t0 1

ϕϕ

ddt

I f g IX·

X· X· X· = + − = + 0 0

ϕ



0

pres

sure

time

e

d

c

b

a

Fig. 4. Typical trends in pressure increase as a function of time during combustion in a static system. Curves a-ecorrespond to increasing values of the initial pressure: a and b slow combustion, c in the presence of one cold flame, d and e in the presence of two cold flames.

0

tem

pera

ture

pressure

blue-flame zone

ignition

slow combustion

one coldflame

two coldflames

ignitionhump

Fig. 5. Example of an ignition diagram.

that, the curves rise sharply and reach a peak pressure point.If the initial pressure is greater, one obtains the trendsillustrated in curves c, d and e, in which there aremomentary bursts of pressure that last for a span of onesecond, during which time the temperature increases byabout 150°C. These bursts are accompanied by a weakluminescence and are called cold flames.

The behaviour of every combustible compound may besummarized in a diagram known as an ignition diagram; byplotting temperatures as ordinates and pressure values asabscissas, different zones can be identified that correspondto the different ways in which combustion occurs. Anexample of an ignition diagram is given in Fig. 5, where asinuous curve is shown that separates the zone on the left,known as slow combustion, from the one on the right whereignition takes place. Some ‘humps’ are also present thatcorrespond to zones in which bursts of cold flames takeplace. The slow combustion zone and the ignition zone are,in turn, separated by a narrower region, known as the zone ofthe blue flames, which are warmer than cold flames.

Based on the experimental data that has becomeavailable, the ignition diagrams of various hydrocarbonshave been drawn. Although they all have the same overallshape, they differ from one another in their detail, inparticular in the number and shape of the humpscharacterizing the cold flames. In a certain sense, thesediagrams describe the morphology of the combustionprocesses that affect distinct substances and constitute usefultools for identifying their characteristics.

Premixed laminar flamesLet us assume that we have a gaseous mixture at

temperature T0 that contains a fuel (for example methane)and oxygen, and that it flows with laminar motion at a rate u0along the z axis of a cylindrical conduit. The fluid’s particlesmove in a regular manner parallel to the axis of the cylinderand with a parabolic rate profile that starts from a value ofzero next to the wall of the cylinder and reaches a peak at itscentre. In a simplified, and idealised, scenario it is preferablenot to take into account the effects of the walls of the conduit

on the fluid, and to assume that motion occurs with a piston-like flow, on the basis of which the radial rate profile isdescribed as a function with a constant value at every point ofthe sections perpendicular to the direction in which the fluidis flowing, except at the walls of the conduit, where it is nil.The formation of a flame is characterized by a sudden andsharp increase in the temperature along the z axis, as shownin Fig. 6, and by an equally fast and sharp decrease in theconcentration of the reagents that are then transformed intocombustion products. This means that a frontal part of theflame that is only a few millimetres thick may be identifiedwhere the temperature peaks to values as high as 2,000-3,500K. The front propagates at a rate Su in the direction oppositeto that of the flow of the gas. Under stationary conditions, thevalues of the two velocities are the same, and the position ofthe flame front, therefore, stays the same.

In order for the combustion process in the initial mixtureto start, the mixture must be brought to ignition temperatureTi. Only then will the retroactive mechanism take placewhereby the heat released in the zone of the reaction, whichflows in the direction opposite to that of the flow of the gas,preheats the mixture to the temperature required for theexothermic reaction of combustion to take placespontaneously. Over a century ago, Mallard and Le Châteliergave a tentative description of the process which, thoughsomewhat approximate, captures the phenomenon’s essentialaspects. If the thickness of the zone in which the reactiontakes place is indicated by d the equivalence between theheat flow that moves backwards and the heat absorbed by thecurrent that feeds the reaction leads to the followingequation:

[44]

where Tf is the temperature of the flame, kT is the thermalconductivity of the fluid mixture, cp its average specific heatand G the mass flow per surface unit which, in turn, can beexpressed as the product of the rate of flow of the gas and itsdensity, Gur. The product of the average reaction rate rand the average molecular weight of the mixture M, yieldsthe reactive mass transformed per unit of time and ofvolume. If it is divided by the density r, the inverse of theaverage reaction time is obtained which when multiplied bythe thickness d of the zone in which the reaction occursyields the rate of the flame propagation, Su:

[45]

whose value under stationary conditions is the same as thatof the rate of flow of the fluid. If one derives d from thepreceding expression and replaces it in equation [44], thefollowing equation results:

[46]

The fact that Su is proportional to the square root of therate of the reaction is the most significant result that hasbeen achieved by the theory of premixed flames, even if thepreceding equation actually produces only approximatesolutions.

A more refined model to describe the behaviour ofpremixed flames, known as the ZSF model after the initials

Sk T T rM

c T TuT f i

p i

=−( )−( )

1

0r

S rMu =

δr

kT T

Gc T TTf i

p i

−= −( )δ 0



intermediatestem

pera

ture

, con

cent

rati

on

z

reactants

T0 H2O

q

temperature

Tf

˙

Fig. 6. Changes in temperature as a function of the coordinate z in a premixed laminar flame. The figure also shows the curves that indicate the changes in the concentration of reactants, intermediates and waterproduced by the reaction. Finally, the curve indicated by .q represents the amount of heat released by the combustionreaction per unit of time.

of the Russian scientists who developed it, Zeldovich,Semenov and Frank-Kameneskij, was produced in the firsthalf of the Twentieth century. If one derives the heat balanceof a reactive mixture over a volume whose cross-section isequal to that of the mixture’s average flow and whosethickness is infinitesimal, with an axis aligned with thedirection of the flow of the gas, it can be easily demonstratedthat, under stationary conditions, the following equationholds true:

[47]

where q is the heat flow per unit of time and per surface unitperpendicular to the direction of the flow, which can beexpressed as the sum of two factors, i.e. the amount oftransport associated with the convection of the fluid currentand the amount of thermal conductivity:

[48]

A constant average value has been attributed to themixture’s specific heat in the expression of the first term,while the second contributing factor is provided by theproduct of the coefficient of thermal conductivity kT and thetemperature gradient, and moves in the direction opposite tothat of the convective flow. By substituting [48] in [47] andassuming a kinetic equation with an order of zero as the rateof the reaction one arrives at the following equation:

[49]

The preceding is a differential equation of the secondorder, non-linear because of the presence of Arrhenius’sexponential term, which prevents an analytic expression todescribe variations in temperature as a function of z frombeing obtained. It is, therefore, preferable to identify a valuefor the coordinate z0 somewhere along the curve of theflame’s temperature, as illustrated in Fig. 7, where thetemperature is assumed to reach Ti. Values of z0 are in thepreheating zone where the reaction, for all intents andpurposes, does not take place, so that equation [49] can besimplified by omitting its third term. In this form, theequation can easily be integrated by setting TTi as limitconditions for z0 and that the temperature will tendtowards T0 when z. By setting aG cp kT , thefollowing is obtained:

[50]

which describes the trend of the temperature in the preheatingzone. Values of z0 are in the reaction zone where thetemperature increases from Ti to the final value Tf and theamount of heat released is much greater than that associatedwith the convective flow. The latter, therefore, can be ignoredand equation [49] can be approximated to the following:

[51]

The preceding equation is also impossible to resolveanalytically. It can be observed, however, that thecorresponding temperature behaviour that connects Ti withTf depends exclusively on the physical properties of themixture kT and QR as well as on the kinetic parameters A andE, so that, if these are defined, the curve presents a uniquetrend, as shown in Fig. 7 B. If z0, on the other hand, thereexists a whole family of curves, each of which correspondsto a different value of the parameter a. For z0, the curvescorresponding to the preheating zone and to the reactionzone, in addition to intersecting when the temperature is Ti,will also necessarily have the same tangent. This condition,as the Figure illustrates, can be attained only for a particularvalue of a which, since cp and kT are given, will determine awell-defined value of G, i.e. the rate of the gas flow, whichtherefore becomes an eigenvalue imposed upon [49] by itslimit conditions. By simplification the following is obtained:

[52]

By means of the preceding formula the rate ofpropagation of a laminar flame can be calculated startingfrom the physico-chemical parameters of the combustiblecompound. Generally, Su values are higher for the alkenesthan for the alkanes, and higher for the alkines than for thealkenes, and decrease when the ramifications in the structureincrease.

General treatment of combustion phenomenaIn the ZSF model, the combustion process is described

by means of a single reaction, whereas in reality it takesplace through complex reaction schemes in which a largenumber of intermediate species participate. As we now haveavailable a variety of data on the chemical characteristicsand kinetic parameters of these reactions, thanks tosupercomputers it has become possible to carry out highly

Skc C

AT T

RTEu

T

p

E RT

f

f=−

2 1

0 0

2

r

e

k d Tdz

Q AT RE RT

2

2= −e

T T T Tiz= + −( )0 0

eα

k d Tdz

c G dTdz

Q AT p RE RT

2

20− − =−e

q c GT k dTdzp T= −

dqdz

Q rR=



z

z

B

A

0

0

zone II

zone I

T0

T0

TiTf

Tf

Ti

T

T

Fig. 7. Temperature profile of a premixed flame. In A two zones are visible: the first corresponds to preheating (I)and extends to the ignition temperature Ti, while the secondcorresponds to the proper reaction (II) and continues from that temperature. As shown in B, when Ti is reached, the curves relative to the two zones must have a commontangent, whose slope enables the a parameter in equation [50] to be derived.

detailed calculations to simulate the evolution in theconcentration space, due to chemical reactions and processesof diffusion, of the combined species involved.

As already seen, in a combustion process the varioustransformations at work take place at the same time, andproceed along complex patterns of successive and parallelreactions, such as the one pertaining to methane illustrated inFig. 2. It is, therefore, useful to distinguish each reaction bymeans of an index k, identifying with vi,k the stoichiometriccoefficient of the generic component i. In this manner, therate at which a given component is formed as a result of theeffect of all the reactions in which it participates can beexpressed by the equation:

[53]

Once this has been established, the general treatment ofcombustion processes in gaseous systems subject to laminarmotion proceeds through the application of the laws of theconservation of motion, energy and matter, whose values perunit of volume will be generically indicated with the symbolf. The symbol FF, instead, indicates the amount of flow persurface unit, associated both with the laminar motion of thegas and with transport processes which, in this approach,pertain solely to chaotic molecular motion. It should beobserved that this flow constitutes a vectorial quantity as itdepends both on the rate u and on the local values of thegradients of the intensive quantities that characterise thephysical state of the system, such as the molarconcentrations, Ci of the various components and thetemperature. The balance of f is obtained by equating itsbuild-up in a container of a given volume to the differencebetween incoming and outgoing flows plus any amount of itthat may have been generated per unit of time and volume,which is indicated with the symbol s

.. In the analysis that

follows the flow of the generic component i measured inmoles per unit of time and surface is indicated with Ni,which multiplied by the molecular weight Mi yields the flowin terms of mass Gi per unit of surface. The composition ofthe gas, moreover, will be characterized, respectively, bothby its molarity Ci and by the mole fractions yi or massfractions wi, which are directly correlated to one another(wiyiMi j yjMj). Molarities and mole fractions are relatedto each other via the simple formula Ciryi, where r is themolar density that, in general, is calculated using theequation of state of perfect gasses rPRT. That beingstated, it is easy to demonstrate that the general balanceequation of f will have the following form:

[54]

In stationary cases the first term of the first member isobviously nil. This equation can easily be applied to the totalmass, to the various components and to the energy by usingthe flow formulas summarized in Table 4.

For reasons of simplicity, in the definition of the energybalance only the contribution associated with the thermalstate has been taken into account, while those associatedwith the flow of radiation and the work due to viscous stresshave instead been ignored. The first of these latter terms, infact, is small for non-luminous flames and significant onlyfor those that are fuel-rich and contain blazing particles ofcoal that radiate energy. The second term is instead oftennegligible if the rate of flow of the gas does not reach veryhigh values. The parameter u*(i Ni)P/RT represents themolar mean velocity derived by expressing the density bymeans of the equation of state of perfect gasses, while Di,m isthe average diffusion coefficient of the component i in themixture, which can be determined starting from the values ofthe binary diffusion coefficients Di, j:

[55]

Actually, the preceding equation provides only approximatevalues since it was derived assuming that the values of thevarious binary coefficients are comparable to one another.

The balance in the quantity of motion is to be achievedbearing in mind that its flow is made up of convectionassociated with the motion of the fluid, of a contributionfrom pressure forces, and of a dissipative term connectedwith the internal resistance resulting from the viscosity ofthe fluid itself. Said balance leads to the Navier-Stokesequation, whose integration, carried out for the most partnumerically, provides the profile of the rate of flow of a gasas a function of time.

The values of typical quantities of molecular transportthat recur in the preceding equations, such as viscosity m,thermal conductivity kT and the binary diffusion coefficientsDi, j for each pair of components i and j, can be accuratelycalculated, on the basis of data on the size of molecules andon the intermolecular forces at work between them by meansof the kinetic theory of molecular transport, that in its mostadvanced form links back to the integration of Boltzmann’sequation. As a first approximation, if the the kinetic theoryof gasses is applied, it follows that (kT cpr)(mr)Dlc,where l is the mean free path of the molecules and c is theirmean velocity as a result of thermal motion.

It is worth noting that if one assumes as a firstapproximation iGi cp,iGcp, the following energy balanceequation is easily derived by applying [54] to the heat flow q:

D yyD

i mi

i

i jj i

,

,

= −

≠∑

1

∂∂+ ∇ ⋅ =f

tΦΦ σ.

r ri k i kk

=∑ ν,



f FF s

Total mass r ru 0

i Component Ci

Energy

Table 4. Variables used in balance equation [54]

N ui= − ∇*

,C D Ci i m i rk i k

k

ν,∑

q G= −( ) − ∇∑ i p ii

Tc T T k T, 0rc T Tp −( )0

−( )∑ ∆ f i k i kk

H r ν,

[56]

For a one-dimensional stationary system in which apremixed gas is present, if it is assumed that only onereaction takes place, the preceding equation becomes equalto [49].

If [54] is applied to the total mass, the matter continuityequation is obtained. Analogously, this equation may beapplied to the flow of the individual components; however,since the average diffusion coefficients obtained from [55]are only approximate, it is preferable to maintain the balanceof each component in the form:

[57]

and to deal with the thorny problem presented by themutual inter-diffusion of the various species present inthe mixture by means of the Stefan-Maxwell equation,which links the local gradients of the compoundsexpressed using the mole fractions yi to the flows of theindividual components:

[58]

Actually, these equations only determine the relativeflows of the various components and, therefore, belong to thecategory of boot-strap equations that are mutuallyinterdependent without having any explicit connection withthe fluid-dynamic behaviour of the system as a whole. Theymust, therefore, be associated with a further equation that iscompatible with an adequate reference system moving atmolar mean velocity in the mixture and for which theformula iNi0 is valid.

As a whole, it becomes a system of non-lineardifferential equations. Given the initial and the limitconditions, their integration can only be carried outnumerically using extremely powerful computers andemploying the kinetic parameters of the various elementaryreactions and coefficients of molecular transport.

In the particular case of a premixed flame, the results ofthe calculations have been compared to the experimentallyverified trends of the concentrations of reagents along theaxis of a flow reactor. As Fig. 8 illustrates, the results ofcalculations performed for the case of methane are verymuch in agreement with the experimental data evaluatingboth the temperature (Fig. 8 A) and the chemical behaviour(Fig. 8 B), a fact which highlights how our understanding ofthe phenomena associated with the homogeneouscombustion of hydrocarbons has reached a satisfactory level.Analogous confirmations based on experimental data havealso been obtained for other hydrocarbons such as ethane,propane and so on.

Premixed turbulent flamesTurbulence is characterized by the irregular random

motion of the particles of a fluid that occurs over and beyondthe average uniform motion of the gaseous current. Adescription of the phenomenon, advanced by the NobelPrize-winning physicist, Lev Landau, traces its origin to thepresence of broad, stable vortices with anisotropiccharacteristics, that iron themselves out, splinter into piecesand give rise to smaller vortices that are both faster andisotropic; the latter then become smaller and smaller untiltheir energy dissipates into the kinetic energy of molecularmotion. The transition from laminar to turbulent motiontakes place abruptly when the dimensional group known asthe Reynolds number, expressed by the equation:

[59]

where l represents a typical dimension of the system, exceedsa certain critical value, which for a fluid flowing in an emptytubular conduit is in the order of 2,300. Although many ofthe phenomenological aspects of turbulence are known, thereis still no satisfactory general theory on the subject, so thattransport processes can only be described with the aid ofadequate models. In the most commonly used of these, thefield of turbulent velocity is split into a slower componentthat acts on a large scale, and a faster, fluctuating component that acts on a small scale. The effect of the faster component on the field of slow motion creates turbulentviscosity, whose mechanism is similar to that of molecularcollisions in laminar motion. Unfortunately, there is no clearcriterion according to which the two scales in question may be distinguished from each other. To overcome this limitation,Ludwig Prandtl introduced the concept of mixing length,analogous to that of the mean free path of molecules,

Re = ulrµ

∇ =−( )

∑y

y y

DRTPi

i j j i

i jj

N N

,

∂∂+∇ ⋅ = =∑C

tr ri

i i k i kk

N ν,

∇⋅ ∇( ) − ∇ + −( ) = ∂∂∑k T c T H r c TtT p f i k i k

kpG ∆ ν

,r



y i y O2

0.20

0.05

0.10

0.15

0.90

0.75

0.80

0.85

0 2 41 3

0 2 41 3A

B

CH4

H2O

CO2

CO

z (mm)

z (mm)

T (

K)

2,000

1,000

1,500

500

O2

Fig. 8. Comparison between the experimental values (represented by the dots) and the corresponding calculatedcurves of the temperature (A) and chemical composition (B)along the z axis of a flow reactor for a mixture of methane and oxygen that undergoes combustion.

on the basis of which it is assumed that vortices travel atypical distance before becoming integrated in, and tradingtheir specific properties with, the surrounding fluid.

According to a more modern approach, justified by thehypothesis that there exists an invariance of scale inturbulent vortices, the motion of the fluid is described bymeans of the usual conservation equations but employingtransport parameters that reflect the motion’s turbulentcharacteristics. In particular, in a model denominated e-k,viscosity is expressed as a function of two parameters thatrepresent, respectively, the energy’s rate of dissipation andturbulent kinetic energy.

The study of combustion in gaseous mixtures subject toa turbulent flux is very important both scientificallyspeaking and for its applications. Turbulence increases aflame’s rate of propagation and intensifies combustionprocesses. Moreover, it facilitates the exchange of heatreleased by the flame front, and as a result also improves theignition process.

Since a laminar flame presents a flat front thatpropagates at a rate Su, it is important to understand in depththe factors that render it unstable and create the conditionsnecessary for its degeneration towards a turbulent state. Thefirst physico-chemical analysis of the stability of laminarflames was carried out by Landau with the aim ofinvestigating the response of the flat front of the flame tosmall wrinkles of amplitude x that perturb its surface. Themethod applied constitutes a classic example of analysis ofthe stability of dynamic systems, in which the temporalevolution of the wrinkling is expressed by means of anexponential function of the type:

[60]

where f (x,y) is a function of the two coordinates that definethe flame’s plane, whose specific characteristics are of noconcern. An important role is instead played by the sign thatpertains to the parameter g: when it is positive, x increasesexponentially and the flame is unstable; when it is negative,x tends asymptotically towards zero and the flame is stablesince, after having experienced perturbation, it returns to itsoriginal configuration. In Landau’s analysis, onlyfluid-dynamic aspects were taken into account and processesof exchange of heat and matter were ignored. Flames weredescribed in terms of a discontinuity in the density of thegas, which is greater for an unburnt mixture at a lowtemperature and, instead, less for a combusted mixture thathas reached a high temperature and that moves at rate Su. Inthis approach, the g parameter was always positive, which isin blatant contradiction with experimental data thatirrefutably confirm the existence of laminar flames. Thisshows that a mere fluid-dynamic analysis is inadequate to

tackle the problem, because it ignores the effects ofphenomena relating to the transport of matter and heat,which are the real protagonists of combustion processes. Inan approach that is diametrically opposed to the preceding,Zeldovich arbitrarily assumed for the sake of simplicity thatin the passage from unburnt to combusted gasses the densityremained the same, though taking explicitly into accountprocesses of exchange of matter and heat. The model’smathematical developments highlight the fact that a laminarflame will contain zones of stability because its wrinklinglocally increases the rate of reaction and depletes thereagent. A diffusive retroactive effect thus occurs that tendsto give the flame a flat configuration once again.

If the characteristics of unstable solutions are examinedfurther, it can be seen that the system’s evolution gives riseto cellular configurations such as those illustrated in Fig. 9.These configurations were obtained by simulating thebehaviour of unstable flames with the aid of a computerfollowing a model that is more general than the precedingmodel and in which both fluid-dynamic andthermo-diffusion phenomena are taken into account. Theexistence of such cellular structures has been experimentallyconfirmed in the flames of various hydrocarbons. Theirformation represents the beginning of an evolution towardsthe chaotic behaviour that leads to turbulence.

There does not yet exist an easily employablephysical-mathematical approach that can describe thebehaviour of a mixture subject to combustion in a fluid inturbulent motion. The simulation of such phenomenarepresents an real challenge that presumably will be metthanks to the use of ever-more powerful computers, forexample through methods of direct simulation, whoseapplication, however, is currently limited to very simplesituations.

In a simplified semi-empirical approach it is assumedthat the flame front has a wrinkled configuration. It is alsoassumed that the front moves at rate ut, which is greater thanrate ul belonging instead to the corresponding laminar flame.If one indicates with Wf the surface of the wrinkled flamefront and with W that of the laminar flat flame, following

ξ γ= ( )f x y e t,



Fig. 9. Characteristic examples of the structure of cellular flames derived from model-based calculations.

high temperature(2,200 °C)

low temperature(100 °C)

Fig. 10. Geometrical and thermal configuration of a hydrogen-based laminar diffusion flame, simulated by numerically integrating material and thermal balanceequations. The temperature increases from the value it has in the environment (blue) to the maximum value of about2,200°C at the core of the flame (red).

Damkoehler it can be assumed that:

[61]

Therefore, computing ut is the same as computing Wf .Various criteria for evaluating the relation expressed in thepreceding equation have been suggested; the most generalcriteria is related to the description of turbulent motionthrough the concept of mixing length. More recently, it has been suggested that Wf can becalculated by assimilating the turbulent flame front to afractal object whose characteristics, when viewed ondifferent scale, remain unvaried.

Diffusion flamesA typical diffusion flame is generated by a jet of gaseous

fuel injected into a current of air that might also beoxygen-enriched. The peculiar characteristic of these flamesis the fact that the rate at which fuel is consumed isdetermined by the rate at which it is diffused when it isbrought into contact with oxygen. To illustrate the process,Fig. 10 shows a simulated geometric and thermalconfiguration of a hydrogen flame. The configuration of theflame front, where the gaseous current acquires an intenseluminosity, is characterized by a cylindrical symmetry.Injection systems of this type are very common as they areemployed both in the burners of industrial furnaces and inreaction engines. Internal combustion engines of the Dieseltype also operate through a series of transitory injections offuel into air that has previously been compressed in acylinder. Up until a few years ago, the study of diffusionflames had elicited less attention than that of premixedflames, especially because of the greater difficulties that areencountered in carrying out quantitative experimentalinvestigations of these systems. Moreover, their descriptionruns into the additional difficulty of having to establish aparticular rate of propagation, along the same lines as thatwhich characterizes premixed flames.

A simplified approach is to refer to a stationary laminarjet injected into an air current having the configurationshown in Fig. 11, where u stands for the axial component ofthe rate of propagation. It is also assumed that pressureremains uniform and that effects of the forces of gravity andflotation can be ignored. If a theoretical solution to theproblem is attempted, it is best to refer to what is known asthe SCRS (Simple Chemically Reacting System) model, inwhich the fuel and oxygen combine in fixed proportions. It

is also assumed that the specific heat of all the componentsis the same and that at every point of the mixture, which inany case is not uniform, transport properties satisfy thefollowing equation derived from the kinetic theory of gasses:

[62]

The velocities of consumption of the fuel rF and of theoxygen rO, furthermore, are to be bound by the followingequation:

[63]

where gst is the stoichiometric molar relation correspondingto the combustion reaction. In this approach, it can be provedthat the conservation equations [54], when written incylindrical coordinates and referring to a stationary state,will acquire the same form (where v is the radial componentof velocity):

[64]

where f indicates the specific values, as they referre to theunit of mass, of the axial component of the quantity either ofmotion u or of thermal energy cp(TT0), respectively. As faras the two reagents are concerned, a single quantity is used,x, where:

[65]

The limit conditions are defined by the values of thecomposition and of the temperature at entry, along with the:

[66]

If we indicate by Re0u0d0mr the value of the Reynoldsnumber at the jet’s point of entry, the general solution of [64]can be expressed as follows:

[67]

where r0 is the radius of the fuel feeding conduit, z is theaxial coordinate, while rf is the value of the radial coordinatethat locally defines the position of the flame front.

In an approximate formulation it is taken into accountthat the rate of the combustion reaction is very high, so thatit is assumed that the combination of oxygen and fuel takes

ξzr

T TT T rf0 0 0 0

0

2

1 1 0 1875

1 0 0117Re Re Re=

−−

=+

∞

∞

.

. 22 22

z( )

ru

T T→∞

===

∞

0

0ξ

ξ γ= −Y YF st O

∂∂( ) + ∂∂ ( ) = ∂

∂∂∂

z

urfr

rfr

r fr

r rv µ

r rF st= γ O

µr

r= =D

ckp

T

uu

SS

t

l

t

u

f= =ΩΩ



fuel

air

air

r

z

u0

u

0

r0

Fig. 11. Schematic diagram of the currents relative to the inflow of fuel throughthe nozzle and to the lateralinflow of air into a burner.

place instantaneously in those points where the relationbetween the two components coincides with thestoichiometric relation required for the combustion reaction,in other words where yF yOgst. In this approach, the rate atwhich the process takes place is determined entirely by thefuel’s rate of diffusion in the internal part of the flame andthe oxygen’s rate of diffusion in its external part, both in thedirection of the combustion front. The radius of the flame rffor every particular value of z is expressed by the followingequation:

[68]

where:

[69]

(F/O)st being the stoichiometric relation between the mass offuel and of oxygen.

The length of the flame is determined, obviously, byassuming that rf0, so that one obtains:

[70]

which can be approximately expressed as follows:

[71]

where QV is the volumetric capacity of the fuel current andD its diffusion coefficient. The preceding equation can beapplied even if the flame is turbulent, as long as D isreplaced by the value of a coefficient of turbulent diffusionthat reflects the mixing process resulting from thefluctuations of the fluid aggregates and can be determined,for example, by applying the concept of mixing length. Itcan thus be shown that it is approximately equal to u0r0, andso it follows that:

[72]

Essentially, one establishes that the height of a turbulentflame is proportional to the radius of the nozzle, butindependent of the rate of flow and volumetric capacity ofthe fuel. This is a significant result whose validity has beenverified experimentally and which has important applicativeimplications.

The variation of the height of a diffusion flame is thusexpressed by [71] if the flame is laminar and by [72] if it isturbulent.

Formation of carbonaceous particlesIt has already been pointed out in this treatment how

combustion can give rise to the formation of pollutingsubstances such as nitrogen oxides. Another source ofpollution is to be found in the production of particles of coalthat have diameters as small as a few dozen nanometres andshow up in the form of soot. Their formation is facilitated infuel-rich flames which acquire a pronounced luminosityprecisely because of the presence of these particles at hightemperatures. At times the particles will have a high contentof condensed aromatic polynuclear hydrocarbons, which arepotentially carcinogenic.

The formation of the coal particles occurs via a complexprocess that is not yet understood in detail, but which in anycase involves a series of stages that have been identified asfollows: nucleation, growth, oxidation and coagulation.

The least understood of these stages is the first, whichprobably occurs through the formation of small molecules ofunsaturated hydrocarbons such as acetylene, that thencondense to form cyclic molecules. A possible mechanismfor the increase of the rings entails a series of reactions, thefirst of which leads to the formation of an aromatic radicalAr, and then to the reaction of a hydrogen atom with amolecule of the arene ArH:

[73] ArHHArH2

The subsequent reaction of the Ar radical with anacetylene molecule leads to the formation of an unsaturatedproduct which, after further cyclisation, forms apolyaromatic hydrocarbon:

The evolution of the process gives rise to the formationof polyaromatic hydrocarbons of ever-increasing size whichcoagulate into clusters a few nanometres in size that make upthe soot. In an oxidizing atmosphere, the particles may befurther transformed as they tend to react with O2, O andOH.

The kinetics of the particle coagulation process can be described using a second-order function, on thebasis of which the rate of the formation of particles of acertain size is proportional to the product of the particleconcentrations that engender it. Through this growthprocess the particles attain sizes of the order of severaldozen nanometres.

There are experimental data in the literature that refer tothe tendency of various hydrocarbons to form suchparticulate matter. Their correlation has been achieved bydifferentiating premixed flames from diffusion flows anddefining for both a numerical index that characterizes thethreshold of the particulate matter formation, known as TSI(Threshold Soot Index).

7.1.2 Heterogeneous combustion

Evaporation and combustion of liquid dropsVarious devices and plants of applicative importance,

such as industrial kilns, furnaces and Diesel engines, aresupplied with liquid hydrocarbon fuels introduced in theform of a shower of droplets. In such cases the combustionprocess can be subdivided into two stages, the evaporation ofthe liquid and its ensuing combustion in the gaseous phase.It is clear, therefore, that the size of the droplets plays a basicrole in the rate of the global process, as the rate of theevaporation process is proportional to the surface area of thedrop (in particular, therefore, to the square of its diameter ifthe drop is assumed to be round).

Commonly the liquid is dispersed by means of anatomizer, or spray, in which a rapid formation of drops takes

C CHC2H2

C

CHC

zr ur u

rft∝ =0

2

0

0 0

0

zQD

r uDf

V∝ ≈π

0

2

0

zz ff

st0

3

16= Re

fF OF Ost

st

st

=( )+( )1

rz

rzf

f

st

=

−

9 237

0 1875 10

0 0

1 2

..

/

ReRe

1 2/



place, and equally rapidly they become remixed with thesurrounding air. There is a wide variety of vaporizers whoseoperation has been studied in relation to the size of the holesand the flow of liquid supplied. However, it is not easy toforesee the characteristics of the dispersion thus obtained,above all due to the processes of disintegration of the dropsand their coagulation. So, these assessments are mostlyconducted exploiting empirical relations and correlating thedata by means of three parameters or dimensionless groups,which are usually shown by the letter j, meaning that a jet ofliquids is referred to:

[74] Reynolds number

[75] Weber number

[76] Ohnesorge number

where dj is the diameter of the drops before redistributionamong the various diameters has taken place, s is the surfacetension, u is the velocity of the gas, while the indices g and lassociated with the density (r) and viscosity (m) valuesindicate the gaseous and the liquid phase, respectively.

On the basis of the values assumed by the aforesaiddimensionless groups, it is possible to identify various types

of dispersions, as illustrated in Fig. 12. In particular, threezones are identified which may be characterized as follows:zone I, or the Rayleigh zone, in which the droplets are highlyunstable and tend to disintegrate; zone II, in which helicoidalwaves are observed before the disintegration of the dropletsstarts; and zone III, in which disintegration into dropletstakes place close to the orifices, due essentially to theturbulent motion of the fluid.

The distribution of the dimensions thus obtained ismainly described by means of empirical relations, including:

[77]

in which CVFi is the cumulative volume of all the drops thatare smaller in diameter than di, d0 is a reference diametercorresponding to a value of CVFi equal to 0.632 and q is anempirical parameter.

Calculation of the velocity of evaporation of a dispersionof liquid drops is mostly conducted following the tendenciesof each of them. The simplest case is that of a spherical dropsurrounded by a hot gaseous current. An important factor isthe transfer of heat in the liquid; to evaluate this transfer it isnecessary to analyze the motions that take place within thedrops. Here it will be assumed that these effects are rapidenough to be able to attribute a constant mean temperatureof Tl to the drop. If ml is the mass of the drop and cpl is itsspecific heat, the energy balance may be put as:

[78]

where T is the temperature of the gaseous mass, L is the heatof evaporation of the liquid, a is the surface area of thesphere and h is the coefficient of heat transport from thesurface of the drop to the gaseous mass. For a sphere havinga diameter of d it results, neglecting thermal expansion forthe sake of simplicity, that the term of variation in time ofthe mass of the liquid drop may be expressed as follows:

[79]

If evaporation takes place in stationary conditions, theheat of evaporation absorbed by the evaporating drop and theheat transferred to the drop from the outside are equal, sothat:

[80]

This shows that, with reasonable approximations, wemay put:

ha T T dmdt

Lll−( ) = −

dmdt

d tdt

d tdt

l l l= ( ) = ( )r rπ πδ δ δ δ2 2

2 4

m c dTdt

ha T T dmdt

Ll pll

ll= −( )+

CVFi i

q

= − −

1 10

exp δδ

Oh l

l j

= µσδr

Weu

jg j=

r 2δσ

Rejl j

l

u=

r δµ



III

III

1 102 10510 103 104

Reynolds number

Ohn

esor

ge n

umbe

r10

0.1

1

0.01

0.001

Fig. 12. Atomization zones of the droplets coming from a nozzle calculated as a function of the Ohnesorge and Reynolds numbers. I, unstable droplets; II, helicoidal waves are present; III, disintegration into droplets takes place close to the orifices of the nozzle.

fuel

air

air

z0

Fig. 13. Schematic diagram of a cylindrical chamberwhere the combustion of liquid droplets takes place.

[81]

where:

[82]

Tm(T+Tl) / 2 being a mean temperature and D the diffusioncoefficient of the vapour in the gas.

Making [79] equal to [81] it is seen that:

[83]

which, integrated, supplies the following expression of timeof evaporation:

[84]

The description of the combustion process that takesplace in a generic burner supplied by a dispersion ofdrops of a fuel obviously depends on the fluid-dynamicconditions that develop in the combustion chamber.These aspects fall outside of the present treatment. Forthe sake of simplicity and for purposes of illustration,the case considered here will be that of an atomizer thatproduces a shower of monodispersed drops, i.e. with auniform diameter, propagated with a piston-type flow ina current of air along a cylindrical chamber, as illustratedin Fig. 13.

If n0 indicates the number of drops of diameter d0 perunit of volume at point z0 of the axial coordinate, it isdemonstrated that the fuel/air ratio ( f ) may be expressed as:

[85]

and therefore:

[86]

If it is assumed that the number of drops remainsunchanged, although drops are subject to a process ofevaporation that diminishes their diameter, and that thedensity of the gas remains constant, it is reasonable to applythe following expression:

[87]

Integratring [83], we obtain the following expression,which gives us the diameter of the particles as a function oftime:

[88]

The heat balance of the gaseous current can then beformulated by making the heat liberated by combustion fromthe evaporated gas equal to that absorbed by the gaseouscurrent:

[89]

where LHV (Lower Heating Value) indicates the combustionheat per unit of mass of liquid fuel. Taking the precedingrelations into account, in particular [81] and [87], thefollowing is therefore obtained:

[90]

If this equation is integrated, remembering [88] weobtain lastly the law that expresses the course of the gastemperature as a function of the diameter of the drops:

[91]

T0 being the gas temperature corresponding to z0.

Combustion of solidsThe combustion of solid particles will now be

considered, paying particular attention to coal and wood,even if the analysis conducted is of general validity and maybe applied to any other combustible material. If a particle ofcoal is exposed to a hot gaseous current containing oxygen,for example air, it undergoes a process in which thefollowing three stages can be identified: loss of water bydrying; loss of mass by the release of volatile gases; and truecombustion.

Concerning the first stage, it should be remembered thatthe water present in the particle can be considered assubdivided into two forms: the free water contained in thepores of the solid and the water absorbed on the external andinternal surfaces of the particle. The heat balance of this firststage can be formulated as:

[92]

where the subscripts w and df (dry fuel) indicate the waterand the dry solid, respectively; q indicates the heat that istransferred to the solid per unit of time and of mass. Thevalue of q depends on the characteristics of the heatingsystem; if the process is carried out in a furnace, most of theheat is exchanged by radiation; q is in any case given by thesum of two terms, as in addition to radiation there is also acontribution due to convection. Bearing these aspects inmind, by integrating the preceding equation it is possible toassess the time required to dry the fuel.

The second stage is associated with a pyrolyticprocess through which volatile products are liberated andburn when they come into contact with the oxygen of theair. The velocity of the process depends naturally on thetemperature, but it is also very sensitive to the nature andthe characteristics of the fuel. For example, in lignite theprocess starts at 300-400°C, freeing carbon monoxide anddioxide, and vapours of light hydrocarbons and hydrogen.The velocity of the process, at a first approximation, isdescribed by means of a first-order kinetic equation with

ddt

m c m c T dmdt

L qw pw df pdf+( )

= +

T T ffLHVcp

− =+

−

00

3

11 δ

δ

dTdt

ff

LHVcp

= −+6

1 03

βδ

δ

rc dTdt

n dmdt

LHVpl=

δ δ β= −02 4 t

n n= 0

0

r

r

n ff l

00

031

6=+

r

r πδ

fnn

o l

l

=−

r

r r

ππδδ

03

0 0 03

66

//

tev =δβ02

4

d tdtδ β

2

4( ) = −

β ∼ 2D Tm( )

dmdt

C Mli i= −π βδ



Table 5. Kinetic parameters for some solid fuels

A0,pyr

s1

Epyr

kJ/mol

Lignite 280 47.3

Bituminous carbon 700 49.4

Wood 7107 129.7

constant kpyr, which when integrated supplies thefollowing expression:

[93]

in which the mass of the volatile substance is expressed asthe difference between the mass of the particle (mp, initialvalue mpi), the mass of the carbonaceous fuel (mc, fromchar) and the mass of the residual ash (ma, from ash). Thedependence of the constant of the velocity of reaction on thetemperature is expressed by means of the Arrhenius law; thetypical values of the Arrhenius parameters for a number ofsolid fuels are set out in Table 5.

We now consider the combustion of impoverished drycoal, i.e. without its less stable substances. The reactiontakes place through the interaction of the oxygen moleculescoming from the gaseous phase with the surface carbonatoms, and leads to the formation of carbon monoxide anddioxide through the reactions:

[94] C 1/2O2 CO

[95] C O2 CO2

Moreover, if water is present the following reaction takesplace:

[96] C H2O CO H2

The most important of these reactions is [94], whichconditions the global process. At a first, reasonableapproximation its velocity is expressed by means of a first-order kinetic equation, in which the values of the Arrheniusparameters for a number of types of coal are set out in Table 6. Allowing for the dimensions of the constant of thevelocity reaction (k), the balance of a coal particle, assumedto be round with a diameter d, can be put:

[97]

in which apd2 is the surface of the sphere p(s)O2

, is thepartial pressure of the oxygen corresponding to the solidsurface and MC /MO is the relation between the atomicweights of carbon and of oxygen (12/16).

In stationary conditions the quantity of oxygenconsumed through the effect of the chemical reaction equalsthe quantity of oxygen associated with the diffusive flowfrom the heart of the gaseous phase to the surface of theparticle, so that:

[98]

in which kc is the coefficient of transfer of material, referredto the concentrations. If this is divided by the factor RT, it ispossible to express the concentration of oxygen by means ofthe partial pressures. The theory of the transport processessupplies the following correlation through which the value ofkc can be obtained:

[99]

in which Sh is the Sherwood number, D is the coefficient ofdiffusion of oxygen in the air, Rep is the Reynolds numberevaluated on the basis of the size of the particles and Sc isequal to m/rD; /, lastly, is a factor between 0.6 and 1 whichallows for the influence of the gases produced bycombustion.

If reference is made to situations corresponding to lowvalues of the Reynolds number, in which / may be attributeda unitary value, it follows that kc2D/d and resolving [98]with respect to p(s)

O2we obtain:

[100]

Substituting [100] in [97] and remembering that apd2

we obtain:

[101]

To integrate this equation and to obtain the time requiredto burn the particle it is necessary to find a link between themass of the particle and its diameter. The simplest case isone in which it is assumed that the combustion processoccurs at the surface of the particle, leaving the density ofthe solid constant. In this case, for any value of the diameterthe relationmcpd3rc 6 applies, so that d(6mc prc)

13.If the operating conditions are those in which the

velocity of the global process is limited by the velocity ofthe surface reaction, so that 2DdRTk MO, we obtain:

[102]

from which the following expression of the combustion timeis obtained:

[103]

Vice versa, if the velocity of the global process is limitedby the diffusion, we obtain:

[104]

from which the following expression of the combustion timeis obtained:

[105]

Bibliography

Barnard J.A., Bradley J.N. (1984) Flame and combustion, London,Chapman & Hall.

Borman G.L., Ragland K.W. (1998) Combustion engineering, Boston(MA), McGraw-Hill.

tDp

RTM

c

O c

= 0 785 02

. rδ

2

dmdt

m MDpRT

c c

cc

O= −

2 61 3

ππr

/

2

tkpc

O2

= rδ0

1 5,

dmdt

m kpc c

cO2

= −

π

π6 12

16

2 3

r

/

dmdt

k MM

D RT pk M D RT

c C

O

O

O

= −( )

( )+ ( )πδδ

δ2 2

2/

/ /2

pD RT p

k M D RTOs O

O2

2( ) // /

=( )

( )+ ( )2

2δ

δ

ShkD

Sccp= = +( )δ

2 0 6 1 2 1 3. / /Re ϕ

kM

pkRT

p pO

Os c

O Os

2 2 2

( ) ( )= −( )

dmdt

ka MM

pc C

OOs2

= − ( )

lnm m mm m m

k tp c a

pi c apyr

− −− −

= −



Table 6. Kinetic parameters for some carbons

A0gO2

(cm2satm O2)–1

EkJ/mol

Anthracite 20.4 79.5

Volatile bituminous 66 85.2

Highly volatile bituminous 60 71.8

Subbituminous 145 83.6

Glassman I. (1996) Combustion, San Diego (CA), Academic Press.

Jones J.C. (1993) Combustion science: principles and practice,Newtown, Millennium.

Kanury M.A. (1985) Introduction to combustion phenomena, NewYork, Gordon & Breach.

Kuo K.K. (1986) Principles of combustion, New York, John Wiley.

Lewis B., Elbe G. von (1987) Combustion flames and explosions ofglass, New York, Academic Press.

Williams F.A. (1985) Combustion theory: the fundamental theory ofchemically reacting flow systems, Reading (MA), Persuy.

List of symbols

Boldface type indicates a vector, while ∇ stands for thedivergence operator. A line drawn above a generic quantity

3

gindicates its average value

A pre-exponential factor of the reaction-rate constanta adimensional heat-exchange coefficientCi molar concentration of the component iC (s) molar concentration at surfacecp specific heatcp molar heat capacityDi, j binary diffusion coefficient between components i

and jDi,m diffusion coefficient of the component i in a

gaseous mixtureE activation energyG mass flow per surface unitGi mass flow of component ih heat-exchange coefficient (calories per unit of time

and surface)k reaction-rate constantkc coefficient of transfer of materialkT thermal conductivityMi molecular weight of component iNi molar flow of component i per surface unitpi partial pressure of component i

QR heat of reactionQ heat released or absorbed per unit of timeR gas constantr rate of reaction (moles transformed per units of

volume and time)Su flame propagation ratet timeT thermodynamic temperatureu axial component of a gas rate of flowu* molar mean velocityv radial component of a gas rate of flowV volumeyi molar fraction of component iYi mass fraction of component iz axial coordinate

Greek lettersDf Hi molar enthalpy of formationDH° variation in standard enthalpy associated with a

reactionDrG° variation in standard free energy associated with a

reactionFF flow per unit of surface of a generic sizem viscosityd drop or particle diameterd0 initial valuevi,k stoichiometric coefficient of component i in kth

reactionr density of a fluidr molar densityÿ adimensional temperaturet adimensional time

Sergio Carrà

Dipartimento di Chimica, Materiali e Ingegneria chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy



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