Kinetically Controlled Kinetically Controlled Combustion PhenomenaCombustion Phenomena

Categorisation of Categorisation of Combustion PhenomenaCombustion Phenomena

Most of the practical combustion phenomena Most of the practical combustion phenomena belong to one of the following three belong to one of the following three

categories:categories:i.i. Phenomena which are primarily controlled Phenomena which are primarily controlled

by by chemical kineticschemical kineticsii.ii. Phenomena which are primarily controlled Phenomena which are primarily controlled

by diffusion, convection and other by diffusion, convection and other physical physical mixing processesmixing processes

iii.iii. Phenomena in which the roles played by Phenomena in which the roles played by chemical kineticschemical kinetics and and physical mixingphysical mixing are are more or less of equal importance.more or less of equal importance.

• Ignition, explosion, extinction and Ignition, explosion, extinction and quenchingquenching of flames serve as examples of of flames serve as examples of kinetically controlled phenomena kinetically controlled phenomena

• The burning of a gaseous fuel jet, of a The burning of a gaseous fuel jet, of a liquid fuel spill, spray, or drop, of a carbon liquid fuel spill, spray, or drop, of a carbon sphere and of a candle in which fuel and sphere and of a candle in which fuel and oxidant are contacted by oxidant are contacted by diffusiondiffusion illustrate the illustrate the diffusionally controlled diffusionally controlled combustion phenomenacombustion phenomena. .

• Flames in a gasoline engine, a Bunsen Flames in a gasoline engine, a Bunsen burner and other situations in which the burner and other situations in which the fuel and oxidant are fuel and oxidant are premixedpremixed belong to belong to the third category.the third category.

• Let A, B and C denote respectively the fuel, Let A, B and C denote respectively the fuel, oxidant and product of combustion which oxidant and product of combustion which are all in gas phase and are are all in gas phase and are uniformly uniformly distributeddistributed in a combustion chamber in a combustion chamber

• Such a "uniform" distribution may be Such a "uniform" distribution may be accomplished by increasing the molecular accomplished by increasing the molecular mixing or turbulent intensity, say, by mixing or turbulent intensity, say, by mechanical stirringmechanical stirring

• Kinetically controlled phenomenaKinetically controlled phenomena are those are those in which the reaction rate is slow compared in which the reaction rate is slow compared to the rates of heat and species diffusion to the rates of heat and species diffusion so that the species and temperature have so that the species and temperature have adequate time available to smooth out any adequate time available to smooth out any spatial non-uniformities.spatial non-uniformities.

• When When reactions are very fastreactions are very fast, the spatial , the spatial nonuniformities of composition and nonuniformities of composition and temperature fail to be washed out in the short temperature fail to be washed out in the short available time. As a result, gradients of species available time. As a result, gradients of species and temperature are established in space. and temperature are established in space.

• Such gradients cause conduction of heat and Such gradients cause conduction of heat and diffusion of species towards the regions of diffusion of species towards the regions of lower temperatures and concentrations lower temperatures and concentrations respectively. respectively.

• The The reactants diffuse into the flamereactants diffuse into the flame zone zone whereas the whereas the products andproducts and heat diffuse away heat diffuse away from the flamefrom the flame zone. Such a poorly mixed zone. Such a poorly mixed combustion is said to be combustion is said to be diffusionally controlleddiffusionally controlled. . Explicitly, diffusion controlled phenomena are Explicitly, diffusion controlled phenomena are those in which the rate of reaction is much those in which the rate of reaction is much faster than the rate of diffusion.faster than the rate of diffusion.

• In a given combustion phenomenon, the fuel In a given combustion phenomenon, the fuel and oxidant have to be supplied (by flow, and oxidant have to be supplied (by flow, diffusion and mixing) to a station where they diffusion and mixing) to a station where they react chemically. Heat and products have then react chemically. Heat and products have then to be removed from this station physically.to be removed from this station physically.

• There are two characteristic rates involved in There are two characteristic rates involved in this problem, this problem, the rate of supplythe rate of supply and and the rate of the rate of consumptionconsumption. The lowest of these two rates . The lowest of these two rates governs the overall speed of the process. governs the overall speed of the process.

• In In kinetically controlled combustionkinetically controlled combustion phenomena, the rate of consumption of the phenomena, the rate of consumption of the fuel and oxidant by chemical reaction is much fuel and oxidant by chemical reaction is much smaller than the rate of supply by flow, smaller than the rate of supply by flow, diffusion or mixing. diffusion or mixing.

• In In diffusion controlled combustiondiffusion controlled combustion phenomena, phenomena, the rate of flow, diffusion and mixing is much the rate of flow, diffusion and mixing is much smaller than the chemical reaction rate.smaller than the chemical reaction rate.

• In kinetically controlled phenomena, the In kinetically controlled phenomena, the flame flame occurs more or less occurs more or less uniformly in theuniformly in the entireentire reaction space reaction space

• In diffusionally controlled phenomena, In diffusionally controlled phenomena, flame is locatedflame is located at some distinct station in at some distinct station in the space. the space.

• This point of distinction is schematically This point of distinction is schematically represented in Figure 4. 1.represented in Figure 4. 1.

• Let the reacting gas mixture of the species A Let the reacting gas mixture of the species A (fuel) and B (oxygen) be confined by a wall in (fuel) and B (oxygen) be confined by a wall in such a way that the characteristic thickness such a way that the characteristic thickness of the body of gas is of the body of gas is ..

• Let YLet YAWAW be the mass fraction of A at the wall be the mass fraction of A at the wall and Y be some characteristic mean mass and Y be some characteristic mean mass fraction of A in the reacting body of gas. fraction of A in the reacting body of gas.

• The amount of species A transferred from the The amount of species A transferred from the wall to the gaswall to the gas phase can be written in terms phase can be written in terms of a mass transfer coefficient hof a mass transfer coefficient hDD (which is (which is proportional to proportional to ρρD (density x diffusion coeff) D (density x diffusion coeff) asas

ŴŴAA""= h= hDD(Y(YAWAW ‑ Y ‑ YAA)) gm/cmgm/cm22 /sec /sec• The amount of A consumed in the gas phase The amount of A consumed in the gas phase

reactionreaction is given by a simple rate law which is given by a simple rate law which we assume as of order unity and of Arrhenius we assume as of order unity and of Arrhenius typetype

ŴŴAA" " = k= k11 Y YAA e e-E/(RT)-E/(RT) gm/cm gm/cm22 /sec /sec

• Solving for YSolving for YAA,,

• The ratio in the denominator is known as The ratio in the denominator is known as Damkohler number, DaDamkohler number, Da..

Case (a): Kinetically controlled regime:Case (a): Kinetically controlled regime:• Da Da → → 00, then , then hhDD >> k >> k11 e e-E/RT-E/RT and and YYAA YYAWAW

• The composition remains nearly uniform The composition remains nearly uniform throughout the reaction space. throughout the reaction space.

• The rate of depletion of the fuel, then, is given by The rate of depletion of the fuel, then, is given by the reaction rate the reaction rate ŴŴAA

" " k k11 Y YAWAW e e-E/(RT)-E/(RT) (4.4)(4.4)• The regime arises when the mixing is high, the The regime arises when the mixing is high, the

diffusion coefficient is high, the gas body diffusion coefficient is high, the gas body thickness is small, the pre‑exponential collision thickness is small, the pre‑exponential collision factor is small, the activation energy is large and factor is small, the activation energy is large and the gas body temperature is low. the gas body temperature is low.

11 4 2

E/ RT

A AWD

k eY Y / ( . )

h

Case (b): Diffusionally (or Diffusion or Flow or Physically)Case (b): Diffusionally (or Diffusion or Flow or Physically)Controlled RegimeControlled Regime::• Da Da , , hD <<hD << kk11 e e-E/RT-E/RT

• The physical rate (of supply flow, mixing and diffusion) The physical rate (of supply flow, mixing and diffusion) is much smaller than the chemical rate (of the is much smaller than the chemical rate (of the reaction)reaction)

• Equation 4.2 then givesEquation 4.2 then gives

i.e. the gas body mass fraction of species A is i.e. the gas body mass fraction of species A is negligible compared with the mass fraction at the wall. negligible compared with the mass fraction at the wall.

• The rate of fuel depletion is given by the mass transfer The rate of fuel depletion is given by the mass transfer rate asrate asŴŴAA

" " h hDD Y YAWAW (4.5)(4.5)• These phenomena arise when the mixing is poor, the These phenomena arise when the mixing is poor, the

flow and diffusion are slow, the gas body thickness is flow and diffusion are slow, the gas body thickness is large and the chemical reaction is fastlarge and the chemical reaction is fast

1

D AW

A AWE/ RT

h YY Y

k e

• Diffusion flamesDiffusion flames can be can be converted into kinetic converted into kinetic flamesflames by either increasing the chemical time by either increasing the chemical time or decreasing the physical time. or decreasing the physical time.

• For example, if we blow out a match or a For example, if we blow out a match or a candle, or send a burning droplet into a fast candle, or send a burning droplet into a fast moving air stream, the flame is extinguished moving air stream, the flame is extinguished due to the chemical reaction rate << the due to the chemical reaction rate << the increased rate of diffusion of the fuel and increased rate of diffusion of the fuel and oxygen into the reaction zone (oxygen into the reaction zone (fuel is fuel is diluteddiluted). ). Kinetically controlledKinetically controlled

• Similarly, if we increase the flow velocity of Similarly, if we increase the flow velocity of fuel in any diffusion flame, a point is reached fuel in any diffusion flame, a point is reached where the flame lifts away from the tube where the flame lifts away from the tube through which the fuel issues (through which the fuel issues (heat is heat is “swallowed”)“swallowed”)

IgnitionIgnition

• Most of the energy released in a combustion Most of the energy released in a combustion reaction is reaction is in thermal formin thermal form while a fraction while a fraction is released is released in the form of lightin the form of light. .

• Emission of light is either due to Emission of light is either due to incandescent solid particles such as carbon incandescent solid particles such as carbon in the flames (hot light = in the flames (hot light = incandescenceincandescence) or ) or due to some unstable (excited) due to some unstable (excited) intermediate species (colt light = intermediate species (colt light = chemiluminencechemiluminence). ).

• Of the heat generated, part is lost from the Of the heat generated, part is lost from the reacting mixture and part is retained by it.reacting mixture and part is retained by it.

(a) (a) Thermal IgnitionThermal Ignition

• Under certain conditions of heating brought Under certain conditions of heating brought about by an external source of energy such about by an external source of energy such as a spark, hot vessel walls, compression, as a spark, hot vessel walls, compression, etc., there is always some temperature of etc., there is always some temperature of the reacting mixture at which the reacting mixture at which the rate of the rate of heat generation exceeds the loss rateheat generation exceeds the loss rate. .

• The excess heat increases the mixture The excess heat increases the mixture temperature which in turn leads to higher temperature which in turn leads to higher reaction rate. The mixture temperature reaction rate. The mixture temperature rises continuously and acceleratively until a rises continuously and acceleratively until a high heat evolution rate is attained. high heat evolution rate is attained. IgnitionIgnition is then said to have occurred.is then said to have occurred.

• In reality, the accelerative rise of In reality, the accelerative rise of temperature is quite abrupt; the temperature is quite abrupt; the previously invisible slow reaction suddenly previously invisible slow reaction suddenly becomes visible and measurable.becomes visible and measurable.

• An uncontrollably fast reaction is known as An uncontrollably fast reaction is known as an an explosion.explosion. Closed vessel explosions are Closed vessel explosions are very common in practice. very common in practice.

• At ignition, any combustion reaction At ignition, any combustion reaction seems as though it were an explosion. For seems as though it were an explosion. For this reason, superficially, the terms this reason, superficially, the terms ""explosionexplosion" and "" and "ignitionignition" are used " are used synonymsynonymously in the combustion ously in the combustion literature.literature.

(b) (b) Chemical Chain IgnitionChemical Chain Ignition• If the combustion reaction involves If the combustion reaction involves

intermediate chain carriersintermediate chain carriers, ignition is , ignition is possible even under isothermal conditions. possible even under isothermal conditions. If the rate of chain carrier generation If the rate of chain carrier generation exceeds the rate of their terminationexceeds the rate of their termination, the , the reaction becomes progressively fast and reaction becomes progressively fast and subsequently leads to ignition. subsequently leads to ignition.

• The chain initiation itself may require an The chain initiation itself may require an external source of thermal or photon external source of thermal or photon energy. Once the chain is initiated, the energy. Once the chain is initiated, the external source may be removed and external source may be removed and ignition may be expected if the above ignition may be expected if the above criterion of criterion of positive chain carrier balancepositive chain carrier balance is is fulfilled.fulfilled.

• Determination of the conditions under which Determination of the conditions under which a given combustible mixture ignites, is an a given combustible mixture ignites, is an important topic in the design of combustion important topic in the design of combustion engines as well as in fire prevention.engines as well as in fire prevention.

(c) Scope of the Present Chapter(c) Scope of the Present Chapter

• A major part of the rest of this chapter A major part of the rest of this chapter deals with ignition and extinction from deals with ignition and extinction from a a thermal viewpointthermal viewpoint. .

• The concepts of The concepts of ignition delay, ignition delay, flammability limits, and minimum ignition flammability limits, and minimum ignition energyenergy are presented through this thermal are presented through this thermal theory. theory.

(d) (d) Two Types of IgnitionTwo Types of Ignition• Experience shows that there are two general Experience shows that there are two general

modes of ignition ‑ modes of ignition ‑ spontaneousspontaneous and and forcedforced..Spontaneous ignitionSpontaneous ignition• Spontaneous ignitionSpontaneous ignition is sometimes called as is sometimes called as

auto‑ignition or self‑ignition. Spontaneous auto‑ignition or self‑ignition. Spontaneous ignition occurs as a result of raising the ignition occurs as a result of raising the temperature of a considerable volume of a temperature of a considerable volume of a combustible gas mixture by containing it in combustible gas mixture by containing it in hot boundarieshot boundaries or by subjecting it to or by subjecting it to adiabatic compressionadiabatic compression. .

• Because the Because the heat generation rateheat generation rate is a is a strong strong exponential function of temperatureexponential function of temperature whereas whereas the the heat loss rate is a simple linear functionheat loss rate is a simple linear function, , even a slight increase in the temperature of even a slight increase in the temperature of the reacting mixture would greatly increase the reacting mixture would greatly increase the rate of its temperature rise. the rate of its temperature rise.

• As a consequence, once As a consequence, once the generation rate the generation rate exceeds the loss rateexceeds the loss rate, ignition occurs in the , ignition occurs in the whole volume almost instantaneously. whole volume almost instantaneously.

• The reaction then The reaction then proceeds by itselfproceeds by itself without without any further external heating.any further external heating.

Forced ignitionForced ignition• Forced ignitionForced ignition occurs as a result of occurs as a result of local local

energy addition from an energy addition from an external sourceexternal source such such as an electrically heated wire, an electric as an electrically heated wire, an electric spark, an incandescent particle, a pilot flame, spark, an incandescent particle, a pilot flame, etc. etc.

• A flame is A flame is initiated locallyinitiated locally near the ignition near the ignition source and it source and it propagates into the restpropagates into the rest of the of the mixture. mixture.

• There are many instances in which a fuel and There are many instances in which a fuel and an oxidant are rapidly an oxidant are rapidly mixed at a high mixed at a high temperaturetemperature which can result in a which can result in a spontaneous ignitionspontaneous ignition. .

• For example, a spray of diesel fuel into the hot For example, a spray of diesel fuel into the hot compressed air is in part vaporized and mixed compressed air is in part vaporized and mixed with the air in a very short time. Following a with the air in a very short time. Following a definite delay, the reaction would proceed definite delay, the reaction would proceed rapidly enough to be considered a flame. rapidly enough to be considered a flame.

• There are technically important instances of There are technically important instances of spontaneous ignitionspontaneous ignition where it is where it is not vvantednot vvanted, , i.e. i.e. fire fire such as that occuring on oil splashed such as that occuring on oil splashed on on hot surfaceshot surfaces and the and the knockknock in a gasoline in a gasoline engine.engine.

SpontaneousSpontaneous IgnitionIgnition

Spontaneous Ignition DelaySpontaneous Ignition Delay

(a) The Criterion(a) The Criterion

Consider a vessel of Consider a vessel of volume Vvolume V and and surface surface SS containing a combustible mixture. Let T containing a combustible mixture. Let T00 be the initial temperature of the mixture. be the initial temperature of the mixture. Assume that the temperature at any later Assume that the temperature at any later time in the mixture is spatially uniform. time in the mixture is spatially uniform. Let Let the vessel walls be kept at Tthe vessel walls be kept at T00 for all for all times. times.

• Eq. (4.7) applies to Eq. (4.7) applies to adiabaticadiabatic combustion combustion systemsystem

• Eq. (4.8) takes into account the Eq. (4.8) takes into account the heat heat transfertransfer from flame to the wall from flame to the wall

• First term represents First term represents heat generationheat generation by by reactionsreactions

• Second term represents Second term represents accumulation of accumulation of heatheat in the vessel in the vessel

• Third term represents Third term represents heat transferheat transfer

0 4 8 dTodt

Vq CV hS T T ( . )

dTVq CV 0 (4.7)

dt

• If the heat transfer coefficient is constant, If the heat transfer coefficient is constant, at at a very low pressurea very low pressure the reaction rate will be the reaction rate will be small because small because

the system then practically remains at the system then practically remains at T = TT = Too

• At a very high pressureAt a very high pressure, the heat generation , the heat generation overwhelmsoverwhelms the loss term (i.e. the system the loss term (i.e. the system approaches adiabatic conditions). The approaches adiabatic conditions). The temperature and the reaction rate, then, temperature and the reaction rate, then, enhance one another until enhance one another until spontaneous spontaneous ignition occursignition occurs. .

• Therefore, it is reasonable to expect that Therefore, it is reasonable to expect that there exists there exists a critical pressurea critical pressure below whichbelow which the reaction behaves more like an the reaction behaves more like an isothermal (isothermal (non-explosivenon-explosive) process and ) process and above whichabove which it behaves like a it behaves like a spontaneously spontaneously explodingexploding (abrupt temperature) adiabatic (abrupt temperature) adiabatic process. process.

n nA Aq W C P

(b) Ignition delay(b) Ignition delay• The criterion of positive heat balance is The criterion of positive heat balance is

required for spontaneous ignitionrequired for spontaneous ignition

• When the When the limiting case of adiabaticity islimiting case of adiabaticity is approachedapproached, Eq. 4.8 with 3, Eq. 4.8 with 3rdrd term = 0 can term = 0 can be used to deduce the concept of be used to deduce the concept of ignition ignition delaydelay (sometimes called ignition lag, (sometimes called ignition lag, induction time, or ignition time).induction time, or ignition time).

• The strong influence of temperature on a The strong influence of temperature on a simple thermal reaction ratesimple thermal reaction rate is often is often expressed by the expressed by the conventional Arrhenius conventional Arrhenius exponentialexponential or by or by a simple power lawa simple power law..

A 0HVW hS T T (4.14)

• The power m is of the order 20 to 30 for The power m is of the order 20 to 30 for most combustible mixtures whereas the most combustible mixtures whereas the activation energy E is of the order 20 to 60 activation energy E is of the order 20 to 60 kilocalories per mole. kilocalories per mole.

• Incorporating Incorporating Eq. 4.15 into Eq. 4.7Eq. 4.15 into Eq. 4.7 and and integrating, the time history of integrating, the time history of temperature is obtained astemperature is obtained as

n mA n AW k C T (4.15)

n E/ RTA n AW k C e (4.15a)

1

m 1

o0

n mn A0 0

T t1 (4.16)

T C T

m 1 Hk C T

• Equation 4.16 indicates that as the timeEquation 4.16 indicates that as the time

the the temperature T of the reacting temperature T of the reacting mixture rises very steeplymixture rises very steeply. .

• The critical timeThe critical time

is called is called ignition delayignition delay

0

n mn A0 0

C Tt

m 1 Hk C T

0i n m

n A0 0

C Tt (4.17)

m 1 Hk C T

• Eq. 4.17 shows that the ignition Eq. 4.17 shows that the ignition delay is short if the mixture has a delay is short if the mixture has a low low volumetric heat capacityvolumetric heat capacity, high , high temperature dependence of the rate, temperature dependence of the rate, high heat of combustion and high high heat of combustion and high initial reaction rate.initial reaction rate.

• Incorporating Incorporating Eq. 4.15a into Eq. 4.7Eq. 4.15a into Eq. 4.7 and and integrating, the time history of integrating, the time history of temperature is obtained as with the temperature is obtained as with the assumption of negligible reactant assumption of negligible reactant consumption during the ignition delay, the consumption during the ignition delay, the time history of temperature istime history of temperature is

wherewhere

2

0 00 i

T texp E/ RT T / T 1 1 (4.16a)

T t

02 E/ RT0

i nn A0

R T et C (4.17a)

E Hk C

• Figure 4.5Figure 4.5

illustrates illustrates

mixture mixture

temperaturetemperature

at fraction of at fraction of

delay timedelay time

for variousfor various

values of E/RTvalues of E/RT00

Semenov Theory of Spontaneous IgnitionSemenov Theory of Spontaneous Ignition

• Eq. 4.8 can be rewritten in the following eq.Eq. 4.8 can be rewritten in the following eq.

• Keeping the pressure fixed, (T)Keeping the pressure fixed, (T) will be a will be a steeply rising function as shown in Figure 4.6(a)steeply rising function as shown in Figure 4.6(a)

• (T) is(T) is a linear function of Ta linear function of T with a slope of hS with a slope of hS cal/(sec K). Keeping hS fixed, three different cal/(sec K). Keeping hS fixed, three different (T) functions are also shown in Figure 4.6(a) (T) functions are also shown in Figure 4.6(a) corresponding to three values of the corresponding to three values of the wall wall temperature Ttemperature T00. . T is mixture temperatureT is mixture temperature..

• The right hand side ofThe right hand side of Eq. 4.8 as a function of T Eq. 4.8 as a function of T is shown in Figure 4.6(b) for the is shown in Figure 4.6(b) for the three values of three values of TT00

4 8 dTg ld t

CV q q ( . )

gq

lq

lq

Case: Wall temperature relatively lowCase: Wall temperature relatively low

• When When TT00 = T = T0303, the curve and line , the curve and line intersect at two points, a and b; at intersect at two points, a and b; at a a and b and b thus dT/dt =thus dT/dt = zero. zero.

• If the starting (i.e., T at time t = 0) reacting If the starting (i.e., T at time t = 0) reacting mixture temperature mixture temperature T < TT < Taa, (and by , (and by Eq. 4.8, dT/dt) > 0 and Eq. 4.8, dT/dt) > 0 and < 0. So, < 0. So, a mixture with T < Ta mixture with T < Taa slowly heats up until slowly heats up until TTa a (or dT/dt > 0), at a rate which (or dT/dt > 0), at a rate which continuously decreases with time (dcontinuously decreases with time (d2 2 T/dtT/dt2 2 < < 0). Curves of such heating for four different 0). Curves of such heating for four different values of the starting temperature are values of the starting temperature are shown in Figure 4.7(a).shown in Figure 4.7(a).

g lq q

g ld q q / dt

gq

lq

• If the starting temperature of the mixture If the starting temperature of the mixture is is TTa a < T < T< T < Tbb, both and < 0 , both and < 0

so that the mixture so that the mixture cools down to Tcools down to Taa at a at a

continuously decreasing rate. Such cooling continuously decreasing rate. Such cooling curves are shown in Figure 4.7(a) for three curves are shown in Figure 4.7(a) for three different values of the starting different values of the starting temperature.temperature.

• If the starting temperature of the mixture If the starting temperature of the mixture T T > T> Tbb, both dT/dt and d, both dT/dt and d22T/dtT/dt22 > 0 so that the > 0 so that the

temperature of the reacting gases temperature of the reacting gases increases at an accelerating rate as shown increases at an accelerating rate as shown by four curves in Figure 4.7(a). by four curves in Figure 4.7(a).

g lq q

g ld q q / dt

Case: Wall temperature relatively highCase: Wall temperature relatively high• When TWhen Too = T = To1o1, curve and line never , curve and line never

intersect. Thus, intersect. Thus, is always > 0. As is always > 0. As shown in Figure 4.7(c), the temperature of shown in Figure 4.7(c), the temperature of the the gases increases acceleratively.gases increases acceleratively.

Case: Wall temperature moderate:Case: Wall temperature moderate:• As the wall temperatures progressively As the wall temperatures progressively > >

TT0303 are considered, the points a and are considered, the points a and b b approach one another when ultimately they approach one another when ultimately they coincide at the point c corresponding to a coincide at the point c corresponding to a critical wall temperature Tcritical wall temperature T0202 in Figure 4.6(a). in Figure 4.6(a).

• The heat balance curve for this situation is The heat balance curve for this situation is shown in 4.6(b). The heating curves are shown in 4.6(b). The heating curves are shown in Figure 4.7(b).shown in Figure 4.7(b).

gq

lq

g lq q

Important criteria concerning spontaneousImportant criteria concerning spontaneousignitionignition• The wall temperature The wall temperature TT0202 is a limiting one is a limiting one

beyond which the reaction progressively beyond which the reaction progressively accelerates. The corresponding accelerates. The corresponding temperature temperature TTcc is called spontaneous is called spontaneous ignition temperatureignition temperature of the reactant gas of the reactant gas mixture in the given vessel. mixture in the given vessel.

• TTcc is not a fundamental property of the is not a fundamental property of the given fuel/oxidant mixture. The vessel in given fuel/oxidant mixture. The vessel in which such a mixture is contained has which such a mixture is contained has quite a strong influence on it.quite a strong influence on it.

• At the critical point, c, the curve and the At the critical point, c, the curve and the line are tangential. The interrelationship line are tangential. The interrelationship between pressure, temperature and between pressure, temperature and composition at the ignition threshold hence composition at the ignition threshold hence is given by the following two equations.is given by the following two equations.

• In the above analysis, In the above analysis, the pressure (i.e. the the pressure (i.e. the reaction) and the heat transfer coefficient reaction) and the heat transfer coefficient are kept fixedare kept fixed and the critical wall and the critical wall temperature is deducedtemperature is deduced

g l cc

q q (4.18)

g l

cc

dq dq(4.19)

dT dT

• Keeping the reaction (=pressure) and wall Keeping the reaction (=pressure) and wall temperature fixed while temperature fixed while seeking for the seeking for the critical heat transfer coefficientcritical heat transfer coefficient results in results in Fig. 4.8Fig. 4.8

• Keeping the wall temperature and the heat Keeping the wall temperature and the heat transfer coefficient fixed and while seeking transfer coefficient fixed and while seeking for the for the critical reaction (=critical pressure)critical reaction (=critical pressure) results in Fig. 4.9results in Fig. 4.9

Application of Semenov Theory to Application of Semenov Theory to Predict Predict

Ignition RangeIgnition Range

• The The critical point c in Figure 4.6(a) marks critical point c in Figure 4.6(a) marks the transition of a slow stable reaction into the transition of a slow stable reaction into one that is explosiveone that is explosive. Assuming Arrhenius . Assuming Arrhenius type rate law, Eqs. 4.18 and 4.19 becometype rate law, Eqs. 4.18 and 4.19 become

cE/ RTnn Ac c 0cH V k C e hs T T (4.18a)

cE/RTn

n Ac 2c

EH V k C e hS (4.19a)

RT

• Eliminating Eliminating H VkH Vknn C CAcAcnn/hS/hS from these two from these two

equations and with an assumption that the equations and with an assumption that the amount of reactant consumed in the ignition amount of reactant consumed in the ignition delay is negligible,delay is negligible,

This quadratic has two roots; the lower one This quadratic has two roots; the lower one applies to ignition and the upper one to applies to ignition and the upper one to extinctionextinction. .

2c

c 0c

RTT T (4.20)

E

• Substituting Eq. 4.20 in Eq. 4.18(a) we obtain for a Substituting Eq. 4.20 in Eq. 4.18(a) we obtain for a simple simple second ordersecond order thermal reaction ( thermal reaction (applies to applies to most of HC/air reactionsmost of HC/air reactions))

• Since RTSince RTcc/E << 1,/E << 1,

• If the gases are assumed perfect and if PIf the gases are assumed perfect and if Pcc and P and PAA are respectively the total pressure and the species are respectively the total pressure and the species A partial pressure,A partial pressure,

where Xwhere XAA is the mole fraction of the species A. is the mole fraction of the species A.

2c cE/ R T RT / E2

2 Ac cH V k C e hSRT / E 4.22

cE/ RT22 Ac cH V k C e hSRT / E 4.22a

A c A c cAc

c c

P X PC

RT RT

• Equation 4.22 thus becomes,Equation 4.22 thus becomes,

• If the composition X, is kept fixed, Eq. 4.23 If the composition X, is kept fixed, Eq. 4.23 relates the critical pressure with the critical relates the critical pressure with the critical temperature.temperature.

• Logarithmically,Logarithmically,

c2 E/RT

2 Ac c c cH V k X P / RT e hSRT / E 4.23

c

2 3E/RTc

4 2c 2 A

P hSRe

T HVk X E

0.52 3c4 2

cc 2 A

P hSR Eln ln (4.24)

2RTT HVk X E

• Eq. 4.24 is known as Eq. 4.24 is known as Semenov EquationSemenov Equation. . PlottingPlotting

ln (Pln (Pcc/T/Tcc22) on the y‑axis and (I/T) on the y‑axis and (I/Tcc) on the ) on the

x‑axis, Eq. 4.24 gives a straight line with a x‑axis, Eq. 4.24 gives a straight line with a slope of E/2R (see Figure 4.10).slope of E/2R (see Figure 4.10).

• PPcc ‑ T ‑ Tcc plane, as shown in Figure 4.11, delineates plane, as shown in Figure 4.11, delineates the ignitable from non-ignitable conditions.the ignitable from non-ignitable conditions.

• At low pressures, very high temperatures are At low pressures, very high temperatures are needed to accomplish ignition and vice versa.needed to accomplish ignition and vice versa.

• Equation 4.24 can also be used to Equation 4.24 can also be used to construct the ignition ranges on a construct the ignition ranges on a T ‑ XT ‑ XAA plane at a fixed total pressureplane at a fixed total pressure and on a and on a P ‑ P ‑ XXAA plane at a fixed temperature plane at a fixed temperature (see (see Figures 4.12 and 4.13)Figures 4.12 and 4.13)

• In general, these graphs are U‑shaped. In general, these graphs are U‑shaped. The conditions lying inside the U result in The conditions lying inside the U result in an ignition whereas those lying outside, do an ignition whereas those lying outside, do not. not.

• Several inferences can be drawn from Several inferences can be drawn from these figures (see T ‑ Xthese figures (see T ‑ XAA relation given by relation given by Figure 4.12)Figure 4.12)

1.1. Firstly, there exist a lower and an upper Firstly, there exist a lower and an upper concentration limits for ignition; concentration limits for ignition; if the if the mixture is too fuel‑lean or too fuel‑rich, mixture is too fuel‑lean or too fuel‑rich, ignition is not possibleignition is not possible no matter what no matter what the temperature is. The critical fuel the temperature is. The critical fuel concentration, below which ignition is concentration, below which ignition is impossible, is known as the impossible, is known as the lower limit of lower limit of ignitabilityignitability; ; and that above which ignition and that above which ignition is impossible, is known as the is impossible, is known as the upper limit.upper limit.

2.2. Secondly, as the temperature is lowered, Secondly, as the temperature is lowered, these two limits approach one another, these two limits approach one another, thus narrowing the range of ignition.thus narrowing the range of ignition.

3.3. Thirdly, if the temperature is very low, Thirdly, if the temperature is very low, ignition is impossible at any composition.ignition is impossible at any composition.

Forced IgnitionForced IgnitionSome Preliminary ConceptsSome Preliminary Concepts• When a cold reactant mixture is rapidly and When a cold reactant mixture is rapidly and

locallylocally heated by a heat source (an heated by a heat source (an incandescent solid particle, a heated incandescent solid particle, a heated electrical filament or a spark, a pocket of hot electrical filament or a spark, a pocket of hot gas, or a pilot flame), a flame can be gas, or a pilot flame), a flame can be initiated in the vicinity of the heat source initiated in the vicinity of the heat source and propagated into the rest of the cold and propagated into the rest of the cold mixture. Such mixture. Such an an initiation initiation of a of a propagating propagating flame is defined as forced ignitionflame is defined as forced ignition. .

• Both spontaneous and forced ignition share Both spontaneous and forced ignition share the common virtue of self acceleration and the common virtue of self acceleration and auto‑catalytic behavior motivated by auto‑catalytic behavior motivated by thermal and/or chain branching reactionsthermal and/or chain branching reactions. .

• Initial provocation by an external source of Initial provocation by an external source of energy is necessary for both types of energy is necessary for both types of ignition.ignition.

• The phenomenon of forced ignition is The phenomenon of forced ignition is considered in this section in a considered in this section in a thermal thermal view‑pointview‑point in order to systematically develop in order to systematically develop some simple relationships between the critical some simple relationships between the critical physico‑chemical properties of the system.physico‑chemical properties of the system.

• Let us examine the definition of forced Let us examine the definition of forced ignition by a hot particle. Consider as shown ignition by a hot particle. Consider as shown in Figure 4.17 an incandescently hot metal in Figure 4.17 an incandescently hot metal particle (whose temperature is particle (whose temperature is TTww) located in ) located in an infinite combustible mixture (whose an infinite combustible mixture (whose temperature temperature TT00 < T < Tww). ). TTww: wall temperature of : wall temperature of hot particle.hot particle.

Case: TCase: Tww is moderate (see Fig. 4.17) is moderate (see Fig. 4.17)

• A steady state temperature profile T = T(x) is A steady state temperature profile T = T(x) is established in such a way that established in such a way that the steepest the steepest temperature gradients are confined to a thin temperature gradients are confined to a thin boundary layerboundary layer around the particle. around the particle.

• Profile (Profile (aa) when the mixture is ) when the mixture is noncombustiblenoncombustible and ( and (bb) when it is ) when it is combustiblecombustible. .

• The difference between these two profiles is The difference between these two profiles is due to the heat release in chemical reaction. due to the heat release in chemical reaction. In terms of the temperature gradients at the In terms of the temperature gradients at the wall, the wall, the heat flow from the wall to the heat flow from the wall to the mixture is lower when the gases react mixture is lower when the gases react exothermicallyexothermically than when they are inert. than when they are inert.

Case : TCase : Tww is high (see Fig. 4.18) is high (see Fig. 4.18)

• The The higher Thigher Tww, , the lower the heat flux from the the lower the heat flux from the wallwall (the rate of reaction is even higher). (the rate of reaction is even higher).

• At a critical particle temperature At a critical particle temperature TTcc, the heat , the heat flux from the wall to the reactive mixture is flux from the wall to the reactive mixture is zero. zero.

• If the particle temperature is even slightly > TIf the particle temperature is even slightly > Tcc, , the enhanced reaction in the mixture shows the enhanced reaction in the mixture shows a a maxima of temperature (Tmaxima of temperature (Tmaxmax) of the flame) of the flame a a small distance away from the particle surface. small distance away from the particle surface. Heat flows then Heat flows then partly partly to to the particle and mostly the particle and mostly to the reservoir of gasesto the reservoir of gases. A steady state . A steady state temperature profile is impossible because Ttemperature profile is impossible because Tmaxmax continuously moves away from the particle wall. continuously moves away from the particle wall.

• The higher TThe higher Tww, the higher the exothermic heat , the higher the exothermic heat of the reaction renderingof the reaction rendering one time one time TTreactionreaction > T > Tww..

• When When dT/dx at and normal to the igniting dT/dx at and normal to the igniting particle surface = 0particle surface = 0, the reaction layer of gas , the reaction layer of gas (i.e., the (i.e., the flameflame)) begins to propagate begins to propagate into the into the unburnt mixture. unburnt mixture.

• The onset of this propagation is conveniently The onset of this propagation is conveniently considered as the criterion for forced ignition.considered as the criterion for forced ignition.

• dT/dx in a reacting mixture are always dT/dx in a reacting mixture are always accompanied by composition gradients (dC/dx)accompanied by composition gradients (dC/dx)

• In the In the immediate vicinity of the particleimmediate vicinity of the particle, the , the concentration of the reactants will be the concentration of the reactants will be the lowest and that of the products will be the lowest and that of the products will be the highesthighest

• Far awayFar away from the particle, from the particle, the product the product concentration will be zeroconcentration will be zero and the and the reactant reactant concentration will be equal to the initial concentration will be equal to the initial valuevalue

• As a result of the concentration gradients, As a result of the concentration gradients, the products diffuse away from the particle the products diffuse away from the particle whereas the reactants diffuse towards the whereas the reactants diffuse towards the particle to replenish the depleted mixture particle to replenish the depleted mixture with fresh new reactants.with fresh new reactants.

Propagation of flame (see Fig. 4.19)Propagation of flame (see Fig. 4.19)

• The The fundamental fundamental flame speed flame speed (u(uoo)) is defined is defined as the speed at which the flame front travels as the speed at which the flame front travels in a direction normal to itself and in a direction normal to itself and relativerelative to to the speed of the unburnt mixture. the speed of the unburnt mixture.

• The temperature across the flame front The temperature across the flame front varies from the flame temperature Tvaries from the flame temperature Tff behind behind the front to the initial mixture temperature the front to the initial mixture temperature TT00 ahead of the front. ahead of the front.

• The flame thickness The flame thickness ff is defined as the ratio is defined as the ratio of the maximum temperature of the maximum temperature differencedifference to to the maximum temperature the maximum temperature gradientgradient

Phenomena of propagation• Heat is generated by combustion in a layer

of gas mixture of thickness f.• Due to the temperature gradient, the heat

so generated is transferred by conduction to the unburnt gases.

• Utilizing this heat, the unburnt mixture heats up so that the combustion front progresses forward at a velocity u0.

f 0f

max

T T4.33

dT / dx

• The The heat generation rateheat generation rate is given by is given by

where W"' is the rate of combustion averaged where W"' is the rate of combustion averaged over the flame thickness, over the flame thickness, f f and the area of cross and the area of cross section considered, A. section considered, A.

• The The rate of conduction heat transferrate of conduction heat transfer is given is given approximately byapproximately by

• K = thermal conductivity of K = thermal conductivity of unburntunburnt gas gas

g fq H.W . .A 4.34

k f 0 fq K A T T / 4.35

• The The rate of heatingrate of heating to raise the to raise the temperature of a unit mass of the unburnt temperature of a unit mass of the unburnt mixture from Tmixture from T00 to T to Tff is C (T is C (Tff ‑ T ‑ T00) cal/gm.) cal/gm.

where where ρρ00 is the density of the unburnt is the density of the unburnt mixture and C is the mixture and C is the specific heat of specific heat of unburnt gasunburnt gas..

• Equating andEquating and ,,

h 0 0 f 0q u AC T T 4.36

gq

Kq

1/ 2

f 0f

K(4

T T

H W.37)

• Eq. 4.37 shows that the Eq. 4.37 shows that the flame thickness is flame thickness is largerlarger if the if the mixture conductivity is largermixture conductivity is larger, , (T(Tff ‑ T ‑ T00)) is greateris greater and the and the average heat average heat production rate is lowerproduction rate is lower

• Equating and and recalling the Equating and and recalling the definition of thermal diffusivity definition of thermal diffusivity

• Equating andEquating and ,,

Kq

hq 0 0K / C

0

f0

(4.38)u

0 0 f 0

f

u C T T

H W4.39

gq

hq

• Eliminating Eliminating ff from Eqs. 4.37 and 4.39, from Eqs. 4.37 and 4.39,

The The flame speed is greaterflame speed is greater if the if the mixture mixture conductivity is greaterconductivity is greater, mean , mean heat production heat production raterate HW"'HW"' is is greatergreater, the volumetric specific , the volumetric specific heat of the original mixture is lower and heat of the original mixture is lower and (T(Tff ‑ T ‑ T00) ) is loweris lower

• Effect of pressure on the flame speedEffect of pressure on the flame speed

For most HC fuels burning in air or OFor most HC fuels burning in air or O22, , n is n is 2 2 so that the fundamental so that the fundamental flame speed is nearly flame speed is nearly independent of the total pressureindependent of the total pressure

• So, a good fire extinguisher is that has So, a good fire extinguisher is that has low low thermal conductivitythermal conductivity and and high heat capacityhigh heat capacity

1/ 2

00 f 0

1 K H Wu 4.40

C T T

(n 2) / 20u P (4.41)

The most conducive conditions for ignition The most conducive conditions for ignition (u(u00

high) are provided byhigh) are provided by• A low flame temperatureA low flame temperature• A high initial temperatureA high initial temperature• A high heat of combustionA high heat of combustion• A high mean reaction rateA high mean reaction rate• A low volumetric heat capacityA low volumetric heat capacity• A high thermal conductivityA high thermal conductivity• A high total pressureA high total pressure• A mixture whose composition is nearly A mixture whose composition is nearly

stoichiometricstoichiometric• A low gas velocity (for flame stabilisation)A low gas velocity (for flame stabilisation)• A low intensity of turbulence, if the flow is A low intensity of turbulence, if the flow is

turbulentturbulent

Range of ignitionRange of ignition• Fig. 4.24 indicates that Fig. 4.24 indicates that EEminmin is the least is the least for a for a

mixture whose composition is mixture whose composition is stoichiometricstoichiometric (or (or nearly stoichiometricnearly stoichiometric). ).

• If the mixture gets "leaner" or "richer," the EIf the mixture gets "leaner" or "richer," the Eminmin increases first gradually and then abruptly. increases first gradually and then abruptly.

• The abrupt rise of EThe abrupt rise of Eminmin suggests that when the suggests that when the mixture is mixture is too "lean"too "lean" (i.e. if the fuel content < (i.e. if the fuel content < l l % in Figure 4.24) or too % in Figure 4.24) or too "rich ""rich " (i.e. if the fuel (i.e. if the fuel content > content > rr % in Figure 4.24) ignition is % in Figure 4.24) ignition is possible only if "infinite" amount of energy is possible only if "infinite" amount of energy is supplied through the igniter (supplied through the igniter (practically not practically not possiblepossible).).

Some relevant matters on Fig. 4.24Some relevant matters on Fig. 4.24

• All the energies and compositions All the energies and compositions corresponding to the area lying corresponding to the area lying inside the inside the UU result in result in an ignitionan ignition whereas those lying whereas those lying outside do not.outside do not.

• Ignition is impossible if the mixture is too Ignition is impossible if the mixture is too lean or too rich. The upper and lower limits lean or too rich. The upper and lower limits of flammability (corresponding to ”r” of flammability (corresponding to ”r” and ”l” in Figure 4.24) are characteristic of and ”l” in Figure 4.24) are characteristic of the nature of fuel/oxidant combination. the nature of fuel/oxidant combination.

• For any given energy EFor any given energy Eigig, all mixtures, lying , all mixtures, lying between an upper (B) and a lower (A) limits between an upper (B) and a lower (A) limits of composition, can be ignited. All the of composition, can be ignited. All the mixtures lying outside these limits cannot mixtures lying outside these limits cannot be ignited.be ignited.

• The range of ignition‑AB‑is The range of ignition‑AB‑is narrowednarrowed if if EEigig is is chosen smallerchosen smaller. If E. If Eigig < E < Eminmin, ignition is , ignition is impossible for any mixture.impossible for any mixture.

• Flammability curves found extensively in Flammability curves found extensively in the combustion literature delineate the the combustion literature delineate the ignitible and nonignitible domains on ignitible and nonignitible domains on a a pressure (or temperature) and pressure (or temperature) and composition plotcomposition plot. These curves are similar . These curves are similar to to Figures 4.12 and 4.13Figures 4.12 and 4.13. It was found that . It was found that as the pressure is decreasedas the pressure is decreased, the , the U‑curve U‑curve of Figure 4.24 progressively becomes of Figure 4.24 progressively becomes narrowernarrower, the upper and lower limits , the upper and lower limits ultimately coinciding as the minimum ultimately coinciding as the minimum pressure is approached.pressure is approached.

• If the If the initial temperature of the mixture is initial temperature of the mixture is raisedraised, it is well‑known that the , it is well‑known that the ignition ignition range is broadened forrange is broadened for most HC/air most HC/air mixtures. That is, the lower limit is mixtures. That is, the lower limit is decreased and the upper limit is raised.decreased and the upper limit is raised.

• Addition of Addition of inert gasesinert gases to the combustible to the combustible mixtures mixtures narrows the ignition rangenarrows the ignition range, such , such a narrowing being mainly due to a narrowing being mainly due to a a decrease in the upper limitdecrease in the upper limit. The maximum . The maximum amount of the inert gas necessary to amount of the inert gas necessary to eliminate ignition altogether is a matter of eliminate ignition altogether is a matter of importance in the extinguishment of importance in the extinguishment of accidental fires. accidental fires.