Lewis Number Effects in Distributed Flames

A. J. Aspden, M. S. Day and J. B. Bell

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Abstract

Recent computational studies have simulated a mode of distributed premixed combustion where turbulent mixingplays a significant role in the transport of mass and heat near the reaction zone. Under these conditions, moleculartransport processes play a correspondingly smaller role. A consequence of burning in this regime is that a changein the gas mixture composition can occur within the flame zone, which modifies the burning rate. The compositiondepends on the Lewis number (ratio of molecular heat to mass diffusivity), and so the response to the transitionto distributed burning will be different for fuels with different Lewis numbers. In this paper, we examine the roleof Lewis number on flames in the distributed burning regime. We use high-resolution three-dimensional flamesimulations with detailed transport models to explore the turbulent combustion of a range of fuels, specificallylean premixed hydrogen, methane and propane mixtures. The response of the burning rate is found to be morepronounced in hydrogen than in the other fuels.

Keywords: turbulent premixed combustion, low Mach number flow, adaptive mesh refinement

1. Introduction

There has been considerable recent interest in fuel-flexible lean premixed combustion systems. Burningunder lean conditions reduces the exhaust gas tem-peratures, and consequently, thermal NOx emissions.Low-swirl burner technology, introduced by [1] as atool for studying the fundamental properties of leanpremixed turbulent flames has proven to be an effec-tive stabilization mechanism for a wide range of fuels,see [2] and [3]. However, high-swirl injectors, oftenused in gas turbines, can generate much higher levelsof turbulence. For example, [4] have measured turbu-lent intensities as high as 30 m/s.

There is little work in the literature that deals withpremixed flames under highly turbulent conditions.Peters [5] argues that at sufficiently high Karlovitznumbers, the Kolmogorov scale becomes smaller thanthe inner layer of the flame, turbulence enhances heatloss to the preheat zone, and the chemistry breaksdown. The conclusion is that a premixed flame is un-able to survive in the distributed burning regime (orbroken reaction zone), which is taken to mean largeKarlovitz number (KaL

>∼100), for

Ka2L =

u3

s3L

lLl

(1)

where u and l are the turbulent rms velocity and inte-gral length scale, respectively, and sL and lL are theflat laminar flame speed and width, respectively.

Aspden et al. [6] considered turbulence-flameinteractions in carbon-burning thermonuclear (pre-mixed) flames in type Ia supernovae, where the Lewisnumber (the ratio of thermal diffusivity to the defi-cient species diffusivity, Le = κ/D) was very high.Once the Karlovitz number was sufficiently large, acategorically different mode of burning was observed:a distributed flame. Global extinction was not ob-served under the conditions studied. Diffusive pro-cesses were dominated by turbulent mixing, result-ing in an effective unity Lewis number flame. Thismeant that fuel and temperature were advected mate-rially and mixed by turbulence on time scale fasterthan diffusive mixing. The burning occurred in asingle turbulently-broadened flame, where the localburning rate was much lower than the laminar flame.However, the volume of burning was significantly en-hanced, and so the turbulent flame speed was approx-imately 5 or 6 times faster than the laminar flamespeed.

Motivated by [6], two groups began to study dis-tributed flames with terrestrial fuels. Poludnenko[7] reported a distributed flame in a stoichiomet-ric hydrogen-air mixture with simple chemistry, butfound that subsonic turbulence could not penetratethe internal flame structure. Aspden et al. [8] alsoexamined premixed hydrogen-air mixtures, but undermuch leaner conditions, which allowed the flame tobecome distributed at much lower Mach numbers, andfound substantial flame broadening was possible. De-

tailed examination of the fuel species scalar fluctua-tion equation demonstrated that the turbulent mixingwas the same order of magnitude as the reaction andmolecular dissipation terms, but molecular diffusionremained an important process. However, once again,global extinction was not observed under the condi-tions studied, and the turbulent flame speed was fasterthan the laminar flame.

It should be emphasized that there are several dif-ferences between the distributed flames mentionedabove and the high Karlovitz number flames that havebeen observed to extinguish experimentally. The dis-tributed flames of [6–8] are confined by periodic lat-eral boundary conditions, which means there is apool of hot fluid that is mixed with the fuel by fully-developed turbulence. The Bunsen flames of [9–12]and the single vortex-flame interactions of [13, 14]that extinguish at high Karlovitz number, all expe-rience large-scale global mean stretch, which is thelikely explanation for the difference in behavior.

In contrast to the very high Lewis number of the su-pernovae flames of [6], lean premixed hydrogen hasa Lewis number of approximately 0.35 to 0.365 forequivalence ratios between ϕ = 0.31 and 0.4 (therange studied in [8]). Methane and propane flames,each at an equivalence ratio of ϕ = 0.7, have Lewisnumbers of 0.96 and 1.95, respectively. Following[6, 8], this paper examines the effect of Lewis num-ber on distributed flames using high-resolution three-dimensional simulations of lean hydrogen, methaneand propane flames with detailed chemistry at highKarlovitz number.

A consequence of the effective unity Lewis num-ber due to strong turbulent mixing in the distributedburning regime is that there is a shift in composi-tion through the flame, which changes how intenselythe flame burns. Consider the distribution of fuelmole fraction and temperature (each of which canbe thought of as an approximate progress variable)through flames with different Lewis numbers. Fig-ure 1 shows the flat laminar flame distribution of fuelmole fraction and temperature for four fuels with dif-ferent Lewis numbers (hydrogen, methane, propaneand carbon in SNe), normalized by the fuel mole frac-tion and the temperature where the mole fraction goesto zero. The cold fuel starts in the top-left of the plot,and burns to hot products in the bottom-right. In themethane flame, the diffusion coefficient of methaneis essentially the same as the thermal conductivity,suitably normalized, reflecting the near unity Lewisnumber of the flame. For this case, the relation-ship between T andX(CH4) is essentially linear overmuch of the normalized temperature range except fora turnover in the hot region of the flame where there issignificant heat release. For the propane flame and thesupernova flame, the diffusion coefficient of the fuelis significantly smaller than the thermal conductiv-ity, allowing the fuel to be preferentially heated morerapidly than it diffuses into the flame zone. This re-sults in a shift in the T versus fuel curve in figure 1so that it lies above the methane curve. Analogously,

the diffusion coefficient of H2 is higher, resulting en-hanced diffusion of fuel into the flame and a shiftdownward in the T versus X(H2) plot. It was shownin [6, 8] that as the turbulent intensity was increased,turbulent diffusion began to dominate molecular dif-fusion and the T versus fuel curves became similarto to the methane versus fuel curve in figure 1, so weexpect that the propane curve will do the same. Sincethe burning rate increases as fuel concentration andtemperature increase, when the flame becomes dis-tributed, it can be expected that the burning rate de-creases in the high Lewis number case, and increasesin the low Lewis number case.

Despite the decrease in the burning rate in the su-pernova study, there was significant broadening of theflame, and so the overall flame speed increased, andglobal extinction was not observed. The low Lewisnumber of lean hydrogen leads to an increase in thelocal burning rate as the flame becomes distributed,making global extinction less likely. However, thereremains the possibility that the Lewis number can af-fect the flame response to intense turbulent mixing,and a lean propane flame (Le > 1) may extinguishat high Karlovitz number. Another important differ-ence between terrestrial and supernovae flames is thelack of intermediate species in the latter. It may alsobe possible to disrupt the chemical pathways in a leanpropane flame, which could lead to global extinction.

2. Computational Methodology

The simulations presented here are based on a lowMach number formulation of the reacting flow equa-tions. The methodology treats the fluid as a mixtureof perfect gases. We use a mixture-averaged modelfor differential species diffusion and ignore Soret, Du-four, gravity and radiative transport processes. Withthese assumptions, the low Mach number equationsfor an open domain are

∂(ρu)

∂t+∇ · (ρuu) = −∇π +∇ · τ,

∂(ρYi)

∂t+∇ · (ρYiu) = ∇ · (ρDi∇Yi)− ωi,

∂(ρh)

∂t+∇ · (ρhu) = ∇ ·

„λ

cp∇h«

+

Xi

∇ ·»hi

„ρDi −

λ

cp

«∇Yi

–,

where ρ is the density, u is the velocity, Yi is the massfraction of species i, h is the mass-weighted enthalpyof the gas mixture, T is the temperature, and ωi is thenet destruction rate for species i due to chemical re-actions. Also, λ is the thermal conductivity, τ is thestress tensor, cp is the specific heat of the mixture, andhi(T ) and Di are the enthalpy and species mixture-averaged diffusion coefficients of species i, respec-tively. These evolution equations are supplementedby an equation of state for a perfect gas mixture.

The basic discretization combines a symmet-ric operator-split treatment of chemistry and trans-port with a density-weighted approximate projectionmethod. The projection method incorporates theequation of state by imposing a constraint on the ve-locity divergence. The resulting integration of the ad-vective terms proceeds on the time scale of the rela-tively slow advective transport. Faster diffusion andchemistry processes are treated time-implicitly. Thisintegration scheme is embedded in a parallel adap-tive mesh refinement algorithm framework based ona hierarchical system of rectangular grid patches. Thecomplete integration algorithm is second-order ac-curate in space and time, and discretely conservesspecies mass and enthalpy. The reader is referred to[15] for details of the low Mach number model andits numerical implementation, and to [16] for previousapplications of this methodology to the simulation ofpremixed turbulent flames.

The detailed chemistry and transport models usedfor the hydrogen flames was taken from the GRI-Mech 2.11 mechanism [17], which has 9 species and27 fundamental reactions. The methane mechanismwas DRM19 [18], with 19 species and 84 fundamen-tal reactions. The propane mechanism was that of[19], with 71 species and 469 fundamental reactions.

The overall numerical scheme is known to con-verge with second-order accuracy, and the ability ofthe scheme to perform direct numerical simulationwas examined in [20]. The non-oscillatory finite-volume approach used remains stable even whenunder-resolved, and so some care is required to en-sure that simulations are well-resolved. It was shownin [20] that a marginally resolved viscous simulationhas an effective Kolmogorov length scale, ηe and ef-fective viscosity that are slightly larger than that spec-ified. Dimensional analysis was used to characterizethe inviscid extreme, known as Implicit Large EddySimulation (ILES), through an analogy with the the-ory of [21]. From this analysis, an expression forthe effective viscosity νe was obtained that describesthe transition from well-resolved to pure ILES, νe =νu + ν∆x exp(−νu/2ν∆x), from which the effectiveKolmogorov length scale can be derived, where ν∆x

is the effective viscosity of a simulation with a cellwidth ∆x and zero diffusion, and νu is the physicalviscosity. In the regime considered here, resolutionof the turbulence places more stringent requirementson the simulation than does the chemistry. Detailedsimulations, discussed in Aspden et al. [8] show theif ηe/η < 1.5, which is satisfied for the simulationpresented here, then the turbulence is well-resolved.

3. Turbulent Flame Simulations

Three-dimensional simulations were conducted ofthree downward-propagating flames in a high aspectratio domain (8:1), with periodic lateral boundaryconditions, a free-slip base and outflow at the top. Aturbulent background velocity field was maintainedthrough a source term in the momentum equations

following [20], consisting of a superposition of long-wavelength Fourier modes. This results in a time-dependent zero-mean turbulent velocity field. It wasshown in [20] that this approach gives approximately10 integral length scales across the domain width. Aninert calculation was run to establish the turbulenceat reduced expense, and the reacting flow simulationwas initialized by superimposing a laminar flame so-lution onto the turbulent velocity field. The base gridin each case was 64×64×256, with two levels of re-finement used once the flame had become established,giving an effective resolution of 256×256×2048.

The simulations have been configured to have thesame Karlovitz and Damkohler number in each case,where freely-propagating values have been used to ac-count for the thermodiffusively unstable nature of thehydrogen flame, see Aspden et al. [22] for details.The domain size in each case is approximately 7.6times the (freely-propagating) laminar flame width,and the turbulent intensity is approximately 50 timesthe (freely-propagating) laminar flame speed. Thesimulation properties are shown in table 1, where theeffective Kolmogorov length scale of the simulationis given using the expression derived in [20], whichcompares well with the estimates of the expected Kol-mogorov length scale.

Figure 2 shows two-dimensional slices of den-sity, burning rate and temperature for the three fu-els (note only five-eighths of the domain height isshown). The density and temperature are normal-ized by the laminar flame values, the hydrogen burn-ing rate has been normalized by three times thefreely-propagating flame value, and the methane andpropane flames have been normalized by twice thelaminar value. It is clear that the flames are very simi-lar in structure, and present the characteristics typicalof a distributed flame: there is a single turbulently-broadened flame, where the burning occurs at the hightemperature end of the distribution.

There is some variation in the relative widths of thedensity distributions (figure 2), but they all appear tobe distributed flames. Here, we consider two charac-teristics of distributed flames that were identified in[8]. First, an exponential distribution was found inthe probability density function (PDF) of |∇ρ|. Thisis characteristic of turbulent scalar mixing, see [23–25] for examples. When the flame was not distributed,there was a sharp interface between fuel and products,and the normalized PDF presented more rapid decaythan exponential. Second, the local equivalence ratioacross the flame was found to be close to constant.This was demonstrated using joint probability den-sity functions of fuel mole fraction and temperature(again, used as a rough measure of progress).

Figure 3 shows PDFs of |∇ρ| normalized by the re-spective standard deviation for the three cases. Thedata have been conditioned such that only regionswhere the density is between 10% and 90% of thelaminar extremes contribute to the PDF. The distribu-tions are all very similar, and close to the exponentialdistribution shown by the straight dotted line. This is

indicative that turbulence is a dominant mixing pro-cess, and characteristic of a distributed flame.

Figure 4 shows the JPDF of local equivalence ratioagainst temperature for each fuel, where the equiv-alence ratio is defined as ϕ = (4CC + CH)/2CO,where Ci denotes the molar concentration of atom i.Taking the temperature as a measure of progress, theflame burns from left-to-right. The laminar distribu-tions are shown by the solid red lines. In the laminarhydrogen flame, the fuel diffuses downstream lead-ing to a decrease in equivalence ratio. The laminarmethane flame experiences little change in equiva-lence ratio because the fuel Lewis number is closeto unity, but there is some effect of hydrogen diffu-sion. The laminar propane flame becomes richer atfirst due to oxygen diffusion in the downstream di-rection. In each case, the turbulent distribution is in-deed closer to being constant than the laminar distri-butions, with much less variation. This is an interest-ing consequence of burning in the distributed regime.Because turbulent mixing gives rise to essentially thesame (turbulent) diffusion coefficient for all species,there is little change in local equivalence ratio. Alloxygen-containing molecules are essentially advectedin packets with all hydrogen-containing moleculesand carbon-containing molecules. This demonstratesfurther that the flames are in the distributed burn-ing regime. Some variation remains, however, whichsuggests that species diffusion is still having someeffect on the large scales. This observation high-lights an interesting difference between the methaneflame, which has a Lewis number of approximately0.96, and the effective unity Lewis number distributedflame. Individual species have different diffusion co-efficients, which is why there can be variation in thelocal equivalence ratio, as shown by the laminar dis-tribution. In the distributed flame, species diffusion isdictated by the turbulent eddy viscosity, and so is thesame for all species.

The main difference between the three cases is inthe burning rate. In hydrogen (see figure 2), the burn-ing is both broader and relatively more intense thanthe other two cases; recall the hydrogen has been nor-malized by three times the freely-propagating flamevalue, whereas the other two cases have been normal-ized by just twice the laminar value, and yet appearto be burning less intensely. This is the main dif-ference between distributed flames at different Lewisnumbers. Figure 5 shows a probability density func-tion (PDF) of the burning rate for each case, normal-ized by the corresponding peak laminar value, andthe first moment has been taken to remove effects ofnon-burning regions near the origin. For methane andpropane, the main contribution to the burning is closeto the corresponding laminar value, whereas in hy-drogen, a significant portion of the burning occurs be-tween 1.5 and 2 times the peak laminar rate.

The different burning rates can be accounted forby considering the distribution of fuel mole fractionagainst temperature, a joint probability density func-tion (JPDF) of which is shown in figure 6 for the three

cases. The cold fuel starts in the top-left of the plot,and burns to hot products in the bottom-right, and thesolid red line denotes the laminar distribution (recallthe laminar profiles were shown in figure 1). In eachcase there is only a small amount of spread in the tur-bulent flame distribution, which is close to linear ineach case, again consistent with the distributed flamesof [6, 8]. Note in lower turbulence cases, the distribu-tion is close to the laminar distribution, and in moder-ate turbulence cases, the distribution is broad, due toboth differential diffusion and turbulent mixing, see[6, 8] for more detail. Because the laminar distribu-tion for hydrogen lies below the linear distribution,the burning becomes enhanced in the distributed burn-ing regime. The methane distribution is close to thelaminar flame, which is to be expected because theLewis number of the flame is close to unity, and sois largely unaffected by the transition to distributedflame. The JPDF of the turbulent propane flame is alsoclose to linear, which suggests turbulence has affectedthe mixing at low-to-intermediate temperatures. Forboth methane and propane, the high-temperature endof the distribution (where the burning was observedto take place, see figure 2) is close to the laminarflame distribution, which explains why the burningrate did not vary from the laminar rate significantly(again see 2). Despite these variations with Lewisnumber, global extinction was not observed in thesesimulations.

Figure 7 shows the integral of the burning rate asa function of both temperature and fuel mole frac-tion. Specifically, the rate of destruction of fuelis integrated Q(T,X) =

RρωF dV (T,X), where

V (T,X) is the volume of fluid with temperature Tand fuel mole fraction X , and Q(T,X) is then aver-aged in time. Note the axes are broken to focus onthe high temperature regions where the burning takesplace. The solid black line denotes the flat laminarflame distribution of fuel mole fraction versus tem-perature. In the turbulent hydrogen flame, almost allof the burning is seen to take place close to the adi-abatic flame temperature, whereas in the other fuels,the burning occurs over a broader range of tempera-tures, centered around the flat laminar flame distribu-tion. We speculate that the differing behavior is be-cause the turbulent time scale is faster than the burn-ing time scale in hydrogen, and so the fuel is mixedbefore it burns, and so only occurs at the highest tem-perature attained. For the other fuels the burning timescale is faster so they can burn over a broader rangeof temperatures.

4. Discussion and Conclusions

The effects of Lewis number on distributed burninghave been examined through high-resolution three-dimensional simulations of lean premixed hydrogen,methane and propane flames. The simulations wereconfigured to have a Karlovitz number of 410 (wherea freely-propagating flame speed and width wereused to account for the thermodiffusively-unstable na-

ture of the hydrogen flame, following [22]). Two-dimensional slices of the density, burning rate andtemperature showed that all of the flames appearedvery similar in structure, reinforcing the normaliza-tion approach of [22]. All three cases were shown toshare characteristics typical of distributed flames: thenormalized PDFs of |∇ρ| were exponential in distri-bution (indicative of turbulent scalar mixing) and theJPDFs of local equivalence ratio against temperatureshowed little variation in equivalence ratio (indicativethat mixing was dominated by turbulence).

The main effect of Lewis number was found to bein the response of the burning rate. It was shown in [8]that the transition to distributed burning led to an en-hanced burning rate in low Lewis number hydrogenflames, and a decrease was reported in high Lewisnumber carbon-burning supernova flames in [6]. Inthe present paper, the burning rates of both methaneand propane were found to be larger unaffected by thetransition to distributed burning, remaining close tothe corresponding laminar values. JPDFs of fuel molefraction against temperature showed that the methanedistribution was close to the laminar distribution (asexpected). The propane distribution was only affectedat low temperatures, where there was little burning,and the distribution at high temperatures was close tothe laminar profile. Since the burning rate is a strongfunction of fuel and temperature, these JPDFs explainthe observed burning behavior.

Global extinction was not observed under the con-ditions investigated, but the present study does not ex-clude that possibility in other fuels. The distributionof fuel mole fraction against temperature in a flat lam-inar propane flame is close to the distribution in thedistributed flame, and so little change in local burn-ing rate was observed. Another high Lewis numberfuel may have a distribution closer to that of the su-pernova flame. Consequently, the transition to dis-tributed burning may result in a sufficient reductionin local burning rate for global extinction to occur. Itshould also be noted that the turbulent intensity herewas only approximately 10m/s, so the possibility re-mains that even higher levels of turbulence could ex-tinguish the flame.

Acknowledgments

AJA was supported by a Glenn T. Seaborg Fel-lowship at LBNL; JBB was supported by the DOEApplied Mathematics Research Program; MSD wassupported by the DOE SciDAC Program. Simula-tions were performed on Lawrencium at LBNL, andon Franklin (under an INCITE award) and Hopperat NERSC. All support is provided by the U.S. De-partment of Energy under Contract No. DE-AC02-05CH11231.

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Case H2 CH4 C3H8

ϕ 0.40 0.70 0.70sL (m/s) 0.47 0.205 0.195lL (mm) 0.41 0.60 0.60u (m/s) 23.9 10.3 9.8l (mm) 0.314 0.46 0.45L (mm) 3.14 4.6 4.5∆x (µm) 13 18 18η (µm) 3.7 7.0 7.2ηe (µm) 5.1 8.8 8.8

Table 1: Simulation properties.

List of Figure CaptionsFigure 1. (52 = 115 words) Distribution of normalized tem-perature and fuel mole fraction for four fuels with differentLewis numbers.

Figure 2. (75 = 330 words double column) Two-dimensionalslices of density, burning rate and temperature for hydrogen,methane and propane, respectively. Note only five-eighthsof the domain height is shown. The density and temperatureare normalized by the laminar flame values, the hydrogenburning rate has been normalized by three times the lami-nar value, and the methane and propane flames have beennormalized by twice the laminar value.

Figure 3. (53 = 117 words) Probability density functions of|∇ρ|.

Figure 4. (154 = 339 words) JPDFs of local equivalence ratioagainst temperature.

Figure 5. (57 = 126 words) Normalized first moment of theprobability density function of normalized burning rate.

Figure 6. (154 = 339 words) JPDFs of fuel mole fractionagainst temperature.

Figure 7. (154 = 339 words) Integrated fuel consumptionrate as a function of fuel mole fraction and temperature.

Total words from text: 4*900 + 346 (157mm) = 3946 words

Total words from figures: 1705 words

Words from table: 46 = 102 words

Total words: 5753

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Temperature

Nor

mal

ized

Fue

l Mol

e F

ract

ion

HydrogenMethanePropaneSNIa

Fig. 1: Distribution of normalized temperature and fuel molefraction for four fuels with different Lewis numbers.

Hydrogen Methane Propane

Den

sity

Bur

ning

Rat

e

Tem

pera

ture

Fig. 2: Two-dimensional slices of density, burning rate andtemperature for hydrogen, methane and propane, respec-tively. Note only five-eighths of the domain height is shown.The density and temperature are normalized by the laminarflame values, the hydrogen burning rate has been normal-ized by three times the laminar value, and the methane andpropane flames have been normalized by twice the laminarvalue.

0 1 2 3 4 5 610

−4

10−3

10−2

10−1

100

|∇ρ |/σ

σ p(

|∇ρ|)

PropaneMethaneD40Exponential

Fig. 3: Probability density functions of |∇ρ|.

Temperature

Loca

l Equ

ival

ence

Rat

io

400 600 800 1000 1200 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Hydrogen

Temperature

Equ

ival

ence

Rat

io

400 600 800 1000 1200 1400 1600 1800 20000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Methane

Temperature

Equ

ival

ence

Rat

io

400 600 800 1000 1200 1400 1600 1800 20000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Propane

Fig. 4: JPDFs of local equivalence ratio against temperature.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

Normalized Burning Rate

Nor

mal

ized

Firs

t Mom

ent

HydrogenMethanePropane

Fig. 5: Normalized first moment of the probability densityfunction of normalized burning rate.

Temperature

Fue

l Mol

e F

ract

ion

400 600 800 1000 1200 1400

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Hydrogen

Temperature

CH

4 Mol

e F

ract

ion

400 600 800 1000 1200 1400 1600 1800 20000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Methane

Temperature

C3H

8 Mol

e F

ract

ion

400 600 800 1000 1200 1400 1600 1800 20000

0.005

0.01

0.015

0.02

0.025

0.03

Propane

Fig. 6: JPDFs of fuel mole fraction against temperature.

Temperature

H2 M

ole

Fra

ctio

n

1200 1250 1300 1350 1400 1450

0.005

0.01

0.015

0.02

0.025

Hydrogen

Temperature

CH

4 Mol

e F

ract

ion

1300 1400 1500 1600 1700 1800

2

4

6

8

10

12

14

16x 10

−3

Methane

Temperature

C3H

8 Mol

e F

ract

ion

1300 1350 1400 1450 1500 1550 1600 1650 1700

0.5

1

1.5

2

2.5

3

3.5

4

x 10−3

Propane

Fig. 7: Integrated fuel consumption rate as a function of fuelmole fraction and temperature.