An Approach of Droplet Ignition Delay Time

Jun Chen and Xiaofeng Peng Laboratory of Phase Change and Interfacial Transport Phenomena, Department of Thermal

Engineering, Tsinghua University, Beijing, China

When a droplet is suddenly injected into a high-temperature environment, thedroplet self-ignition phenomenon occurs. A simple model, based on the temperaturehistory of target gas mixture of which the equivalent ratio is equal to 1, was proposedto predict the droplet ignition delay time in this paper. This approach clearly dividesthe droplet self-ignition delay into two parts, the physical delay and the chemical delay.The predicted droplet ignition times agree well with the experimental data and numericalsimulation results. In addition, the influence of droplet diameter on the droplet ignitiondelay was discussed in detail using this approach. © 2008 Wiley Periodicals, Inc. HeatTrans Asian Res, 38(2): 73–82, 2009; Published online 30 December 2008 in WileyInterScience (www.interscience.wiley.com). DOI 10.1002/htj.20240

Key words: fuel droplets, self-ignition, ignition delay time

1. Introduction

When a fuel droplet is suddenly injected into a high-temperature air environment, the dropletis heated to raise its temperature, and the gasification fuel vapor from the droplet surface mixes withhigh-temperature air. Once the equivalent ratio and temperature of the fuel-air mixture reachappropriate values or their threshold, the combustion chemical reaction becomes fast and violent. Thischemical reaction further influences the flow and thermal fields in the gas phase, and eventually,self-ignition takes place. This time period between the droplet injection and self-ignition occurrenceis termed droplet ignition delay time.

The combustion of a single fuel droplet is of great interest in the fundamental understandingof both droplet burning and complicated spray combustion. Droplet ignition has an important role ondesign of diesel engines, especially for spray combustors. In the early 60s, Faeth and Olson [1, 2]employed microgravity experiments to study the droplet ignition delay time and obtained a lot ofexperimental data. Their experimental data revealed that the environment temperature had animportant role on droplet ignition delay time and the logarithmic delay time was directly proportionalto the logarithmic reciprocal of the environment temperature. Recently, Tsue et al. [3] carried outexperiments on the self-ignition of single fuel droplets in lean fuel-air mixtures and investigated theinfluence of fuel gas existence in the ambience, which was considered to be close to the actualcombustor environment, on the droplet ignition behavior.

© 2008 Wiley Periodicals, Inc.

Heat Transfer—Asian Research 38 (2), 2009

Contract grant sponsor: National Natural Science Foundation of China (No. 50628607), the Specialized Research Fund for theDoctoral Program of High Education (No. 20060003011), and Tsinghua University (No. Jcpy2005050).

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The ignition of fuel droplets is governed by the complex interaction of droplet heating,vaporization, and chemical kinetics. The reaction mechanism of higher hydrocarbons in the gas phaseis characterized by a large number of chemical species and elementary reactions. It was not untilrecently, thanks to growing computer technology, that numerical simulation became possible incor-porating detailed reaction mechanisms for self-ignition and the combustion of single fuel droplets.Stauch et al. [4] simulated the self-ignition of a single n-heptane droplet using detailed transport modeland chemical mechanisms. The influence of different physical parameters, such as ambient pressure,droplet radius, and initial conditions, on the ignition delay time and the location of ignition wasinvestigated and their results showed the environment temperature was the main parameter dominat-ing the droplet’s ignition process. Stauch and Mass [5] further investigated the self-ignition ofn-heptane/iso-octane droplets in air by numerical simulation using detailed chemical mechanisms.Below an ambient temperature of 1000 K, the ignition delay time was found to increase with anincreasing volume fraction of the iso-octane. Above 1000 K, the ignition delay time appeared to bealmost independent of the mixture composition of the droplet.

However, there are very few theoretical models on droplet ignition delay in the availableliterature [6]. In this investigation, an attempt was made to propose a simple model describing dropletignition delay time. The present model can very well reveal the essential process of droplet ignitionand be used for fast calculation of the droplet ignition delay time.

Nomenclature

Bv: (T~∞ − T~

s) / H~

C: ln(1 – YF,st) / ln(1 + Bv)Cp: specific heatD: mass diffusivityd: droplet diameterfv: defined in Eq. (6)H: effect latent heat of evaporationH~

: non-dimensional effect latent heat of evaporationL: specific latent heat of evaporationm: mass evaporation ratem~: non-dimensional mass evaporation rateRs: rs / rs0

r: radial distance from droplet centerr~: non-dimensional radial distanceT: temperatureTcritical: critical temperatureTst: temperature of the target gas mixtureT~

: non-dimensional temperaturet: timetchem: chemical ignition delay timetdelay: droplet ignition delay timetphy: physical ignition delay timet~: non-dimensional timeY: mass fraction

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Greek Symbols

λ: 3(ρgs / ρl)(CpTs / L)ρ: density

Subscripts

F: fuel

g: gas phase

l: liquid phase

s: droplet surface

0: initial state at t = 0

∞: ambiance

2. Determination of Droplet Ignition Delay

2.1 Physical process

Away from the droplet surface, the fuel vapor concentration decreases continuously. Thereshould always be a position where the air is in stoichiometric proportion. The mixture composed ofair and fuel vapor at this position was chosen as the investigative object, called the target gas mixturein this paper. The temperature of the gas mixture also changes continuously away from the dropletsurface, or increases close to the environment temperature away from the droplet center. Under theeffect of Stephen Convection, the target gas mixture moves away from the droplet surface and itstemperature becomes larger or approaches the environmental one. In the early stage, the target gasmixture is at a low temperature and the chemical reaction is so weak that it can be omitted. When thetemperature of the target gas mixture increases, the chemical reaction begins to play an important roleand the radicals in the target gas mixture accumulate continuously until droplet ignition finally occurs.So the droplet ignition process can be divided into two parts, physical delay and chemical delay. Inthe physical delay stage, the temperature increasing process of the target gas mixture is dominantwithout involving chemical reaction, while at the chemical delay stage, the chemical reaction orchemical ignition is dominant. There is a critical temperature for the target gas mixture, which isdefined in Section 2.4, to distinguish between the physical delay and the chemical delay. The physicaldelay is the time needed for the target gas mixture to reach the critical temperature, while the chemicaldelay is the delay time of the target gas mixture creating chemical ignition at the critical temperature.

2.2 Concentration and temperature field

The fuel evaporation at the droplet surface is dominant before droplet self-ignition occurs. Soit is very important to quantitatively describe the fuel vapor concentration and mixture temperatureevolution.

We first define the following non-dimensional parameters (which are non-dimensional massevaporation rate, radial distance from droplet center, temperature, effective latent heat of vaporization,and time, respectively):

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m~ = m / (4πρgDrs), r~ = r / rs, T~

= CpT / L, H~

= H / L, t~ = ρgD

ρlr02 t (1)

According to the unsteady droplet evaporation model of Law et al. [7], the fuel vapor massfraction and mixture temperature can be calculated as the following:

YF = 1 − 1 − YF,∞

(1 + Bv)(x−x∞) / (1−x∞)(2)

T~

= (T~s − H~) + H

~(1 + Bv)(1−x) / (1−x∞) (3)

where x = 1 / r~, H~

= (T~

∞ − T~

s) / ((1 − YF,∞) / (1 − YF,s) − 1), Bv = (T~

∞ − T~

s) / H~

, and YF,∞ is the fuel vapormass fraction, r~∞ is the diffusion front where YF = YF,∞, and YF,s is the fuel vapor mass fraction at thedroplet surface, which is related to the droplet surface temperature. Introducing the following threeimportant relations [7]:

Rs−3 = exp

∫ T~

s,0

T~

sdT

~s

H~

− 1

= 1 + λfv

(4)

t~ = − ∫ 1

Rs

(Rsg

/ m~)dRsg (5)

fv = ∫ 1

r~∞

r~2

YF

T~ −

YF,∞

T~

∞

dr~

(6)

where m~ = (ln (1 + Bv)) / (1 − x∞). For a given r~∞, both Rs and t~ can be obtained from Eqs. (4) to (6),and then fuel mass fraction YF and temperature T

~ are determined, as described in Ref. 7. In principle,

other unsteady droplet evaporation models can also be employed to describe the fuel vapor concen-tration and temperature field evolution, and this has little influence on the present droplet ignitionmodel.

Using the above unsteady droplet evaporation model, the temperature evolution of the targetgas mixture can be easily obtained. Rewrite Eq. (2) as

and set YF,∞ = 0,

x = x∞ +

ln 1 − YF,∞

1 − YF,st

ln (1 + Bv) (1 − x∞) (7)

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x = x∞ − C(1 − x∞) = (1 + C)x∞ − C (8)

where C = (ln (1 − YF,st)) / (ln (1 + Bv)). Then the non-dimensional temperature of the target gasmixture T

~st can be derived by combining Eqs. (8) and (3), or

T~

st = (T~

s − H~

) + H~

(1 + Bv)1+C (9)

All terms in Eq. (9) are related to Ts. After the relation between Ts and t~ is determined fromthe unsteady droplet evaporation model, the relation between Tst (target gas mixture temperature) andt~ is then obtained.

2.3 Chemical ignition delay time

The target gas mixture, in which the fuel vapor is in stoichiometric proportion mixing withair, is a homogenous mixture. At a specified mixture temperature, the chemical ignition delay timeof the target gas mixture is unique and can be obtained from experimental data or chemical reactionmechanism in the literature. In this paper, the 56-step skeletal mechanism of Peters et al. [8] wasadopted to calculate the chemical ignition delay time of n-heptane in atmospheric pressure. Therelation between the chemical ignition delay time and the temperature of the target gas mixture wasfinally established as follows

lgtchem = 8734.8

T − 9.8863 (10)

For other fuels besides n-heptane, the same method can be used to obtain the relation betweenchemical ignition delay time and the temperature of the target gas mixture. Also, the relation betweenthe chemical ignition delay time and environment pressure is found to impact the droplet ignitiondelay time at different pressures. However, it is not applicable for the droplet ignition at a supercriticalstate.

2.4 Critical temperature

From the model, the temperature of the target gas mixture Tst(t~) changing with time t~ was

calculated, as a dashed curve shown in Fig. 1. Suppose that the target gas mixture has the possibilityto ignite at any physical time t~, the total droplet ignition delay time t~delay(t~) is the physical time t~ plusthe chemical ignition delay time t~chem(Tst) at Tst(t

~). Actually, the droplet ignition will occur when thetotal delay time reaches the minimum value.

t~delay = min (t~ + t~chem(Tst)) (11)

Represented in Fig. 1 by a solid line, there is obviously a minimum value of the total delaytime which is the physical time t~ plus the chemical ignition delay time t~chem(Tst). When the minimumvalue is found at time t~phy, the corresponding temperature of the target gas mixture at that time is atthe critical temperature Tcritical. Then the minimum value predicted is the droplet ignition delay time

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t~delay, which is composed of two parts, namely, the physical delay time t~phy and the chemical delaytime t~chem.

Apparently, the temperature of the target gas mixture Tst(t~) changing with time is first predicted

by the model, and then the total droplet ignition delay time t~delay(t~) is estimated by adding the chemicalignition delay time t~chem(Tst) to the physical time t~. Finally, find the critical time where the total dropletignition delay time has a minimum value t~delay, which is the droplet ignition delay time needed.

3. Model Validation

As shown in Fig. 2, the ignition curves of n-heptane droplets are much different from eachother at various environmental temperatures; dashed curves represent the target gas mixture tempera-ture and solid curves the total delay time as in Fig. 1. The black dots in Fig. 2 represent the minimumvalue locations or the predicted droplets ignition delay time. Very similar to the experimental results,the droplet ignition delay time decreases with the increasing environmental temperature. Furthermore,the physical delay plays a more important role at lower environmental temperatures because the target

gas mixture needs more time to reach the critical temperature.

As shown in Fig. 3, the predicted droplet ignition delay times agree very well with theexperimental results of Faeth and Olson [1] and the numerical simulation of Marchese [9]. In a wider

temperature range, the predicted droplet ignition delay times in Table 1 also agree well with thenumerical simulation results.

Fig. 1. Physical delay and chemical delay.

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Table 1. Theoretical and Numerical Results of Droplet Ignition Delay Time

Fig. 3. Comparison of droplet ignition delay time.

Fig. 2. Influence of environment temperature on ignition curve.

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4. Influence of Droplet Diameter

Droplet diameter has no influence on the relation between Tst and t~, because this parameterdoes not exist in the non-dimensional equations of the unsteady droplet evaporation model. As shownin Fig. 4 (solid curves), the temperature of the target gas mixture changing with time is the same fordifferent droplet diameters. By adding the non-dimensional chemical ignition delay time, the ignitioncurves for two droplets with different diameters are illustrated as dashed lines in Fig. 4 and black dots

Fig. 4. Influence of droplet diameter on ignition curve.

Fig. 5. Influence of droplet diameter on droplet ignition time.

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represent the non-dimensional droplet ignition delay times. The influence of droplet diameter on bothnon-dimensional and dimensional droplet ignition delay time is presented in Fig. 5.

The following conclusions can be reached from Figs. 4 and 5.

(1) With decreasing droplet diameter, the non-dimensional droplet ignition delay timeincreases, while the dimensional droplet ignition delay time decreases.

(2) For a smaller droplet, the critical temperature is higher, which means decreased chemicalignition delay time and increased non-dimensional physical delay time.

(3) The chemical delay plays a more important role for smaller droplets.

5. Discussion

An important assumption in this simple model on droplet ignition delay time is that the ignitionoccurs at the point where the equivalent ratio is equal to 1. However, the detailed numerical simulationresults indicated that the ignition positions were usually on the lean fuel side [4]. So further discussionis needed on whether the predicted results of the model are sensitive to this assumption.

If the droplet ignition is supposed to occur at different equivalent ratios, the droplet ignitiondelay times can still be calculated using the simple model. Only the value of C in Eq. (9) and thechemical ignition delay time in Eq. (10) need to be modified. Especially, the chemical ignition delaytimes corresponding to different equivalent ratios are different. Table 2 shows the calculated resultsby presuming the ignition position at different equivalent ratios. These results indicate that the dropletignition delay time is not very sensitive to the equivalent ratio at the ignition point, especially in thelow-temperature environment region. The assumption in the simple model, with the equivalent ratioequal to 1 at the droplet ignition point, is reasonable.

6. Conclusions

A simple model was proposed to predict the ignition delay time of a liquid fuel droplet in thisinvestigation. This model employs an unsteady droplet evaporation model to predict the temperatureevolution of the target gas mixture away from the droplet surface. This model has a clear physicalmeaning and can be used for fast estimation of the droplet ignition delay time. Also, the model is

Table 2. Sensitivity of Equivalent Ratio

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suitable for theoretically investigating the influence of important parameters, such as environmentaltemperature, pressure, and droplet diameter, on the droplet ignition delay.

Acknowledgment

This research is currently supported by the National Natural Science Foundation of China(No. 50628607), the Specialized Research Fund for the Doctoral Program of High Education (No.20060003011), and Tsinghua University (No. Jcpy2005050).

Literature Cited

1. Faeth GM, Olson DR. The ignition of hydrocarbon fuel droplets in air. SAE Trans1968;77:1793–1802.

2. Bernard JW, Willis AR Jr. An experimental study of fuel droplet ignition. AIAA J1969;7:2288–2292.

3. Tsue M, Ishimaru R, Ukita T et al. Spontaneous ignition of fuel droplets in lean fuel-airmixtures. J Propulsion Power 2006;22:1339–1347.

4. Stauch R, Lipp S, Mass U. Detailed numerical simulation of the autoignition of singlen-heptane droplets in air. Combustion and Flame 2006;145:533–542.

5. Stauch R, Mass U. The ignition of single n-heptane/iso-octane droplets. Int J Heat Mass Transf2007;50:3047–3053.

6. Law CK. Theory of thermal ignition in fuel droplet burning. Combustion and Flame1978;31:285–296.

7. Law CK, Chung SH, Srinivasan N. Gas-phase quasi-steadiness and fuel vapor accumulationeffects in droplets burning. Combustion and Flame 1980;38:173–198.

8. Peters N, Paczko G, Seiser R, Seshadri K. Temperature cross-over and non-thermal runawayat two-stage ignition of n-heptane. Combustion and Flame 2002;128:38–59.

9. Marchese AJ. Numerical modeling of isolated n-alkane flames: initial comparisons withground and space-based microgravity experiments. Combustion and Flame 1999;116:432–459.

"F F F"

Originally published in Sciencepaper Online (www.paper.edu.cn), 2008, 1–8.Translated by Jun Chen, Laboratory of Phase Change and Interfacial Transport Phenomena, Depart-

ment of Thermal Engineering, Tsinghua University, Beijing 100084, China.

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