Paper # 070IC-0133 Topic: Internal combustion and gas turbine engines

*Corresponding author: myen@purdue.edu

8th

U. S. National Combustion Meeting

Organized by the Western States Section of the Combustion Institute

and hosted by the University of Utah

May 19-22, 2013

Modeling Lifted Diesel Jets: Insights into the Correlation

between Flame Lift-Off Height and Soot Formation

May Yen

1* John Abraham

1,2

1School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA 2School of Mechanical Engineering, University of Adelaide, Adelaide, South Australia 5005,

Australia

Lift-off in reacting diesel jets is of interest because soot concentration in the jet has a suggested correlation

with it. It has been suggested that larger lift-off height enables greater fuel/air mixing upstream of the lift-off

height which, in turn, leads to reduced formation rates of soot. In this work, computations of reacting diesel

jets, including soot and NO, are carried out for a wide range of conditions by employing a RANS model in

which an unsteady flamelet progress variable (UFPV) submodel is employed to represent

turbulence/chemistry interactions. The conditions selected reflect changes in injection pressure, chamber

temperature, oxygen concentration, and density, and orifice diameter. As reported in prior work, the UFPV

model predicts the ignition delay and flame lift-off height within about 25% of reported measurements. The

flame is found to stabilize at the location where the ignition scalar dissipation rate is equal to the local scalar

dissipation rate at the stoichiometric mixture fraction. For all cases, except the cases with different orifice

diameter and ambient density the soot concentration is found to correlate with the lift-off height. Analysis of

the entrained mass upstream of the lift-off height confirms that this correlation arises from variation in

entrained mass.

1. Introduction

Understanding soot formation in reacting sprays as it relates to lift-off height Lf is important as

the behavior of Lf can then be a predictor of soot formation in engines. It has been shown that the

premixing of fuel and air upstream of the Lf has a significant effect on the formation of soot in the

jet. The larger the Lf, the more air is entrained into the jet upstream of Lf. As the ratio of air

entrainment rate to fuel mass flow rate increases, the less soot is formed (Siebers, 2001; Pickett and

Siebers, 2003). Experiments have shown that changes in Lf caused by changes in ambient conditions

and injection pressure affect the fuel-air mixture at the lift-off height (Picket and Siebers, 2004).

The work of Bajaj et al. (2013) is of special relevance to the present work. Bajaj et al. computed

ignition delays and lift-off heights for a set of measured conditions. In their work, n-heptane jets

were computed for cases which reflected changes in injection pressure, chamber temperature,

oxygen content, orifice diameter, and chamber density. The model they employed will be discussed

in the next section. Table 1 lists the measured conditions they computed. In Table 1, the parameters

dnoz, Pinj, Pamb, Tfuel, Tambient, ρambient, and O2 % represent the nominal injector orifice diameter, the

2

injection pressure, the pressure in the chamber, the temperature of the fuel, the chamber temperature,

the chamber density, and the percentage of oxygen in the chamber, respectively. The structure of

vaporizing diesel sprays in conventional diesel engines under high pressure and high temperature

conditions has been shown to be momentum-controlled and it can be well-approximated using vapor

jets with the same mass and momentum flow rates as the liquid spray (Iyer and Abraham 1997;

Abraham and Pickett, 2010; Bajaj et al., 2011). In Table 1, dgas is the equivalent diameter of an

injector that injects the vapor.

Table 1. Computed conditions

Case dnoz

(mm)

dgas

(mm)

Pinj

(mm)

Pamb

(mm)

Tfuel

(mm)

Tambient

(mm)

ρambient

(mm) O2%

1 0.1 0.199 150 42.66 373 1000 14.8 21

2 0.1 0.199 60 42.66 373 1000 14.8 21

3 0.1 0.1745 150 55.45 373 1300 14.8 21

4 0.1 0.2097 150 38.39 373 900 14.8 21

5 0.1 0.199 150 43.02 373 1000 14.8 15

6 0.1 0.199 150 43.2 373 1000 14.8 12

7 0.1 0.199 150 43.45 373 1000 14.8 8

8 0.18 0.3858 140 42.66 373 1000 14.8 21

9 0.1 0.1397 150 86.47 373 1000 30.0 15

Table 2 shows the computed ignition delays and lift-off heights for the 9 cases of Table 1. It can

be seen that in general the lift-off heights agree within 25% and ignition delays within 30% of

measured values. The normalized lift-off height Lf in Table 2 will be explained later. The present

work is an extension of the work of Bajaj et al. and computes the soot and NO in the same 9 jets of

Table 1. The next section will discuss the computational model employed. Results and discussion

will follow. The paper will close with summary and conclusions.

Table 2. Computed and measured ignition delay and lift-off height

Case

Ignition Delay (ms) Lift-off Height (Lf)

Measured Computed Measured (cm) Computed (cm) Normalized

(Lf*)

1 0.53 0.542 17.00 18.50 929.65

2 -- 0.615 13.50 15.05 756.28

3 0.26 0.209 7.70 8.05 461.32

4 0.79 0.89 25.50 23.30 1111.11

5 0.73 0.56 23.20 22.90 1150.75

6 0.947 1.225 29.20 27.30 1371.86

7 1.52 2.17 42.30 52.88 2657.29

8 0.57 0.65 23.97 25.80 668.74

9 0.38 0.175 11.90 12.00 1222.97

2. The Computational Model

The REC code employed by Bajaj et al. (2013) is used in this work. Turbulence is modeled using

the k-ε model with boundary layers modeled using wall functions. The REC model has been used in

computing diesel jets in many prior studies (for example, Iyer and Abraham, 1997; Abraham and

Pickett, 2010). Turbulence/chemistry interactions are modeled using the unsteady flamelet progress

3

variable (UFPV) model employed by Bajaj et al. (2013). In the model, the averaged chemical source

terms are determined using the local temperature T, local mixture fraction Z, and local scalar

dissipation rate χ. Instantaneous (non-averaged) chemical source terms are tabulated in libraries as a

function of mixture fraction Z, stoichiometric scalar dissipation rate χst, and the stoichiometric

progress of reaction variable Cst. The instantaneous source terms are obtained by solving the flamelet

equations

, (1)

where is a vector that represents all of the reactive scalars: temperature and species mass fraction.

is the rate of change of the reactive scalar, is the source term of reactive scalar, is the local

scalar dissipation rate defined as

| | , (2)

where D is the molecular diffusivity. In mixing layers may be assumed to depend on Z. An error-

function profile, given by the expression below, is often employed to represent this dependence

(Peters, 1984):

( )

( )

, (3)

where and are the stoichiometric scalar dissipation rate and mixture fraction, respectively.

These error function profiles for various specified values of are employed in Eq. (1) when

generating the flamelet profiles and tabulated libraries. The progress of reaction variable C is defined

as

, (4)

where the local temperature T, adiabatic flame temperature Ta, and unburned temperature Tu are

dependent on the local mixture fraction Z.

Note that tabulating the entries in the library as a function of all values of C at all values of Z

generates a very large library. This need for tabulation can be simplified if an assumption is made

that C(Z) can be characterized by the progress variable Cst at the stoichiometric mixture fraction.

Unlike for , an analytical expression does not exist for the C(Z) profile; but, this profile can be

obtained from tabulated values of C as a function of Z. Hence, Z, Cst, and are the independent

parameters in the library. The instantaneous source terms are tabulated as a function of these

independent variables. In REC, the averaged source terms are required. These are obtained by

convolving the instantaneous variable with the joint probability density function (PDF) of the

independent variables, i.e.

∭ ( ) , (5)

where is the Fávre averaged source term.

The soot is modeled using a kinetic mechanism (Frenklach Wang, 1991; Appel et al., 2000). In

this model, the polycyclic aromatic hydrocarbons (PAH), from which soot forms, are formed by

Hydrogen-Abstraction-Carbon-Addition (HACA). The method of moments is then used to solve for

the soot volume fraction and the soot number density (Frenklach and Harris, 1987; Gopalakrishnan

and Abraham, 2004). In the method of moments, particle mass is expressed as

(6)

4

where j is the number of mass units per particle and is the smallest unit of mass in the particle

which in this model is C2 for the reason that C2H2 is the precursor to soot (and ignoring the mass of

the hydrogen). The particle number density is expressed as

, (7)

where ρ is the density, Yi is the mass fraction of the species i, MWj is the molecular weight for

particle size j. The statistical moment of the distribution of soot particle sizes is

∑ , where r = 0,…, ∞. (8)

One can see that if r is set to 0, the first moment M0 becomes the total soot number density. When r

is set to 1, the second moment M1 becomes the total soot in units of m1 per unit volume is obtained.

This can then be used to calculate the volume fraction of soot, i.e.

, (9)

where the density of soot, ρs, is assumed to be 1.8 g/cm3

. NO formation is modeled using a sub-

mechanism taken from GRI Mech 3.0 (Smith, 1999; Gopalakrishnan and Abraham, 2004) . When extending the UFPV model to compute soot and NO, the following approach is used: the

NO mass fraction, soot volume fraction, and soot number density are tabulated as a function of the

same three variables as for the chemical source terms described earlier. However, since soot

variables do not reach equilibrium values, unlike temperature and species mass fractions, time is

employed as the progress variable for the soot variables.

A 44-species, 185-step reaction mechanism is employed to model n-heptane oxidation (Bajaj et

al., 2013). This mechanism is not suitable for the soot and NO kinetics considered in this study. For

this purpose, a 160-species, 1995- step reaction mechanism is employed. The use of these different

mechanisms gives rise to a dilemma. When using the RANS model, the average scalar dissipation

rate is modeled as (Jones and Whitelaw, 1982)

, (10)

(missing overbar on variance) where Cχ is a constant and Z”2 is the variance of the mixture fraction.

The choice of Cχ determines the numerical value of the scalar dissipation rate. In particular, the

choice will determine the physical distribution of the scalar dissipation rates in the jet. Bajaj et al.

(2013) concluded that the lift-off height was at the location where the ignition scalar dissipation rate

matched the local scalar dissipation rate, i.e. the predicted lift-off height will depend on Cχ. The two

reaction mechanisms employed have different ignition and extinction scalar dissipation rates. Hence,

the value of Cχ which predicts measured parameters will be different for variables computed

employing the two mechanisms. For the Lf and ignition delay predictions using the 44-species, 185

reaction mechanism for n-heptane for Cχ was found to be 6.5 (Bajaj et al., 2013).This constant is,

however, unlikely to be applicable when the 160-species 1995-step mechanism is employed. In

practice, the value has to be selected to match computed and measured results, i.e. it is an adjustable

constant. For the initial set of results presented below, a value of 325 was employed. The sensitivity

of the results to the choice of the constant will be discussed.

3. Results and Discussion

Figures 1 (a) – (e) show the development of flooded mixture fraction contours at various times

after start of injection (ASI) for Case 1 of Table 1 and Figs. 2 (a) – (e) show the development of the

corresponding flooded temperature contours (Bajaj thesis; Bajaj et al., 2013). At about 0.55 ms,

ignition occurs about 3.4 cm downstream of the orifice, near the leading tip of the jet (Fig. 2(a)). As

described by Bajaj et al. (2013), an ignition front propagates outwards from the point of ignition

5

toward the stoichiometric surface (Fig. 2(b)) followed by flame front propagation upstream along the

stoichiometric surface (Fig. 2 (c, d)). Meanwhile the jet penetrates farther into the chamber (compare

Fig. 2(c) with Fig. 2(b)). The flame that propagates upstream stabilizes at a lift-off height of 1.8 cm

where the ignition scalar dissipation rate matches the local scalar dissipation rate (Fig. 2 (d)). As the

reacting jet continues to develop, the change in lift-off height is negligible (Fig. 2 (e)).

(a) (b) (c) (d) (e)

Figure 1. Development of mixture fraction contours in jet, case 1

(b) (b) (c) (d) (e)

Figure 2. Development of temperature contours in the jet, case 1

Now we will present the computed results of NO and soot in the jet, starting with the evolution

of soot and NO for Case 1 at various times after ignition at the same 5 instants as Fig. 2. Figures 3

and 4 show the soot volume fraction and NO mass fraction, respectively, in the jet. Obviously, the

selection of the cut-off values for the volume fraction and mass fraction will affect the visual results.

For the selected contour values, soot is not noticeable in the first 3 time instants. Subsequently, the

soot volume fraction increases with time and the peak value is observed at increasing axial distances

as time increases (compare Figs. 3(d) and (e)). This reflects the combined effect of the soot being

advected downstream and additional soot being generated in the jet with increasing time. The peak

soot concentrations are confined to the center of the jet along the axis near the head vortex of the jet.

NO is produced soon after ignition (Fig. 4(b)). Peak NO is located where mixture fraction is at or

slightly leaner than stoichiometric and temperatures are high.

(a) (b) (c) (d) (e)

Figure 3. Development of soot volume fraction, Case 1

6

(a) (b) (c) (d) (e)

Figure 4. Development of NO mass fraction, Case 1

Figure 5. Soot volume fraction for the 9 cases of Table 1 at 4 ms ASI.

Figure 6. NO mass fraction for the 9 cases of Table 1 at 4 ms ASI.

7

Case 1 Case 6

Case 5 Case 9

Figure 7. Quasi-steady soot volume fraction distributions (www.sandia.gov/ecn/).

Figures 5 and 6 show the soot and NO distribution, respectively, for the 9 cases of Table 1 at 4

ms ASI. Figure 7 shows the measured values. Quasi-steady experimental results of soot volume

fraction found by combining time averaged line-of-extinction data and soot profile LII imaging are

available for Cases 1, 5, 6, 7, and 9 (Engine Combustion Network, Sandia). The time averaging is

carried out across several time intervals based on the axial and cross stream position. For instance,

along the axis between 30-55 mm downstream of the orifice the averaging is carried out from 1.5 to

6.5 ms ASI and averaged between 2 to 7 ms for axial distances past 60 mm for Case 1 and averaged

between 3.5 to 7 ms ASI for sections of Case 6. The specific times and their corresponding axial and

cross stream positions can be found at www.sandia.gov/ecn. Note that the computed results are

shown at 4 ms ASI. Hence, it is expected that there will be some differences. Nevertheless, there is

qualitative agreement of the measured and computed results in terms of location of peak values and

distribution. Table 3 shows the mass of soot msoot and mass of NO mNO in the chamber at 4 ms ASI

for the 9 cases. Also in Table 3 are msoot* and mNO* which are values of soot and NO which have

been normalized by the total mass of fuel injected during 4 ms.

Table 3. Soot and NO mass at 4 ms ASI

Case msoot (g) mNO (g) msoot * mNO *

1 6.01E-08 1.16E-06 2.23E-05 4.29E-04

2 5.51E-08 7.18E-07 3.24E-05 4.22E-04

3 9.63E-08 3.12E-06 3.57E-05 1.15E-03

4 3.81E-08 7.73E-07 1.41E-05 2.86E-04

5 1.43E-08 2.83E-07 5.73E-06 1.13E-04

6 3.22E-09 1.77E-07 1.29E-06 7.09E-05

7 1.35E-14 1.71E-08 5.41E-12 6.86E-06

8 3.62E-07 3.67E-06 3.96E-05 4.01E-04

9 1.26E-08 2.48E-07 4.66E-06 9.19E-05

Next, the correlation of the msoot with and entrained mass at the will be examined. It

seems reasonable to assume that the more the air entrained, the lower the amount of soot formed. In

8

particular, it appears reasonable to suggest that the more the air entrained upstream of the lift-off

height, the lower the amount of soot formed downstream of the (Siebers and Higgins, 2001). The

air entrained downstream of the lift-off height is reacted in the flame front. The rate of entrained

mass flow rate normalized by the injected mass flow rate is given by the following

expression (Abraham, 1996):

(

)

, (11)

where K is a constant, is the mass flow rate of entrained air, is the mass flow rate of fuel

injected, x is the axial distance from the orifice, d is the diameter of the nozzle, is the density of

ambient chamber air, and is the density of the injected fuel. In fact, the combination of variables

(1/d)( / )1/2

can be considered to be a normalizing variable for the distance. In Table 2, the

normalized values of lift-off heights are shown in the last column where

(

)

(

), (12)

where, for reference, ρb is taken to be the density of the baseline Case 1. It is interesting to compute

the ratio of entrained to injected mass flow rate at the lift-off height for the 9 cases. Results for

reacting and non-reacting jets are given in Table 4 along with the results from Eq. (11) where K is

chosen to be 0.32 for quasi-steady jets.

Table 4. Computed and calculated at lift-off height

CASE

Computed Calculated

Reacting Non

react Eq. (1)

1 10.56 9.81 9.78

2 9.18 8.24 7.96

3 4.13 3.96 4.30

4 14.89 14.04 8.89

5 14.40 12.78 12.11

6 16.24 15.48 14.43

7 29.19 30.48 27.96

8 7.33 6.77 7.04

9 7.08 8.89 9.03

Comparing the lower injection pressure Case 2 with its baseline Case 1, Table 2 shows that the

Lf* is shorter in Case 2 although the ignition scalar dissipation rates are the same for both cases,

because the local scalar dissipation for Case 2 is lower as a result of the lower injection velocity.

Equation (11) shows that at the same axial distance, ⁄ is the same; but, because the Lf is

shorter, the ratio is smaller in Case 2 as shown in Table 4. Comparing the normalized value of soot

for Case 2 with Case 1, Case 2 is higher.

In Case 3 the ambient temperature has been increased to 1300 K which increases the ignition

scalar dissipation rate therefore decreasing the lift-off height. This decreases the ⁄ at the lift-

off height and increases the normalized soot compared to Case 1. Cases 4-7 follow the same

argument as Case 3 where a decrease in ignition scalar dissipation rate as a result of lower

9

temperature in Case 4 and progressively lower oxygen concentrations in Cases 5-7 increases the Lf*

and therefore increases the ⁄ leading to decreased amounts of normalized soot.

Case 8 is an interesting case because the nozzle diameter is 1.8 times greater compared to Case

1. The ignition scalar dissipation rate of Case 1 and Case 8 are equal. The results suggest that the

increase in diameter increases the local scalar dissipation rate and therefore increases the Lf. When

normalized, however, Lf* is shorter than Case 1 as seen in Table 2. Equation (11) supports this by

showing that the effect of the increase in diameter results in ⁄ being less than that of Case 1,

increasing normalized soot.

When the chamber density increases in Case 9, the ignition scalar dissipation rate increases. Note

that the oxygen content of Case 9 is 15% and the results must be compared with Case 5. Increasing

the ignition scalar dissipation rate allows the flame to travel farther upstream before the flame

stabilizes. When Lf is normalized from the effects of increased chamber density and decreased

effective diameter, Lf*is higher than the Lf* of Case 5 which explains the decrease in normalized

soot.

In the case of the mass of NO, no discernible trend with lift-off height is observed. For the three

cases with the same ignition scalar dissipation rate, Cases 1, 2, and 8, but with different injected

mass of fuel, the normalized NO is relatively the same suggesting that the differences in NO are on

account of the differences in fuel mass injected. When temperature increases (Case 3) the NO

increases and similarly when the temperature decreases (Case 4) the NO decreases. In Cases 5, 6,

and 7 where the oxygen content is reduced to 15%, 12%, and 8 %, reducing oxygen will reduce the

temperature and, hence, the mass of NO. The predominant effect, not surprisingly, is that of

temperature. The mass of NO is less in Case 9 than Case 5 because the mass of fuel injected is the

same but there is more inert in the chamber as a result of the higher density thus lowering the

temperature.

(a) (b)

Figure 8. Actual (a) and normalized (b) soot mass at 4 ms vs lift-off height (actual in (a) and

normalized in (b)).

Figure 8 summarizes the results showing mass of soot as a function of the lift-off height. Before

normalization (Fig. 8(a)), the correlation is evident, but has obvious outliers in cases 3, 4, 8, and 9.

When the cases are normalized by injected mass flow rate, and nozzle diameter and chamber

density, we see clearly that the normalized soot decreases for increasing normalized lift-off height.

The increased diameter case is an outlier in Fig. 8 (b). This needs to be investigated further.

0 10 20 30 40 50 600

1

2

3

4x 10

-7

Lf (cm)

so

ot

(g)

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

0 500 1000 1500 2000 2500 30000

1

2

3

4x 10

-5

Lf*

soot*

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

10

(a) (b)

Figure 9. NO mass at 4 ms vs lift-off height (actual in (a) and normalized in (b))

Figure 9 shows the corresponding results for the mass of NO. When normalized, the NO* vs Lf*

plots show that generally as lift-off height increases, the NO* decreases. This is primarily because

the cases where the lift-off height increases are the same cases where the flame temperature

decreases. Cases 1, 2, and 8 have approximately the same amount of NO* because the flame

temperature is about the same for the three cases.

The value of Cχ in Eq. (10) employed for obtaining the soot and NO results above was 325 and

6.5, respectively. It is interesting to assess the sensitivity of the conclusions to the value of the

constant. Figure 11 shows the normalized results for soot and NO when the constants selected are

both 19.5. The trends and conclusions have not changed. While the trends are preserved, the Cχ

value is important in determining the distribution of soot and NO in the chamber as can be seen

when comparing Fig. 5 and Fig. 13.

(a) (b)

Figure 10. Soot mass at 4 ms vs lift-off height for Cχ of 19.5

0 10 20 30 40 50 600

1

2

3

4x 10

-6

Lf (cm)

NO

(g)

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2x 10

-3

Lf*

NO

*

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2x 10

-6

Lf (cm)

so

ot

(g)

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

-4

Lf*

so

ot*

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

11

(a) (b)

Figure 11. NO mass at 4 ms vs lift-height for Cχ of 19.5

Figure 13. Soot volume fraction for the 9 cases of Table 1 at 4 ms ASI for Cχ of 19.5.

4. Summary and Conclusions

The earlier work of Bajaj et al. (2013) on modeling flame lift-off in diesel jets is extended to

include the modeling of soot and NO in the lifted jets. Kinetic models are employed to compute the

soot and NO. The unsteady flamelet progress variable (UFPV) model employed by Bajaj et al. is

employed. The computed distribution of soot in the jet is found to be qualitatively similar to

measured distributions. When the soot mass and lift-off heights are appropriately normalized, the

results show that the normalized mass of soot correlates well with the normalized lift-off height, i.e.

higher lift-off height results in lower soot mass. No such correlation is evident for NO where the

predominant correlation is with flame temperature.

It is important to note that these results and conclusions are applicable only during the period of

injection. The soot-NO trade-off in an engine is dependent on oxidation characteristics of soot once

injection ends, i.e. during the expansion stroke. In fact, the oxidation effects may be the dominant

controlling factor. Hence, extending the results of this study to explain engine exhaust behavior is

not possible.

0 10 20 30 40 50 600

0.5

1

1.5

2x 10

-6

Lf (cm)

NO

(g)

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

0 500 1000 1500 2000 2500 30000

1

2

3

x 10-4

Lf*

NO

*

case 1

case 2

case 3

case 4

case 5

case 6

case 7

case 8

case 9

12

Acknowledgements

The authors are grateful to Caterpillar, Inc. for providing the funding for this work.

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