**Ignition Process in a Non-Homogeneous Mixture**

Hiroshi Kawanabe

*Graduate School of Energy Science, Kyoto University, Japan* 

### **1. Introduction**

154 Chemical Kinetics

Yamashita, D. Kweon, S. Sato, S. & Iida, N. (2005). The Study on Auto-ignition and

Sjoberg, M. John E. D. & Cernansky, N. P. (2005). Potential of Thermal Stratification and

Kumano K. & Iida, N. (2004). Analysis of the Effect of Charge Inhomogeneity on HCCI

Luz A.E. Rupley F. & Miller J.A. (1989). CHEMKIN-II:A FORTRAN ChemicalKineticsPacage

Luz A.E. Kee R.J. & Miller J.A. (1988). SENKIN:A FORTRAN Program for Predicting

Curran, H.J. Pitz, W.J. Westbrook, C.K. Dagaut, P.B. Boettner, J-C & Cathonnet, M. (1998).

Multi-Zone Modeling and Experiments, *SAE,* SAE paper 2005-01-0113 Kwon, O. & Lim O. (2009). Effect of the Boost Pressure on Thermal Stratification on

Engines,*Transaction of JSAE*, Vol.36, No.6, pp85-90

*National Laboratories Report*, SAND87-8248

*Chemical Kinetics*, 30-3, 229-241 GRI-Mech3.0, http://www.me.berkeley.edu/gri\_mech

Combustion Process of the fuel Blended with Methane and DME in HCCI

Combustion Retard for Reducing Pressure-Rise Rates in HCCI Engines, Basedon

HCCI Engine using multi-zone modeling. *Trans. of the KSME* (B)Vol.33,

Combustion by Chemiluminescence Measurement, SAE paper 2004-01-

for the Analysis of Gas-Phase Chemical Kinetics. *Sandia National LaboratoriesReport*,

Homogeneous Gas Phase Chemical Kinetics With Sensitivity Analysis, *Sandia* 

AWide Range Modeling Study of Dimethyl Ether Oxidation, *International Journal* 

**6. References** 

No.3(accepted)

SAND89-8009B

1902

An auto-ignition process of a non-homogeneous mixture is investigated using a numerical calculation based on chemical kinetics and the stochastic approach. This type of autoignition phenomenon is considered as a fundamental process of the initial stage of diesel combustion (Ishiyama, et al., 2003) or homogeneous charged compression ignition (Shimasaki, et al., 2003). In order to investigate these combustion processes, many numerical calculations have been performed and for many of those it has been assumed that the ignition process is dominated by the turbulent mixing (Kong and Reitz, 2000).

However, the fuel-air mixing and chemical reaction progress happen simultaneously for these types of combustion processes. Due to the long ignition delay time and high homogeneity of the mixture, combustion characteristics, such as ignition delay and combustion duration, could be affected equally by non-homogeneity of the mixture, turbulent mixing rate and chemical reaction rate. Therefore, the understanding of those combustion mechanisms is incomplete due to the complexity of the phenomena in which the mixing and chemical processes interact with each other. The main purpose of this chapter is to estimate these effects quantitatively using a numerical method.

Here, n-heptane is assumed as a fuel and its reaction process is calculated by means of a reduced mechanism (Seiser et al., 2000). The non-uniform states of turbulent mixing are statistically described using probability density functions and the stochastic method, which was newly developed from Curl's model (Curl, 1963). Focusing on the effects of mixture heterogeneity on combustion characteristics, such as ignition delay and combustion intensity (rate of temperature rise), the evolution of chemical reactions was calculated for the mixture in which variance in fuel-mass fraction decreases at given rates from the initial value under a fixed mean fuel-mass fraction and a constant pressure. The results show that the start timing of the low-temperature oxidation and ignition delay period are hardly affected by the equivalence-ratio variation, however, combustion duration increases with increasing variance. Furthermore, the combustion duration is mainly affected by the non-homogeneity at the ignition and is not much affected by the mixing rate.

Ignition Process in a Non-Homogeneous Mixture 157

Fig. 1. Course of normalized variance *vm*/*vmi* against non-dimensional time *t*\*

Fig. 2. Temporal change in PDF of fuel-mass fraction *mf*

Here, n-heptane is assumed as a fuel and a chemical reaction system is described by a semidetailed kinetic mechanism (Seiser et al., 2008[Web]). This system consists of 160 chemical

By means of the above-mentioned procedure, an ignition process of non-homogeneous mixture with a constant mixing rate is calculated. For simplicity, the calculation is performed under a constant pressure. The initial temperature *Ti* and pressure *p* are set to

species and 1540 elemental reactions, and is selected for reducing computation time.

**2.2 Chemical reaction model** 

**3. Results and discussion** 

#### **2. Numerical models**

#### **2.1 Mixing model**

In order to describe the homogenization process of a fuel-air mixture by turbulent mixing, the statistical state of the mixture is expressed by means of probability density function (PDF) and the dissipation process is described by a particle-interaction model. In the particle-interaction model, the statistical state *f* of scalar at time *t* is represented by the *N* delta functions ;

$$f\left(\phi;t\right) = \frac{1}{N} \sum\_{n=1}^{N} \delta\left(\phi - \phi\left(n\right)\right) \tag{1}$$

Here, \*(*n*) indicates value on the nth particle. In the coalescence/dispersion model proposed by Curl (1963), the state of *f* at *t* + *t* is calculated from the collision frequency . The compositions of a pair of particles, which are selected at random (denoted by *n*1 and *n*2), change as;

$$\left. \phi^\* \left( n\_1 \right) \right|\_{t=t+\Lambda t} = \phi^\* \left( n\_2 \right) \Big|\_{t=t+\Lambda t} = \frac{1}{2} \left( \phi^\* \left( n\_1 \right) \Big|\_{t=t} + \phi^\* \left( n\_2 \right) \Big|\_{t=t} \right) \tag{2}$$

Then, within the time interval *t*, this operation is repeated *tN*times.

The change in the value of \* is sometimes large, especially in the early stages of mixing, because it is calculated as the average of significantly different quantities of two particles. This tends to cause unrealistic change in the progress of chemical reactions. The modified Curl's model (Janicka, Kolbe and Kollmann, 1979) is one possible method to mitigate this tendency. In this method, the scalar exchange of a pair of particles (*n*1 and *n*2) is calculated from a uniform random number *C* between 0 and 1;

$$\begin{aligned} \left. \phi^\* \left( \boldsymbol{n}\_1 \right) \right|\_{t=t + \Delta t} &= \phi^\* \left( \boldsymbol{n}\_1 \right) \big|\_{t} + \frac{1}{2} \mathbb{C} \Big( \phi^\* \left( \boldsymbol{n}\_2 \right) \big|\_{t=t} - \phi^\* \left( \boldsymbol{n}\_1 \right) \big|\_{t=t} \Big) \\ \left. \phi^\* \left( \boldsymbol{n}\_2 \right) \big|\_{t=t + \Delta t} &= \phi^\* \left( \boldsymbol{n}\_2 \right) \big|\_{t} - \frac{1}{2} \mathbb{C} \Big( \phi^\* \left( \boldsymbol{n}\_2 \right) \big|\_{t=t} - \phi^\* \left( \boldsymbol{n}\_1 \right) \big|\_{t=t} \Big) \end{aligned} \tag{3}$$

This operation is repeated 3/2*tN* times and can describe statistically the same process as Curl's model. However, the value of \**t*+*<sup>t</sup>* still changes significantly if *C* is near 1. Here, in order to avoid a significant difference between \**t* and \**t*+*<sup>t</sup>*, *C* is fixed to a value sufficiently smaller than unity and the replacement procedures are repeated *tN*/(2*C*) times. In order to confirm the consistency between the present model and Curl's model, a mixing process was calculated starting from the initial state in which fuel and air are perfectly separated.

Time change of variance *vm* for fuel-mass fraction mf distribution is shown in Fig. 1. Here, *vm* is normalized with respect to initial variance *vmi* and dimensionless time *t*\* = *t* is used. *C* = 0.05 and *N* = 1000 are adopted. As shown, the time changes of *vm*/*vmi* calculated by these three methods are completely equivalent. Meanwhile, Figure 2 shows the time change in *mf*-PDF. The distributions at smaller *t* calculated by the present model are similar to Gaussian rather than to the PDFs by the Curl's and modified Curl's models.

In order to describe the homogenization process of a fuel-air mixture by turbulent mixing, the statistical state of the mixture is expressed by means of probability density function (PDF) and the dissipation process is described by a particle-interaction model. In the

> 1

 

*N*

*n f t n N*

The compositions of a pair of particles, which are selected at random (denoted by *n*1 and *n*2),

1 2 12 <sup>1</sup> \* \* \*\* <sup>2</sup> *tt t tt t t t t t*

*n n nn*

*t*, this operation is repeated

because it is calculated as the average of significantly different quantities of two particles. This tends to cause unrealistic change in the progress of chemical reactions. The modified Curl's model (Janicka, Kolbe and Kollmann, 1979) is one possible method to mitigate this tendency. In this method, the scalar exchange of a pair of particles (*n*1 and *n*2) is calculated

*tt t t t t t t*

times and can describe statistically the same process as

*<sup>t</sup>* still changes significantly if *C* is near 1. Here, in

*tN*

*<sup>t</sup>*, *C* is fixed to a value sufficiently

*tt t t t t t t*

1 1 21

*n n Cn n*

<sup>1</sup> \* \* \*\* <sup>2</sup>

<sup>1</sup> \* \* \*\* <sup>2</sup>

2 2 21

*n n Cn n*

\**t* and \**t*+

to confirm the consistency between the present model and Curl's model, a mixing process was calculated starting from the initial state in which fuel and air are perfectly separated.

Time change of variance *vm* for fuel-mass fraction mf distribution is shown in Fig. 1. Here, *vm*

0.05 and *N* = 1000 are adopted. As shown, the time changes of *vm*/*vmi* calculated by these three methods are completely equivalent. Meanwhile, Figure 2 shows the time change in *mf*-PDF. The distributions at smaller *t* calculated by the present model are similar to Gaussian

1

;

 

> 

 

> \**t*+

is normalized with respect to initial variance *vmi* and dimensionless time *t*\* =

*tN*

smaller than unity and the replacement procedures are repeated

rather than to the PDFs by the Curl's and modified Curl's models.

from a uniform random number *C* between 0 and 1;

order to avoid a significant difference between

Curl's model. However, the value of

This operation is repeated 3/2

value on the nth particle. In the coalescence/dispersion model

(2)

*tN*times.

\* is sometimes large, especially in the early stages of mixing,

(1)

*t* is calculated from the collision frequency

at time *t* is represented by the *N*

.

(3)

/(2*C*) times. In order

*t* is used. *C* =

**2. Numerical models** 

;

\*(*n*) indicates

Then, within the time interval

The change in the value of

particle-interaction model, the statistical state *f* of scalar

proposed by Curl (1963), the state of *f* at *t* +

**2.1 Mixing model** 

delta functions

Here, 

change as;

Fig. 1. Course of normalized variance *vm*/*vmi* against non-dimensional time *t*\*

Fig. 2. Temporal change in PDF of fuel-mass fraction *mf*

#### **2.2 Chemical reaction model**

Here, n-heptane is assumed as a fuel and a chemical reaction system is described by a semidetailed kinetic mechanism (Seiser et al., 2008[Web]). This system consists of 160 chemical species and 1540 elemental reactions, and is selected for reducing computation time.

#### **3. Results and discussion**

By means of the above-mentioned procedure, an ignition process of non-homogeneous mixture with a constant mixing rate is calculated. For simplicity, the calculation is performed under a constant pressure. The initial temperature *Ti* and pressure *p* are set to

Ignition Process in a Non-Homogeneous Mixture 159

Fig. 4. Temporal changes of Ta due to low temperature oxidation

(a) *mfa* = 3.83×10-2 (

Fig. 5. Effect of mean mass-fraction *mfa*

similar to the result of

result of

*a* = 0.6) (b) *mfa* = 1.17×10-1 (

*<sup>a</sup>* = 1.0 in Fig. 3. For the lean case (a), due to the longer delay of hot

is fixed to *vmi* = 4.34×10-5. For the two conditions, histories of the mean temperature *Ta* are

flame heat release, the mixture is thought to be more homogeneous and to provide rapid temperature rise. However, the temperature-rising rate at ignition is lower compared to the

large, whereas the temperature rise at the ignition is steep, similar to the homogeneous case.

*<sup>a</sup>* = 1. Meanwhile, for the rich case (b), the variance *vm* at the ignition is comparably

*<sup>a</sup>* = 2.0)

*Ti* = 900K and *p* = 4MPa. The initial distribution of fuel-mass fraction is Gaussian with a mean value *mfa* and a variance *vmi*. Figure 3 shows time histories of mean temperature *Ta* and variance *vm* for the initial condition of *mf* = 6.22×10-2 and *vmi* = 4.34×10-5, corresponding to a mean equivalence ratio a of unity and the standard deviation of of approximately 0.1. The collision frequency is varied within the range of 0 s-1 to 10000 s-1. For comparison, the result for the case of the homogeneous mixture is also shown. Collision frequency is usually given by = *Cm* /*k* for the PDF model combined with CFD using the *k*-turbulence model.

Fig. 3. Courses of mean-temperature *Ta* and *vm* for the non-homogeneous mixture (*<sup>a</sup>* = 1.0)

Here, *k* and represent turbulence energy and its dissipation rate, respectively. *Cm* is fixed at 2, therefore, = 2000 s-1 corresponds approximately to turbulence intensity of 5 m/s and scale of 3 mm. For these calculations, the number of fluid particles is fixed to *N* = 100. Generally, as collision frequency increases, the start of heat release by hot flame delays and maximum heat release rate increases simultaneously to approach the result of a homogeneous mixture. Meanwhile, the time when the temperature reaches 50% of adiabatic flame temperature is approximately constant regardless of the collision frequency . The reason will be discussed later in Fig. 8. Furthermore, Fig. 4 shows a magnified view of Fig. 3 around the starting points of heat release by cool flame (marked as 'A'). The change in mean temperature *Ta* is exactly the same, in spite of the change of .

Next, in order to clarify the effect of mean equivalence ratio, calculations are performed for lean and rich cases. Fig. 5 shows *Ta* and *vm* under the conditions of (a) *mfa* = 3.87×10-2 (*<sup>a</sup>* = 0.6) and (b) *mfa* = 1.17×10-1 (*a* = 2.0) for the same as in Fig. 3. Here, initial value of variance

*Ti* = 900K and *p* = 4MPa. The initial distribution of fuel-mass fraction is Gaussian with a mean value *mfa* and a variance *vmi*. Figure 3 shows time histories of mean temperature *Ta* and variance *vm* for the initial condition of *mf* = 6.22×10-2 and *vmi* = 4.34×10-5, corresponding to a

a of unity and the standard deviation of

for the case of the homogeneous mixture is also shown. Collision frequency

/*k* for the PDF model combined with CFD using the *k*-

Fig. 3. Courses of mean-temperature *Ta* and *vm* for the non-homogeneous mixture (

flame temperature is approximately constant regardless of the collision frequency

*a* = 2.0) for the same

temperature *Ta* is exactly the same, in spite of the change of

scale of 3 mm. For these calculations, the number of fluid particles is fixed to *N* = 100. Generally, as collision frequency increases, the start of heat release by hot flame delays and maximum heat release rate increases simultaneously to approach the result of a homogeneous mixture. Meanwhile, the time when the temperature reaches 50% of adiabatic

reason will be discussed later in Fig. 8. Furthermore, Fig. 4 shows a magnified view of Fig. 3 around the starting points of heat release by cool flame (marked as 'A'). The change in mean

Next, in order to clarify the effect of mean equivalence ratio, calculations are performed for lean and rich cases. Fig. 5 shows *Ta* and *vm* under the conditions of (a) *mfa* = 3.87×10-2 (

represent turbulence energy and its dissipation rate, respectively. *Cm* is fixed at

= 2000 s-1 corresponds approximately to turbulence intensity of 5 m/s and

.

as in Fig. 3. Here, initial value of variance

is varied within the range of 0 s-1 to 10000 s-1. For comparison, the result

of approximately 0.1. The

is usually given

*<sup>a</sup>* = 1.0)

> . The

> > *<sup>a</sup>* =

turbulence model.

mean equivalence ratio

collision frequency

by = *Cm* 

Here, *k* and

2, therefore,

0.6) and (b) *mfa* = 1.17×10-1 (

Fig. 4. Temporal changes of Ta due to low temperature oxidation

Fig. 5. Effect of mean mass-fraction *mfa*

is fixed to *vmi* = 4.34×10-5. For the two conditions, histories of the mean temperature *Ta* are similar to the result of *<sup>a</sup>* = 1.0 in Fig. 3. For the lean case (a), due to the longer delay of hot flame heat release, the mixture is thought to be more homogeneous and to provide rapid temperature rise. However, the temperature-rising rate at ignition is lower compared to the result of *<sup>a</sup>* = 1. Meanwhile, for the rich case (b), the variance *vm* at the ignition is comparably large, whereas the temperature rise at the ignition is steep, similar to the homogeneous case.

Ignition Process in a Non-Homogeneous Mixture 161

In each fluid particle, hot flame occurs with rapid temperature rise when mf approaches *mfa*. For every case of mean fuel-mass fraction, the hot flame ignition delay varies over a wide range, because the mixture contains a variety of fuel-mass fractions. However, the particleto-particle variation of the hot flame start time is not determined by the variation in fuel-

In order to examine the relation between non-homogeneity and temperature rise rate quantitatively, some characteristic times of temperature rise process are defined as shown in Fig. 7. Firstly, the time at 50% temperature rising between the initial and adiabatic

*<sup>s</sup>* are defined as below;

50 and that at 95% as

*s* = <sup>95</sup> −

Based on these values, the ignition processes of non-homogeneous and homogeneous

calculated for homogeneous mixtures are drawn. Within this *mf*-condition, ignition delay

of the homogeneous mixture becomes shorter with increasing *mf*. At the same time,

becomes larger for the leaner and richer sides. Here, ignition delay time of each fluid particle varies due to the variation of equivalence ratio at ignition, therefore, the temperature rise period increases. In addition, *mf* dispersion becomes larger for the richer condition, due to shorter ignition delay time. The change in ignition delay against mf

50 is also displayed for each mixture. In addition, the curves of

*s* and combustion duration

*i* 

*<sup>a</sup>* of 0.6, 1.0, and 2.0. In Fig. 8, PDF

*s* and 95

> *s*

> > *i*

*i* (marked by an arrow) are displayed for non-

*i* = 2 × (<sup>50</sup> −

95. Next, using these values, temperature

95) (4)

*<sup>i</sup>* (5)

*a* = 2.0) at *p* = 4 MPa, *T* = 900 K and

*<sup>a</sup>* = 0.6), (b) *mfa* =

= 2000 s-1.

20 fluid particles, which are selected randomly, for (a) *mfa* = 3.83×10-2 (

*a* = 1.0) and (c) *mfa* = 1.17×10-1 (

*i* and ignition delay time

Fig. 7. Definition of ignition delay time

*s*, 95 and

homogeneous mixtures with average equivalence ratios

mixture are discussed.

for mf at *t* =

6.22×10-2 (

mass fraction.

rise period

temperatures is expressed as

In order to clarify the reason why the temperature-rising rate of the non-homogeneous mixture becomes smaller than in the homogenous case, the temporal change of temperature *T* and mass fraction of fuel mf in each fluid particle are examined. Fig. 6 shows the data of

Fig. 6. Time history of *T* and *mf* for each stochastic particle

In order to clarify the reason why the temperature-rising rate of the non-homogeneous mixture becomes smaller than in the homogenous case, the temporal change of temperature *T* and mass fraction of fuel mf in each fluid particle are examined. Fig. 6 shows the data of

(c) *mfa* = 1.17×10-1 (

*<sup>a</sup>* = 2.0)

*a* = 0.6) (b) *mfa* = 6.22×10-1 (

*<sup>a</sup>* = 1.0)

(a) *mfa* = 3.83×10-2 (

Fig. 6. Time history of *T* and *mf* for each stochastic particle

20 fluid particles, which are selected randomly, for (a) *mfa* = 3.83×10-2 (*<sup>a</sup>* = 0.6), (b) *mfa* = 6.22×10-2 (*a* = 1.0) and (c) *mfa* = 1.17×10-1 (*a* = 2.0) at *p* = 4 MPa, *T* = 900 K and = 2000 s-1. In each fluid particle, hot flame occurs with rapid temperature rise when mf approaches *mfa*. For every case of mean fuel-mass fraction, the hot flame ignition delay varies over a wide range, because the mixture contains a variety of fuel-mass fractions. However, the particleto-particle variation of the hot flame start time is not determined by the variation in fuelmass fraction.

In order to examine the relation between non-homogeneity and temperature rise rate quantitatively, some characteristic times of temperature rise process are defined as shown in Fig. 7. Firstly, the time at 50% temperature rising between the initial and adiabatic temperatures is expressed as 50 and that at 95% as 95. Next, using these values, temperature rise period *i* and ignition delay time *<sup>s</sup>* are defined as below;

$$
\tau\_i = \mathcal{Q} \times (\tau\_{50} - \tau\_{95}) \tag{4}
$$

$$
\tau\_s = \tau\_{95} - \tau\_i \tag{5}
$$

Fig. 7. Definition of ignition delay time *s* and combustion duration *i* 

Based on these values, the ignition processes of non-homogeneous and homogeneous mixture are discussed. *s*, 95 and *i* (marked by an arrow) are displayed for nonhomogeneous mixtures with average equivalence ratios *<sup>a</sup>* of 0.6, 1.0, and 2.0. In Fig. 8, PDF for mf at *t* = 50 is also displayed for each mixture. In addition, the curves of *s* and 95 calculated for homogeneous mixtures are drawn. Within this *mf*-condition, ignition delay *s* of the homogeneous mixture becomes shorter with increasing *mf*. At the same time, *i* becomes larger for the leaner and richer sides. Here, ignition delay time of each fluid particle varies due to the variation of equivalence ratio at ignition, therefore, the temperature rise period increases. In addition, *mf* dispersion becomes larger for the richer condition, due to shorter ignition delay time. The change in ignition delay against mf

Ignition Process in a Non-Homogeneous Mixture 163

Fig. 9. Effect of collision frequency

(a)

at *t* = 50 *i*-

Fig. 10. Correlations between combustion duration

on combustion duration

*i* - collision rate

*<sup>i</sup>* -*vm*50 Correlation

*<sup>i</sup>* - mass variation

 and 

Correlation (b)

around *a* = 2.0 is small so that *<sup>i</sup>* also becomes smaller in spite of the wide distribution of *mf*. On the other hand, for the leaner side around *<sup>a</sup>* = 0.6, the distribution width is narrow, whereas *s*-change against mf becomes larger than in the stoichiometric case. Then *i* becomes longer.

Fig. 8. Changes of ignition delay time *s* and combustion duration *i*

In order to confirm the effect of mixing rate on the combustion process, temperature histories are compared for different mixing rates at the same *mf*-variance in the middle of hot flame temperature rise, as shown in Fig. 9. The calculation starts with = 2000 s-1, then is suddenly increased to 10000 s-1 just before the ignition. The obtained temperature history differs from that of = 10000 s-1 and is similar to that of = 4000 s-1, whose variance of *mf*-PDF at 50 is nearly equal to this jumping-up case. This result shows that the rate of temperature rise is strongly affected by the *mf*-distribution at ignition rather than the mixing rate. Fig. 10 shows the correlation plots of (a) *i* and (b) *ivm*50. Here, *vm*50 represents the variance of *mf*-PDF at 50. In this case these plots are calculated for wide ranges of *vmi*, *Ti* and with fixed *p* = 4.0 MPa and *mfa* = 6.22×10-2 (*<sup>a</sup>* = 1.0). For the plots on (a), *i* scatters widely even at the same , however, for (b), the plots distribute on a certain curve. This indicates the great influence of *mf* variance on combustion duration.

A similar calculation is performed for the case of longer ignition delay and lower temperature rise rate, which is set by *mfa* = 1.95×10-2 (*<sup>a</sup>* = 0.3), *p* = 2.0MPa and *Ti* = 900K.

In this case, the mixture is comparably lean and ambient pressure is low so that *s* and *i* become much longer than in the case shown in Fig. 3. Fig. 11 shows the results for = 0 s-1, 200 s-1, 400 s-1, 1000 s-1 and 2000 s-1. Here, the ignition delay time is longer, therefore, mixture at ignition becomes more homogeneous for the case of > 2000 s-1. The ignition delay time *<sup>s</sup>* and temperature rise rate become larger with increasing mixing rate, which is similar to the results shown in Fig. 3. Also, 50 is almost constant against .

*s*-change against mf becomes larger than in the stoichiometric case. Then

*s* and combustion duration

is suddenly increased to 10000 s-1 just before the ignition. The obtained

In order to confirm the effect of mixing rate on the combustion process, temperature histories are compared for different mixing rates at the same *mf*-variance in the middle of

50 is nearly equal to this

shows that the rate of temperature rise is strongly affected by the *mf*-distribution at

*i*

= 10000 s-1 and is similar to that of

with fixed *p* = 4.0 MPa and *mfa* = 6.22×10-2 (

*<sup>a</sup>* = 0.3), *p* = 2.0MPa and *Ti* = 900K.

> 2000 s-1. The ignition delay time

. *i*and (b)

jumping-up case. This result

50. In this case these plots are

, however, for (b), the plots

= 2000 s-1,

= 4000 s-1,

*<sup>a</sup>* =

*s* and *i*

= 0 s-1,

*<sup>i</sup>* also becomes smaller in spite of the wide distribution of *mf*.

*<sup>a</sup>* = 0.6, the distribution width is narrow,

*i*

around

whereas

then 

becomes longer.

*a* = 2.0 is small so that

Fig. 8. Changes of ignition delay time

temperature history differs from that of

calculated for wide ranges of *vmi*, *Ti* and

whose variance of *mf*-PDF at

1.0). For the plots on (a),

the results shown in Fig. 3. Also,

combustion duration.

On the other hand, for the leaner side around

hot flame temperature rise, as shown in Fig. 9. The calculation starts with

ignition rather than the mixing rate. Fig. 10 shows the correlation plots of (a)

*ivm*50. Here, *vm*50 represents the variance of *mf*-PDF at

temperature rise rate, which is set by *mfa* = 1.95×10-2 (

at ignition becomes more homogeneous for the case of

In this case, the mixture is comparably lean and ambient pressure is low so that

become much longer than in the case shown in Fig. 3. Fig. 11 shows the results for

*i* scatters widely even at the same

distribute on a certain curve. This indicates the great influence of *mf* variance on

A similar calculation is performed for the case of longer ignition delay and lower

200 s-1, 400 s-1, 1000 s-1 and 2000 s-1. Here, the ignition delay time is longer, therefore, mixture

*<sup>s</sup>* and temperature rise rate become larger with increasing mixing rate, which is similar to

50 is almost constant against

Fig. 9. Effect of collision frequency on combustion duration

Fig. 10. Correlations between combustion duration *i* - collision rate and *<sup>i</sup>* - mass variation at *t* = 50

Ignition Process in a Non-Homogeneous Mixture 165

4. Ignition delay time of each fluid particle varies due to the variation of equivalence ratio at ignition, therefore, the temperature rise period increases. In addition, the temperature rise rate becomes larger with decreasing variance of fuel-mass fraction

Shimazaki, N., Tsurushima, T. and Nishimura, T., Dual-Mode Combustion Concept with

Ishiyama, T., Shioji, M. and Ihara, T., Analysis of Ignition Processes in a Fuel Spray Using an

Kong, S–C., Reitz, R. D., Modeling HCCI Engine Combustion Using Detailed Chemical

Curl, R. L., Dispersed Phase Mixing: I. Theory and Effects in Simple Reactors, *A. I. Ch. En. J.*,

*of Engine Research*, Vol. 4, No. 3 (2003), 155-162.

Premixed Diesel combustion by Direct Injection near Top Dead Center, SAE Paper,

Ignition Model Including Turbulent Mixing and Reduced Chemical Kinetics, *Int. J.* 

Kinetics with Combustion of Turbulent Mixing Effects, ASME Paper 2000-ICE-306,

distribution.

**5. Nomenclature** 

*t*: Time

*p*: Pressure *T*: Temperature *Ta*: Mean temperature *Ti*: Initial temperature *vm*: Variance of *mf vm*50: vm at *τ*<sup>50</sup>

*C*: Coefficient of modified Curl's model *Cm*: Coefficient of turbulent mixing *f*: Statistical state function *mf*: Fuel-mass fraction *mfa*: Mean value of *mf*

*N*: Total number of fluid particles

*t*\*: Dimensionless time

*vmi*: Initial value of *vm*

*<sup>a</sup>*: Mean equivalence ratio

*<sup>i</sup>*: Combustion duration

: Collision frequency

2003-01-0742.

Vol. 9, No. 2 (1963), 175-181.

2000.

50: Time at 50% temperature rise

95: Time at 95% temperature rise

*<sup>s</sup>*: Ignition delay of hot flame

: Delta function

: Scalar

**6. References** 

Fig. 11. Courses of mean-temperature *Ta* and *vm* for PCCI-like condition

#### **4. Conclusion**

An auto-ignition process of a non-homogeneous mixture in fuel concentration was fundamentally investigated by means of a numerical calculation based on chemical kinetics and the stochastic approach. The auto-ignition process of n-heptane is calculated by means of a semi-detailed mechanism and the non-uniform state of turbulent mixing is statistically described by means of probability density functions and the stochastic method. The following conclusions are derived from the results:


4. Ignition delay time of each fluid particle varies due to the variation of equivalence ratio at ignition, therefore, the temperature rise period increases. In addition, the temperature rise rate becomes larger with decreasing variance of fuel-mass fraction distribution.

#### **5. Nomenclature**

164 Chemical Kinetics

Fig. 11. Courses of mean-temperature *Ta* and *vm* for PCCI-like condition

following conclusions are derived from the results:

ratio than the cool flame delay.

initial temperature and pressure.

An auto-ignition process of a non-homogeneous mixture in fuel concentration was fundamentally investigated by means of a numerical calculation based on chemical kinetics and the stochastic approach. The auto-ignition process of n-heptane is calculated by means of a semi-detailed mechanism and the non-uniform state of turbulent mixing is statistically described by means of probability density functions and the stochastic method. The

1. For the auto-ignition process of a non-homogeneous mixture during the mixing process, ignition delay time of the cool flame is almost constant against the mixing rate. On the other hand, ignition delay time of hot flame becomes longer with increasing mixing rate. This is because the hot flame ignition delay is more sensible to equivalence

2. For the temperature rising process of hot flame, start points of heat release vary depending on equivalence ratio in a non-homogeneous mixture. Therefore, the rise period increases with increasing non-homogeneity. Also, the temperature rise rate due

3. The tendencies described above are the same for the case of changing equivalence ratio,

to heat release of hot flame increases with increasing mixing rate.

**4. Conclusion** 


#### **6. References**


**Part 3** 

**Chemical Kinetics and Phases** 


**Part 3** 

**Chemical Kinetics and Phases** 

166 Chemical Kinetics

Janicka, J., Kolbe W, and Kollmann, W., Closure of the Transport Equation for the

Seiser, H., et al., Extinction and Autoignition of n-Heptane in Counterflow Configuration,

*Proc. of the Combustion Institute* 28 (2000), pp. 2029-2037.

*Thermodynamics*, Vol. 4 (1979), 47.

Probability Density Function of Turbulent Scalar Fields, *J. of Non-equilibrium* 

**8** 

*Italy* 

**Chemical Kinetics in Cold Plasmas** 

The chapter deals with the chemical kinetics in the gas-phase of cold plasmas. After a section concerning cold plasmas and the modeling scheme of the chemical kinetics happening in their gas-phases, the text will focus on hydrocarbon plasma chemistry. After a general review of the chemical kinetic pattern, we discuss several features presenting applications to two main fields, carbon film deposition by radiofrequency low pressure plasmas (discussing a typical Plasma Enhanced Chemical Vapour Deposition process) and

As for the first topic, methane and other light hydrocarbon plasmas are of great interest in industrial applications, in particular in the chemical vapor deposition processes. Amorphous carbon and diamond-like thin films, suitable for mechanical and electronic applications can be prepared using low pressure discharges of hydrocarbon gases . The research in this field is mainly devoted to the understanding of the nature of the film growing mechanism but in spite of intense experimental and theoretical work it is not yet

New data concerning the interaction of ions and hydrocarbon radicals allow to have a quite complete mapping of the relevant reaction rates in an Ar/CH4 plasma by now and it is possible to investigate the effect of the chemical kinetics in such a system and eventually to identify the gaseous precursor of the chemical species incorporated in the deposited film. We have modeled the gas-phase chemistry of a typical radio frequency CH4/Ar plasma used for the deposition of diamond and diamond-like carbon films, with the aim of

As for the second topic, hydrogen reforming from methane is up to now the most viable source for large scale as well as localized production of hydrogen for fuel cell systems. Conventional reforming is carried out thermally in oven with oxygen and steam or using catalytic beds, but the development of more compact devices is actively pursued too. Plasma reformers based on different kind of discharges, as arcs, microwave plasmas and dielectric barrier discharges have been investigated so far. We present recent results concerning the viability of a hydrocarbon plasma reforming process to produce a hydrogen enriched gas-mixture based on atmospheric pressure discharges operating in

fully understood which species are responsible for the deposition process.

understanding the effects of the chemical kinetics of argon ions and metastables.

hydrogen reforming by atmospheric pressure plasmas.

**1. Introduction** 

the spark regime.

Ruggero Barni and Claudia Riccardi *Department of Physics G. Occhialini, University of Milano-Bicocca, Milan,* 
