**Chemical Kinetics and Inverse Modelling Problems**

Victor Martinez-Luaces

*Electrochemistry Engineering Multidisciplinary Research Group, University of the Republic of Uruguay, Montevideo, Uruguay* 

#### **1. Introduction**

60 Chemical Kinetics

Zeldovich, Ja. B. (1938). Proof of the unique solution to equation of the mass action law.

Russian).

*Russian Journal of Physical Chemistry A*, Vol.11, No.5, pp. 658–687, ISSN 0044-4537 (in

Late last century and early this century, the Electrochemistry Engineering Multidisciplinary Research Group (Grupo Multidisciplinario de Ingenieria Electroquimica, in Spanish) at Montevideo, Uruguay studied the adsorption of Carbon Dioxide on Platinum surfaces (Zinola et al., 1997; Martinez-Luaces et al., 2001; Martinez-Luaces, 2001, Martinez-Luaces & Guineo, 2002; Guineo & Martinez-Luaces, 2002).

The research found three different adsorbates E1, E2, and E3 and measured the surface concentrations by conventional electrochemical techniques.

The initial objective of the group was to propose a mechanism with the closest fit to the experimental curves (Zinola et al., 1997). Nevertheless, as the research progressed, the focus shifted to explain, or try to explain the double inflexion that consistently appeared in the experimental curves of surface concentration versus time for one of the adsorbates (Martinez-Luaces et al., 2001; Martinez-Luaces, 2001, Martinez-Luaces & Guineo, 2002; Guineo & Martinez-Luaces, 2002).

From a theoretical point of view, these are both inverse problems incorporating chemical and mathematical modelling at the same time. In previous papers and books, these problems were described as Inverse Modelling Problems (Martinez-Luaces, 2007, 2008, 2009a, 2009b, 2009c, 2009d).

In this paper, both inverse problems and the associated modelling are analysed from a theoretical point of view and subsequently solved. The main results will then be generalised to other Chemical Kinetics applications.

#### **2. Theoretical framework. Inverse problems**

Direct problems, according to Groestch (Groestch, 1999, 2001), can be regarded as those that provide the necessary information to follow a well-defined, stable procedure leading to a single correct solution.

Inverse problems, in contrast, are both more difficult and more interesting, largely due to their either having multiple solutions, or not being capable of being solved at all (Bunge,

Chemical Kinetics and Inverse Modelling Problems 63

In this case, both the causes and the effects are known, and what must be determined is the

Both types of problems are common in the different branches of Chemistry. For example, in Qualitative Chemical Analysis one is frequently faced with a solution containing several unknown cations and anions. The analyst must carry out a pre-established series of reactions, and according to the results must rule out or confirm the presence or absence of the more usual cations and anions. This procedure allows the analyst, depending on the results (formation of precipitates, turbidity, colour reactions in the solution, etc.), to deduce the composition of the problem sample. Clearly, this is an example of a causation problem, because the procedure is pre-established, the results are in plain view and the objective is to know what solution composition is compatible with the results obtained (Martinez-

In Inorganic Chemistry, on the other hand, the problem is more frequently to synthesize a given salt from simpler substances (oxides, hydroxides, anhydrides, etc.). A typical example of this is the Solvay process for obtaining sodium carbonate *Na CO* 2 3 and sodium bicarbonate *NaHCO*<sup>3</sup> , used in the glass, paper and soap industries among others, from sodium chloride *NaCl* and carbon dioxide *CO*<sup>2</sup> which are much more abundant and easily obtained. The problem here does not lie in identifying the reactants and/or products, but in knowing and correctly carrying out the process that will lead to the formation of the desired products, starting from cheap and easily available reagents. This is clearly a

Finally, in Organic Chemistry, double inverse problems (involving causation and specification) tend to be posed simultaneously. For instance, in organic synthesis, four different ways of preparing acetone are usually presented, starting from four different reagents: ethyl acetate, acetonitrile, acetaldehyde or 2-methylpropene. Obviously the reactants are not predetermined, much less the process, and only the final product or target

Problem solving, modelling and applications are not synonymous, although they are obviously related. For instance, the Discussion Document preparatory to the ICMI Study 14 (Blum et al., 2002), mentions that the term "modelling" focuses in the direction that goes from the real world towards basic sciences, while the term "applications" goes in the

Fig. 3. Schema of specification problems

Luaces, 2011).

process that leads from the former to the latter.

specification problem (Martinez-Luaces, 2011).

molecule is known (Martinez-Luaces, 2011).

**3. Modelling and applications** 

2006). They frequently crop up in the practice of several professions and careers. For instance, when treating a patient for a particular illness, listing the symptoms is a simple, direct problem that has already been solved and can be looked up in any medical textbook. On the other hand, diagnosing a patient's illness from his or her symptoms is not always a straightforward task, and requires an experienced doctor.

Inverse problems have not always been properly studied, and a quote from Bunge (Bunge, 2006) is appropriate: "The fact that almost all philosophers have ignored the peculiarities of inverse problems poses another inverse problem: to guess the reasons for this huge oversight on the part of philosophers."

In principle there are two different types of inverse problems, but in order to characterise them correctly, let us begin with a schematization of direct problems, adapting a study by Groetsch (Groestch, 1999, 2001). His scheme for a direct problem is like this:

Fig. 1. Schema of direct problems

In the scheme in Figure 1, data and a given procedure are available, and the answer is sought; for instance, the reagents for a certain chemical reaction are given, and the conditions of temperature, pH, etc of the reaction are known, and we wish to know the products of the reaction.

Now we can change the schema to obtain two inverse problems. The first is the causation problem, schematized in Figure 2:

Fig. 2. Schema for causation problems

In this first type of inverse problem (causation problem), the results are known, as well as the process that produces them, but the causes are unknown. Finding the causes is the problem to be solved.

The other inverse problem generally encountered is the specification problem, schematized in Figure 3:

2006). They frequently crop up in the practice of several professions and careers. For instance, when treating a patient for a particular illness, listing the symptoms is a simple, direct problem that has already been solved and can be looked up in any medical textbook. On the other hand, diagnosing a patient's illness from his or her symptoms is not always a

Inverse problems have not always been properly studied, and a quote from Bunge (Bunge, 2006) is appropriate: "The fact that almost all philosophers have ignored the peculiarities of inverse problems poses another inverse problem: to guess the reasons for this huge

In principle there are two different types of inverse problems, but in order to characterise them correctly, let us begin with a schematization of direct problems, adapting a study by

In the scheme in Figure 1, data and a given procedure are available, and the answer is sought; for instance, the reagents for a certain chemical reaction are given, and the conditions of temperature, pH, etc of the reaction are known, and we wish to know the

Now we can change the schema to obtain two inverse problems. The first is the causation

In this first type of inverse problem (causation problem), the results are known, as well as the process that produces them, but the causes are unknown. Finding the causes is the

The other inverse problem generally encountered is the specification problem, schematized

Groetsch (Groestch, 1999, 2001). His scheme for a direct problem is like this:

straightforward task, and requires an experienced doctor.

oversight on the part of philosophers."

Fig. 1. Schema of direct problems

problem, schematized in Figure 2:

Fig. 2. Schema for causation problems

problem to be solved.

in Figure 3:

products of the reaction.

Fig. 3. Schema of specification problems

In this case, both the causes and the effects are known, and what must be determined is the process that leads from the former to the latter.

Both types of problems are common in the different branches of Chemistry. For example, in Qualitative Chemical Analysis one is frequently faced with a solution containing several unknown cations and anions. The analyst must carry out a pre-established series of reactions, and according to the results must rule out or confirm the presence or absence of the more usual cations and anions. This procedure allows the analyst, depending on the results (formation of precipitates, turbidity, colour reactions in the solution, etc.), to deduce the composition of the problem sample. Clearly, this is an example of a causation problem, because the procedure is pre-established, the results are in plain view and the objective is to know what solution composition is compatible with the results obtained (Martinez-Luaces, 2011).

In Inorganic Chemistry, on the other hand, the problem is more frequently to synthesize a given salt from simpler substances (oxides, hydroxides, anhydrides, etc.). A typical example of this is the Solvay process for obtaining sodium carbonate *Na CO* 2 3 and sodium bicarbonate *NaHCO*<sup>3</sup> , used in the glass, paper and soap industries among others, from sodium chloride *NaCl* and carbon dioxide *CO*<sup>2</sup> which are much more abundant and easily obtained. The problem here does not lie in identifying the reactants and/or products, but in knowing and correctly carrying out the process that will lead to the formation of the desired products, starting from cheap and easily available reagents. This is clearly a specification problem (Martinez-Luaces, 2011).

Finally, in Organic Chemistry, double inverse problems (involving causation and specification) tend to be posed simultaneously. For instance, in organic synthesis, four different ways of preparing acetone are usually presented, starting from four different reagents: ethyl acetate, acetonitrile, acetaldehyde or 2-methylpropene. Obviously the reactants are not predetermined, much less the process, and only the final product or target molecule is known (Martinez-Luaces, 2011).

#### **3. Modelling and applications**

Problem solving, modelling and applications are not synonymous, although they are obviously related. For instance, the Discussion Document preparatory to the ICMI Study 14 (Blum et al., 2002), mentions that the term "modelling" focuses in the direction that goes from the real world towards basic sciences, while the term "applications" goes in the

Chemical Kinetics and Inverse Modelling Problems 65

*d E kE kE k E*

*d E kE kE k E*

*x x k k* 0 0 *y k kk y z z k kk*

It is interesting to observe that if the three rows of the Ordinary Differential Equations (ODE) system are added, the result is zero. This property remains true with other proposed mechanisms where only the solution will change. For example, consider the following

*K*

1

 

> 1 2

 

1 2

*E E*

*K K*

*K K*

2 3

 

2 3

*E E*

*K*

where all the possible reactions between the three adsorbates are considered.

3

For such a mechanism, the mathematical model (i.e., the Ordinary Differential Equations

13 1 3 1 12 2 3 2 23

 

*x x kk k k y k kk k y z z k k kk*

As in the previous case, the surface concentrations *E*<sup>1</sup> ,*E*<sup>2</sup> and *E*<sup>3</sup> were replaced by *x* ,

Once again it is easy to observe that the result of adding the three rows of this new matrix is the null vector. This fact could be considered just a mathematical curiosity, though in the

The first proposed mechanism was the better when fitting the theoretical curves to the experimental values of surface concentration versus time. Nevertheless, one of the

next section it will be the main observation to solve the second proposed problem.

1 3

*E E*

11 32 33

(2)

(3)

(4)

(5)

 

21 32 33

1 33 2 33

121

1 2

*d E k kE*

1

*dt*

 

 

or equivalently:

mechanism:

if notation is simplified.

system) will be as follows:

*y* and *z* in order to simplify notation.

**5. The second problem. The double inflexion** 

*dt*

*dt*

2

3

opposite direction, that is, from science towards the real world. In addition, the term "modelling" emphasizes the process that is taking place, while the word "applications" stresses the object involved (particularly in areas of the real world that are susceptible to a given mathematical treatment). The same document uses the term "problem" in a broad sense, including, therefore, not only practical problems, but also abstract problems, or those that attempt to describe, explain, understand, or even design parts of the real world.

Obviously, solving problems and modelling are not the same thing (one can model even in the absence of a concrete problem to be solved; one would simply be giving a mathematical description of a given phenomenon), and of course, "pure" mathematics problems, which do not require any kind of modelling, can be solved.

In the light of the above, we can arrive at the following schema:

Fig. 4. Scheme of modelling and applications

Finally, it is worth mentioning that a more extensive discussion of modelling, applications and problem solving, and their teaching in university courses and secondary education in Latin America is available in previous papers (Martinez-Luaces, 2005, 2009a, 2011).

#### **4. The first problem. The mechanism proposal**

As it was mentioned before, the main objective of our research group was to study the adsorption of Carbon Dioxide on Platinum surfaces. The research found three different adsorbates E1, E2, and E3 and measured the surface concentrations by conventional electrochemical techniques.

On completion of the trials of the experiment, several mechanisms were proposed and they were assessed by comparing the theoretical curves with the experimental ones. The result of this process was that the best fit was obtained by the following mechanism:

$$\begin{array}{l} \mathrm{E\_1 \xrightarrow{k\_1} \mathrm{E\_2}}\\ \mathrm{E\_1 \xrightarrow{k\_2} \mathrm{E\_3}}\\ \mathrm{E\_2 \xrightleftharpoons} \mathrm{E\_3} \end{array} \tag{1}$$

If *E*<sup>1</sup> ,*E*<sup>2</sup> and *E*<sup>3</sup> represent the adsorbates' surface concentrations and 1 *k* , <sup>2</sup> *k* , <sup>3</sup> *k* and 3 *k* the kinetic constants, then the mathematical model corresponding to this mechanism is the following:

$$\begin{cases} \frac{d\left[E\_1\right]}{dt} = -\left(k\_1 + k\_2\right)\left[E\_1\right] \\ \frac{d\left[E\_2\right]}{dt} = k\_1\left[E\_1\right] - k\_3\left[E\_2\right] + k\_{-3}\left[E\_3\right] \\ \frac{d\left[E\_3\right]}{dt} = k\_2\left[E\_1\right] + k\_3\left[E\_2\right] - k\_{-3}\left[E\_3\right] \end{cases} \tag{2}$$

or equivalently:

64 Chemical Kinetics

opposite direction, that is, from science towards the real world. In addition, the term "modelling" emphasizes the process that is taking place, while the word "applications" stresses the object involved (particularly in areas of the real world that are susceptible to a given mathematical treatment). The same document uses the term "problem" in a broad sense, including, therefore, not only practical problems, but also abstract problems, or those

Obviously, solving problems and modelling are not the same thing (one can model even in the absence of a concrete problem to be solved; one would simply be giving a mathematical description of a given phenomenon), and of course, "pure" mathematics problems, which do

Finally, it is worth mentioning that a more extensive discussion of modelling, applications and problem solving, and their teaching in university courses and secondary education in

As it was mentioned before, the main objective of our research group was to study the adsorption of Carbon Dioxide on Platinum surfaces. The research found three different adsorbates E1, E2, and E3 and measured the surface concentrations by conventional

On completion of the trials of the experiment, several mechanisms were proposed and they were assessed by comparing the theoretical curves with the experimental ones. The result of

> 1 2 3

 

E E E E

*k k k*

1 2 1 3

(1)

3

2 3

E E

*k*

If *E*<sup>1</sup> ,*E*<sup>2</sup> and *E*<sup>3</sup> represent the adsorbates' surface concentrations and 1 *k* , <sup>2</sup> *k* , <sup>3</sup> *k* and 3 *k* the kinetic constants, then the mathematical model corresponding to this mechanism is the

Latin America is available in previous papers (Martinez-Luaces, 2005, 2009a, 2011).

this process was that the best fit was obtained by the following mechanism:

that attempt to describe, explain, understand, or even design parts of the real world.

not require any kind of modelling, can be solved.

Fig. 4. Scheme of modelling and applications

electrochemical techniques.

following:

**4. The first problem. The mechanism proposal** 

In the light of the above, we can arrive at the following schema:

$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix}' = \begin{pmatrix} -k\_1 - k\_2 & 0 & 0 \\ k\_1 & -k\_3 & k\_{-3} \\ k\_2 & k\_3 & -k\_{-3} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \tag{3}
$$

if notation is simplified.

It is interesting to observe that if the three rows of the Ordinary Differential Equations (ODE) system are added, the result is zero. This property remains true with other proposed mechanisms where only the solution will change. For example, consider the following mechanism:

$$\begin{aligned} \underbrace{E\_1 \xleftarrow{K\_1}}\_{K\_{-1}} E\_2 \\ E\_2 \xleftarrow{K\_2} E\_3 \\ E\_1 \xleftarrow{K\_3} E\_3 \end{aligned} \tag{4}$$

where all the possible reactions between the three adsorbates are considered.

For such a mechanism, the mathematical model (i.e., the Ordinary Differential Equations system) will be as follows:

$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix}' = \begin{pmatrix} -k\_1 - k\_3 & k\_{-1} & k\_{-3} \\ k\_1 & -k\_{-1} - k\_2 & k\_{-2} \\ k\_3 & k\_2 & -k\_{-2} - k\_{-3} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \tag{5}
$$

As in the previous case, the surface concentrations *E*<sup>1</sup> ,*E*<sup>2</sup> and *E*<sup>3</sup> were replaced by *x* , *y* and *z* in order to simplify notation.

Once again it is easy to observe that the result of adding the three rows of this new matrix is the null vector. This fact could be considered just a mathematical curiosity, though in the next section it will be the main observation to solve the second proposed problem.

#### **5. The second problem. The double inflexion**

The first proposed mechanism was the better when fitting the theoretical curves to the experimental values of surface concentration versus time. Nevertheless, one of the

Chemical Kinetics and Inverse Modelling Problems 67

1

1 Glucose Glucose *k*

*K*

*k A B*

*d A KA kB*

*dt*

*K k <sup>M</sup> K k* 

This process led to a second order equation *K k Kk*

*v K* <sup>1</sup> and *v*<sup>2</sup>

the solution of the ODE problem is the following linear combination:

*x t k*

*y t K*

( ) ( )

*K k* 0 and the eigenvalues are: 1

*i*

*<sup>k</sup>*

*dt*

Once again, if *x A* and *y B* the ODE system can be written in matrix form, as follows:

*x Kk x y K k y*

det det 0

0 and

*M I K k*

the corresponding eigenvectors can be easily computed and the results are the following:

In this case, the system matrix is equivalent to a diagonal form (Martinez-Luaces, 2009e) and

 11 22 ( ) <sup>1</sup> exp exp exp ( ) <sup>1</sup> *i ii*

1 2

1 2

*xt Ck Ce y t CK Ce*

*C tv C t C t*

*K k*

*d B KA kB*

 

(6)

*dt*<sup>y</sup>

(11)

can be written in the

(12)

(14)

0 that can be written as:

. (15)

 

<sup>2</sup> *K k* . For these eigenvalues,

(17)

*d B dt*

(13)

and its eigenvalues can easily be obtained from the

 

1 1 

(16)

( )

 

*K kt K kt*

( )

*k*

In this case, a mathematical model, in terms of *d A*

in a simpler form as:

following form:

The system matrix is

<sup>2</sup>

characteristic equation:

 

and the final solution is:

experimental curves consistently showed two inflexion points. Initially these inflexions were regarded as the uncertainty of the experimental measures, but each time the experiments were repeated, the results were always the same, i.e., one of the curves showed two inflexion points (Martinez-Luaces et al., 2001; Martinez-Luaces, 2001, Martinez-Luaces & Guineo, 2002; Guineo & Martinez-Luaces, 2002).

This led to other mechanisms being proposed, but none of them could explain this behaviour. As previously mentioned, all proposed mechanisms led to an O.D.E. system, for which the associated matrix had the property that the sum of all the matrix rows was zero. This observation is very useful because in all cases the matrix determinant must be zero. Finally, a null matrix determinant implies that one of the matrix eigenvalues is always zero and this determines the behaviour of the theoretical curves.

In order to explain these ideas, we shall consider a simpler chemical kinetics problem, with a simpler mechanism. For this purpose a well known process will be considered: the mutarotation of Glucose (Guerasimov, 1995; Martinez-Luaces, 2009e). In this mechanism there are only two simple uni-molecular reactions:

$$a \text{--Glucose} \xleftarrow[\_{\overline{k\_{-1}}}]{} \beta \text{--Glucose} \tag{6}$$

This mechanism can be expressed mathematically through a simple ODE. In fact, if *x* is the concentration of - Glucose and *y* that of - Glucose, then it is easy to observe that the - Glucose concentration can be obtained from the following ODE:

$$\frac{d\mathbf{x}}{dt} = k\_1 \mathbf{y} - k\_{-1} \mathbf{x} \tag{7}$$

Letting *a* be the initial - Glucose concentration and the corresponding initial concentration of - Glucose being zero, then the variable *y* can be replaced by *a x* and the ODE can be written as:

$$\frac{d\mathbf{x}}{dt} = k\_1.(a - \mathbf{x}) - k\_{-1}\mathbf{x} \tag{8}$$

Finally, very simple algebraic manipulations lead to the following linear, first order equation:

$$\frac{d\mathbf{x}}{dt} = k\_1 \mathbf{a} - \left(k\_1 + k\_{-1}\right) \mathbf{x} \tag{9}$$

The solution for this ODE is the following:

$$\mathbf{x}(t) = \frac{k\_1 a}{k\_1 + k\_{-1}} \left\{ 1 - \exp\left[ -\left(k\_1 + k\_{-1}\right)t \right] \right\} \tag{10}$$

This is the typical approach for studying the Glucose Mutarotation problem (Guerasimov, 1995; Martinez-Luaces, 2009e), but not the best one for this article devoted to inverse problems. In order to get a different point of view, we shall write the mechanism:

$$a\text{-Glucose} \xleftarrow[k\_{\cdot \cdot}]{k\_{\cdot \cdot}} \quad \beta\text{-Glucose} \tag{6}$$

in a simpler form as:

66 Chemical Kinetics

experimental curves consistently showed two inflexion points. Initially these inflexions were regarded as the uncertainty of the experimental measures, but each time the experiments were repeated, the results were always the same, i.e., one of the curves showed two inflexion points (Martinez-Luaces et al., 2001; Martinez-Luaces, 2001, Martinez-Luaces &

This led to other mechanisms being proposed, but none of them could explain this behaviour. As previously mentioned, all proposed mechanisms led to an O.D.E. system, for which the associated matrix had the property that the sum of all the matrix rows was zero. This observation is very useful because in all cases the matrix determinant must be zero. Finally, a null matrix determinant implies that one of the matrix eigenvalues is always zero

In order to explain these ideas, we shall consider a simpler chemical kinetics problem, with a simpler mechanism. For this purpose a well known process will be considered: the mutarotation of Glucose (Guerasimov, 1995; Martinez-Luaces, 2009e). In this mechanism

1

1 Glucose Glucose *k*

This mechanism can be expressed mathematically through a simple ODE. In fact, if *x* is the

1 1 . *dx <sup>k</sup> <sup>y</sup> k x*

1 1 .( ) *dx k a x kx*

Finally, very simple algebraic manipulations lead to the following linear, first order

1 11 . . *dx ka k k x*

*k a x t kkt*

This is the typical approach for studying the Glucose Mutarotation problem (Guerasimov, 1995; Martinez-Luaces, 2009e), but not the best one for this article devoted to inverse

( ) 1 exp

*dt*

*k k* 1

1 1

problems. In order to get a different point of view, we shall write the mechanism:

 

(6)


*dt* (7)

*dt* (8)

(9)



1 1

(10)


*k*

Guineo, 2002; Guineo & Martinez-Luaces, 2002).

and this determines the behaviour of the theoretical curves.


Glucose concentration can be obtained from the following ODE:

there are only two simple uni-molecular reactions:

concentration of

concentration of

equation:

The solution for this ODE is the following:

Letting *a* be the initial

the ODE can be written as:

$$A \xleftarrow[\xleftarrow{k}]{}\_{B} \tag{11}$$

In this case, a mathematical model, in terms of *d A dt*<sup>y</sup> *d B dt* can be written in the following form:

$$\begin{cases} \frac{d\begin{bmatrix} A \\ \end{bmatrix}}{dt} = -K\begin{bmatrix} A \end{bmatrix} + k\begin{bmatrix} B \end{bmatrix} \\\\ \frac{d\begin{bmatrix} B \end{bmatrix}}{dt} = K\begin{bmatrix} A \end{bmatrix} - k\begin{bmatrix} B \end{bmatrix} \end{cases} \tag{12}$$

Once again, if *x A* and *y B* the ODE system can be written in matrix form, as follows:

$$
\begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}' = \begin{pmatrix} -\mathbf{K} & \mathbf{k} \\ \mathbf{K} & -\mathbf{k} \end{pmatrix} \cdot \begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix} \tag{13}
$$

The system matrix is *K k <sup>M</sup> K k* and its eigenvalues can easily be obtained from the characteristic equation:

$$\det\begin{pmatrix} M - \lambda I \end{pmatrix} = \det\begin{pmatrix} -K - \lambda & k \\ K & -k - \lambda \end{pmatrix} = 0 \tag{14}$$

This process led to a second order equation *K k Kk* 0 that can be written as: <sup>2</sup> *K k* 0 and the eigenvalues are: 1 0 and <sup>2</sup> *K k* . For these eigenvalues, the corresponding eigenvectors can be easily computed and the results are the following:

$$
\vec{v}\_1 = \begin{pmatrix} k \\ K \end{pmatrix} \text{ and } \quad \vec{v}\_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}. \tag{15}
$$

In this case, the system matrix is equivalent to a diagonal form (Martinez-Luaces, 2009e) and the solution of the ODE problem is the following linear combination:

$$
\begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \sum\_{i} \mathbb{C}\_{i} \exp\left(\mathcal{A}\_{i} t\right) \ \vec{v}\_{i} = \mathbb{C}\_{1} \exp\left(\mathcal{A}\_{1} t\right) \begin{pmatrix} k \\ K \end{pmatrix} + \mathbb{C}\_{2} \exp\left(\mathcal{A}\_{2} t\right) \begin{pmatrix} 1 \\ -1 \end{pmatrix} \tag{16}
$$

and the final solution is:

$$
\begin{pmatrix} \varkappa(t) \\ \varkappa(t) \end{pmatrix} = \begin{pmatrix} \mathcal{C}\_1 k + \mathcal{C}\_2 e^{-(K+k)t} \\ \mathcal{C}\_1 K - \mathcal{C}\_2 e^{-(K+k)t} \end{pmatrix} \tag{17}
$$

Chemical Kinetics and Inverse Modelling Problems 69

This formula explains why the sum of all the equations of the ODE system adds up to zero. From this observation, it follows that the sum of all the rows of the system matrix will

For the same reasons, all the possible mechanisms involving unimolecular reactions between all the adsorbates will show the same property. As a consequence of this fact, the system matrix has a null determinant for all the proposed mechanisms. This observation can be easily proved, because if 123 *row row row* 0 , then 3 12 *row row row* and since each of the rows is a linear combination of the others, it follows that det 0 *A* , where *A* is the

Therefore, if det 0 *A* , then matrix *A* will have a null eigenvalue, which is independent of the mechanism proposed. For each mechanism the ODE system will have three

the solutions of ODE systems and their qualitative behaviour. In order to get qualitative

All these cases will be analysed in order to discover whether the two inflexion points can be

1 2 **,** <sup>1</sup> 0 **,** <sup>2</sup> 0 **)** 

, where *<sup>i</sup> <sup>c</sup>* represents the coefficients,

the associated eigenvectors (Martinez-Luaces, 2009e). In fact, this

1 2 and by differentiating we obtain:

 

exp 0 *t* , which only happens if

0 , all the surface concentrations have this

 2 1 , so

<sup>1</sup> *<sup>t</sup>* ln

 where *<sup>i</sup>* the

.

 

 

 

In this case, if *E EEE* <sup>123</sup> , , , a well known formula – valid if all the eigenvalues are

formula was used to solve the mutarotation problem, as in (16). The same formula can be

Then, in this case, only one inflexion point can be explained and so, two inflexion points are

explained by mechanisms involving only electrochemical unimolecular reactions.

*i ii* exp

0 and 3

 

 

*E c tv* 

0 . This observation has important consequences in

always be the null vector (Martinez-Luaces, 2007, 2009e).

matrix associated with the ODE system.

results, three different cases will be investigated:

**5.1 First case: Three different eigenvalues (**

different – expresses that

eigenvalues and *<sup>i</sup> v*

 

not possible.

2

*dt*

2 <sup>2</sup> <sup>0</sup> *<sup>i</sup> d E*

Taking into account that 1

form: *Et t <sup>i</sup>*

 

 exp exp 

> 

*dt* if and only if 2 2

There are three possibilities under this case:

<sup>2</sup> *R* and 3

2. two eigenvalues are the same and the other is a simple one

3

1

2 2 2 2 <sup>2</sup> 1 1 2 2 1 12 exp exp exp exp *<sup>i</sup> d E t tt t*

*i*

used for other diagonal forms with repeated eigenvalues.

 

 

1 2

**5.2 Second case: A double eigenvalue and a simple one** 

 

 0 , 2 

3. there exists an unique eigenvalue with algebraic multiplicity three.

<sup>1</sup> *R* ,

1. all the eigenvalues are different,

eigenvalues:

More important than the solution itself – at least for this article – is to observe that one of the eigenvalues was zero (and the sum of both matrix rows was the null vector, as in the adsorption problem). This fact has important consequences for the asymptotic behaviour of the solutions. Moreover, as time approaches infinity:

$$\lim\_{\begin{array}{c}t \to +\infty \\ t \to +\infty \end{array}} \begin{pmatrix} \mathbf{x}(t) \\ y(t) \end{pmatrix} = \lim\_{\begin{array}{c}t \to +\infty \\ t \to +\infty \end{array}} \begin{pmatrix} \mathbf{C}\_{1}k + \mathbf{C}\_{2}e^{-(K+k)t} \\ \mathbf{C}\_{1}K - \mathbf{C}\_{2}e^{-(K+k)t} \end{pmatrix} = \begin{pmatrix} \mathbf{C}\_{1}k \\ \mathbf{C}\_{1}K \end{pmatrix} \tag{18}$$

The qualitative result is that both concentrations tend to finite, non-zero real values, which from a chemical point of view are the equilibrium concentrations.

Similar ideas can be explored when three or even more substances are involved. In order to explain this idea, we shall consider a very simple mechanism involving three different chemical substances *E*<sup>1</sup> , *E*2 and *E*3 , like the following one:

$$E\_1 \xrightarrow{k\_1} E\_2 \xrightarrow{k\_2} E\_3 \tag{19}$$

In this case, the ODE system (Martinez-Luaces, 2009e) is as follows:

$$\begin{cases} \frac{d\left[E\_1\right]}{dt} = -k\_1 \left[E\_1\right] \\ \frac{d\left[E\_2\right]}{dt} = k\_1 \left[E\_1\right] - k\_2 \left[E\_2\right] \\ \frac{d\left[E\_3\right]}{dt} = k\_2 \left[E\_2\right] \end{cases} \tag{20}$$

or equivalently:

$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix}' = \begin{pmatrix} -k\_1 & 0 & 0 \\ k\_1 & -k\_2 & 0 \\ 0 & k\_2 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \tag{21}
$$

where, as in the previous example, adding the rows the result is a null vector.

Applying the mass conservation law to the mechanism:

$$E\_1 \xrightarrow{k\_1} E\_2 \xrightarrow{k\_2} E\_3 \tag{19}$$

it follows that *EEE* <sup>123</sup> remains constant.

Finally, if *E E EC* <sup>123</sup> is differentiated with respect to *t* , we obtain the following equation:

$$\frac{d\left[E\_1\right]}{dt} + \frac{d\left[E\_2\right]}{dt} + \frac{d\left[E\_3\right]}{dt} = 0\tag{22}$$

More important than the solution itself – at least for this article – is to observe that one of the eigenvalues was zero (and the sum of both matrix rows was the null vector, as in the adsorption problem). This fact has important consequences for the asymptotic behaviour of

*t y t t CK Ce C K*

The qualitative result is that both concentrations tend to finite, non-zero real values, which

Similar ideas can be explored when three or even more substances are involved. In order to explain this idea, we shall consider a very simple mechanism involving three different

> 1 2 123

*d E k E*

1

*dt*

*dt*

2

3

where, as in the previous example, adding the rows the result is a null vector.

Applying the mass conservation law to the mechanism:

it follows that *EEE* <sup>123</sup> remains constant.

*x x k y k k y z z k* 

*d E k E dt*

*d E kE kE*

2 2

11 22

0 0 0

0 0

1 2 123

Finally, if *E E EC* <sup>123</sup> is differentiated with respect to *t* , we obtain the following

 <sup>123</sup> <sup>0</sup> *dE dE dE dt dt dt*

1 1

*x t Ck Ce C k*

( ) 1 2 1 ( ) <sup>1</sup> 1 2

 

*K kt K kt*

*k k EEE* (19)

*k k EEE* (19)

(22)

(20)

(21)

(18)

the solutions. Moreover, as time approaches infinity:

lim ( ) lim ( )

from a chemical point of view are the equilibrium concentrations.

In this case, the ODE system (Martinez-Luaces, 2009e) is as follows:

 

 

or equivalently:

equation:

chemical substances *E*<sup>1</sup> , *E*2 and *E*3 , like the following one:

This formula explains why the sum of all the equations of the ODE system adds up to zero. From this observation, it follows that the sum of all the rows of the system matrix will always be the null vector (Martinez-Luaces, 2007, 2009e).

For the same reasons, all the possible mechanisms involving unimolecular reactions between all the adsorbates will show the same property. As a consequence of this fact, the system matrix has a null determinant for all the proposed mechanisms. This observation can be easily proved, because if 123 *row row row* 0 , then 3 12 *row row row* and since each of the rows is a linear combination of the others, it follows that det 0 *A* , where *A* is the matrix associated with the ODE system.

Therefore, if det 0 *A* , then matrix *A* will have a null eigenvalue, which is independent of the mechanism proposed. For each mechanism the ODE system will have three eigenvalues: <sup>1</sup> *R* , <sup>2</sup> *R* and 3 0 . This observation has important consequences in the solutions of ODE systems and their qualitative behaviour. In order to get qualitative results, three different cases will be investigated:


All these cases will be analysed in order to discover whether the two inflexion points can be explained by mechanisms involving only electrochemical unimolecular reactions.

#### **5.1 First case: Three different eigenvalues (** 1 2 **,** <sup>1</sup> 0 **,** <sup>2</sup> 0 **)**

In this case, if *E EEE* <sup>123</sup> , , , a well known formula – valid if all the eigenvalues are different – expresses that 3 1 *i ii* exp *i E c tv* , where *<sup>i</sup> <sup>c</sup>* represents the coefficients, *<sup>i</sup>* the eigenvalues and *<sup>i</sup> v* the associated eigenvectors (Martinez-Luaces, 2009e). In fact, this formula was used to solve the mutarotation problem, as in (16). The same formula can be used for other diagonal forms with repeated eigenvalues.

Taking into account that 1 0 , 2 0 and 3 0 , all the surface concentrations have this form: *Et t <sup>i</sup>* exp exp 1 2 and by differentiating we obtain: 2 2 2 2 2 <sup>2</sup> 1 1 2 2 1 12 exp exp exp exp *<sup>i</sup> d E t tt t dt* where 2 1 , so 2 <sup>2</sup> <sup>0</sup> *<sup>i</sup> d E dt* if and only if 2 2 1 2 exp 0 *t* , which only happens if 2 1 2 2 <sup>1</sup> *<sup>t</sup>* ln .

Then, in this case, only one inflexion point can be explained and so, two inflexion points are not possible.

#### **5.2 Second case: A double eigenvalue and a simple one**

There are three possibilities under this case:

Chemical Kinetics and Inverse Modelling Problems 71

*J*

*X PY* , all the surface concentrations are second order polynomials, such as:

As a consequence of this analysis, two inflexion points cannot be explained by mechanisms

Finally, as mentioned in other papers, this double inflexion is explained in terms of

2

 

2 2 3 *K*

13 1 3 1 12 2 3 2 23

 

*x x kk k k y k kk k y z z k k kk*

In this problem – as in other models, corresponding to other mechanisms – the result of the sum of all the rows is the null vector, and this fact implies the existence of a null eigenvalue.

*K E E* 

where all the possible reactions between the three adsorbates are considered. As it was mentioned before, for such a mechanism, the mathematical model is:

electrocatalytic reactions (Martinez-Luaces, 2001: Guineo & Martinez-Luaces, 2002).

. Finally, double differentiation of *E t <sup>i</sup>* with respect to *t* , gives:

000 100 010

*y c ct c* and taking into account the change of variables

or equivalently 1 *y* 0 , 2 1 *y y* and 3 2 *y y* . Then 1 1 *y c* ,

*K*

*K E E* 3

3 1 3 

(4)

(5)

. Then, in this sub-case, only one inflexion point can be explained.

 0 **)** 

2 2 *<sup>i</sup> d E*

*dt* is zero only for

2 <sup>2</sup> <sup>2</sup> *<sup>i</sup> d E dt*

and the new ODE system is in

this expression can be zero only for a unique *t* value, concretely

**5.3 Third case: A unique eigenvalue (** <sup>123</sup>

2 31 2 3 2 *t*

In this last case the Jordan canonical form is:

1 1 2 2 3 3

*y y y y y y* 

000 100 010

, so in this case there are no inflexion points.

involving electrochemical uni-molecular reactions.

As an example, let us consider again the following mechanism:

1

 

1 1 2 *K*

*K E E* 

> 

2 2 <sup>2</sup> *<sup>t</sup>* 

this case:

 

21 2 *y ct c* and

<sup>2</sup> *Et t <sup>i</sup>* 

 

**6. Other qualitative results** 

 

#### **5.2.1 First sub-case (** <sup>1</sup> 0 **and** 2 3 0 **)**

In this sub-case, the ODE system matrix *A* is not equivalent to a diagonal matrix *D* , but it is equivalent to a triangular matrix *J* (the Jordan canonical form). More precisely, there exists a matrix *P* (it is important to note that in this case not all the columns of *P* are eigenvectors) such that <sup>1</sup> *P AP J* and as a consequence, the ODE system *X AX* can be easily converted in the following one *Y JY* by a simple change of variables *X PY* .

As a consequence of this fact, the new ODE system to solve is: 1 1 1 2 2 3 3 0 0 0 00 0 10 *y y y y y y* then

1 11 *y y* , 2 *y* 0 and 3 2 *y y* . The solution of this new ODE problem is *y*11 1 *c t* exp , 2 2 *y c* and 32 3 *y ct c* . Finally, since *X PY* , it follows that all the *Ei* are linear combinations of these functions 1 *y* , <sup>2</sup> *y* and 3 *y* and so, all of them are of the form: *E tt <sup>i</sup>* exp <sup>1</sup> and differentiating twice with respect to time, the result is: 2 2 <sup>2</sup> 1 1 exp 0 *<sup>i</sup> d E t tR dt* .

As a consequence of this fact, there are no inflexion points in this sub-case.

#### **5.2.2 Second sub-case (** 1 3 0 **and** <sup>2</sup> 0 **)**

This case is the same as the previous one, except for a change in the order of the eigenvalues, so, the main result is the same: there are no inflexion points in this sub-case.

#### **5.2.3 Third sub-case (** 1 2 0 **and** <sup>3</sup> 0 **)**

In this new sub-case, the corresponding Jordan canonical form has the following form:

$$J = \begin{pmatrix} \lambda & 0 & 0 \\ 1 & \lambda & 0 \\ 0 & 0 & 0 \end{pmatrix}, \text{ where } \lambda = \lambda\_1 = \lambda\_2 \text{ and the new ODE system is: } \mathbf{1}$$

$$
\begin{pmatrix}
\dot{y}\_1 \\ \dot{y}\_2 \\ \dot{y}\_3
\end{pmatrix} = \begin{pmatrix}
\lambda & 0 & 0 \\ 1 & \lambda & 0 \\ 0 & 0 & 0
\end{pmatrix} \begin{pmatrix}
y\_1 \\ y\_2 \\ y\_3
\end{pmatrix} \quad \text{or equivalently} \quad \dot{y}\_1 = \lambda y\_1 \quad \text{ } \quad \dot{y}\_2 = y\_1 + \lambda y\_2 \quad \text{and} \quad \dot{y}\_3 = 0. \text{ Then, the}
$$

solution of the first and the third ODE are *y*1 1 *c t* exp and 3 3 *y c* . Finally, the second ODE can be written as *y*2 21 *yc t* exp and the solution is *y*2 12 *ct c t* exp.

Finally, taking into account the change of variables *X PY* , all the surface concentrations are of the following form: *Et t <sup>i</sup>* exp . Once again, if *E t <sup>i</sup>* is differentiated twice with respect to *t* , the result is: 2 2 2 <sup>2</sup> 2 exp *<sup>i</sup> d E t t dt* As in previous cases,

In this sub-case, the ODE system matrix *A* is not equivalent to a diagonal matrix *D* , but it is equivalent to a triangular matrix *J* (the Jordan canonical form). More precisely, there exists a matrix *P* (it is important to note that in this case not all the columns of *P* are eigenvectors) such that <sup>1</sup> *P AP J* and as a consequence, the ODE system *X AX* can be easily converted in the following one *Y JY* by a simple change of variables *X PY* .

, 2 *y* 0 and 3 2 *y y* . The solution of this new ODE problem is *y*11 1 *c t* exp

<sup>1</sup> and differentiating twice with respect to time, the result is:

2 2 *y c* and 32 3 *y ct c* . Finally, since *X PY* , it follows that all the *Ei* are linear combinations of these functions 1 *y* , <sup>2</sup> *y* and 3 *y* and so, all of them are of the form:

1 1 1 2 2 3 3

*y y y y y y*

 

 

then

,

**5.2.1 First sub-case (** <sup>1</sup>

1 11 *y y* 

> exp

2

*dt*

*J*

 

*E tt <sup>i</sup>*

2

 

.

 

 

solution of the first and the third ODE are *y*1 1 *c t* exp

are of the following form: *Et t <sup>i</sup>*

 

<sup>2</sup> 1 1 exp 0 *<sup>i</sup> d E t tR*

 

**5.2.2 Second sub-case (** 1 3

**5.2.3 Third sub-case (** 1 2

, where

ODE can be written as *y*2 21

1 1 2 2 3 3

 

*y y y y y y*

 

 0 **and** 2 3 0 **)** 

As a consequence of this fact, the new ODE system to solve is:

As a consequence of this fact, there are no inflexion points in this sub-case.

0 **and** <sup>2</sup>

so, the main result is the same: there are no inflexion points in this sub-case.

0 **)** 

0 **and** <sup>3</sup>

or equivalently 1 1 *y*

*yc t* exp

twice with respect to *t* , the result is: 2

*dt*

 

0 **)** 

This case is the same as the previous one, except for a change in the order of the eigenvalues,

In this new sub-case, the corresponding Jordan canonical form has the following form:

1 2 and the new ODE system is:

Finally, taking into account the change of variables *X PY* , all the surface concentrations

 

2 2 <sup>2</sup> 2 exp *<sup>i</sup> d E t t*

 

*y* , 21 2 *y y y*

and the solution is *y*2 12 *ct c t* exp

exp . Once again, if *E t <sup>i</sup>* is differentiated

 

and 3 *y* 0 . Then, the

As in previous cases,

.

and 3 3 *y c* . Finally, the second

this expression can be zero only for a unique *t* value, concretely 2 2 *<sup>i</sup> d E dt* is zero only for

$$t = \frac{-\lambda^2 \beta - 2\lambda a}{\lambda^2 a}. \text{ Then, in this sub-case, only one information point can be explained.}$$

#### **5.3 Third case: A unique eigenvalue (** <sup>123</sup> 0 **)**

In this last case the Jordan canonical form is: 000 100 010 *J* and the new ODE system is in

this case: 1 1 2 2 3 3 000 100 010 *y y y y y y* or equivalently 1 *y* 0 , 2 1 *y y* and 3 2 *y y* . Then 1 1 *y c* , 2

21 2 *y ct c* and 31 2 3 2 *t y c ct c* and taking into account the change of variables *X PY* , all the surface concentrations are second order polynomials, such as: <sup>2</sup> *Et t <sup>i</sup>* . Finally, double differentiation of *E t <sup>i</sup>* with respect to *t* , gives: 2 <sup>2</sup> <sup>2</sup> *<sup>i</sup> d E dt* , so in this case there are no inflexion points.

As a consequence of this analysis, two inflexion points cannot be explained by mechanisms involving electrochemical uni-molecular reactions.

Finally, as mentioned in other papers, this double inflexion is explained in terms of electrocatalytic reactions (Martinez-Luaces, 2001: Guineo & Martinez-Luaces, 2002).

#### **6. Other qualitative results**

As an example, let us consider again the following mechanism:

$$E\_1 \xleftarrow[E\_1 \xleftarrow[E\_{-1}]{E\_2}]{E\_2} \quad E\_2 \xleftarrow[E\_{-2}]{E\_3} \quad \quad \quad E\_1 \xleftarrow[E\_3]{E\_3} \xrightarrow[E\_{-3}]{E\_3} \tag{4}$$

where all the possible reactions between the three adsorbates are considered.

As it was mentioned before, for such a mechanism, the mathematical model is:

$$
\begin{pmatrix} \dot{\mathbf{x}} \\ \dot{y} \\ \dot{z} \end{pmatrix} = \begin{pmatrix} -k\_1 - k\_3 & k\_{-1} & k\_{-3} \\ k\_1 & -k\_{-1} - k\_2 & k\_{-2} \\ k\_3 & k\_2 & -k\_{-2} - k\_{-3} \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ y \\ z \end{pmatrix} \tag{5}
$$

In this problem – as in other models, corresponding to other mechanisms – the result of the sum of all the rows is the null vector, and this fact implies the existence of a null eigenvalue.

Chemical Kinetics and Inverse Modelling Problems 73

1 33 2 33

 

13 3 2 33

 

*k kk*

0 0

 0 , 2 12 

(20)

(25)

,

 

*k k EEE* (19)

0 . Again, in this case, solutions are stable, but not

(23)

(24)

*k k* 0 and

*k kk*

*k k* 0 0

 

1 2

1 2

*k k AI k k k*

det det 0

*k k* 0 *.* Then, in this case two eigenvalues are negative and the other one is zero

1 2 123

*d E k E*

1

*dt*

 

 

*dt*

2

3

In this simpler case it is not necessary to solve the characteristic equation, i.e. det 0 *A I*

because *A* is a lower triangular matrix and so, the eigenvalues are the diagonal elements:

asymptotically, and so, experimental errors in the initial conditions tend to remain small and

As a final example, let us consider again the following mechanism involving all the possible

*k Ak k*

*d E k E dt*

*d E kE kE*

2 2

11 22

0 0 0

0 0

*k*

1 1

*A k kk*

and the corresponding ODE system matrix is:

In order to obtain the eigenvalues, the characteristic equation:

must be solved.

3 33

 

and the matrix system is:

 *k* 0 , 2 2 

*k* 0 and 3

reactions between the three adsorbates:

bounded, but they will not tend to disappear with time.

1 1 

In this case, the ODE system is as follows:

Algebraic manipulations lead to the following equation:

*kk kk* 12 3 3 0 and the solutions are 1

Let us turn again to the simpler mechanism, already studied:

and so, solutions of the ODE system are stable, but not asymptotically.

In the previous section, this fact was utilised to prove the impossibility of two inflexion points in the surface concentration curves. Now, in this section, these facts will be used to reach other conclusions about qualitative behaviours, particularly in terms of stability of the solutions and several inverse problems related with this sequence of reactions.

#### **6.1 Stability of the solutions**

From a mathematical viewpoint when an ODE solution is altered slightly for instance, changing the initial conditions, a set of new curves may or may not show a different behaviour from the original one. From this fact, the heuristic basis for the concepts of stability and instability may be found. Moreover, a third possibility – asymptotically stable solutions – can arise when the altered solutions tend to the original one over a period of time, i.e., time nullifies any changes made to original solution (Martinez-Luaces, 2009e).

From a chemical point of view, the last option is the most desirable one, because it means that small errors due to measurements can be expected to diminish and almost disappear as reaction time proceeds.

In these cases, as in other mathematical models corresponding to ODE linear systems, stability and/or asymptotical stability strongly depend on the eigenvalues, particularly on their signs and multiplicity.

In all the unimolecular mechanisms studied there exists a null eigenvalue, and this fact implies that the solutions cannot be asymptotically stable (Martinez-Luaces, 2007, 2009e). As a consequence of chemical kinetics, small errors in the initial surface concentrations cannot be expected to vanish as the reaction tends to completion. Moreover, in the best hypothesis, we can expect these errors to be small and bounded, if the other eigenvalues are negative (i.e., this best possibility takes place only if 1 0 , 2 0 and , or any other similar combination)*.*  0 3

Let us return to the first mechanism considered in this article:

$$\begin{array}{l} \mathrm{E\_1 \xrightarrow{k\_1} \mathrm{E\_2}}\\ \mathrm{E\_1 \xrightarrow{k\_2} \mathrm{E\_3}}\\ \mathrm{E\_2 \xrightleftharpoons} \mathrm{E\_3} \end{array} \tag{1}$$

As mentioned before, the mathematical model corresponding to this mechanism is the following one:

$$\begin{cases} \frac{d\left[E\_1\right]}{dt} = -\left(k\_1 + k\_2\right)\left[E\_1\right] \\ \frac{d\left[E\_2\right]}{dt} = k\_1\left[E\_1\right] - k\_3\left[E\_2\right] + k\_{-3}\left[E\_3\right] \\ \frac{d\left[E\_3\right]}{dt} = k\_2\left[E\_1\right] + k\_3\left[E\_2\right] - k\_{-3}\left[E\_3\right] \end{cases} \tag{2}$$

and the corresponding ODE system matrix is:

$$A = \begin{pmatrix} -k\_1 - k\_2 & 0 & 0 \\ k\_1 & -k\_3 & k\_{-3} \\ k\_2 & k\_3 & -k\_{-3} \end{pmatrix} \tag{23}$$

In order to obtain the eigenvalues, the characteristic equation:

$$\det\begin{pmatrix} A - \lambda I \end{pmatrix} = \det\begin{pmatrix} -k\_1 - k\_2 - \lambda & 0 & 0\\ k\_1 & -k\_3 - \lambda & k\_{-3} \\ k\_2 & k\_3 & -k\_{-3} - \lambda \end{pmatrix} = 0 \tag{24}$$

must be solved.

72 Chemical Kinetics

In the previous section, this fact was utilised to prove the impossibility of two inflexion points in the surface concentration curves. Now, in this section, these facts will be used to reach other conclusions about qualitative behaviours, particularly in terms of stability of the

From a mathematical viewpoint when an ODE solution is altered slightly for instance, changing the initial conditions, a set of new curves may or may not show a different behaviour from the original one. From this fact, the heuristic basis for the concepts of stability and instability may be found. Moreover, a third possibility – asymptotically stable solutions – can arise when the altered solutions tend to the original one over a period of time, i.e., time nullifies any changes made to original solution (Martinez-Luaces, 2009e).

From a chemical point of view, the last option is the most desirable one, because it means that small errors due to measurements can be expected to diminish and almost disappear as

In these cases, as in other mathematical models corresponding to ODE linear systems, stability and/or asymptotical stability strongly depend on the eigenvalues, particularly on

In all the unimolecular mechanisms studied there exists a null eigenvalue, and this fact implies that the solutions cannot be asymptotically stable (Martinez-Luaces, 2007, 2009e). As a consequence of chemical kinetics, small errors in the initial surface concentrations cannot be expected to vanish as the reaction tends to completion. Moreover, in the best hypothesis, we can expect these errors to be small and bounded, if the other eigenvalues are negative

> 0 , 2

1 2 3

 

E E E E

*k k k*

1 2 1 3

3

As mentioned before, the mathematical model corresponding to this mechanism is the

*d E kE kE k E*

*d E kE kE k E*

11 32 33

21 32 33

121

*d E k kE*

1

*dt*

 

 

*dt*

*dt*

2

3

2 3

E E

*k*

0 and , or any other similar

0 3

(1)

(2)

solutions and several inverse problems related with this sequence of reactions.

**6.1 Stability of the solutions** 

reaction time proceeds.

combination)*.* 

following one:

their signs and multiplicity.

(i.e., this best possibility takes place only if 1

Let us return to the first mechanism considered in this article:

Algebraic manipulations lead to the following equation:

 *kk kk* 12 3 3 0 and the solutions are 1 0 , 2 12 *k k* 0 and 3 33 *k k* 0 *.* Then, in this case two eigenvalues are negative and the other one is zero and so, solutions of the ODE system are stable, but not asymptotically.

Let us turn again to the simpler mechanism, already studied:

$$E\_1 \xrightarrow{k\_1} E\_2 \xrightarrow{k\_2} E\_3 \tag{19}$$

In this case, the ODE system is as follows:

$$\begin{cases} \frac{d\left[E\_1\right]}{dt} = -k\_1 \left[E\_1\right] \\ \frac{d\left[E\_2\right]}{dt} = k\_1 \left[E\_1\right] - k\_2 \left[E\_2\right] \\ \frac{d\left[E\_3\right]}{dt} = k\_2 \left[E\_2\right] \end{cases} \tag{20}$$

and the matrix system is:

$$A = \begin{pmatrix} -k\_1 & 0 & 0 \\ k\_1 & -k\_2 & 0 \\ 0 & k\_2 & 0 \end{pmatrix} \tag{25}$$

In this simpler case it is not necessary to solve the characteristic equation, i.e. det 0 *A I* , because *A* is a lower triangular matrix and so, the eigenvalues are the diagonal elements: 1 1 *k* 0 , 2 2 *k* 0 and 3 0 . Again, in this case, solutions are stable, but not asymptotically, and so, experimental errors in the initial conditions tend to remain small and bounded, but they will not tend to disappear with time.

As a final example, let us consider again the following mechanism involving all the possible reactions between the three adsorbates:

Chemical Kinetics and Inverse Modelling Problems 75

does any ODE linear system correspond – at least theoretically – to a given mechanism? If not, what condition must be considered to ensure that a given ODE linear system

In order to answer these questions we shall slightly modify the ODE system matrix corresponding to one of the mechanisms previously analysed. For instance, if we consider

*K*

1

 

> 1 2

 

1 2

*E E*

*K K*

*K K*

2 3

 

2 3

(4)

(27)

(28)

cannot be associated

*E E*

*K*

3

13 1 3 1 12 2 3 2 23

 

 

does not correspond to a certain reaction mechanism, because if

and the last formula implies that adding all

instead of the null vector as in the case of matrix

*k k kk*

13 1 3 1 12 2 3 2 23

*k k kk*

*kk k k*

(it is important to note that the first entry 11 1 3 *a kk* was changed to a slightly different

This observation has a dramatic consequence for the inverse modelling problem: *A*

with any of these mechanisms. As mentioned before, this is due to the mass conservation law, which in this case can be written as *E E Ec* <sup>123</sup> . Derivation of this equation

the matrix rows the result must be zero. This criterion can be used in order to know

*kk k k A k kk k*

Now, if just the first entry of this system matrix is modified, the resulting matrix will be:

*A k kk k*

1 3

*E E*

corresponds to a chemical unimolecular reaction mechanism?

again the following mechanism:

and the corresponding ODE system matrix is:

).

new one, *a kk* \*

This new matrix *A*

*A* .

11 1 3

all its rows are added, the result is: ( ,0,0)

gives the following: <sup>0</sup>

corresponds to a given unimolecular reaction mechanism, but *A*

*dA dB dC dt dt dt*

$$\begin{aligned} \xleftarrow{E\_1} \xleftarrow{\x\_1} E\_2\\ E\_2 \xleftarrow{\x\_2} E\_3\\ E\_1 \xleftarrow{\x\_3} E\_3 \end{aligned} \tag{4}$$

The mathematical model for this mechanism, written in a simplified vector form is:

$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix}' = \begin{pmatrix} -k\_1 - k\_3 & k\_{-1} & k\_{-3} \\ k\_1 & -k\_{-1} - k\_2 & k\_{-2} \\ k\_3 & k\_2 & -k\_{-2} - k\_{-3} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \tag{5}
$$

And the characteristic equation is:

$$\det\left(A - \lambda I\right) = \det\begin{pmatrix} -k\_1 - k\_3 - \lambda & k\_{-1} & k\_{-3} \\ k\_1 & -k\_{-1} - k\_2 - \lambda & k\_{-2} \\ k\_3 & k\_2 & -k\_{-2} - k\_{-3} - \lambda \end{pmatrix} = 0 \tag{26}$$

Once again, algebraic manipulations lead to a third order polynomial equation of this form: 3 2 *a b* 0 that can be written as <sup>2</sup> *a b* 0 where 123 1 2 3 0 *<sup>i</sup> i ak k k k k k k* and 12 1 2 1 3 23 2 3 3 1 3 2 1 2 1 3 *b kk kk kk kk kk kk kk k k k k* 0 .

On the other hand, since the eigenvalues are <sup>1</sup> , <sup>2</sup> and <sup>3</sup> 0 , this polynomial equation can be expressed as: 1 2 0 0 which can be written in the alternative form: <sup>2</sup> 1 2 12 () 0 .

Finally, if <sup>2</sup> *a b* 0 and <sup>2</sup> 1 2 12 () 0 are different forms of the same equation, then: 1 2 *a* 0 and 1 2 *b* 0 . These inequalities lead to the following result: 1 0 , 2 0 and <sup>3</sup> 0 , as in the other examples previously analysed and, once again, this fact means that the ODE solutions will be stable, but not asymptotically.

In all the previous examples, results can be summarised as: 1 0 , 2 0 and <sup>3</sup> 0 or any other similar combination. It is important to mention that the null eigenvalue can be obtained simply as a consequence of Lavoisier's law, as shown previously (in section 5), while the others must be obtained from the characteristic equation.

In all cases analysed solutions are just stable (a weak stability, not asymptotic stability) and as a consequence, experimental errors in the initial concentrations tend to remain bounded, but they do not tend to vanish with time.

#### **6.2 Inverse modelling problems**

Each chemical or electrochemical mechanism involving unimolecular reactions leads to a linear ODE system. This is a well known result, but what happens with the inverse question:

*K*

1

 

> 1 2

 

1 2

*E E*

*K K*

*K K*

2 3

 

2 3

(4)

 

0 , this polynomial equation

0 and <sup>3</sup>

0 or

() 0 are different forms of the

*b* 0 . These inequalities lead to the following

(5)

(26)

*a b* 0 where

*E E*

*K*

The mathematical model for this mechanism, written in a simplified vector form is:

And the characteristic equation is:

3 2 

can be expressed as:

 () 0 .

result: 1 0 , 2 

 <sup>2</sup> 1 2 12

Finally, if <sup>2</sup> 

same equation, then: 1 2

123 1 2 3 0 *<sup>i</sup>*

On the other hand, since the eigenvalues are

 

 

but they do not tend to vanish with time.

**6.2 Inverse modelling problems** 

 

0 and <sup>3</sup>

In all the previous examples, results can be summarised as: 1

while the others must be obtained from the characteristic equation.

*ak k k k k k k* and

3

13 1 3 1 12 2 3 2 23

 

1 3 1 3 1 12 2 3 22 3

*k k kk*

*kk k k*

 

 

<sup>2</sup> and <sup>3</sup> 

1 2 0 0 which can be written in the alternative form:

0 , as in the other examples previously analysed and, once

 0 , 2 

*x x kk k k y k kk k y z z k k kk*

det det 0

Once again, algebraic manipulations lead to a third order polynomial equation of this form:

<sup>1</sup> , 

any other similar combination. It is important to mention that the null eigenvalue can be obtained simply as a consequence of Lavoisier's law, as shown previously (in section 5),

In all cases analysed solutions are just stable (a weak stability, not asymptotic stability) and as a consequence, experimental errors in the initial concentrations tend to remain bounded,

Each chemical or electrochemical mechanism involving unimolecular reactions leads to a linear ODE system. This is a well known result, but what happens with the inverse question:

1 2 12

*AI k k k k*

*a b* 0 that can be written as <sup>2</sup>

 

*a b* 0 and <sup>2</sup>

again, this fact means that the ODE solutions will be stable, but not asymptotically.

 

 

*i*

 

*a* 0 and 1 2

12 1 2 1 3 23 2 3 3 1 3 2 1 2 1 3 *b kk kk kk kk kk kk kk k k k k* 0 .

1 3

*E E*

does any ODE linear system correspond – at least theoretically – to a given mechanism? If not, what condition must be considered to ensure that a given ODE linear system corresponds to a chemical unimolecular reaction mechanism?

In order to answer these questions we shall slightly modify the ODE system matrix corresponding to one of the mechanisms previously analysed. For instance, if we consider again the following mechanism:

$$\begin{aligned} \xleftarrow{E\_1} \xleftarrow{\x\_1} E\_2\\ E\_2 \xleftarrow{\x\_2} E\_3\\ E\_1 \xleftarrow{\x\_3} E\_3\\ E\_1 \xleftarrow{\x\_3} E\_3 \end{aligned} \tag{4}$$

and the corresponding ODE system matrix is:

$$A = \begin{pmatrix} -k\_1 - k\_3 & k\_{-1} & k\_{-3} \\ k\_1 & -k\_{-1} - k\_2 & k\_{-2} \\ k\_3 & k\_2 & -k\_{-2} - k\_{-3} \end{pmatrix} \tag{27}$$

Now, if just the first entry of this system matrix is modified, the resulting matrix will be:

$$A\_x = \begin{pmatrix} x - k\_1 - k\_3 & k\_{-1} & k\_{-3} \\ & k\_1 & -k\_{-1} - k\_2 & k\_{-2} \\ & k\_3 & k\_2 & -k\_{-2} - k\_{-3} \end{pmatrix} \tag{28}$$

(it is important to note that the first entry 11 1 3 *a kk* was changed to a slightly different new one, *a kk* \* 11 1 3 ).

This new matrix *A* does not correspond to a certain reaction mechanism, because if all its rows are added, the result is: ( ,0,0) instead of the null vector as in the case of matrix *A* .

This observation has a dramatic consequence for the inverse modelling problem: *A* corresponds to a given unimolecular reaction mechanism, but *A* cannot be associated with any of these mechanisms. As mentioned before, this is due to the mass conservation law, which in this case can be written as *E E Ec* <sup>123</sup> . Derivation of this equation

gives the following: <sup>0</sup> *dA dB dC dt dt dt* and the last formula implies that adding all the matrix rows the result must be zero. This criterion can be used in order to know

Chemical Kinetics and Inverse Modelling Problems 77

Guineo Cobs, G. & Martínez Luaces, V., (2002). Electrocatalytic reactions: An interesting

Martínez Luaces, V., (2001). Reacciones Electroquímicas y Electrocatalíticas: Un problema de

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Martínez Luaces, V., (2007). Inverse-modelling problems in Chemical Engineering courses.

Martínez Luaces, V., (2008). Modelado inverso y Transformada de Laplace en problemas de

Martínez Luaces, V., (2009a). Modelling and inverse-modelling: experiences with

Martínez Luaces,V., (2009b). Problemas de modelado directo e inverso con Ecuaciones

Martínez Luaces,V., (2009c). Modelling and Inverse Modelling with second order P.D.E. in

Martínez Luaces,V., (2009d). Modelling, applications and Inverse Modelling: Innovations in

Martínez Luaces, V., (2009e). *Aplicaciones y modelado: Ecuaciones Diferenciales Ordinarias,* 

Martínez Luaces, V., (2011). Problemas inversos: los casi olvidados de la Matemática

Martínez Luaces, V. & Guineo Cobs, G., (2002). Un problema de Electroquímica y su

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96-348-1, El Calafate, Argentina, November 2007.

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July 2002.

0020-739X.

ISSN 0020-739X.

Africa, November 2009.

607-95306-4-8.

Gordon's Bay, South Africa, November 2009.

*COMAT 95-97-99*, ISBN 959 – 160097 – 6. Cuba.

9974-96-621-5 Montevideo, Uruguay.

pp. 272 – 276, ISSN 0328 – 087X.

2009.

Cuba.

problem of Numerical Calculus, *Communications of the 2nd International Conference on the Teaching of Mathematics at the undergraduate level*. Crete, Greece,

Matemática Aplicada. *Actas COMAT 95-97-99*, ISBN 959 – 160097 – 6. Matanzas,

Modelling: Some Experiences in South American Countries. *International Journal of Mathematical Education in Science and Technology,* Vol 36, No. 2-3, pp. 193-194, ISSN

*Vision and change for a new century. Proceedings of Calafate Delta '07.* ISBN: 978-9974-

diseño de reactores químicos. *Proceedings of EMCI XIV. Educación Matemática en* 

O.D.E. linear systems in engineering courses, *International Journal of Mathematical Education in Science and Technology*, Vol. 40, No. 2, pp. 259-268,

Diferenciales y Transformación de Laplace. *Proceedings of EMCI XV. Educación Matemática en Carreras de Ingeniería*, Tucumán, Argentina, September

Engineering courses*. Proceedings of Southern Right Delta '09.* Gordon's Bay, South

Differential Equations courses. *Proceedings of Southern Right Gordon's Bay Delta '09*,

*Transformación de Laplace, Ecuaciones en Derivadas Parciales.* Matser (Eds.), ISBN 978-

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Modelación Matemática, *Anuario Latinoamericano de Educación Química.* Año 2002,

Computacionales en el estudio de Mecanismos de Reacciones Químicas", *Actas de* 

whether or not a given matrix corresponds to a certain mechanism (Martinez-Luaces, 2007, 2009a, 2009e).

#### **7. Conclusions and observations**

This article describes a combination of chemical and mathematical modelling applied to the adsorption of Carbon Dioxide on Platinum surfaces, but a similar procedure can be applied to any chemical or electrochemical mechanism involving unimolecular reactions. Moreover, mathematical theorems about eigenvalues, eigenvectors, diagonalization, Jordan canonical forms, etc., and chemical laws, particularly Lavoisier's law of mass conservation can be combined to solve inverse causation and stability problems.

It is well known that any given chemical mechanism involving unimolecular reactions leads to a linear ODE system, but the converse is not true. In fact, as previously shown, slightly modifying a given ODE system matrix results in a new matrix that does not correspond to any unimolecular reaction mechanism. Then, this result gives a negative answer to the inverse problem about whether a mechanism exists for every linear ODE system. At the same time, it gives a negative answer to the "inverse stability problem", i.e., a slight change in one of the coefficients produces dramatic changes in the inverse problem. Moreover, there exists a mechanism corresponding to a certain matrix *A* but not for the modified matrix *A* . This is a typical conclusion of inverse problems, where existence, uniqueness and stability do not usually occur.

Finally, other issues like the qualitative behaviour of solutions (stability, instability and asymptotic stability), can be analysed using this approach. Moreover, in all the examples studied, solutions exhibited weak stability, which in terms of the electrochemical problem means that experimental errors in the initial surface concentrations tend to remain limited as the reaction proceeds to completion. Nevertheless, these experimental errors never tend to vanish, and this fact is a consequence of having a null eigenvalue. As observed earlier, all these facts can be proved combining mathematical theorems with chemical laws, so they can be easily generalised to other unimolecular reactions mechanisms.

#### **8. References**


http://grupobunge.wordpress.com/2006/07/20/119.


whether or not a given matrix corresponds to a certain mechanism (Martinez-Luaces,

This article describes a combination of chemical and mathematical modelling applied to the adsorption of Carbon Dioxide on Platinum surfaces, but a similar procedure can be applied to any chemical or electrochemical mechanism involving unimolecular reactions. Moreover, mathematical theorems about eigenvalues, eigenvectors, diagonalization, Jordan canonical forms, etc., and chemical laws, particularly Lavoisier's law of mass conservation can be

It is well known that any given chemical mechanism involving unimolecular reactions leads to a linear ODE system, but the converse is not true. In fact, as previously shown, slightly modifying a given ODE system matrix results in a new matrix that does not correspond to any unimolecular reaction mechanism. Then, this result gives a negative answer to the inverse problem about whether a mechanism exists for every linear ODE system. At the same time, it gives a negative answer to the "inverse stability problem", i.e., a slight change in one of the coefficients produces dramatic changes in the inverse problem. Moreover, there exists a mechanism corresponding to a certain matrix *A* but not for the modified matrix *A*

This is a typical conclusion of inverse problems, where existence, uniqueness and stability

Finally, other issues like the qualitative behaviour of solutions (stability, instability and asymptotic stability), can be analysed using this approach. Moreover, in all the examples studied, solutions exhibited weak stability, which in terms of the electrochemical problem means that experimental errors in the initial surface concentrations tend to remain limited as the reaction proceeds to completion. Nevertheless, these experimental errors never tend to vanish, and this fact is a consequence of having a null eigenvalue. As observed earlier, all these facts can be proved combining mathematical theorems with chemical laws, so they can

Blum, W. et al., (2002). ICMI Study 14: applications and modelling in mathematics education

Bunge, M., (2006), Problemas directos e inversos, In: *Grupobunge. Filosofia y ciencia*, Access:

Groestch, C.W., (1999), *Inverse problems: activities for undergraduates,* Mathematical

Groestch, C.W., (2001). Inverse problems: the other two-thirds of the story, *Quaestiones Mathematicae,* Vol. Suppl. 1, 2001, Supplement, 89-93, ISSN 1607-3606. Guerasimov, Y.A., et al*.*, (1995), *Physical Chemistry,* 2nd Edition, Houghton Mifflin, (Eds.)

– discussion document. *Educational Studies in Mathematics,* Vol. 51, No. 1-2, 149-

.

2007, 2009a, 2009e).

do not usually occur.

**8. References** 

171.

May, 2011, Available from:

Boston, U.S.A.

**7. Conclusions and observations** 

combined to solve inverse causation and stability problems.

be easily generalised to other unimolecular reactions mechanisms.

http://grupobunge.wordpress.com/2006/07/20/119.

Association of America (Eds.), Washington D.C., U.S.A.


**1. Introduction**

system.

form:

**0**

**4**

Terese Løvås

*Norway*

**Model Reduction Techniques for**

To be able to meet the increasing demands for efficiency in energy producing systems (engines, turbines and furnaces), changes in the geometry of combustion devices or fuel composition are necessary. Such changes should be based on a thorough understanding of both the physical and chemical processes involved. Today, computer simulations are one of the most important tools for research to address these issues. This is increasingly employed for testing and characterizing the many features of the combustion process, along with or sometimes replacing the considerably more expensive experiments. Numerous types of reliable software are available for simulations of car engines, gas turbines, heaters and boilers. However, one limitation in using computer simulations is the degree to which one is forced to rely on the available computer capacity. Since most cases of combustion involve turbulent processes and numerous chemical reactions, this can be a serious drawback if one wishes to carry out a detailed simulation. Codes for turbulent flow situations are developed to account for turbulent motion between small regions, often called cells, where combustion within the cell is assumed to be homogenous. To deal with complex interaction between the physical and chemical processes in reactive systems, it is necessary to find methods that simplify modeling

in such a way that it becomes both more comprehensible and practically useful.

*∂***Y**

The starting point for any reduction of chemical model is the detailed mechanism itself. Detailed chemical mechanisms have been investigated extensively during the history of combustion physics since the chemistry involved plays a key role in the outcome of any combustion process. By understanding the chemistry one can predict the rate of production and consumption of different species and thus also predict the total change in enthalpy during the combustion process. This in turn provides the values for the energy output, the end gas temperature and the emission characteristics allowing one to describe the overall combustion

The chemical system consists of a mechanism containing a set of differential equations representing the evolution of the concentrations of the individual species during the combustion process, thus representing the species conservation equation. The system itself is most often described in terms of molar fractions, *Xi*, or mass fractions, *Yi*, of the following

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> **P(Y,u,**T**)** <sup>+</sup> *<sup>ω</sup>***(Y,**T**)** (1)

**Chemical Mechanisms**

*Department of Energy and Process Engineering Norwegian University of Science and Technology*

Zinola, F., Méndez, E. & Martínez Luaces, V., (1997). Modificación de estados adsorbidos de Anhídrido Carbónico reducido por labilización electroquímica en superficies facetadas de platino. *Proceedings of X Congreso Argentino de Fisicoquímica*, Tucumán, Argentina, April 1997.

## **Model Reduction Techniques for Chemical Mechanisms**

#### Terese Løvås

*Department of Energy and Process Engineering Norwegian University of Science and Technology Norway*

#### **1. Introduction**

78 Chemical Kinetics

Zinola, F., Méndez, E. & Martínez Luaces, V., (1997). Modificación de estados adsorbidos de

Argentina, April 1997.

Anhídrido Carbónico reducido por labilización electroquímica en superficies facetadas de platino. *Proceedings of X Congreso Argentino de Fisicoquímica*, Tucumán,

> To be able to meet the increasing demands for efficiency in energy producing systems (engines, turbines and furnaces), changes in the geometry of combustion devices or fuel composition are necessary. Such changes should be based on a thorough understanding of both the physical and chemical processes involved. Today, computer simulations are one of the most important tools for research to address these issues. This is increasingly employed for testing and characterizing the many features of the combustion process, along with or sometimes replacing the considerably more expensive experiments. Numerous types of reliable software are available for simulations of car engines, gas turbines, heaters and boilers. However, one limitation in using computer simulations is the degree to which one is forced to rely on the available computer capacity. Since most cases of combustion involve turbulent processes and numerous chemical reactions, this can be a serious drawback if one wishes to carry out a detailed simulation. Codes for turbulent flow situations are developed to account for turbulent motion between small regions, often called cells, where combustion within the cell is assumed to be homogenous. To deal with complex interaction between the physical and chemical processes in reactive systems, it is necessary to find methods that simplify modeling in such a way that it becomes both more comprehensible and practically useful.

> The starting point for any reduction of chemical model is the detailed mechanism itself. Detailed chemical mechanisms have been investigated extensively during the history of combustion physics since the chemistry involved plays a key role in the outcome of any combustion process. By understanding the chemistry one can predict the rate of production and consumption of different species and thus also predict the total change in enthalpy during the combustion process. This in turn provides the values for the energy output, the end gas temperature and the emission characteristics allowing one to describe the overall combustion system.

> The chemical system consists of a mechanism containing a set of differential equations representing the evolution of the concentrations of the individual species during the combustion process, thus representing the species conservation equation. The system itself is most often described in terms of molar fractions, *Xi*, or mass fractions, *Yi*, of the following form:

$$\frac{\partial \mathbf{Y}}{\partial t} = \mathbf{P}(\mathbf{Y}\_{\prime}\mathbf{u}\_{\prime}\mathbf{T}) + \omega(\mathbf{Y}\_{\prime}\mathbf{T}) \tag{1}$$

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called *time scale separation methods*. The most common methods are those of (i) *Intrinsic Low Dimensional Manifolds* (ILDM), (ii) *Computational Singular Perturbation*(CSP), and (iii) *level of importance* (LOI) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. In Figure (1) a schematic overview of the different reduction paths is laid out. As can be seen, different paths and possibilities for reducing the mechanisms are possible. Which method is adopted is determined mainly by the nature of the problem to be solved. The starting point is a detailed mechanism generated by hand or by the software available and validated for the current conditions. These mechanisms can contain up to hundreds of active species and thus need to be reduced for practical use. However, smaller detailed mechanisms can successfully be applied in stationary combustion situations, and in situations in which speed-up is of less concern. This could be the development of flamelet libraries or simple zone models, for example. Larger mechanisms are the object of reaction flow and sensitivity analysis in which minor and major pathways of the mechanism are identified. Species that are only involved in the minor paths and do not appear as species sensitive to the governing parameters, are selected for removal from the system. The result is a smaller mechanism containing the major pathways and the sensitive species, termed a skeletal mechanism. This mechanism is complete in the sense that it can be applied in combustion simulations. For reasons of speed-up it is favorable for use in stationary combustion situations. If the mechanism is small enough, it can also be applied in non-stationary situations such as in the calculation of interactive flamelets and transported probability density functions (PDF) for turbulent motion simulations. However, these calculations require very compact mechanisms and in most cases a further reduction through a lifetime analysis is necessary. This reduces the number of active species in the mechanism and accounts for the contributions of the remaining species selected as steady state species. The resulting reduced mechanism can be successfully applied in flow codes, and is applicable in simpler systems for speed-up reasons as well. The lifetime analysis and the resulting reduced mechanism is employed when on-line reduction is carried out, since

Model Reduction Techniques for Chemical Mechanisms 81

the number of active species involved can change during the computational run.

constrained equilibria (RCCE) approach.

**2. Static and dynamic analysis tools**

In the following, reaction flow analysis, sensitivity analysis and the directed relation graph method will be presented as static and dynamic reduction procedures. Thereafter will the main features of ILDM (including extensions such as flamelet generated manifolds (FGM) and reaction-diffusion manifolds (REDIM)), CSP and the LOI be discussed, including the fundamentals of the quasi steady state elimination procedure and the rate-controlled

The static and dynamic analysis tools available provide important information that is useful for reduction procedures and includes reaction flow and sensitivity analysis procedures. Reaction flow analysis detects minor and major flows in the reaction system, where the species involved in the minor flows can be neglected in further investigation. Under given physical

where **P(Y,u,**T**)** represents the spatial differential operator (advection, convection and diffusion; it is thus depending on the velocity field**, u**, and the temperature**,** T) and *ω***(Y,**T**)** represents the chemical source term, which includes chemical production and consumption of the species. **Y** is the N*S*-dimensional vector of mass fractions of a mechanism containing N*<sup>S</sup>* species.

A long list of different detailed mechanisms have been presented in literature, ranging from simple hydrogen-oxygen mechanisms (Lewis & von Elbe, 1987);(Li et al., 2004) to complex fuel mechanisms which involve hundreds of species that interact in thousands of reactions (Côme et al., 1996) or even thousand of species reacting in tens of thousands of reactions (Westbrook et al., 2009). Discussions related to the nature of these detailed mechanisms in the literature is dating back to the beginning of combustion science. Some of these mechanisms have been developed and extended by hand, the validation of the detailed mechanism coming through experimental measurements, a very time consuming procedure. One very well known mechanisms for methane kinetics is the GRI-Mech 3.0 mechanism (Bowman et al., 2004). This mechanism is the result of collaboration between several groups and is under constant revision. This mechanism contains 49 species and considers 277 reactions for the C1 chain only.

Detailed mechanisms can also be obtained through an automatic generation process carried out by computer software. One way to automatically generate detailed mechanisms is described in works by Blurock (Blurock, 1995); (Blurock, 2000). Large and detailed chemical mechanisms including species larger than C12-hydrocarbons are generated by the use of *reaction classes* associated with the reacting substructure in the species and a set of reaction constants. As Blurock (Blurock, 2000) indicates this "allows for a wide variety of arbitrarily branched and substituted species far beyond straight-chained hydrocarbons". This is an advantage when performing numeric computations of the combustion process, since only the reacting part of a class of species is of importance for the computations. This allows a symbolically generated mechanism to be studied that is substantially larger and more detailed than mechanisms previously carried out by hand. Mechanism generation tools are also freely available, such as the "Reaction Mechanism Generator" (RGM) developed by W. Green and co-workers (Green et al., 2011). This is a very extensive tool which generates mechanisms based on knowledge on how molecules react through elementary reactions.

As much as such tools are needed to develop mechanisms for complex chemical processes such as for e.g. combustion of bio-fuels from various biomass sources, the models become increasingly unmanageable due to their size. The detailed mechanism is therefore the object of different reduction procedures aimed at obtaining smaller mechanisms to be used in more complex simulations of combustion systems. The reduced mechanisms can be and to a large extent have been, worked out by hand on the basis of the total sum of the experience and knowledge available. This can result in compact and accurate mechanisms subsequently validated by experimental data. However, these mechanisms are often restricted to a narrow range of conditions. It is advantageous therefore to find general procedures that can reduce mechanisms automatically by using physical quantities associated with different reactions and species. Through identifying a set of selection criteria it is possible to detect species and reactions that play a minor role in the overall reaction process and can thus be excluded from the mechanism. This can be done by use of *reaction flow* and *sensitivity analysis* or by using *directed relation graph* (DRG) methods, which result in what is hereafter referred to as a skeletal mechanism.

2 Will-be-set-by-IN-TECH

where **P(Y,u,**T**)** represents the spatial differential operator (advection, convection and diffusion; it is thus depending on the velocity field**, u**, and the temperature**,** T) and *ω***(Y,**T**)** represents the chemical source term, which includes chemical production and consumption of the species. **Y** is the N*S*-dimensional vector of mass fractions of a mechanism containing

A long list of different detailed mechanisms have been presented in literature, ranging from simple hydrogen-oxygen mechanisms (Lewis & von Elbe, 1987);(Li et al., 2004) to complex fuel mechanisms which involve hundreds of species that interact in thousands of reactions (Côme et al., 1996) or even thousand of species reacting in tens of thousands of reactions (Westbrook et al., 2009). Discussions related to the nature of these detailed mechanisms in the literature is dating back to the beginning of combustion science. Some of these mechanisms have been developed and extended by hand, the validation of the detailed mechanism coming through experimental measurements, a very time consuming procedure. One very well known mechanisms for methane kinetics is the GRI-Mech 3.0 mechanism (Bowman et al., 2004). This mechanism is the result of collaboration between several groups and is under constant revision. This mechanism contains 49 species and considers 277 reactions for the C1

Detailed mechanisms can also be obtained through an automatic generation process carried out by computer software. One way to automatically generate detailed mechanisms is described in works by Blurock (Blurock, 1995); (Blurock, 2000). Large and detailed chemical mechanisms including species larger than C12-hydrocarbons are generated by the use of *reaction classes* associated with the reacting substructure in the species and a set of reaction constants. As Blurock (Blurock, 2000) indicates this "allows for a wide variety of arbitrarily branched and substituted species far beyond straight-chained hydrocarbons". This is an advantage when performing numeric computations of the combustion process, since only the reacting part of a class of species is of importance for the computations. This allows a symbolically generated mechanism to be studied that is substantially larger and more detailed than mechanisms previously carried out by hand. Mechanism generation tools are also freely available, such as the "Reaction Mechanism Generator" (RGM) developed by W. Green and co-workers (Green et al., 2011). This is a very extensive tool which generates mechanisms

As much as such tools are needed to develop mechanisms for complex chemical processes such as for e.g. combustion of bio-fuels from various biomass sources, the models become increasingly unmanageable due to their size. The detailed mechanism is therefore the object of different reduction procedures aimed at obtaining smaller mechanisms to be used in more complex simulations of combustion systems. The reduced mechanisms can be and to a large extent have been, worked out by hand on the basis of the total sum of the experience and knowledge available. This can result in compact and accurate mechanisms subsequently validated by experimental data. However, these mechanisms are often restricted to a narrow range of conditions. It is advantageous therefore to find general procedures that can reduce mechanisms automatically by using physical quantities associated with different reactions and species. Through identifying a set of selection criteria it is possible to detect species and reactions that play a minor role in the overall reaction process and can thus be excluded from the mechanism. This can be done by use of *reaction flow* and *sensitivity analysis* or by using *directed relation graph* (DRG) methods, which result in what is hereafter referred to as a skeletal

based on knowledge on how molecules react through elementary reactions.

N*<sup>S</sup>* species.

chain only.

mechanism.

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called *time scale separation methods*. The most common methods are those of (i) *Intrinsic Low Dimensional Manifolds* (ILDM), (ii) *Computational Singular Perturbation*(CSP), and (iii) *level of importance* (LOI) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species.

In Figure (1) a schematic overview of the different reduction paths is laid out. As can be seen, different paths and possibilities for reducing the mechanisms are possible. Which method is adopted is determined mainly by the nature of the problem to be solved. The starting point is a detailed mechanism generated by hand or by the software available and validated for the current conditions. These mechanisms can contain up to hundreds of active species and thus need to be reduced for practical use. However, smaller detailed mechanisms can successfully be applied in stationary combustion situations, and in situations in which speed-up is of less concern. This could be the development of flamelet libraries or simple zone models, for example. Larger mechanisms are the object of reaction flow and sensitivity analysis in which minor and major pathways of the mechanism are identified. Species that are only involved in the minor paths and do not appear as species sensitive to the governing parameters, are selected for removal from the system. The result is a smaller mechanism containing the major pathways and the sensitive species, termed a skeletal mechanism. This mechanism is complete in the sense that it can be applied in combustion simulations. For reasons of speed-up it is favorable for use in stationary combustion situations. If the mechanism is small enough, it can also be applied in non-stationary situations such as in the calculation of interactive flamelets and transported probability density functions (PDF) for turbulent motion simulations. However, these calculations require very compact mechanisms and in most cases a further reduction through a lifetime analysis is necessary. This reduces the number of active species in the mechanism and accounts for the contributions of the remaining species selected as steady state species. The resulting reduced mechanism can be successfully applied in flow codes, and is applicable in simpler systems for speed-up reasons as well. The lifetime analysis and the resulting reduced mechanism is employed when on-line reduction is carried out, since the number of active species involved can change during the computational run.

In the following, reaction flow analysis, sensitivity analysis and the directed relation graph method will be presented as static and dynamic reduction procedures. Thereafter will the main features of ILDM (including extensions such as flamelet generated manifolds (FGM) and reaction-diffusion manifolds (REDIM)), CSP and the LOI be discussed, including the fundamentals of the quasi steady state elimination procedure and the rate-controlled constrained equilibria (RCCE) approach.

#### **2. Static and dynamic analysis tools**

The static and dynamic analysis tools available provide important information that is useful for reduction procedures and includes reaction flow and sensitivity analysis procedures. Reaction flow analysis detects minor and major flows in the reaction system, where the species involved in the minor flows can be neglected in further investigation. Under given physical

to the total production of species *i*, and *c<sup>a</sup>*

The number of atoms *n<sup>a</sup>*

for the flow parameters.

*<sup>k</sup>* <sup>=</sup> <sup>∑</sup>*NS <sup>i</sup>*=<sup>1</sup> *<sup>n</sup><sup>a</sup> <sup>k</sup>* (*ν*�� *ik* − *ν*�

homogeneous reactor (Nilsson, 2011).

reaction �*n<sup>a</sup>*

thinner arrows.

*i* to species *j* relative to the total consumption of species *i*:

*f a ij* <sup>=</sup> <sup>∑</sup>*NR*

*ca ij* <sup>=</sup> <sup>∑</sup>*NR*

*<sup>k</sup>*=<sup>1</sup> *rkν*�

Model Reduction Techniques for Chemical Mechanisms 83

∑*NR <sup>k</sup>*=<sup>1</sup> *rkν*� *jk*

*<sup>k</sup>*=<sup>1</sup> *rkν*�� *jkν*� *ik na i* �*n<sup>a</sup> k*

> ∑*NR <sup>k</sup>*=<sup>1</sup> *rkν*�� *jk*

or consumption rate of species *i* in order to achieve a non-dimensional range between 0 and 1

An example of the scheme of such an analysis is given in Figure (2) which shows the flow of C atoms in a reaction mechanism for a CH4-fueled homogeneous reactor. This analysis was performed under specific conditions in which the engine speed was set to 1400 revolutions per minute (RPM), the fuel-air ratio was set to 0.412 and the compression ratio was set to 17.30. In the figure, the major reaction flows are marked by thick arrows and the minor flows by

Fig. 2. Reaction flows of C atoms in a natural-gas mechanism for CH4 as fuel in a

*jkν*�� *ik na i* �*n<sup>a</sup> k*

*<sup>j</sup>* is now normalized to the total number of atoms *a* transported in the

*ik*). The flow parameters are normalized to the total formation

*ij* is the flow of atom *a* by the consumption of species

, (3)

. (4)

Fig. 1. A figure over the reduction methods and their relations. The dashed lines represent suggested uses of the different mechanisms. As shown in the figure, reduced mechanisms are applicable to both static and dynamic problems. Detailed mechanisms, in contrast, are not practical for dynamic systems.

conditions these will represent static properties of the mechanism. However, some species in the minor flows may exhibit a strong influence on the quantities under investigation and should thus not be neglected. These reactions and species are found through sensitivity analysis, in which the dynamic properties of the system are investigated. This can be formalized into one reduction tool called the directed relation graph (DRG) methodology. These three methods will now be discussed below.

#### **2.1 Reaction flow analysis**

Reaction flow analysis describes the importance of reaction paths in the mechanism under specified conditions. It is performed by calculating the transfer rate of certain atomic species such as C,O, N and H between molecular species. The flow of atoms between the reacting molecules is used as a measure of the relevance of the species in the reaction mechanism. The reaction flow of atomic species *a* between species *i* and species *j* can be described very simply by (Nilsson, 2011)

$$f\_{ij}^a = \sum\_{k=1}^{N\_R} r\_k (n\_i^a \nu\_{ik}^{\prime \prime} - n\_j^a \nu\_{jk}^{\prime})\_\prime \tag{2}$$

where the sum is taken over the number of reactions *NR*. *n<sup>a</sup> <sup>i</sup>* and *<sup>n</sup><sup>a</sup> <sup>j</sup>* are the numbers of the atom *a* in molecule *i* and *j*, and *ν*�� *ik* and *ν*� *jk* are the stoichiometric coefficients for the molecules *i* and *j* in reaction *k*. *rk* is the reaction rate, *ν*� *jk* and *ν*�� *ik* are the stoichiometric coefficients of the reactants and the products in reaction *k*.

A more adequate method of determining the reaction flow, is described in more detail by Soyhan et al. (Soyhan et al., 2001); (Soyhan et al., 2001) and optimized further . In this work, forward and backward reactions are treated separately so as to capture reversible reaction pairs in which the flow of atoms is high in both directions but where the net flow is not necessarily high. This procedure results in the two new flow parameters shown below in which *f <sup>a</sup> ij* is the normalized flow of atom *a* by the formation of species *i* from species *j* relative 4 Will-be-set-by-IN-TECH

Fig. 1. A figure over the reduction methods and their relations. The dashed lines represent suggested uses of the different mechanisms. As shown in the figure, reduced mechanisms are applicable to both static and dynamic problems. Detailed mechanisms, in contrast, are

conditions these will represent static properties of the mechanism. However, some species in the minor flows may exhibit a strong influence on the quantities under investigation and should thus not be neglected. These reactions and species are found through sensitivity analysis, in which the dynamic properties of the system are investigated. This can be formalized into one reduction tool called the directed relation graph (DRG) methodology.

Reaction flow analysis describes the importance of reaction paths in the mechanism under specified conditions. It is performed by calculating the transfer rate of certain atomic species such as C,O, N and H between molecular species. The flow of atoms between the reacting molecules is used as a measure of the relevance of the species in the reaction mechanism. The reaction flow of atomic species *a* between species *i* and species *j* can be described very simply

not practical for dynamic systems.

**2.1 Reaction flow analysis**

atom *a* in molecule *i* and *j*, and *ν*��

*i* and *j* in reaction *k*. *rk* is the reaction rate, *ν*�

reactants and the products in reaction *k*.

by (Nilsson, 2011)

which *f <sup>a</sup>*

These three methods will now be discussed below.

*f a ij* =

*ik* and *ν*�

where the sum is taken over the number of reactions *NR*. *n<sup>a</sup>*

*NR* ∑ *k*=1

*rk*(*n<sup>a</sup> <sup>i</sup> ν*�� *ik* <sup>−</sup> *<sup>n</sup><sup>a</sup> j ν*�

*jk* and *ν*��

*ij* is the normalized flow of atom *a* by the formation of species *i* from species *j* relative

A more adequate method of determining the reaction flow, is described in more detail by Soyhan et al. (Soyhan et al., 2001); (Soyhan et al., 2001) and optimized further . In this work, forward and backward reactions are treated separately so as to capture reversible reaction pairs in which the flow of atoms is high in both directions but where the net flow is not necessarily high. This procedure results in the two new flow parameters shown below in

*jk*), (2)

*ik* are the stoichiometric coefficients of the

*<sup>j</sup>* are the numbers of the

*<sup>i</sup>* and *<sup>n</sup><sup>a</sup>*

*jk* are the stoichiometric coefficients for the molecules

to the total production of species *i*, and *c<sup>a</sup> ij* is the flow of atom *a* by the consumption of species *i* to species *j* relative to the total consumption of species *i*:

$$f\_{ij}^a = \frac{\sum\_{k=1}^{N\_R} r\_k \nu\_{jk}^\prime \nu\_{ik}^{\prime\prime} \frac{n\_i^a}{\sum n\_k^a}}{\sum\_{k=1}^{N\_R} r\_k \nu\_{jk}^\prime},\tag{3}$$

$$\mathcal{c}\_{ij}^{a} = \frac{\sum\_{k=1}^{N\_R} r\_k \nu\_{jk}^{\prime\prime} \nu\_{ik}^{\prime} \frac{n\_i^a}{\sum n\_k^a}}{\sum\_{k=1}^{N\_R} r\_k \nu\_{jk}^{\prime\prime}}. \tag{4}$$

The number of atoms *n<sup>a</sup> <sup>j</sup>* is now normalized to the total number of atoms *a* transported in the reaction �*n<sup>a</sup> <sup>k</sup>* <sup>=</sup> <sup>∑</sup>*NS <sup>i</sup>*=<sup>1</sup> *<sup>n</sup><sup>a</sup> <sup>k</sup>* (*ν*�� *ik* − *ν*� *ik*). The flow parameters are normalized to the total formation or consumption rate of species *i* in order to achieve a non-dimensional range between 0 and 1 for the flow parameters.

An example of the scheme of such an analysis is given in Figure (2) which shows the flow of C atoms in a reaction mechanism for a CH4-fueled homogeneous reactor. This analysis was performed under specific conditions in which the engine speed was set to 1400 revolutions per minute (RPM), the fuel-air ratio was set to 0.412 and the compression ratio was set to 17.30. In the figure, the major reaction flows are marked by thick arrows and the minor flows by thinner arrows.

Fig. 2. Reaction flows of C atoms in a natural-gas mechanism for CH4 as fuel in a homogeneous reactor (Nilsson, 2011).

species, of temperature profiles and of other important features. The remaining species are the *redundant species* that are candidates for removal from the mechanisms without loss of important information. Tomlin et al. eliminated a reaction from the mechanisms if the redundant species played simply the role of being reactants in the reaction, since only the reactants are present in the equation for the rate of change of species concentrations. Even if a particular reaction needs to be retained in the mechanism the redundant species can be eliminated from the list of products if it does not violate the conservation of atoms and mass

Model Reduction Techniques for Chemical Mechanisms 85

The approach to finding redundant species described by Tomlin et al. (Tomlin et al., 1997) is to investigate the matrix **J**(*t*) = *∂ω*(t)/*∂***Y** (for homogeneous systems; P≡ 0), where the element *∂ωi*/*∂Yj* describes the change of the rate of production of species *i* caused by a change of concentration of species *j*. A species is considered redundant if a change in its concentration does not affect the rate of production of important species significantly. A slightly different approach is taken by Soyhan et al. who let the sensitivities being transported through the mechanism in the sense that a species is rated according to its own importance and its involvement in producing or consuming important species (Soyhan et al., 1999);(Soyhan et al., 2001). Thus, the method can be considered as representing a simultaneous reaction flow and sensitivity analysis, called *necessity analysis*. First the reaction sensitivity represents a variable *ψ*'s sensitivity towards a chosen reaction coefficient *Ak* in reaction *k* is changed by a

> *Sr <sup>ψ</sup><sup>k</sup>* <sup>=</sup> *∂ψ ∂Ak*

> > *Ss <sup>ψ</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup> *ci ∂ψ ∂α<sup>i</sup>*

Therefore, *α<sup>i</sup>* can be considered an error term for the concentrations which become *c*<sup>∗</sup>

⎛ ⎝

*Ii* = max

The species sensitivity can be taken either as the sum of the reaction sensitivities in which the species are involved, or can be derived directly in the same manner as the reaction sensitivity; investigating how much a variable *ψ* is changed when the concentration of species *i* is changed

*<sup>ψ</sup><sup>i</sup>* contains the information on how sensitive a chosen target variable, *ψ*, such as the

mass fraction of a particular species or temperature, is to the species *i*. The factor in which the concentrations are allowed to change is small and *α<sup>i</sup>* = 1 + *�<sup>i</sup>* where *�<sup>i</sup>* is e.g. 1 percent.

A species is assigned a relative necessity index which represents how important the species is for changes in variable *ψ* in relation to the other species. The redundancy index is based on

max*k*=1,*Ns*

where *Zi* takes the value 1 or 0 depending on wheather the species has been preselected as an important species (fuel, oxidant etc.). Even if a species has a low necessity it can be assigned a high overall necessity index if there is a significant flow of atoms to or from an important

*SS ψi*

� *SS ψk* �, *Zi*

⎞

*Akx* (7)

(8)

*<sup>i</sup>* = *ciαi*.

⎠ (9)

in the calculations.

factor *x* (e.g. 1 percent):

where *S<sup>S</sup>*

by a small factor *�<sup>i</sup>* (again e.g. 1 percent):

the species sensitivity in the following way:

Figure (2) shows that the major paths represents the high-temperature hydrocarbon oxidation of CH4 through the relatively stable radicals C2H5 and CH3 shown in the lower right corner of the figure. Flows with a lower transport rate than 5 <sup>×</sup> <sup>10</sup>−<sup>7</sup> mol/cm3 can be considered minor flows. These are shown for the most part in the upper left corner of Figure (2). These flows can thus be neglected in further calculations. Note that the situation is different if natural gas is used as a fuel, in which some of the fuel components may be selected for removal. Thus some of the species involved in minor flows may be important for the desired results and should be retained in the reaction mechanism. These are selected by the sensitivity analysis, as described below.

#### **2.2 Sensitivity analysis**

Sensitivity analysis involves investigating the change in a quantity of interest due to small changes in the controlling parameters. Investigating this is of interest in itself for gaining insight into the reaction model. In addition, it is a very useful tool for reduction of reaction mechanisms. After the minor flows have been detected by reaction flow analysis, it is important to ensure that the species involved in the minor flows do not influence the result significantly, in which case they should not be excluded in the further calculations. In local sensitivity analysis, this is described by the partial derivative of the investigated quantity with respect to the controlling parameters. Emanating from Equation (1), investigation of how the concentration of a species *i* is influenced by a perturbation **Δ***k* of the rate coefficient *k* can be expressed by differentiating the original set of chemical differential equations with respect to *kj* and expanding the right hand side giving

$$\frac{d}{dt}\frac{\partial \mathbf{Y}}{\partial k\_{\dot{j}}} = \frac{\partial \mathbf{P}}{\partial \mathbf{Y}}\frac{\partial \mathbf{Y}}{\partial k\_{\dot{j}}} + \frac{\partial \mathbf{P}}{\partial k\_{\dot{j}}} + \frac{\partial \omega(\mathbf{t})}{\partial \mathbf{Y}} + \frac{\partial \mathbf{Y}}{\partial k\_{\dot{j}}} \frac{\partial \omega(\mathbf{t})}{\partial k\_{\dot{j}}}, \quad \dot{j} = 1, \dots, N\_{\rm S}. \tag{5}$$

If considering a homogeneous reaction system, P≡ 0, the equation above results in the following expression:

$$\frac{d}{dt}\frac{\partial \mathbf{Y}}{\partial k\_{\dot{j}}} = \mathbf{J}(t)\frac{\partial \mathbf{Y}}{\partial k\_{\dot{j}}} + \frac{\partial \omega(\mathbf{t})}{\partial k\_{\dot{j}}}, \quad \dot{j} = 1, \dots, N\_{\mathbf{S}'} \tag{6}$$

where **J**(*t*) = *∂ω*(t)/*∂***Y** is recognized as the Jacobian matrix and the initial condition for *∂***Y**/*∂kj* (recognized as the sensitivity matrix) is a vector containing only zeros (Tomlin et al., 1997). In order to solve Equation (6), the concentrations in the original set of equations must be known, which is not always the case. The most efficient way of overcoming this problem is to solve the two systems in succession. This method is called the decoupled direct method. It first takes one step in solving Equation (1) and then performs an equal step in solving Equation (6). This can be done since both sets of differential equations have the same Jacobian matrix **J** (Tomlin et al., 1997).

The same method can be applied to investigate the sensitivities of such parameters of the system as the flame temperature for premixed flames or the ignition timing in such ignition systems as in engine simulations as outlined below. The information obtained through this analysis is the basis for reduction of the mechanism. After the sensitivity analysis, the next step is to order the species and select those that are redundant. Tomlin et al. (Tomlin et al., 1997) classifies the species involved in a mechanism into three categories. The *important species* are those needed for the current investigation, such as the reaction products when investigating pollution or the initial reactants defining the fuels. The *necessary species* are those needed to provide for an accurate calculation of the concentration profiles of the important 6 Will-be-set-by-IN-TECH

Figure (2) shows that the major paths represents the high-temperature hydrocarbon oxidation of CH4 through the relatively stable radicals C2H5 and CH3 shown in the lower right corner of the figure. Flows with a lower transport rate than 5 <sup>×</sup> <sup>10</sup>−<sup>7</sup> mol/cm3 can be considered minor flows. These are shown for the most part in the upper left corner of Figure (2). These flows can thus be neglected in further calculations. Note that the situation is different if natural gas is used as a fuel, in which some of the fuel components may be selected for removal. Thus some of the species involved in minor flows may be important for the desired results and should be retained in the reaction mechanism. These are selected by the sensitivity analysis,

Sensitivity analysis involves investigating the change in a quantity of interest due to small changes in the controlling parameters. Investigating this is of interest in itself for gaining insight into the reaction model. In addition, it is a very useful tool for reduction of reaction mechanisms. After the minor flows have been detected by reaction flow analysis, it is important to ensure that the species involved in the minor flows do not influence the result significantly, in which case they should not be excluded in the further calculations. In local sensitivity analysis, this is described by the partial derivative of the investigated quantity with respect to the controlling parameters. Emanating from Equation (1), investigation of how the concentration of a species *i* is influenced by a perturbation **Δ***k* of the rate coefficient *k* can be expressed by differentiating the original set of chemical differential equations with respect to

> *∂***Y** *∂kj*

If considering a homogeneous reaction system, P≡ 0, the equation above results in the

where **J**(*t*) = *∂ω*(t)/*∂***Y** is recognized as the Jacobian matrix and the initial condition for *∂***Y**/*∂kj* (recognized as the sensitivity matrix) is a vector containing only zeros (Tomlin et al., 1997). In order to solve Equation (6), the concentrations in the original set of equations must be known, which is not always the case. The most efficient way of overcoming this problem is to solve the two systems in succession. This method is called the decoupled direct method. It first takes one step in solving Equation (1) and then performs an equal step in solving Equation (6). This can be done since both sets of differential equations have the same Jacobian matrix **J**

The same method can be applied to investigate the sensitivities of such parameters of the system as the flame temperature for premixed flames or the ignition timing in such ignition systems as in engine simulations as outlined below. The information obtained through this analysis is the basis for reduction of the mechanism. After the sensitivity analysis, the next step is to order the species and select those that are redundant. Tomlin et al. (Tomlin et al., 1997) classifies the species involved in a mechanism into three categories. The *important species* are those needed for the current investigation, such as the reaction products when investigating pollution or the initial reactants defining the fuels. The *necessary species* are those needed to provide for an accurate calculation of the concentration profiles of the important

*∂ω*(t) *∂kj*

, *j* = 1, ..., *NS*. (5)

, *j* = 1, ..., *NS*, (6)

as described below.

**2.2 Sensitivity analysis**

*kj* and expanding the right hand side giving

<sup>=</sup> *<sup>∂</sup>***<sup>P</sup>** *∂***Y** *∂***Y** *∂kj* + *∂***P** *∂kj* + *∂ω*(t) *<sup>∂</sup>***<sup>Y</sup>** <sup>+</sup>

*d dt ∂***Y** *∂kj*

= **J**(*t*)

*∂***Y** *∂kj* + *∂ω*(t) *∂kj*

*d dt ∂***Y** *∂kj*

following expression:

(Tomlin et al., 1997).

species, of temperature profiles and of other important features. The remaining species are the *redundant species* that are candidates for removal from the mechanisms without loss of important information. Tomlin et al. eliminated a reaction from the mechanisms if the redundant species played simply the role of being reactants in the reaction, since only the reactants are present in the equation for the rate of change of species concentrations. Even if a particular reaction needs to be retained in the mechanism the redundant species can be eliminated from the list of products if it does not violate the conservation of atoms and mass in the calculations.

The approach to finding redundant species described by Tomlin et al. (Tomlin et al., 1997) is to investigate the matrix **J**(*t*) = *∂ω*(t)/*∂***Y** (for homogeneous systems; P≡ 0), where the element *∂ωi*/*∂Yj* describes the change of the rate of production of species *i* caused by a change of concentration of species *j*. A species is considered redundant if a change in its concentration does not affect the rate of production of important species significantly. A slightly different approach is taken by Soyhan et al. who let the sensitivities being transported through the mechanism in the sense that a species is rated according to its own importance and its involvement in producing or consuming important species (Soyhan et al., 1999);(Soyhan et al., 2001). Thus, the method can be considered as representing a simultaneous reaction flow and sensitivity analysis, called *necessity analysis*. First the reaction sensitivity represents a variable *ψ*'s sensitivity towards a chosen reaction coefficient *Ak* in reaction *k* is changed by a factor *x* (e.g. 1 percent):

$$S^{r}\_{\psi k} = \frac{\partial \psi}{\partial A\_k} A\_k \mathfrak{x} \tag{7}$$

The species sensitivity can be taken either as the sum of the reaction sensitivities in which the species are involved, or can be derived directly in the same manner as the reaction sensitivity; investigating how much a variable *ψ* is changed when the concentration of species *i* is changed by a small factor *�<sup>i</sup>* (again e.g. 1 percent):

$$S^{s}\_{\psi i} = \frac{1}{c\_i} \frac{\partial \psi}{\partial \alpha\_i} \tag{8}$$

where *S<sup>S</sup> <sup>ψ</sup><sup>i</sup>* contains the information on how sensitive a chosen target variable, *ψ*, such as the mass fraction of a particular species or temperature, is to the species *i*. The factor in which the concentrations are allowed to change is small and *α<sup>i</sup>* = 1 + *�<sup>i</sup>* where *�<sup>i</sup>* is e.g. 1 percent. Therefore, *α<sup>i</sup>* can be considered an error term for the concentrations which become *c*<sup>∗</sup> *<sup>i</sup>* = *ciαi*.

A species is assigned a relative necessity index which represents how important the species is for changes in variable *ψ* in relation to the other species. The redundancy index is based on the species sensitivity in the following way:

$$I\_i = \max\left(\frac{S^S\_{\psi i}}{\max\_{k=1, N\_s} \left(S^S\_{\psi k}\right)}, Z\_i\right) \tag{9}$$

where *Zi* takes the value 1 or 0 depending on wheather the species has been preselected as an important species (fuel, oxidant etc.). Even if a species has a low necessity it can be assigned a high overall necessity index if there is a significant flow of atoms to or from an important

**3. Time scale separation methods**

span several orders of magnitude (Nilsson, 2011).

typically terms involving the fastest time scales.

lifetime analysis based on the Level Of Importance (LOI).

This is illustrated in Figure (3).

In the following section the methods based on time scale separation analysis will be discussed. Employing these methods relies on the fact that the chemical system consists of a number of species that react with each other on time scales that range over several orders of magnitude. Thus, some of the reactions can be considered as being fast compared with the physical processes involved, such as diffusion, turbulence and other reactions that are considered slow.

Model Reduction Techniques for Chemical Mechanisms 87

Fig. 3. Comparison of typical chemical and physical time scales. The chemical time scales

A time scale separation method makes use of the fact that the physical and chemical time scales have only a limited range of overlap. The time scales of some of the more rapid chemical processes can thus be decoupled and be described in approximate ways by the Quasi Steady State Assumption (QSSA) or partial equilibrium approximations for the selected species. This reduces the species list to only the species left in the set of differential equations. Also, eliminating the fastest time scales in the system solves the numerical stiffness problem that these time scales introduce. Numerical stiffness arises when the iteration over the differential equations need very small steps as some of the terms lead to rapid variations of the solution,

Three time-scale separation methods will be described below. These are the Intrinsic Low Dimensional Manifold method (ILDM), Computational Singular Perturbation (CSP), and the

Since the methods are all concerned with determining the processes involved on the shortest time scales, they have many similarities. Although it could be argued that they in reality are virtually the same, they have different coatings and different ways of presenting the solutions. In the overall picture here one can describe the differences as follows: the key difference between ILDM and CSP is the sub-space created to describe the new manifold. ILDM employs partial equilibrium approximations for the species selected and creates a species sub-space. CSP employs steady state approximations and creates a reaction sub-space. It does not indicate a preference for QSSA as to opposed to partial equilibrium approximation. Including the partial equilibrium approximation in selecting the "steady state" species represents a more general criterion than using only the QSSA (Goussis, 1996). However, this introduces all the species in the stoichiometry vector, often making the algorithm more complicated. The difference between CSP and lifetime analysis is that CSP includes coupled time scales, whereas lifetime analysis assumes that on a short time scale no coupling is significant, it is

Once the "fast" and the "slow" components are identified, several of these methods use the quasi-steady state assumption (QSSA) as the basis for reduction. Together with a

thus operating with a diagonal Jacobian. Details of this are described below.

species. The overall necessity is determined as follows:

$$I\_{\bar{l}} = \max \left( I\_{\bar{j}} f\_{\bar{i}j'}^a \quad I\_{\bar{j}} c\_{\bar{i}j'}^a \quad I\_{\bar{i}}; \bar{j} = 1, N\_{\bar{s}'} a = 1, N\_{\mathfrak{a}} \right) \tag{10}$$

where *f <sup>a</sup> ij* and *<sup>c</sup><sup>a</sup> ij*, as defined previously, are the flow of atom *a* by formation or consumption of species *i* from or to species *j*. This equation needs to be solved iteratively with a preset value of ¯*Ii*. Species with a low overall ¯*Ii* are considered to be redundant. This procedure has been used extensively to perform reduction on large hydrocarbon mechanisms, e.g. up to C-8 reaction chains where together with a lumping procedure the mechanism was reduced from about 246 species to 47 (Zeuch et al., 2008). Also, this reduction procedure is been made available in reaction kinetics simulation tools such as DARS (DARS, 2011) where the reduction is performed entirely automatically. The computed necessity index ranks the species accordingly, and a user set cut-off limit will generate a skeletal mechanism according to desired accuracy of the resulting skeletal mechanism; i.e. the more species which are excluded from the mechanism the less accurate the mechanism will be to predict the combustion process.

#### **2.3 Directed relation graph methodology**

The step wise reduction procedure described above can been automated by the use of the Directed Relation Graph method (Lu and Law, 2005);(Lu and Law, 2006). This procedure quantifies the coupling between species, and assigns an "pair wise" error which contains the information of how much error is introduced to a species *A* by elimination of a species *B*:

$$r\_{AB} \equiv \frac{\max \mid \nu\_{Ai}\omega\_{i}\delta\_{Bi} \mid}{\max \mid \nu\_{Ai}\omega\_{i} \mid}, \delta\_{Bi} = \begin{cases} 1 \text{ if the } i \text{th reaction involves B.} \\ 0 \text{ otherwise.} \end{cases} \tag{11}$$

Note that the denominator contains the maximum reaction rate contributing to the production of *A*, whereas the numerator contains the maximum reaction rate contributing to the production of *A* that also involves species *B* . The original formulation used a summation over *all* reactions which did not prove to be efficient when e.g. dealing with larger isomergroups (Luo et al., 2010). This formalism is not unsimilar to Equation (9). However, both the reaction flow and an error estimate are included when the species dependence is expressed in the following *graph* notation:

$$A \to B \text{ if } r\_{AB} > \varepsilon. \tag{12}$$

Species *A* is thereby connected to species *B* only if the pair wise error is above a user set error threshold. The starting point if the directed relation graph would be one of the important species, such as the fuel species. When species with low connectivity (below the threshold) is eliminated the result will be a skeletal mechanism similar to what is obtained by a necessity analysis.

This procedure will not be discussed further here, but the DRG methodology has proven to be easy to implement and to fully automate. Hence, it has become very popular to use for reduction of larger mechanisms. In its simplest form DRG has however proved to have some limitations. These have been addressed in extended versions of the procedure. This includes DRG with error propagation (DRGEP) (Pepiot-Desjardins & Pitsch, 2008), DRG aided sensitivity analysis (DRGASA) (Zheng et al., 2007) and DRGEP and sensitivity analysis (Niemeyer et al., 2010). The reader is referred to these works for further discussion.

#### **3. Time scale separation methods**

8 Will-be-set-by-IN-TECH

of species *i* from or to species *j*. This equation needs to be solved iteratively with a preset value of ¯*Ii*. Species with a low overall ¯*Ii* are considered to be redundant. This procedure has been used extensively to perform reduction on large hydrocarbon mechanisms, e.g. up to C-8 reaction chains where together with a lumping procedure the mechanism was reduced from about 246 species to 47 (Zeuch et al., 2008). Also, this reduction procedure is been made available in reaction kinetics simulation tools such as DARS (DARS, 2011) where the reduction is performed entirely automatically. The computed necessity index ranks the species accordingly, and a user set cut-off limit will generate a skeletal mechanism according to desired accuracy of the resulting skeletal mechanism; i.e. the more species which are excluded from the mechanism the less accurate the mechanism will be to predict the combustion

The step wise reduction procedure described above can been automated by the use of the Directed Relation Graph method (Lu and Law, 2005);(Lu and Law, 2006). This procedure quantifies the coupling between species, and assigns an "pair wise" error which contains the information of how much error is introduced to a species *A* by elimination of a species *B*:

Note that the denominator contains the maximum reaction rate contributing to the production of *A*, whereas the numerator contains the maximum reaction rate contributing to the production of *A* that also involves species *B* . The original formulation used a summation over *all* reactions which did not prove to be efficient when e.g. dealing with larger isomergroups (Luo et al., 2010). This formalism is not unsimilar to Equation (9). However, both the reaction flow and an error estimate are included when the species dependence is expressed in the

Species *A* is thereby connected to species *B* only if the pair wise error is above a user set error threshold. The starting point if the directed relation graph would be one of the important species, such as the fuel species. When species with low connectivity (below the threshold) is eliminated the result will be a skeletal mechanism similar to what is obtained by a necessity

This procedure will not be discussed further here, but the DRG methodology has proven to be easy to implement and to fully automate. Hence, it has become very popular to use for reduction of larger mechanisms. In its simplest form DRG has however proved to have some limitations. These have been addressed in extended versions of the procedure. This includes DRG with error propagation (DRGEP) (Pepiot-Desjardins & Pitsch, 2008), DRG aided sensitivity analysis (DRGASA) (Zheng et al., 2007) and DRGEP and sensitivity analysis

(Niemeyer et al., 2010). The reader is referred to these works for further discussion.

*ij*, *Ii*; *j* = 1, *Ns*, *a* = 1, *Na*

1 if the *i*th reaction involves B.

0 otherwise. (11)

*A* → *B* if *rAB > ε*. (12)

*ij*, as defined previously, are the flow of atom *a* by formation or consumption

(10)

species. The overall necessity is determined as follows:

¯*Ii* = max

**2.3 Directed relation graph methodology**

*rAB* <sup>≡</sup> max <sup>|</sup> *<sup>ν</sup>AiωiδBi* <sup>|</sup>

max <sup>|</sup> *<sup>ν</sup>Aiω<sup>i</sup>* <sup>|</sup> , *<sup>δ</sup>Bi* <sup>=</sup>

where *f <sup>a</sup>*

process.

*ij* and *<sup>c</sup><sup>a</sup>*

following *graph* notation:

analysis.

 *Ij f <sup>a</sup> ij*, *Ijc<sup>a</sup>* In the following section the methods based on time scale separation analysis will be discussed. Employing these methods relies on the fact that the chemical system consists of a number of species that react with each other on time scales that range over several orders of magnitude. Thus, some of the reactions can be considered as being fast compared with the physical processes involved, such as diffusion, turbulence and other reactions that are considered slow. This is illustrated in Figure (3).

Fig. 3. Comparison of typical chemical and physical time scales. The chemical time scales span several orders of magnitude (Nilsson, 2011).

A time scale separation method makes use of the fact that the physical and chemical time scales have only a limited range of overlap. The time scales of some of the more rapid chemical processes can thus be decoupled and be described in approximate ways by the Quasi Steady State Assumption (QSSA) or partial equilibrium approximations for the selected species. This reduces the species list to only the species left in the set of differential equations. Also, eliminating the fastest time scales in the system solves the numerical stiffness problem that these time scales introduce. Numerical stiffness arises when the iteration over the differential equations need very small steps as some of the terms lead to rapid variations of the solution, typically terms involving the fastest time scales.

Three time-scale separation methods will be described below. These are the Intrinsic Low Dimensional Manifold method (ILDM), Computational Singular Perturbation (CSP), and the lifetime analysis based on the Level Of Importance (LOI).

Since the methods are all concerned with determining the processes involved on the shortest time scales, they have many similarities. Although it could be argued that they in reality are virtually the same, they have different coatings and different ways of presenting the solutions. In the overall picture here one can describe the differences as follows: the key difference between ILDM and CSP is the sub-space created to describe the new manifold. ILDM employs partial equilibrium approximations for the species selected and creates a species sub-space. CSP employs steady state approximations and creates a reaction sub-space. It does not indicate a preference for QSSA as to opposed to partial equilibrium approximation. Including the partial equilibrium approximation in selecting the "steady state" species represents a more general criterion than using only the QSSA (Goussis, 1996). However, this introduces all the species in the stoichiometry vector, often making the algorithm more complicated. The difference between CSP and lifetime analysis is that CSP includes coupled time scales, whereas lifetime analysis assumes that on a short time scale no coupling is significant, it is thus operating with a diagonal Jacobian. Details of this are described below.

Once the "fast" and the "slow" components are identified, several of these methods use the quasi-steady state assumption (QSSA) as the basis for reduction. Together with a

OH + OH ↔ H2O + O, (x) H + HO2 → OH + OH, (xi) H + HO2 → O2 +H2, (xii) OH + HO2 → H2O + O2. (xiii)

Model Reduction Techniques for Chemical Mechanisms 89

Equation (13) indicated that the time derivatives of all the species concentrations can be expressed in terms of the concentrations of species that consume and that produce that particular species. As noted above, the derivatives not only contain a chemistry-term, but in the case of inhomogeneous systems, can also contain diffusive and convective terms, often being denoted in such cases by the L-operator. Here, only the chemical contribution

> *ω<sup>H</sup>* = −*r*ii + *r*iii + *r*iv − *r*vii − *r*xi − *r*xii, 0 = *ωOH* = *r*ii + *r*iii − *r*iv − 2*r*<sup>x</sup> + 2*r*xi − *r*xiii,

where OH, O and HO2 are all set to steady state and their derivatives are set equal to zero, transforming the corresponding differential equations conveniently into algebraic relations. Since the system is already reduced, the steady state concentrations can be found. However, in obtaining a set of global reactions and optimizing the reduction, the corresponding number of reaction rates can be eliminate from the system. A rule of thumb is for each species to eliminate the fastest reaction rates at which this species is consumed. This is *r*iv for OH, *r*iii for O and *r*xii for HO2, them being placed on the right-hand side of the equations and expressed in terms of the remaining reaction rates on the left-hand side (Peters, 1990). However, the reaction rates that are eliminated from the system are arbitrary and result in the same global

> *ω<sup>H</sup>* = 2*r*ii − 2*r*vii + 2*r*xi, *ωH*<sup>2</sup> = −3*r*ii + *r*vii − 3*r*xi, *ωO*<sup>2</sup> = −*r*ii − *r*xi, *ωH*2*<sup>O</sup>* = 2*r*ii − 2*r*xi.

By arranging the right-hand side so that rates with the same stoichiometric coefficients are

*ω<sup>H</sup>* = 2(*r*ii + *r*xi) − 2*r*vii, *ωH*<sup>2</sup> = −3(*r*ii + *r*xi) + *r*vii, *ωO*<sup>2</sup> = −(*r*ii + *r*xi), *ωH*2*<sup>O</sup>* = 2(*r*ii + *r*xi),

3H2 + O2 = 2H + 2H2O,I 2H + M = H2 + M, II (15)

(16)

(17)

is included, resulting in the following set of equations:

reactions. After some algebra, the system is as follows:

added together one obtains

resulting in the following global reactions:

0 = *ω<sup>O</sup>* = *r*ii − *r*iii + *r*x,

*ωH*<sup>2</sup> = −*r*iii − *r*iv + *r*xii, *ωO*<sup>2</sup> = −*r*ii − *r*vii + *r*xii + *r*xiii,

*ωH*2*<sup>O</sup>* = *r*iv + *r*<sup>x</sup> + *r*xiii, 0 = *ωHO*<sup>2</sup> = *r*vii − *r*xi − *r*xii − *ω*xiii.

chemical equilibrium assumption or as an alternative to it, this procedure makes it possible to mathematically eliminate species from the set of differential equations. They are treated separately through use of approximative algebraic relations. Their concentrations still contribute to the reaction rates of the non-reduced species. The basic idea of developing reduced chemical mechanisms by introducing the assumption of a steady state is not new and has, in fact, been used in chemistry since the 1950s (See (Peters, 1990)) and further references therein). However, not until the early 1990s was this strategy for solving complex chemical problems introduced in combustion physics.

Until rather recently, these reduced schemes were mainly developed by hand using QSSA and an à priori definition of which species were to be considered in steady state, a particularly time-consuming procedure. Before presenting the various time scale separation methods, the basic steps of the QSSA reduction procedure will be presented in the subsection below followed by the theory rate-controlled constraints equilibria (RCCE). Thereafter are the time scale separation methods outlined.

#### **3.1 Manual reduction by means of QSSA**

The underlying assumption for QSSA is that some species can be treated as being in steady state. This means that, if one ignores physical terms as convection and diffusion, their concentrations remain constant. It follows then from Equation (1) that the change in species concentration of species *i* is

$$\frac{dX\_i}{dt} = \sum\_{k=1}^r \nu\_{i,k} r\_k = \omega\_i = 0,\tag{13}$$

where *ω<sup>i</sup>* is the chemical source term and the summation is over all reactions of the product *νi*,*krk*. This equation corresponds to Equation (1) for chemical treatment only, where the reaction rate *rk* can be expressed by the following general definition:

$$r\_k = k\_k \prod\_{j=1}^n X\_j^{\nu\_{jk}'} \tag{14}$$

and the net stoichiometric coefficient being *ν<sup>i</sup>* = *ν*�� *<sup>i</sup>* − *ν*� *i* . The physical meaning of this assumption is that the reactions that consume the species are very much faster than the reactions producing the species. Thus, the concentration remain low just as their time derivative.

The H2-O2 system will be used as an example (Peters, 1990);(Peters et al., 1993). The reactions needed for the present analysis are (following the numbering in the original mechanism)

$$\begin{array}{ll} \mathrm{H} + \mathrm{O}\_{2} & \leftrightarrow \,\mathrm{OH} + \mathrm{O}\_{\prime} \quad (\mathrm{ii})\\ \mathrm{O} + \mathrm{H}\_{2} & \leftrightarrow \,\mathrm{OH} + \mathrm{H}\_{\prime} \quad (\mathrm{iii})\\ \mathrm{OH} + \mathrm{H}\_{2} & \leftrightarrow \,\mathrm{H}\_{2}\mathrm{O} + \mathrm{H}\_{\prime} \,(\mathrm{iv})\\ \mathrm{H} + \mathrm{O}\_{2} + \mathrm{M} & \to \,\mathrm{HO}\_{2} + \mathrm{M}\_{\prime} \quad (\mathrm{vii}) \end{array}$$

where the backward reactions are treated as being separate reactions. Additional reactions in the H2-O2 reaction system are also included in the present example:

10 Will-be-set-by-IN-TECH

chemical equilibrium assumption or as an alternative to it, this procedure makes it possible to mathematically eliminate species from the set of differential equations. They are treated separately through use of approximative algebraic relations. Their concentrations still contribute to the reaction rates of the non-reduced species. The basic idea of developing reduced chemical mechanisms by introducing the assumption of a steady state is not new and has, in fact, been used in chemistry since the 1950s (See (Peters, 1990)) and further references therein). However, not until the early 1990s was this strategy for solving complex chemical

Until rather recently, these reduced schemes were mainly developed by hand using QSSA and an à priori definition of which species were to be considered in steady state, a particularly time-consuming procedure. Before presenting the various time scale separation methods, the basic steps of the QSSA reduction procedure will be presented in the subsection below followed by the theory rate-controlled constraints equilibria (RCCE). Thereafter are the time

The underlying assumption for QSSA is that some species can be treated as being in steady state. This means that, if one ignores physical terms as convection and diffusion, their concentrations remain constant. It follows then from Equation (1) that the change in species

where *ω<sup>i</sup>* is the chemical source term and the summation is over all reactions of the product *νi*,*krk*. This equation corresponds to Equation (1) for chemical treatment only, where the

> *n* ∏ *j*=1 *Xν*� *jk*

assumption is that the reactions that consume the species are very much faster than the reactions producing the species. Thus, the concentration remain low just as their time

The H2-O2 system will be used as an example (Peters, 1990);(Peters et al., 1993). The reactions needed for the present analysis are (following the numbering in the original mechanism)

> H + O2 ↔ OH + O, (ii) O + H2 ↔ OH + H, (iii) OH + H2 ↔ H2O + H, (iv) H + O2+M → HO2+M, (vii)

where the backward reactions are treated as being separate reactions. Additional reactions in

*<sup>i</sup>* − *ν*� *i*

*νi*,*krk* = *ω<sup>i</sup>* = 0, (13)

*<sup>j</sup>* , (14)

. The physical meaning of this

*r* ∑ *k*=1

*rk* = *kk*

*dXi dt* <sup>=</sup>

reaction rate *rk* can be expressed by the following general definition:

the H2-O2 reaction system are also included in the present example:

and the net stoichiometric coefficient being *ν<sup>i</sup>* = *ν*��

problems introduced in combustion physics.

scale separation methods outlined.

concentration of species *i* is

derivative.

**3.1 Manual reduction by means of QSSA**

$$\begin{aligned} \text{OH} + \text{OH} &\leftrightarrow \text{H}\_2\text{O} + \text{O}\_{\prime} \quad (\text{x})\\ \text{H} + \text{HO}\_2 &\to \text{OH} + \text{OH}\_{\prime} \text{ (xi)}\\ \text{H} + \text{HO}\_2 &\to \text{O}\_2 + \text{H}\_2 \quad (\text{xii})\\ \text{OH} + \text{HO}\_2 &\to \text{H}\_2\text{O} + \text{O}\_2.\text{ (xiii)} \end{aligned}$$

Equation (13) indicated that the time derivatives of all the species concentrations can be expressed in terms of the concentrations of species that consume and that produce that particular species. As noted above, the derivatives not only contain a chemistry-term, but in the case of inhomogeneous systems, can also contain diffusive and convective terms, often being denoted in such cases by the L-operator. Here, only the chemical contribution is included, resulting in the following set of equations:

$$\begin{array}{rcl}\omega\_{H} &=& -r\_{\text{ii}} + r\_{\text{iii}} + r\_{\text{iv}} - r\_{\text{vii}} - r\_{\text{xi}i} - r\_{\text{xi}i},\\ 0 = \omega\_{OH} &=& r\_{\text{ii}} + r\_{\text{iii}} - r\_{\text{iv}} - 2r\_{\text{x}} + 2r\_{\text{xi}i} - r\_{\text{xi}i},\\ 0 = \omega\_{O} &=& r\_{\text{ii}} - r\_{\text{iii}} + r\_{\text{x}},\\ \omega\_{H\_{2}} &=& -r\_{\text{iii}} - r\_{\text{iv}} + r\_{\text{xi}i},\\ \omega\_{O\_{2}} &=& -r\_{\text{ii}} - r\_{\text{Vi}} + r\_{\text{x}i} + r\_{\text{x}\text{ii}},\\ \omega\_{H\_{2}O} &=& r\_{\text{iv}} + r\_{\text{x}} + r\_{\text{x}\text{iii}},\\ 0 = \omega\_{HO\_{2}} &= r\_{\text{vii}} - r\_{\text{xi}i} - r\_{\text{x}\text{ii}} - \omega\_{\text{xi}\text{iii}}.\end{array} \tag{15}$$

where OH, O and HO2 are all set to steady state and their derivatives are set equal to zero, transforming the corresponding differential equations conveniently into algebraic relations. Since the system is already reduced, the steady state concentrations can be found. However, in obtaining a set of global reactions and optimizing the reduction, the corresponding number of reaction rates can be eliminate from the system. A rule of thumb is for each species to eliminate the fastest reaction rates at which this species is consumed. This is *r*iv for OH, *r*iii for O and *r*xii for HO2, them being placed on the right-hand side of the equations and expressed in terms of the remaining reaction rates on the left-hand side (Peters, 1990). However, the reaction rates that are eliminated from the system are arbitrary and result in the same global reactions. After some algebra, the system is as follows:

$$\begin{array}{ll}\omega\_{H} &= 2r\_{\text{ii}} - 2r\_{\text{vii}} + 2r\_{\text{xi}} \\ \omega\_{H\_{2}} &= -3r\_{\text{ii}} + r\_{\text{vi}} - 3r\_{\text{xi}} \\ \omega\_{O\_{2}} &= -r\_{\text{ii}} - r\_{\text{xi}} \\ \omega\_{H\_{2}O} &= -2r\_{\text{ii}} - 2r\_{\text{xi}} \end{array} \tag{16}$$

By arranging the right-hand side so that rates with the same stoichiometric coefficients are added together one obtains

$$\begin{array}{ll}\omega\_H &= \mathfrak{L}(r\_{\text{ii}} + r\_{\text{xi}}) - 2r\_{\text{vii}},\\\omega\_{H\_2} &= -\mathfrak{J}(r\_{\text{ii}} + r\_{\text{xi}}) + r\_{\text{vi}},\\\omega\_{O\_2} &= -(r\_{\text{ii}} + r\_{\text{xi}}),\\\omega\_{H\_2O} &= -\mathfrak{L}(r\_{\text{ii}} + r\_{\text{xi}}),\end{array} \tag{17}$$

resulting in the following global reactions:

$$\begin{array}{rcl} \text{3H}\_2 + \text{O}\_2 = 2\text{H} + 2\text{H}\_2\text{O} \text{ , I} \\ \text{2H} + \text{M} &= \text{H}\_2 + \text{M}\_\prime \end{array}$$

where *r*iv,*r*ii and *r*iii all include both forward and backward reactions. The equation can be solved by setting *r*iv, for example, on the left-hand side and inserting all the rate constants and

Model Reduction Techniques for Chemical Mechanisms 91

A very powerful and important step in the reduction process is the *truncation of the steady state relations,* such as that shown in Equation (20). By investigating the reaction rates, both the forward and the backward rates, for a stoichiometric flame over the typical physical range of calculations one can track down the one or two reaction rates that are dominant. Peters (Peters, 1990) could show that in the case of OH only *r*iv is dominant, allowing the other reaction rates to be neglected in the further calculations. The advantage of performing truncation is that the most rapid reaction rates are excluded from the system, reducing the stiffness of the problem considerably and enhancing the accuracy. Thus, Equation (20) can easily be solved,

[OH] = *<sup>k</sup>*iv*b*[H2O][H]

Similar calculations for the other steady state species need to be performed. In the next step then these relations are used to calculate the remaining set of differential equations which govern the non-steady state species. The overall reaction rate for each remaining species is thus determined by means of an *inner iteration loop*, a simple fixed-point iteration procedure is often employed. The remaining set of differential equations can be solved in different ways,

This procedure has proven successful, many authors having presented work in which reduced mechanisms resemble detailed mechanisms very closely (See e.g. work presented in (Peters & Rogg, 1993);(Smooke, 1991)). Similarly reduced mechanisms have been produced for a variety of fuels, for example the three-step mechanism for CO-H2-N2/air diffusion flames by Chen et. al. (Chen et al., 1993) and four-step mechanisms for both ethylene/air and ethane/air flames

Methane is the simplest of the carbon-containing fuels and extensive investigations of the reduced mechanisms for methane-air flames have been carried out, even before hydrogen flames as the C1 chain provides a lower radical level, steady state relations thus being easier justified. The first reduced mechanisms for methane-air flame combustion originated from a skeletal mechanism consisting of 25 reactions for the C1 chain, and 61 reactions including the C2 chain evolved during the 1970-1980s (Kaufman, 1982);(Mauss & Peters, 1993);(Smooke, 1991);(Turns, 2000). A fully detailed mechanism can contain over 40 species and involve some 300 reactions. An example of such a mechanism is the GRI-Mech 3.0 mechanism obtained from Berkeley University (Bowman et al., 2004). Peters (Peters, 1990) and Mauss et al.(Mauss & Peters, 1993) demonstrated the reduction of the skeletal mechanism for methane-air flame combustion by use of the quasi steady state assumption, validating it for the range of lean to stoichiometric mixtures. The mechanism was reduced to a four-step mechanism containing 7 species, the reduction procedure following exactly the same underlying principles as for the

The investigation of higher carbon containing fuels such as acetylene for the purpose of reduction is of considerable importance for soot studies. Ring compounds and their growth through acetylene (C2H2) are important features for formation of soot under fuel-rich conditions (Turns, 2000). Mauss et al. (Mauss & Lindstedt, 1993) presented a 7 step reduced mechanism validated for acetylene-air premixed flames which originates from the skeletal

and several numerical solvers for stiff differential equations are freely available.

*<sup>k</sup>*iv *<sup>f</sup>* [H2] (21)

concentrations.

the resulting [OH] being found to be

by Wang and Rogg (Wang & Rogg, 1993).

H2−O2 system described above.

with the reaction rates being *r*<sup>I</sup> = (*r*ii + *r*xi) and *r*II = *r*vii. This procedure shows how a set of eight elementary reactions can be reduced to just two global reactions and two corresponding reaction rates, which are sufficient to describe the system. The resulting system is also called a two-step mechanism.

The computations above are quite straightforward once a choice is made of which species are to be considered to be in steady state. In the H2-O2 system, the three species OH, O and HO2 are conveniently set to steady state. Although, in the manual reduction procedure the choice of steady state species often stems simply from the experience of which species can be approximately calculated. However, the choice is often based on an analysis of the intermediate species' mole fraction. The argument is that no species at high concentration, and no reactant or product species either, can be a steady state candidate. The algorithm is such that the conservation equations are dealt with only for the species included in detail in the mechanism. The steady state species only appear through their contribution to the reaction rates determining the non-steady state species. Thus, species of high concentrations or high enthalpy content need to be kept in the mechanism so as to not violate the conservation of mass fraction and enthalpy. In the case of the methane-air mechanism that was reduced by Peters (Peters, 1990), the computation of mole fractions in a premixed stoichiometric methane-air flame was the basis for defining the steady state species.

In the flame calculations accounting for the physical processes diffusive and convective terms must also be small in order for the differentials to be set equal to zero. Nevertheless, in the reactive inner layer of the flames, the diffusive terms turn out to be dominant as compared with the convective terms. Consequently, since the species concentrations are strongly influenced by the diffusion, obtaining an order-of-magnitude estimate for selection of the steady state species requires that the mole fractions be weighted to the molecular weight of the inert species N2, *WN*<sup>2</sup> , the resulting weighting factor being

$$
\sqrt{\mathcal{W}\_{\text{N}\_2}/\mathcal{W}\_{\text{i},\text{N}\_2}}\tag{18}
$$

where *Wi*,*N*<sup>2</sup> is given by

$$\mathcal{W}\_{\rm i,N\_2} = \frac{2\mathcal{W}\_{\rm i}\mathcal{W}\_{\rm N\_2}}{\mathcal{W}\_{\rm i} + \mathcal{W}\_{\rm N\_2}}.\tag{19}$$

The H radical, for example, diffuses through the flame quickly and displays a low concentration profile. However, it is not a species in steady state and it is highly sensitive to the desired result. The species for the methane-air flame fall into two distinct groups: those in which the corresponding weighted mole fraction has a value well below 1% and those in which the corresponding weighted mole fraction has a value well above 1%. Thus, the choice is clear, the first group being selected as steady state species candidates.

The final step is to calculate the values of the reaction rates for the species and to then determine their concentrations. The reaction rates are expressed in the form of rate constants, stoichiometric coefficients and concentrations, as defined in Equation (13) and (14). However, the reaction rates also contain concentrations of the steady state species. As shown in Equation (13), these can be calculated from their balance equation with *ω<sup>i</sup>* = 0. The concentration of OH, for example, can be calculated as follows:

$$0 = r\_{\rm ii} + r\_{\rm iii} - r\_{\rm iv} - 2r\_{\rm x} + 2r\_{\rm xi} - r\_{\rm xii} \tag{20}$$

12 Will-be-set-by-IN-TECH

with the reaction rates being *r*<sup>I</sup> = (*r*ii + *r*xi) and *r*II = *r*vii. This procedure shows how a set of eight elementary reactions can be reduced to just two global reactions and two corresponding reaction rates, which are sufficient to describe the system. The resulting system is also called

The computations above are quite straightforward once a choice is made of which species are to be considered to be in steady state. In the H2-O2 system, the three species OH, O and HO2 are conveniently set to steady state. Although, in the manual reduction procedure the choice of steady state species often stems simply from the experience of which species can be approximately calculated. However, the choice is often based on an analysis of the intermediate species' mole fraction. The argument is that no species at high concentration, and no reactant or product species either, can be a steady state candidate. The algorithm is such that the conservation equations are dealt with only for the species included in detail in the mechanism. The steady state species only appear through their contribution to the reaction rates determining the non-steady state species. Thus, species of high concentrations or high enthalpy content need to be kept in the mechanism so as to not violate the conservation of mass fraction and enthalpy. In the case of the methane-air mechanism that was reduced by Peters (Peters, 1990), the computation of mole fractions in a premixed stoichiometric

In the flame calculations accounting for the physical processes diffusive and convective terms must also be small in order for the differentials to be set equal to zero. Nevertheless, in the reactive inner layer of the flames, the diffusive terms turn out to be dominant as compared with the convective terms. Consequently, since the species concentrations are strongly influenced by the diffusion, obtaining an order-of-magnitude estimate for selection of the steady state species requires that the mole fractions be weighted to the molecular weight of

*WN*2/*Wi*,*N*<sup>2</sup> (18)

0 = *r*ii + *r*iii − *r*iv − 2*r*<sup>x</sup> + 2*r*xi − *r*xiii, (20)

. (19)

is clear, the first group being selected as steady state species candidates.

OH, for example, can be calculated as follows:

*Wi*,*N*<sup>2</sup> <sup>=</sup> <sup>2</sup>*WiWN*<sup>2</sup>

The H radical, for example, diffuses through the flame quickly and displays a low concentration profile. However, it is not a species in steady state and it is highly sensitive to the desired result. The species for the methane-air flame fall into two distinct groups: those in which the corresponding weighted mole fraction has a value well below 1% and those in which the corresponding weighted mole fraction has a value well above 1%. Thus, the choice

The final step is to calculate the values of the reaction rates for the species and to then determine their concentrations. The reaction rates are expressed in the form of rate constants, stoichiometric coefficients and concentrations, as defined in Equation (13) and (14). However, the reaction rates also contain concentrations of the steady state species. As shown in Equation (13), these can be calculated from their balance equation with *ω<sup>i</sup>* = 0. The concentration of

*Wi* + *WN*<sup>2</sup>

methane-air flame was the basis for defining the steady state species.

the inert species N2, *WN*<sup>2</sup> , the resulting weighting factor being

a two-step mechanism.

where *Wi*,*N*<sup>2</sup> is given by

where *r*iv,*r*ii and *r*iii all include both forward and backward reactions. The equation can be solved by setting *r*iv, for example, on the left-hand side and inserting all the rate constants and concentrations.

A very powerful and important step in the reduction process is the *truncation of the steady state relations,* such as that shown in Equation (20). By investigating the reaction rates, both the forward and the backward rates, for a stoichiometric flame over the typical physical range of calculations one can track down the one or two reaction rates that are dominant. Peters (Peters, 1990) could show that in the case of OH only *r*iv is dominant, allowing the other reaction rates to be neglected in the further calculations. The advantage of performing truncation is that the most rapid reaction rates are excluded from the system, reducing the stiffness of the problem considerably and enhancing the accuracy. Thus, Equation (20) can easily be solved, the resulting [OH] being found to be

$$\text{[OH]} = \frac{k\_{\text{iv}b}[\text{H}\_2\text{O}][\text{H}]}{k\_{\text{iv}f}[\text{H}\_2]} \tag{21}$$

Similar calculations for the other steady state species need to be performed. In the next step then these relations are used to calculate the remaining set of differential equations which govern the non-steady state species. The overall reaction rate for each remaining species is thus determined by means of an *inner iteration loop*, a simple fixed-point iteration procedure is often employed. The remaining set of differential equations can be solved in different ways, and several numerical solvers for stiff differential equations are freely available.

This procedure has proven successful, many authors having presented work in which reduced mechanisms resemble detailed mechanisms very closely (See e.g. work presented in (Peters & Rogg, 1993);(Smooke, 1991)). Similarly reduced mechanisms have been produced for a variety of fuels, for example the three-step mechanism for CO-H2-N2/air diffusion flames by Chen et. al. (Chen et al., 1993) and four-step mechanisms for both ethylene/air and ethane/air flames by Wang and Rogg (Wang & Rogg, 1993).

Methane is the simplest of the carbon-containing fuels and extensive investigations of the reduced mechanisms for methane-air flames have been carried out, even before hydrogen flames as the C1 chain provides a lower radical level, steady state relations thus being easier justified. The first reduced mechanisms for methane-air flame combustion originated from a skeletal mechanism consisting of 25 reactions for the C1 chain, and 61 reactions including the C2 chain evolved during the 1970-1980s (Kaufman, 1982);(Mauss & Peters, 1993);(Smooke, 1991);(Turns, 2000). A fully detailed mechanism can contain over 40 species and involve some 300 reactions. An example of such a mechanism is the GRI-Mech 3.0 mechanism obtained from Berkeley University (Bowman et al., 2004). Peters (Peters, 1990) and Mauss et al.(Mauss & Peters, 1993) demonstrated the reduction of the skeletal mechanism for methane-air flame combustion by use of the quasi steady state assumption, validating it for the range of lean to stoichiometric mixtures. The mechanism was reduced to a four-step mechanism containing 7 species, the reduction procedure following exactly the same underlying principles as for the H2−O2 system described above.

The investigation of higher carbon containing fuels such as acetylene for the purpose of reduction is of considerable importance for soot studies. Ring compounds and their growth through acetylene (C2H2) are important features for formation of soot under fuel-rich conditions (Turns, 2000). Mauss et al. (Mauss & Lindstedt, 1993) presented a 7 step reduced mechanism validated for acetylene-air premixed flames which originates from the skeletal

constraints:

*μ*0

*<sup>j</sup>* (*T*) + *RT* ln

*nj n*

> *NS* ∑ *j*=1 *ac ij Nr* ∑ *k*=1

particularly attractive in adaptive chemistry systems.

by use of ILDM is taken up, and this is to be referred here further.

*∂ψ*(t)

terms, including the chemical reaction rates, and **<sup>P</sup>**(*ψ***,** <sup>∇</sup>*ψ***,** <sup>∇</sup>**<sup>2</sup>**

of the fastest can be decoupled from the system as desired.

where *Ns* is the number of species, and *P*<sup>0</sup> and *μ*<sup>0</sup>

*dCi dt* <sup>=</sup>

**3.3 Intrinsic Low Dimensional Manifolds 3.3.1 The standard approach to ILDM**

variables (See Equation (1) for comparison):

<sup>+</sup> *RT* ln *<sup>P</sup>*

concentrations of the remaining constrained species where the matrix *a<sup>c</sup>*

*P*0 + *Me* ∑ *i*=1 *λe i ae ij* + *Mc* ∑ *i*=1 *λc i ac*

in the standard state, where the chemical potential is a given function of temperature. These algebraic equations need to be solved simultaneously to compute the concentrations at the constrained equilibrium state. The second set becomes differential equations for the

Model Reduction Techniques for Chemical Mechanisms 93

The algebraic-differential equation system is then solved in much the same manner as for the QSSA approach. Because the formulation of Equation (25) yields for all species, the RCCE is

Intrinsic Low Dimensional Manifolds is a method for reducing a chemical system by using *attractors* for the chemical kinetics involved. It is thought that fast reactions quickly bring the composition down to the attracting manifolds, an equilibrium solution space that fast chemical reactions relax towards, slow reactions moving within these manifolds (Maas, 1998);(Schmidt et al., 1998); (Yang & Pope, 1998). This represents a time separation of the fast chemical processes. In the work of Maas et al. (Maas & Pope, 1994) and developed further in Maas (Maas, 1998), the fundamental formalism behind the decoupling of the short time scales

Reacting flow and its scalar field evolve in time according to an N*u*-dimensional partial differential equation system, where N*<sup>u</sup>* is the number of unknown chemical and physical

*ψ* being the N*u*-dimensional vector of the unknown scalars, *ω*(*ψ*) the vector of the source

operator that governs the contributions of all other physical processes such as diffusion, convection, advection, etc. The time scales governed by *ω*(*ψ*) span over a large range. Some

The N*<sup>u</sup>* variables form an N*u*-dimensional space. A chemical reaction represents a movement along a trajectory from the initial state to the final state for t → **∞.** An example is given in Figure (4), which shows the trajectories in a CO-H**<sup>2</sup>** system as plotted in the CO**2**-H**2**O plane. In Figure (4) several cases involving differing initial conditions are plotted. However, one can clearly see in the figure that the trajectories all move towards an equilibrium condition, in this

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> **<sup>P</sup>**(*ψ*, <sup>∇</sup>*ψ*, <sup>∇</sup>**<sup>2</sup>**

*ij* = 0, (*j* = 1, ...*NS*) (25)

*ij* is now incorporated:

*<sup>j</sup>* are the pressure and chemical potential

*ψ*) + *ω*(*ψ*), (27)

*ψ*) being the spatial vector

*νjkrk*(*n*1, ..., *nNS* , *T*, *ρ*), (*i* = 1, ...*Mc*) (26)

mechanism described in Peters and Rogg (Peters & Rogg, 1993), including all 87 reactions for up to C3−species. With use of the 7 step mechanism, both flame velocities and species concentrations were well predicted as compared to the results for the full mechanism. Some deviations were found in the reaction zone, where a steeper consumption of reactants was obtained for the reduced mechanism. However, this can also result from a steeper temperature profile in the same region, which would enhance the reaction rates. In Lindstedt et al. (Lindstedt & Mauss, 1993) the same mechanism for acetylene formation is applied to a diffusion flame system having 8 additional reaction steps. In the work, a five step reduced mechanism is generated and is validated for the entire range of strain rates up to the extinction point. As in the case of the premixed flame, the temperature profiles and species concentration profiles agree very well with the results of the full mechanism for a selected range of pressures.

Nevertheless, as noted, the calculations above and their validations are time-consuming, and since the demand for larger and more complex mechanisms is very strong from areas related to combustion modeling, automating the procedure is important. This will be discussed in the following section.

#### **3.2 Rate-controlled constrained equilibria**

The Rate-Controlled constrained equilibria (RCCE) approach incorporates the assumption of time scale separation in reactive systems in much the same way as the QSSA described above. However, in the case of RCCE the analytical expressions are derived from equilibrium assumptions rather than steady state. These equilibrium states are constrained by the leading or main species, hence the terminology of constrained equilibria. It means that if e.g. considering a simple chemical system such as CH4-O2, and O2 is constrained, then O2 will occupy all O-atoms in the present form as the amount of O2 can not change, whereas both H and C atoms may be occupied by whatever CH-molecule are thermodynamically stable in the current state. The constraints are thus linear combinations of the species' concentrations. The RCCE method is in large developed by Jones and Rigopoulos for combustion systems (Jones and Rigopoulos, 2005);(Jones and Rigopoulos, 2007), and only briefly outlined here.

In order to derive the equilibrium states, the RCCE makes use of minimizing the Gibbs free energy subject to conservation of enthalpy, elements and mass for each constraint *i*:

$$h = \sum\_{j=1}^{N\_S} n\_j H \mathfrak{j}^0(T) \tag{22}$$

$$E\_i = \sum\_{j=1}^{N\_S} a\_{ij}^\ell n\_{j\prime} \text{ ( $i = 1, \dots M\_\ell$ )}\tag{23}$$

$$\mathbf{C}\_{i} = \sum\_{j=1}^{N\_{\mathbf{S}}} a\_{ij}^{\mathbf{c}} n\_{j\prime} \text{ ( $i = 1, \dots M\_{\mathbf{c}}$ )}\tag{24}$$

where *a<sup>e</sup> ij* is the matrix of number of element *<sup>i</sup>* in the *<sup>j</sup>*th species, *<sup>a</sup><sup>c</sup> ij* is the matrix of linear combination of concentration of the *j* species making up the *i*th constraint, and *nj* is the concentration of the *j*th species.

It can be shown that two sets of equations arises when employing minimization of Gibbs free energy (Jones and Rigopoulos, 2005). One set becomes algebraic equations defining the constraints:

14 Will-be-set-by-IN-TECH

mechanism described in Peters and Rogg (Peters & Rogg, 1993), including all 87 reactions for up to C3−species. With use of the 7 step mechanism, both flame velocities and species concentrations were well predicted as compared to the results for the full mechanism. Some deviations were found in the reaction zone, where a steeper consumption of reactants was obtained for the reduced mechanism. However, this can also result from a steeper temperature profile in the same region, which would enhance the reaction rates. In Lindstedt et al. (Lindstedt & Mauss, 1993) the same mechanism for acetylene formation is applied to a diffusion flame system having 8 additional reaction steps. In the work, a five step reduced mechanism is generated and is validated for the entire range of strain rates up to the extinction point. As in the case of the premixed flame, the temperature profiles and species concentration profiles agree very well with the results of the full mechanism for a selected range of pressures. Nevertheless, as noted, the calculations above and their validations are time-consuming, and since the demand for larger and more complex mechanisms is very strong from areas related to combustion modeling, automating the procedure is important. This will be discussed in

The Rate-Controlled constrained equilibria (RCCE) approach incorporates the assumption of time scale separation in reactive systems in much the same way as the QSSA described above. However, in the case of RCCE the analytical expressions are derived from equilibrium assumptions rather than steady state. These equilibrium states are constrained by the leading or main species, hence the terminology of constrained equilibria. It means that if e.g. considering a simple chemical system such as CH4-O2, and O2 is constrained, then O2 will occupy all O-atoms in the present form as the amount of O2 can not change, whereas both H and C atoms may be occupied by whatever CH-molecule are thermodynamically stable in the current state. The constraints are thus linear combinations of the species' concentrations. The RCCE method is in large developed by Jones and Rigopoulos for combustion systems (Jones

and Rigopoulos, 2005);(Jones and Rigopoulos, 2007), and only briefly outlined here.

energy subject to conservation of enthalpy, elements and mass for each constraint *i*:

*h* = *NS* ∑ *j*=1

*NS* ∑ *j*=1 *ae*

*NS* ∑ *j*=1 *ac*

*ij* is the matrix of number of element *<sup>i</sup>* in the *<sup>j</sup>*th species, *<sup>a</sup><sup>c</sup>*

*Ei* =

*Ci* =

In order to derive the equilibrium states, the RCCE makes use of minimizing the Gibbs free

combination of concentration of the *j* species making up the *i*th constraint, and *nj* is the

It can be shown that two sets of equations arises when employing minimization of Gibbs free energy (Jones and Rigopoulos, 2005). One set becomes algebraic equations defining the

*njHj*<sup>0</sup>(*T*) (22)

*ijnj*, (*i* = 1, ...*Me*) (23)

*ijnj*, (*i* = 1, ...*Mc*) (24)

*ij* is the matrix of linear

the following section.

where *a<sup>e</sup>*

concentration of the *j*th species.

**3.2 Rate-controlled constrained equilibria**

$$
\mu\_j^0(T) + RT \ln \frac{n\_j}{n} + RT \ln \frac{P}{P\_0} + \sum\_{i=1}^{M\_\ell} \lambda\_i^\varepsilon a\_{ij}^\varepsilon + \sum\_{i=1}^{M\_\ell} \lambda\_i^\varepsilon a\_{ij}^\varepsilon = 0,\ (j = 1, \dots \\ N\_\mathcal{S}) \tag{25}
$$

where *Ns* is the number of species, and *P*<sup>0</sup> and *μ*<sup>0</sup> *<sup>j</sup>* are the pressure and chemical potential in the standard state, where the chemical potential is a given function of temperature. These algebraic equations need to be solved simultaneously to compute the concentrations at the constrained equilibrium state. The second set becomes differential equations for the concentrations of the remaining constrained species where the matrix *a<sup>c</sup> ij* is now incorporated:

$$\frac{d\mathbb{C}\_{i}}{dt} = \sum\_{j=1}^{N\_{\mathbb{S}}} a\_{ij}^{\mathbb{C}} \sum\_{k=1}^{N\_{\mathbb{S}}} \nu\_{jk} r\_k(n\_1, \dots, n\_{N\_{\mathbb{S}'}}, T, \rho)\_{\prime} \text{ ( $i = 1, \dots, M\_{\mathbb{C}}$ )}\tag{26}$$

The algebraic-differential equation system is then solved in much the same manner as for the QSSA approach. Because the formulation of Equation (25) yields for all species, the RCCE is particularly attractive in adaptive chemistry systems.

#### **3.3 Intrinsic Low Dimensional Manifolds**

#### **3.3.1 The standard approach to ILDM**

Intrinsic Low Dimensional Manifolds is a method for reducing a chemical system by using *attractors* for the chemical kinetics involved. It is thought that fast reactions quickly bring the composition down to the attracting manifolds, an equilibrium solution space that fast chemical reactions relax towards, slow reactions moving within these manifolds (Maas, 1998);(Schmidt et al., 1998); (Yang & Pope, 1998). This represents a time separation of the fast chemical processes. In the work of Maas et al. (Maas & Pope, 1994) and developed further in Maas (Maas, 1998), the fundamental formalism behind the decoupling of the short time scales by use of ILDM is taken up, and this is to be referred here further.

Reacting flow and its scalar field evolve in time according to an N*u*-dimensional partial differential equation system, where N*<sup>u</sup>* is the number of unknown chemical and physical variables (See Equation (1) for comparison):

$$\frac{\partial \boldsymbol{\psi}(\mathbf{t})}{\partial t} = \mathbf{P}(\boldsymbol{\psi}, \nabla \boldsymbol{\psi}, \nabla^2 \boldsymbol{\psi}) + \boldsymbol{\omega}(\boldsymbol{\psi}), \tag{27}$$

*ψ* being the N*u*-dimensional vector of the unknown scalars, *ω*(*ψ*) the vector of the source terms, including the chemical reaction rates, and **<sup>P</sup>**(*ψ***,** <sup>∇</sup>*ψ***,** <sup>∇</sup>**<sup>2</sup>** *ψ*) being the spatial vector operator that governs the contributions of all other physical processes such as diffusion, convection, advection, etc. The time scales governed by *ω*(*ψ*) span over a large range. Some of the fastest can be decoupled from the system as desired.

The N*<sup>u</sup>* variables form an N*u*-dimensional space. A chemical reaction represents a movement along a trajectory from the initial state to the final state for t → **∞.** An example is given in Figure (4), which shows the trajectories in a CO-H**<sup>2</sup>** system as plotted in the CO**2**-H**2**O plane. In Figure (4) several cases involving differing initial conditions are plotted. However, one can clearly see in the figure that the trajectories all move towards an equilibrium condition, in this

method is based on a more intrinsic study of the chemical reaction process happening in combustion" (Yang & Pope, 1998). The use of attractors is sufficient to find a solution. Since computer simulations of combustion processes involving ILDM reduction speed up calculations by a factor of ten, the method is applicable to numerical problems in particular. The results of the calculations of the manifolds are stored and *tabulated* in a preprocessing step. Since the tabulated data consist of the relevant data (such as the reduced reaction rates) the information is assessed through a table look-up during the computational run. This method however requires significant amounts of storage memory, which becomes problematic in higher dimensional manifold calculations. An alternative storage method is proposed by Niemann et al. (Niemann et al., 1996). Whereas the classic approach employs an interpolation procedure within small tabulation cells and a local mesh refinement, Niemann et al. employs a higher order polynomial approximation within large coarse cells. This decreases the storage requirement considerably; for an H**2**−O**<sup>2</sup>** laminar flat flame calculation the storage requirements were reduced by a factor 200 (Niemann et al., 1996). However, since the method does not provide a set of rate equations, it cannot be related to a set of global reactions that provide information on the underlying kinetics involved (Tomlin et al., 1997). Furthermore, classic ILDM does not account for transport processes which may be important for diffusion flames. The two next sub-sections will be devoted to methods to account for transport in the

Model Reduction Techniques for Chemical Mechanisms 95

The Flamelet Generated Manifolds (FGM) approach aims to provide a tool for simplified treatment of the chemical system for flames where also transport in terms of diffusion is important. This is particularly the case for premixed flames where there are also less reactive parts on one side of the flame zone from which radicals diffuse towards the flame zone. The concept is based on two simplification procedures: one is the flamelet approach where a multi-dimensional flame is considered as an ensemble of one-dimensional flames, and another

van Oijen and co-workers have developed this technique for a set of different flames (van Oijen & de Goey, 2000);(van Oijen & de Goey, 2002). The first approach considers a premixed flame. The technique of FGM make use of premixed flames with detailed chemistry which are pre-calculated and tabulated. Then tables are generated, reflecting the solution of these flamelets as a function of one or more control variables. When solving equations for these control variables by the use of the ILDM, one can use the table of flamelets to retrieve quantities related to the composition that are not known. Since the major part of convection and diffusion processes are included in FGM through the flamelet calculations, the method is more accurate in the low-temperature region of a premixed flame than methods based on

The Reaction-Diffusion Manifolds (REDIM) approach represent an extension to the formulation of the standard ILDM. Where the ILDM is in fact a relaxation of a set of ordinary differential equations (ODE's) describing a homogenous system, the REDIM formulation generalizes for a set of partially differential equation (PDE's) where also the coupling between the reaction and diffusion processes are accounted for. Bykov and Maas (Bykov & Maas, 2007) have performed the full derivation of this generalized system in the framework of ILDM and an optimized tabulation procedure of generalized coordinates. They present the method in

which is the consideration of a low-dimensional manifold in composition space.

ILDM framework.

**3.3.2 Flamelet Generated Manifolds (FGM)**

local chemical equilibria as the classic ILDM.

**3.3.3 Reaction-Diffusion Manifolds (REDIM)**

case at the far right end point of the main manifold. The equilibrium state can represent the chemical system as one single point if the computations do not include the chemical dynamics.

Fig. 4. Trajectories of the chemical reactions progress for a CO-H**<sup>2</sup>** system and its projection into the CO**2**-H**2**O plane (Nilsson, 2011).

The equilibrium process in this case takes about 5 ms If the physical processes are slower or have the same order of magnitude, the equilibrium point offers an approximate solution. This corresponds to neglecting chemical dynamics on time scales shorter than 5 ms. By neglecting chemical dynamics that take place on time scales shorter than say 100*μ*s, the solution can be described by a single variable since the main equilibrium curve remains in the state space, which is a one-dimensional manifold. This is shown by the open circles in Figure (4). It is a consequence of the fast relaxation process leading to the reactions being in partial equilibrium and the species involved being in steady state. One can thus conclude that after the relaxation period of 5 ms the system can be described in an approximate way by the equilibrium value, which is a single point. If time scales shorter than 100*μ*s can be neglected, the system can be described by using a single reaction progress variable, a one-dimensional manifold. If even shorter time scales are needed, several additional reaction progresses need to be included, which results in a higher-dimensional manifold.

After identifying the low-dimensional manifolds in the N*u*-dimensional state space that the thermodynamic state of the system has been relaxed to, the reduced system consisting of *<sup>S</sup>* <sup>=</sup>N*<sup>u</sup>* <sup>−</sup> *<sup>R</sup>* variables (*<sup>R</sup>* is the degree of reduction), can be represented as **<sup>Φ</sup>**=(**Φ1, ..., <sup>Φ</sup>***S*)*T*. Thus the original system *ψ*=*ψ*(**Φ**) can be projected onto the S-dimensional sub-space

$$\frac{\partial \Phi}{\partial t} = \Pi(\Phi, \nabla \Phi, \nabla^2 \Phi) + \Omega(\Phi), \tag{28}$$

where **Ω** and **Π** are the corresponding S-dimensional vectors for the source terms and from the other physical processes, respectively. The low-dimensional manifold described by equation (28) represents the chemical system in a further calculation.

The advantage of the ILDM method is that it requires no information concerning which reactions are to be set in equilibrium or which species are in steady state: "The manifold 16 Will-be-set-by-IN-TECH

case at the far right end point of the main manifold. The equilibrium state can represent the chemical system as one single point if the computations do not include the chemical dynamics.

Fig. 4. Trajectories of the chemical reactions progress for a CO-H**<sup>2</sup>** system and its projection

The equilibrium process in this case takes about 5 ms If the physical processes are slower or have the same order of magnitude, the equilibrium point offers an approximate solution. This corresponds to neglecting chemical dynamics on time scales shorter than 5 ms. By neglecting chemical dynamics that take place on time scales shorter than say 100*μ*s, the solution can be described by a single variable since the main equilibrium curve remains in the state space, which is a one-dimensional manifold. This is shown by the open circles in Figure (4). It is a consequence of the fast relaxation process leading to the reactions being in partial equilibrium and the species involved being in steady state. One can thus conclude that after the relaxation period of 5 ms the system can be described in an approximate way by the equilibrium value, which is a single point. If time scales shorter than 100*μ*s can be neglected, the system can be described by using a single reaction progress variable, a one-dimensional manifold. If even shorter time scales are needed, several additional reaction progresses need to be included,

After identifying the low-dimensional manifolds in the N*u*-dimensional state space that the thermodynamic state of the system has been relaxed to, the reduced system consisting of *<sup>S</sup>* <sup>=</sup>N*<sup>u</sup>* <sup>−</sup> *<sup>R</sup>* variables (*<sup>R</sup>* is the degree of reduction), can be represented as **<sup>Φ</sup>**=(**Φ1, ..., <sup>Φ</sup>***S*)*T*.

where **Ω** and **Π** are the corresponding S-dimensional vectors for the source terms and from the other physical processes, respectively. The low-dimensional manifold described by

The advantage of the ILDM method is that it requires no information concerning which reactions are to be set in equilibrium or which species are in steady state: "The manifold

**Φ**) + **Ω**(**Φ**)**,** (28)

Thus the original system *ψ*=*ψ*(**Φ**) can be projected onto the S-dimensional sub-space

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> **<sup>Π</sup>**(**Φ,** <sup>∇</sup>**Φ,** <sup>∇</sup>**<sup>2</sup>**

into the CO**2**-H**2**O plane (Nilsson, 2011).

which results in a higher-dimensional manifold.

*∂***Φ**

equation (28) represents the chemical system in a further calculation.

method is based on a more intrinsic study of the chemical reaction process happening in combustion" (Yang & Pope, 1998). The use of attractors is sufficient to find a solution. Since computer simulations of combustion processes involving ILDM reduction speed up calculations by a factor of ten, the method is applicable to numerical problems in particular. The results of the calculations of the manifolds are stored and *tabulated* in a preprocessing step. Since the tabulated data consist of the relevant data (such as the reduced reaction rates) the information is assessed through a table look-up during the computational run. This method however requires significant amounts of storage memory, which becomes problematic in higher dimensional manifold calculations. An alternative storage method is proposed by Niemann et al. (Niemann et al., 1996). Whereas the classic approach employs an interpolation procedure within small tabulation cells and a local mesh refinement, Niemann et al. employs a higher order polynomial approximation within large coarse cells. This decreases the storage requirement considerably; for an H**2**−O**<sup>2</sup>** laminar flat flame calculation the storage requirements were reduced by a factor 200 (Niemann et al., 1996). However, since the method does not provide a set of rate equations, it cannot be related to a set of global reactions that provide information on the underlying kinetics involved (Tomlin et al., 1997). Furthermore, classic ILDM does not account for transport processes which may be important for diffusion flames. The two next sub-sections will be devoted to methods to account for transport in the ILDM framework.

#### **3.3.2 Flamelet Generated Manifolds (FGM)**

The Flamelet Generated Manifolds (FGM) approach aims to provide a tool for simplified treatment of the chemical system for flames where also transport in terms of diffusion is important. This is particularly the case for premixed flames where there are also less reactive parts on one side of the flame zone from which radicals diffuse towards the flame zone. The concept is based on two simplification procedures: one is the flamelet approach where a multi-dimensional flame is considered as an ensemble of one-dimensional flames, and another which is the consideration of a low-dimensional manifold in composition space.

van Oijen and co-workers have developed this technique for a set of different flames (van Oijen & de Goey, 2000);(van Oijen & de Goey, 2002). The first approach considers a premixed flame. The technique of FGM make use of premixed flames with detailed chemistry which are pre-calculated and tabulated. Then tables are generated, reflecting the solution of these flamelets as a function of one or more control variables. When solving equations for these control variables by the use of the ILDM, one can use the table of flamelets to retrieve quantities related to the composition that are not known. Since the major part of convection and diffusion processes are included in FGM through the flamelet calculations, the method is more accurate in the low-temperature region of a premixed flame than methods based on local chemical equilibria as the classic ILDM.

#### **3.3.3 Reaction-Diffusion Manifolds (REDIM)**

The Reaction-Diffusion Manifolds (REDIM) approach represent an extension to the formulation of the standard ILDM. Where the ILDM is in fact a relaxation of a set of ordinary differential equations (ODE's) describing a homogenous system, the REDIM formulation generalizes for a set of partially differential equation (PDE's) where also the coupling between the reaction and diffusion processes are accounted for. Bykov and Maas (Bykov & Maas, 2007) have performed the full derivation of this generalized system in the framework of ILDM and an optimized tabulation procedure of generalized coordinates. They present the method in

**h***<sup>r</sup>* = **b***<sup>r</sup>*

shortest chemical time scales and thus not being in steady state.

steady state species could affect the results too much.

value in the reaction zone since *qi*/*qi***,max** there is unity.

following expression:

*pointer*, Di**,** *i* = **1,** *NS:*

form similar to

(**P** + *ω*) and **h***<sup>s</sup>* = **b***s*(**P** + *ω*)**,** (30)

**D** = *diag*[**a1b<sup>1</sup>** + **a2b2** + **...** + **a***R***b***R*]**.** (32)

**.** (31)

*dx***,** (33)

where **P** contains the spatial differential operator, *ω* is the species source term from Equation (1). **a***<sup>r</sup>* and **a***<sup>s</sup>* are the stoichiometric vectors for the modified non-physical mechanism, and **b***<sup>r</sup>* and **b***<sup>s</sup>* form the corresponding inverse set of basis vectors . According to the definition of the basis vectors, the first term on the right hand side in Equation (29) should be assumed to be insignificant because of the steady state assumption that *<sup>h</sup><sup>i</sup>* <sup>≈</sup> **<sup>0</sup>** for *<sup>i</sup>* <sup>=</sup> **1,** *<sup>R</sup>***,** resulting in the

Model Reduction Techniques for Chemical Mechanisms 97

<sup>≈</sup> **<sup>a</sup>***s***h***<sup>s</sup>*

The goal of CSP analysis is thus to find the set of basis vectors that can fulfill this requirement. A CSP analysis is performed on each species at each spatial point, the R elements to be considered as being in steady state are identified by the *NS* diagonal elements of the *CSP*

The pointer takes a value between zero and one. It is a function of the space, thus describing the influence of the R shortest chemical time scales on each of the species i at this particular point in space. When Di takes on a value close to unity, species *i* is completely influenced by the shortest time scales and is a candidate for being set to steady state. The opposite occurs when Di becomes close to zero, the species in question is not being influenced at all by the

Some species have local pointers for which the value can go from zero to unity within the range of the calculation. These species are treated wrongly if the local pointer of a certain point in space is the basis of the reduction. Thus, the local pointers are integrated over the computational domain *L*. The *third step* involves an integration over the space to find the overall value on the influence on the time scale for each species. These integrated pointers are weighted with the species' mole fraction. This is done so that species at high concentrations are not set to steady state, since the resulting errors in calculating their concentrations as

Massias et. al. (Massias et al., 1999b) propose three different integration procedures for capturing the steady state candidates in the most appropriate way. However, they take the

where *qi* is the production rate of species *i***,** *qi***,max** is the maximum production rate over the computational range, and *Xi* is the mole fraction of the species. Caution is in order as both *Xi* and *qi* can take on the value zero (inert species), so that small terms *�***<sup>1</sup>** and *�***<sup>2</sup>** are added to the respective denominators so as to avoid numerical problems. The R species with the highest values for *I<sup>I</sup>* are selected as steady state species. Note that since for most species *qi***,max** is situated in the reaction zone, the largest contribution to the integrated pointer comes from the

The elementary reaction rates are integrated over the space in order to determine the *R* fastest elementary reaction rates, which can then be eliminated from the system. The fastest reactions that consumes a steady state species, as identified in the previous step, are selected as to be the "fast reaction". For each steady state species, the fastest reaction consuming the species is

*qi qi***,max**

*<sup>I</sup><sup>I</sup>* <sup>=</sup> **<sup>1</sup>** *L L* **0** *Di* **1** *Xi*

*∂***Y** *∂t*

three limiting cases: (1) where there is pure mixing and thus no chemical source term, (2) where there is a homogeneous system and thus no transport, and (3) where there is a case of known gradients of the local coordinates. This last limiting case will if the known gradients are found from detailed flame calculations become equivalent to the FGM approach as described in the previous sub section.

It is clear that REDIM represents a generalized form of applying known ideas from invariant manifold theory and have been shown to apply to a wide range of applications from homogeneous systems, premixed flames and diffusion flames (Bykov & Maas, 2009). It is less straight forward to implement, tabulation of the multi-dimensional manifolds need special attention and the interpretation of the results are non-trivial. However, once implemented REDIM represents a very general and powerful reduction tool.

#### **3.4 Computational Singular Perturbation method**

Computational Singular Perturbation (CSP) is an alternative method employing a time scale separation analysis, thereafter use being made of exhausting modes for the approximate treatment. The core of the technique is to rewrite the set of differential equations that govern the system, using a new set of basis vectors so defined that they represent the fast and the slow sub domains. These vectors thus contain a linear combination of the reaction rates involved in the original mechanism. By employing this method, the problem becomes one of an eigenvalue problem that can help in discarding fast "modes" which involve species that are candidates for steady state. Work on development of the CSP method can be examined in detail in a substantial selection of papers (Lam, 1993);(Lam et al., 1994); (Massias et al., 1999a); (Massias et al., 1999b);(Tomlin et al., 1997).

The method as a whole is described in terms of three major steps (Massias et al., 1999a); (Massias et al., 1999b). The first step concerns as usual identification of the detailed chemical and physical system and setting the limits for reduction. The next step involves finding the steady state species, and constructing the reduced mechanism that involve the simpler algebraic operations. The final step is to optimize the reduced mechanism through truncation of the algebraic relations or of the remaining differential equations and of the resulting global rates. This is done to enhance the computations and also provide a more accurate solution.

The *second* step which is of interest here consists of choosing the number of global steps that are desired for the reduced mechanism, noted here by *G*, and determining the *local CSP pointers* for each species. The definition of the size of the reduced mechanism is being based on experience and empirical considerations. A method is yet not presented that can determine the size of the reduced mechanism *a priori* based on the relevant physical situation. The number of steady state species, *R*, is given by *R* = *NS* − *E* − *G***,** where *NS* is the total number of species in the detailed mechanism and E is the number of elements in the mechanism, such as C, O, H and N.

The local CSP pointers are generated through representation of the chemical system, Equation (1), by a set of basis vectors, **a**, defined such that (Massias et al., 1999b):

$$\frac{\partial \mathbf{Y}}{\partial t} = \mathbf{a}\_{\prime} \mathbf{h}^{\prime} + \mathbf{a}\_{s} \mathbf{h}^{s},\tag{29}$$

where **a***<sup>r</sup>* is an R-dimensional vector for the R reduced species, **a***<sup>s</sup>* is an S-dimensional vector for the remaining species, and **h***<sup>r</sup>* and **h***<sup>s</sup>* are the corresponding vectors of the form

18 Will-be-set-by-IN-TECH

three limiting cases: (1) where there is pure mixing and thus no chemical source term, (2) where there is a homogeneous system and thus no transport, and (3) where there is a case of known gradients of the local coordinates. This last limiting case will if the known gradients are found from detailed flame calculations become equivalent to the FGM approach as described

It is clear that REDIM represents a generalized form of applying known ideas from invariant manifold theory and have been shown to apply to a wide range of applications from homogeneous systems, premixed flames and diffusion flames (Bykov & Maas, 2009). It is less straight forward to implement, tabulation of the multi-dimensional manifolds need special attention and the interpretation of the results are non-trivial. However, once implemented

Computational Singular Perturbation (CSP) is an alternative method employing a time scale separation analysis, thereafter use being made of exhausting modes for the approximate treatment. The core of the technique is to rewrite the set of differential equations that govern the system, using a new set of basis vectors so defined that they represent the fast and the slow sub domains. These vectors thus contain a linear combination of the reaction rates involved in the original mechanism. By employing this method, the problem becomes one of an eigenvalue problem that can help in discarding fast "modes" which involve species that are candidates for steady state. Work on development of the CSP method can be examined in detail in a substantial selection of papers (Lam, 1993);(Lam et al., 1994); (Massias et al., 1999a);

The method as a whole is described in terms of three major steps (Massias et al., 1999a); (Massias et al., 1999b). The first step concerns as usual identification of the detailed chemical and physical system and setting the limits for reduction. The next step involves finding the steady state species, and constructing the reduced mechanism that involve the simpler algebraic operations. The final step is to optimize the reduced mechanism through truncation of the algebraic relations or of the remaining differential equations and of the resulting global rates. This is done to enhance the computations and also provide a more accurate solution. The *second* step which is of interest here consists of choosing the number of global steps that are desired for the reduced mechanism, noted here by *G*, and determining the *local CSP pointers* for each species. The definition of the size of the reduced mechanism is being based on experience and empirical considerations. A method is yet not presented that can determine the size of the reduced mechanism *a priori* based on the relevant physical situation. The number of steady state species, *R*, is given by *R* = *NS* − *E* − *G***,** where *NS* is the total number of species in the detailed mechanism and E is the number of elements in the mechanism, such

The local CSP pointers are generated through representation of the chemical system, Equation

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> **<sup>a</sup>***r***h***<sup>r</sup>* <sup>+</sup> **<sup>a</sup>***s***h***<sup>s</sup>*

where **a***<sup>r</sup>* is an R-dimensional vector for the R reduced species, **a***<sup>s</sup>* is an S-dimensional vector

**,** (29)

(1), by a set of basis vectors, **a**, defined such that (Massias et al., 1999b):

*∂***Y**

for the remaining species, and **h***<sup>r</sup>* and **h***<sup>s</sup>* are the corresponding vectors of the form

REDIM represents a very general and powerful reduction tool.

**3.4 Computational Singular Perturbation method**

(Massias et al., 1999b);(Tomlin et al., 1997).

as C, O, H and N.

in the previous sub section.

$$\mathbf{h}^{\prime} = \mathbf{b}^{\prime}(\mathbf{P} + \omega) \qquad \text{and} \quad \mathbf{h}^{s} = \mathbf{b}^{s}(\mathbf{P} + \omega) \, \text{,} \tag{30}$$

where **P** contains the spatial differential operator, *ω* is the species source term from Equation (1). **a***<sup>r</sup>* and **a***<sup>s</sup>* are the stoichiometric vectors for the modified non-physical mechanism, and **b***<sup>r</sup>* and **b***<sup>s</sup>* form the corresponding inverse set of basis vectors . According to the definition of the basis vectors, the first term on the right hand side in Equation (29) should be assumed to be insignificant because of the steady state assumption that *<sup>h</sup><sup>i</sup>* <sup>≈</sup> **<sup>0</sup>** for *<sup>i</sup>* <sup>=</sup> **1,** *<sup>R</sup>***,** resulting in the following expression:

$$\frac{\partial \mathbf{Y}}{\partial t} \approx \mathbf{a}\_s \mathbf{h}^s. \tag{31}$$

The goal of CSP analysis is thus to find the set of basis vectors that can fulfill this requirement. A CSP analysis is performed on each species at each spatial point, the R elements to be considered as being in steady state are identified by the *NS* diagonal elements of the *CSP pointer*, Di**,** *i* = **1,** *NS:*

$$\mathbf{D} = \operatorname{diag} [\mathbf{a}\_1 \mathbf{b}^1 + \mathbf{a}\_2 \mathbf{b}^2 + \dots + \mathbf{a}\_R \mathbf{b}^K]. \tag{32}$$

The pointer takes a value between zero and one. It is a function of the space, thus describing the influence of the R shortest chemical time scales on each of the species i at this particular point in space. When Di takes on a value close to unity, species *i* is completely influenced by the shortest time scales and is a candidate for being set to steady state. The opposite occurs when Di becomes close to zero, the species in question is not being influenced at all by the shortest chemical time scales and thus not being in steady state.

Some species have local pointers for which the value can go from zero to unity within the range of the calculation. These species are treated wrongly if the local pointer of a certain point in space is the basis of the reduction. Thus, the local pointers are integrated over the computational domain *L*. The *third step* involves an integration over the space to find the overall value on the influence on the time scale for each species. These integrated pointers are weighted with the species' mole fraction. This is done so that species at high concentrations are not set to steady state, since the resulting errors in calculating their concentrations as steady state species could affect the results too much.

Massias et. al. (Massias et al., 1999b) propose three different integration procedures for capturing the steady state candidates in the most appropriate way. However, they take the form similar to

$$I^I = \frac{1}{L} \int\_0^L D\_i \frac{1}{X\_i} \frac{q\_i}{q\_{i,\text{max}}} d\mathbf{x},\tag{33}$$

where *qi* is the production rate of species *i***,** *qi***,max** is the maximum production rate over the computational range, and *Xi* is the mole fraction of the species. Caution is in order as both *Xi* and *qi* can take on the value zero (inert species), so that small terms *�***<sup>1</sup>** and *�***<sup>2</sup>** are added to the respective denominators so as to avoid numerical problems. The R species with the highest values for *I<sup>I</sup>* are selected as steady state species. Note that since for most species *qi***,max** is situated in the reaction zone, the largest contribution to the integrated pointer comes from the value in the reaction zone since *qi*/*qi***,max** there is unity.

The elementary reaction rates are integrated over the space in order to determine the *R* fastest elementary reaction rates, which can then be eliminated from the system. The fastest reactions that consumes a steady state species, as identified in the previous step, are selected as to be the "fast reaction". For each steady state species, the fastest reaction consuming the species is

al., 2009). However, the method has been adopted into automatic reduction and optimization tools by e.g. Pepiot-Desjardins and Pitsch (Pepiot-Desjardins & Pitsch, 2008) and Shekar et al.

Model Reduction Techniques for Chemical Mechanisms 99

Since a chemical system is strongly non-linear, carrying out a lifetime analysis requires that the system is linearized around a starting point **Y0,** which results in the following equation,

where the limit at which **Y** − **Y0** → **0** and **J** is the Jacobian matrix with respect to the source terms in *ω*. The Jacobian matrix holds information regarding the rate of change in the source terms of the species when a change in species concentrations occur. The error introduced by

*dt* (**<sup>Y</sup>** <sup>−</sup> **Y0**) = *<sup>ω</sup>***<sup>0</sup>** <sup>+</sup> **<sup>J</sup>**(**<sup>Y</sup>** <sup>−</sup> **Y0**) (35)

**dt.** (36)

*<sup>A</sup>***,***iϑi***,** (38)

*Di DN***<sup>2</sup>**

**.** (39)

**,** (37)

corresponding to Equation (1) where the spatial operator is neglected:

for a species, results in a small error in the calculated concentration.

*<sup>τ</sup><sup>i</sup>* <sup>=</sup> **1/** *∂ω<sup>i</sup> ∂ci*

*<sup>ϑ</sup><sup>i</sup>* <sup>=</sup> *<sup>τ</sup>DW* 

 *τDW i*

*i*

**<sup>2</sup>** <sup>+</sup> (*τ<sup>F</sup>* )

**2**

*d*

the steady state approximation to a species **Δ***Yi* is calculated as (Tomlin et al., 1997)

**<sup>Δ</sup>***Yi* <sup>=</sup> **<sup>1</sup>** *Jii* **d***Yi*

The fact that the dimensions of *Jii* is 1/t[sec] means that the inverse of the Jacobian elements, **1/***Jii***,** can be interpreted as the characteristic timescale of the species in question. From Equation (36) it can be seen that a short lifetime, i.e. a small **1/***Jii***,** or a slow rate of change

In line with the argumentation above, and expanding the Jacobi elements accordingly, the

<sup>=</sup> *ci* ∑*NR k*=**1**(*ν*�

where *ω<sup>i</sup>* represents the species source term in terms of concentrations, *ci* is the species concentration, *νik* is the stoichiometric coefficient, the prime denotes the reactant values, the double prime denotes the product values, and *rk* is the reaction rate at which *k* is the Arrhenius reaction coefficient. The chemical lifetime can be understood in these terms as being a measure of how fast a particular species is consumed after being produced. Hence the species with the shortest lifetimes can be selected as steady state species alone. However, species with long lifetimes can still be insensitive to the desired result and can thus be approximated by steady state. On the other hand, species with short lifetimes can still be sensitive and should not be set to steady state. In order to capture these species a combined lifetime and species sensitivity measure termed the *level of importance,* LOI*,* is defined:

(**LOI**)*<sup>i</sup>* = *S<sup>S</sup>*

*i* towards some parameter of interest, such as for example the burning velocity *v* or the temperature *T*. *ϑ<sup>i</sup>* can be chemical lifetime itself, *τ*, or a wegithed function of the chemical

*<sup>A</sup>***,***<sup>i</sup>* is the relative species sensitivity, as defined in the previous section, of species

where *τDW*

*<sup>i</sup>* = *τ<sup>i</sup>*

*ik* − *ν*�� *ik*)*ν*� *ikr*� *k*

*dt* <sup>=</sup> *<sup>ω</sup>*(*Y*) =<sup>⇒</sup>

*d***Y**

chemical lifetime can be expressed as

where *S<sup>S</sup>*

lifetime:

(Shekar et al, 2011).

found through use of the integral (Massias et al., 1999b):

$$H\_k^l = \frac{1}{L} \int\_0^L r\_{k\prime} \tag{34}$$

where *rk* is the reaction rate of the *k th* reaction. The remaining reactions are considered to be the "slow reactions", their reaction rates being retained in the mechanism. This procedure is carried out in order to optimize the computations when a reduced mechanism is applied. This step is not needed for accuracy.

Finally, the rates of the global reactions and their stoichiometric values are determined. The resulting global reactions consist of the major species found in the previous steps, their rates being determined by a linear combination of the rates of the "slow reactions", which depend both on the steady state species and in the non-steady state species.

Obtaining the solutions to the algebraic equations can still require a significant amount of computational time, and the system can still suffer from stiffness problems. If the result obtained in using the reduced mechanisms is found to be discrepant with the results of the detailed mechanism, a truncation is performed. An importance analysis of the reaction rates then can lead to some reaction rates being omitted. This can be understood as being similar to a reaction flow and reaction sensitivity analysis, where for each species the participation rate, *<sup>P</sup>***,** of each elementary reaction and its importance rate, *<sup>I</sup>***,** are ordered, the corresponding reaction rates for the elementary steps that produce negligible values of both *<sup>P</sup>* and *<sup>I</sup>* being truncated from the steady state relations. In the work of Massias et al. (Massias et al., 1999a) the number of species that needs to be calculated in greater detail after the CSP analysis is said to be small, often only two or three truncations needing to be carried out.

The advantage of CSP calculations in a simple eigenvalue analysis is that it provides information about which species and reactions are associated with the fastest modes. However, the method does not always represent a computationally efficient technique for repeated reduction, since the time saved by eliminating the slowest modes may be outweighed by the time required for recomputing the basis for each time step (Tomlin et al., 1997). The terminology employed in the CSP method to some extent hides the chemical information if this is not analyzed in detail. Skevis et al. (Skevis et al., 2002) provides an excellent presentation of the physical and chemical meaning of the CSP data. In this paper there are tables that show the contribution of the major elementary reactions to the CSP reactions, the slowest reactions moving the trajectories of the physical processes along the manifold created by the fast reactions. In the work, diffusion and convection processes were also included, demonstrating the CSP methodology being as much a tool for characterization of the combustion process as a tool for reduction of chemical mechanisms.

#### **3.5 Level of importance**

The success of the reduced mechanism depends on the reliability of the selection procedure in selecting steady state species. Whereas the CSP method selects species that take part in the fastest reactions, which is done by solving an eigenvalue problem, the level of importance (LOI) method concerns the individual chemical lifetime of a species - or a *function* of its lifetime - as the selection parameter. The line of argument used is that some species are in steady state as a result of that the reactions consuming the species are very much faster than those producing them. The species, when formed, are thus very short lived and are low in concentration. This approach has been developed by Løvås et al. (Løvås et al., 2000)-(Løvås et 20 Will-be-set-by-IN-TECH

be the "slow reactions", their reaction rates being retained in the mechanism. This procedure is carried out in order to optimize the computations when a reduced mechanism is applied.

Finally, the rates of the global reactions and their stoichiometric values are determined. The resulting global reactions consist of the major species found in the previous steps, their rates being determined by a linear combination of the rates of the "slow reactions", which depend

Obtaining the solutions to the algebraic equations can still require a significant amount of computational time, and the system can still suffer from stiffness problems. If the result obtained in using the reduced mechanisms is found to be discrepant with the results of the detailed mechanism, a truncation is performed. An importance analysis of the reaction rates then can lead to some reaction rates being omitted. This can be understood as being similar to a reaction flow and reaction sensitivity analysis, where for each species the participation rate, *<sup>P</sup>***,** of each elementary reaction and its importance rate, *<sup>I</sup>***,** are ordered, the corresponding reaction rates for the elementary steps that produce negligible values of both *<sup>P</sup>* and *<sup>I</sup>* being truncated from the steady state relations. In the work of Massias et al. (Massias et al., 1999a) the number of species that needs to be calculated in greater detail after the CSP analysis is said

The advantage of CSP calculations in a simple eigenvalue analysis is that it provides information about which species and reactions are associated with the fastest modes. However, the method does not always represent a computationally efficient technique for repeated reduction, since the time saved by eliminating the slowest modes may be outweighed by the time required for recomputing the basis for each time step (Tomlin et al., 1997). The terminology employed in the CSP method to some extent hides the chemical information if this is not analyzed in detail. Skevis et al. (Skevis et al., 2002) provides an excellent presentation of the physical and chemical meaning of the CSP data. In this paper there are tables that show the contribution of the major elementary reactions to the CSP reactions, the slowest reactions moving the trajectories of the physical processes along the manifold created by the fast reactions. In the work, diffusion and convection processes were also included, demonstrating the CSP methodology being as much a tool for characterization

The success of the reduced mechanism depends on the reliability of the selection procedure in selecting steady state species. Whereas the CSP method selects species that take part in the fastest reactions, which is done by solving an eigenvalue problem, the level of importance (LOI) method concerns the individual chemical lifetime of a species - or a *function* of its lifetime - as the selection parameter. The line of argument used is that some species are in steady state as a result of that the reactions consuming the species are very much faster than those producing them. The species, when formed, are thus very short lived and are low in concentration. This approach has been developed by Løvås et al. (Løvås et al., 2000)-(Løvås et

*rk***,** (34)

*th* reaction. The remaining reactions are considered to

*Hi <sup>k</sup>* <sup>=</sup> **<sup>1</sup>** *L L* **0**

both on the steady state species and in the non-steady state species.

to be small, often only two or three truncations needing to be carried out.

of the combustion process as a tool for reduction of chemical mechanisms.

found through use of the integral (Massias et al., 1999b):

where *rk* is the reaction rate of the *k*

This step is not needed for accuracy.

**3.5 Level of importance**

al., 2009). However, the method has been adopted into automatic reduction and optimization tools by e.g. Pepiot-Desjardins and Pitsch (Pepiot-Desjardins & Pitsch, 2008) and Shekar et al. (Shekar et al, 2011).

Since a chemical system is strongly non-linear, carrying out a lifetime analysis requires that the system is linearized around a starting point **Y0,** which results in the following equation, corresponding to Equation (1) where the spatial operator is neglected:

$$\frac{d\mathbf{Y}}{dt} = \omega(\mathbf{Y}) \Longrightarrow \frac{d}{dt}(\mathbf{Y} - \mathbf{Y\_0}) = \omega\_0 + \mathbf{J}(\mathbf{Y} - \mathbf{Y\_0}) \tag{35}$$

where the limit at which **Y** − **Y0** → **0** and **J** is the Jacobian matrix with respect to the source terms in *ω*. The Jacobian matrix holds information regarding the rate of change in the source terms of the species when a change in species concentrations occur. The error introduced by the steady state approximation to a species **Δ***Yi* is calculated as (Tomlin et al., 1997)

$$
\Delta \mathbf{Y}\_{l} = \frac{\mathbf{1}}{J\_{li}} \frac{\mathbf{d} \mathbf{Y}\_{l}}{\mathbf{d} \mathbf{t}} \tag{36}
$$

The fact that the dimensions of *Jii* is 1/t[sec] means that the inverse of the Jacobian elements, **1/***Jii***,** can be interpreted as the characteristic timescale of the species in question. From Equation (36) it can be seen that a short lifetime, i.e. a small **1/***Jii***,** or a slow rate of change for a species, results in a small error in the calculated concentration.

In line with the argumentation above, and expanding the Jacobi elements accordingly, the chemical lifetime can be expressed as

$$\pi\_i = \mathbb{1}' \frac{\partial \omega\_i}{\partial c\_i} = \frac{c\_i}{\sum\_{k=1}^{N\_R} (v\_{ik}' - v\_{ik}'') v\_{ik}' r\_k'} \,\tag{37}$$

where *ω<sup>i</sup>* represents the species source term in terms of concentrations, *ci* is the species concentration, *νik* is the stoichiometric coefficient, the prime denotes the reactant values, the double prime denotes the product values, and *rk* is the reaction rate at which *k* is the Arrhenius reaction coefficient. The chemical lifetime can be understood in these terms as being a measure of how fast a particular species is consumed after being produced. Hence the species with the shortest lifetimes can be selected as steady state species alone. However, species with long lifetimes can still be insensitive to the desired result and can thus be approximated by steady state. On the other hand, species with short lifetimes can still be sensitive and should not be set to steady state. In order to capture these species a combined lifetime and species sensitivity measure termed the *level of importance,* LOI*,* is defined:

$$(\mathsf{LOL})\_i = \mathbb{S}\_{A,i}^{\mathsf{S}} \mathfrak{d}\_{i\prime} \tag{38}$$

where *S<sup>S</sup> <sup>A</sup>***,***<sup>i</sup>* is the relative species sensitivity, as defined in the previous section, of species *i* towards some parameter of interest, such as for example the burning velocity *v* or the temperature *T*. *ϑ<sup>i</sup>* can be chemical lifetime itself, *τ*, or a wegithed function of the chemical lifetime:

$$\mathfrak{d}\_{i} = \frac{\tau\_{i}^{DW}}{\sqrt{\left(\tau\_{i}^{DW}\right)^{2} + \left(\tau\_{F}\right)^{2}}} \qquad \text{where} \qquad \tau\_{i}^{DW} = \tau\_{i} \frac{D\_{i}}{D\_{N\_{2}}}.\tag{39}$$

contained in the LOI has clearly shown to counteract the high weighted lifetimes found at low temperatures (plot b). The weighted lifetime measure retains the species O, OH and CH3 as active species, H being set to steady state with a maximum lifetime in the order of 10−**2**. In employing the LOI measure, all the species except H are set to steady state. The high LOI for hydrogen is to a large extent caused by weighting the lifetime to the diffusion. Although H has a relatively high diffusion coefficient as compared with the other species, is nevertheless

Model Reduction Techniques for Chemical Mechanisms 101

Fig. 6. The evolution of species' LOI as a function of the residence time, shown for certain

For comparison, Figure 6 shows the evolution of the LOI for an ignition process in a reactor sequence as function of the residence time for a given equivalence ratio in the PSR (Løvås et al., 2001). As the residence time increases, species such as HO**2**, H**2**O**<sup>2</sup>** and H show a significantly high LOI and are thus retained the mechanism. The effect of the low equivalence ratio is also evident in the dominant role of O. Investigation of the evolution of the LOI as a function of the residence time becomes important when reduced mechanisms are applied in CFD calculations. Many of the CFD simulations model each computational cells as a small perfectly stirred reactor. Thus, the validity of the reduced mechanism needs to be considered over the range set for the CFD calculations. The time scale for the turbulent mixing needs to be included since it affects the chosen residence time for the specific computational cells. Although in the unburned regions, the residence time is much longer than the turbulent mixing times, in the reaction zone the turbulent mixing time is dominant. Thus, varying residence times need to be accounted for when reduced mechanisms are developed for this

By thorough investigation the optimum conditions for determining the level of importance can be found. For a simple methane flame, the LOI has been calculated from (1) the maximum

selected species at a given equivalence ratio in PSR, *φ* = **0.6.** (Løvås et al., 2001)

an important species to retain in the mechanism.

purpose.

The lifetime is here weighted to both the diffusion time in a manner similar to which is described for the manual reduction procedure by Equation (19), and the flame time, *τ<sup>F</sup>* suitable for premixed flames. It can also be weighted to the scalar dissipation rate, *a* , in diffusion flames.

Fig. 5. Selection parameter profiles over the flame zone for a set of species (H, O, OH and CH**3**): (a) element mass fraction as a function of the dimensionless flame coordinate x\*, (b) LOI as a function of the dimensionless flame coordinate x\*, and (c) weighted lifetime as a function of the dimensionless flame coordinate x\*. (Løvås et al., 2000)

In Figure (5) the profiles of the selection parameters for various important radicals (H, O, OH and CH**3**) in a stoichiometric methane flame have been plotted as functions of the dimensionless flame coordinate x∗ (Løvås et al., 2000). The upper plot shows the element mass fractions as a function of x\*. After the reaction zone, the species all have high element mass fractions. The threshold limit is most commonly set to 1, leaving only O slightly under the threshold. In the two lower plots, b and c, in Figure (5), the effect of including the sensitivity in the selection procedure is evident. Since during and after the flame zone, the weighted lifetimes are generally short as compared with the cold unburned mixture in which the reactions have not yet started (plot c). An accumulated weighted lifetime over the entire flame zone favors the high values found in the pre-heat zone. However, the sensitivity 22 Will-be-set-by-IN-TECH

The lifetime is here weighted to both the diffusion time in a manner similar to which is described for the manual reduction procedure by Equation (19), and the flame time, *τ<sup>F</sup>* suitable for premixed flames. It can also be weighted to the scalar dissipation rate, *a* , in

Fig. 5. Selection parameter profiles over the flame zone for a set of species (H, O, OH and CH**3**): (a) element mass fraction as a function of the dimensionless flame coordinate x\*, (b) LOI as a function of the dimensionless flame coordinate x\*, and (c) weighted lifetime as a

In Figure (5) the profiles of the selection parameters for various important radicals (H, O, OH and CH**3**) in a stoichiometric methane flame have been plotted as functions of the dimensionless flame coordinate x∗ (Løvås et al., 2000). The upper plot shows the element mass fractions as a function of x\*. After the reaction zone, the species all have high element mass fractions. The threshold limit is most commonly set to 1, leaving only O slightly under the threshold. In the two lower plots, b and c, in Figure (5), the effect of including the sensitivity in the selection procedure is evident. Since during and after the flame zone, the weighted lifetimes are generally short as compared with the cold unburned mixture in which the reactions have not yet started (plot c). An accumulated weighted lifetime over the entire flame zone favors the high values found in the pre-heat zone. However, the sensitivity

function of the dimensionless flame coordinate x\*. (Løvås et al., 2000)

diffusion flames.

contained in the LOI has clearly shown to counteract the high weighted lifetimes found at low temperatures (plot b). The weighted lifetime measure retains the species O, OH and CH3 as active species, H being set to steady state with a maximum lifetime in the order of 10−**2**. In employing the LOI measure, all the species except H are set to steady state. The high LOI for hydrogen is to a large extent caused by weighting the lifetime to the diffusion. Although H has a relatively high diffusion coefficient as compared with the other species, is nevertheless an important species to retain in the mechanism.

Fig. 6. The evolution of species' LOI as a function of the residence time, shown for certain selected species at a given equivalence ratio in PSR, *φ* = **0.6.** (Løvås et al., 2001)

For comparison, Figure 6 shows the evolution of the LOI for an ignition process in a reactor sequence as function of the residence time for a given equivalence ratio in the PSR (Løvås et al., 2001). As the residence time increases, species such as HO**2**, H**2**O**<sup>2</sup>** and H show a significantly high LOI and are thus retained the mechanism. The effect of the low equivalence ratio is also evident in the dominant role of O. Investigation of the evolution of the LOI as a function of the residence time becomes important when reduced mechanisms are applied in CFD calculations. Many of the CFD simulations model each computational cells as a small perfectly stirred reactor. Thus, the validity of the reduced mechanism needs to be considered over the range set for the CFD calculations. The time scale for the turbulent mixing needs to be included since it affects the chosen residence time for the specific computational cells. Although in the unburned regions, the residence time is much longer than the turbulent mixing times, in the reaction zone the turbulent mixing time is dominant. Thus, varying residence times need to be accounted for when reduced mechanisms are developed for this purpose.

By thorough investigation the optimum conditions for determining the level of importance can be found. For a simple methane flame, the LOI has been calculated from (1) the maximum

Fig. 7. Results of employing reduced mechanisms for simulating a methane/air counterflow diffusion flame with *χ* = **0.54** at 1Bar. The detailed mechanism containing 46 species is reduced to one involving 12 species yet still able to reproduce the features of the flame.

Model Reduction Techniques for Chemical Mechanisms 103

For the simulation of emissions from a staged combustor, the reaction mechanism is extended to include 69 species that interact in 770 different reactions. In order to model NO emissions from such a device, the reactor is fueled with a mixture of ethylene (C**2**H**4**) doped with monomethylamine (CH**3**NH**2**) (Kantak et al., 1997); (Klaus et al., 1997). As stated in the previous section, the steady state species selected varies depending upon whether the lifetime alone or an extended lifetime which includes the species sensitivity is used. A reduction of the mechanism through fully automatic selection in terms of the species' LOI towards NO in the PSR has been carried out and presented by Løvås et al. (Løvås et al., 2001). Figure (8) shows the calculated NO concentrations after the PSR as function of the equivalence ratio. The results were obtained using a reduced mechanism containing as little as 31 of the detailed mechanism, the achieved accuracy of the reduced scheme being within an acceptable level of accuracy as compared with the detailed reaction mechanism. Three of the mechanisms are based on a selection according to chemical lifetime only, whereas the mechanism reduced the

most is obtained with the LOI towards NO concentration.

**4.2 Reactor sequence**

accumulated values for the lifetime and the sensitivity, (2) the LOI in the reaction zone and (3) the LOI at the point where the species have a maximum mass fraction. The overall conclusion to be drawn is that the maximum values obtained for the lifetime or for LOI accumulated over the entire range investigated are reliable to some extent, the range involved being over the equivalence ratios, the mixture fractions for the flame calculations, or the time range for the ignition process. However, for the flame calculations a more accurate selection of steady state species is obtained if the values in the reaction zone or the values at maximum species mass fraction are used (Løvås et al., 2009). In ignition situations, the integrated LOI up to the point of ignition, defined as the point of maximum temperature gradient, is found to represent the most appropriate selection parameter ( Løvås et al., 2002c). This result is of interest in comparing the LOI with Equation (33), which defines the selection criteria used in the CSP method. The integrated pointers consist of the CSP pointer, a factor weighting it by the species mass fraction and a factor including the production rate of the species. The production rate factor, *qi***/***qi***,max**, where *qi* is the production rate and *qi***,max** is the maximum production rate over the integration range, is unity for most of the species in the reaction zone. Thus, the largest contribution to the integrated CSP pointer comes from that zone, corresponding to the finding that the LOI in the reaction zone is the most appropriate selection parameter.

#### **4. Results**

In the present section, the results from applications of reduced mechanisms mainly based on skeletal mechanisms and LOI analysis will be presented. For whatever system is chosen the degree of reduction is variable and user defined. It has been found that the strongly reduced mechanisms can be developed for simple premixed flames. Simulating an ignition process however requires a larger set of species in order to predict ignition timing and heat release in an adequate way. If emission rates are of primary concern, however, ignition timing is not necessarily an important feature to investigate. In this section results of developing reduced mechanisms for (i) a diffusion flame configuration, (ii) a reactor sequence for modeling NO emissions and (iii) an ignition process in a SI engine will be presented.

#### **4.1 Diffusion flame configuration**

In the case of a counterflow diffusion flame, the detailed mechanism in the present example containing 46 species was successfully reduced to 12 species, completely automatically through the LOI and placing restrictions on the element mass fraction, as described above. The chemical kinetic model involves a detailed C**1**-C**2**- mechanism (Bowman et al., 2004). The chemistry of H**2**-O**2**-CO-CO**<sup>2</sup>** combustion stems from Yetter et al.(Yetter et al., 1991). In the present case, the LOI was based on the values in the reaction zone. The results of applying reduced mechanisms with varying degrees of reduction to the simulation of a counterflow flame in a mixture fraction space of constant scalar dissipation rate and unity Lewis number is shown in Figure (7). According to the upper plot in the figure, the temperature profile over the mixture fraction range is very accurately reproduced. Most species profiles were in close agreement with the detailed mechanism, but some of the C**2**-species showed a discrepancy as a result of all but a few of the C**2**-species being set to steady state. The errors introduced into the computations of these species' concentrations have a knock-on effect on the species that are retained in the mechanism. This is shown in the lower plot in Figure (7). As can be seen for both the temperature profile and the species concentration profiles, the reduced mechanism perform very well as compared with the results from the detailed mechanism.

24 Will-be-set-by-IN-TECH

accumulated values for the lifetime and the sensitivity, (2) the LOI in the reaction zone and (3) the LOI at the point where the species have a maximum mass fraction. The overall conclusion to be drawn is that the maximum values obtained for the lifetime or for LOI accumulated over the entire range investigated are reliable to some extent, the range involved being over the equivalence ratios, the mixture fractions for the flame calculations, or the time range for the ignition process. However, for the flame calculations a more accurate selection of steady state species is obtained if the values in the reaction zone or the values at maximum species mass fraction are used (Løvås et al., 2009). In ignition situations, the integrated LOI up to the point of ignition, defined as the point of maximum temperature gradient, is found to represent the most appropriate selection parameter ( Løvås et al., 2002c). This result is of interest in comparing the LOI with Equation (33), which defines the selection criteria used in the CSP method. The integrated pointers consist of the CSP pointer, a factor weighting it by the species mass fraction and a factor including the production rate of the species. The production rate factor, *qi***/***qi***,max**, where *qi* is the production rate and *qi***,max** is the maximum production rate over the integration range, is unity for most of the species in the reaction zone. Thus, the largest contribution to the integrated CSP pointer comes from that zone, corresponding to the

finding that the LOI in the reaction zone is the most appropriate selection parameter.

emissions and (iii) an ignition process in a SI engine will be presented.

perform very well as compared with the results from the detailed mechanism.

In the present section, the results from applications of reduced mechanisms mainly based on skeletal mechanisms and LOI analysis will be presented. For whatever system is chosen the degree of reduction is variable and user defined. It has been found that the strongly reduced mechanisms can be developed for simple premixed flames. Simulating an ignition process however requires a larger set of species in order to predict ignition timing and heat release in an adequate way. If emission rates are of primary concern, however, ignition timing is not necessarily an important feature to investigate. In this section results of developing reduced mechanisms for (i) a diffusion flame configuration, (ii) a reactor sequence for modeling NO

In the case of a counterflow diffusion flame, the detailed mechanism in the present example containing 46 species was successfully reduced to 12 species, completely automatically through the LOI and placing restrictions on the element mass fraction, as described above. The chemical kinetic model involves a detailed C**1**-C**2**- mechanism (Bowman et al., 2004). The chemistry of H**2**-O**2**-CO-CO**<sup>2</sup>** combustion stems from Yetter et al.(Yetter et al., 1991). In the present case, the LOI was based on the values in the reaction zone. The results of applying reduced mechanisms with varying degrees of reduction to the simulation of a counterflow flame in a mixture fraction space of constant scalar dissipation rate and unity Lewis number is shown in Figure (7). According to the upper plot in the figure, the temperature profile over the mixture fraction range is very accurately reproduced. Most species profiles were in close agreement with the detailed mechanism, but some of the C**2**-species showed a discrepancy as a result of all but a few of the C**2**-species being set to steady state. The errors introduced into the computations of these species' concentrations have a knock-on effect on the species that are retained in the mechanism. This is shown in the lower plot in Figure (7). As can be seen for both the temperature profile and the species concentration profiles, the reduced mechanism

**4. Results**

**4.1 Diffusion flame configuration**

Fig. 7. Results of employing reduced mechanisms for simulating a methane/air counterflow diffusion flame with *χ* = **0.54** at 1Bar. The detailed mechanism containing 46 species is reduced to one involving 12 species yet still able to reproduce the features of the flame.

#### **4.2 Reactor sequence**

For the simulation of emissions from a staged combustor, the reaction mechanism is extended to include 69 species that interact in 770 different reactions. In order to model NO emissions from such a device, the reactor is fueled with a mixture of ethylene (C**2**H**4**) doped with monomethylamine (CH**3**NH**2**) (Kantak et al., 1997); (Klaus et al., 1997). As stated in the previous section, the steady state species selected varies depending upon whether the lifetime alone or an extended lifetime which includes the species sensitivity is used. A reduction of the mechanism through fully automatic selection in terms of the species' LOI towards NO in the PSR has been carried out and presented by Løvås et al. (Løvås et al., 2001). Figure (8) shows the calculated NO concentrations after the PSR as function of the equivalence ratio. The results were obtained using a reduced mechanism containing as little as 31 of the detailed mechanism, the achieved accuracy of the reduced scheme being within an acceptable level of accuracy as compared with the detailed reaction mechanism. Three of the mechanisms are based on a selection according to chemical lifetime only, whereas the mechanism reduced the most is obtained with the LOI towards NO concentration.

Fig. 9. Calculated temperatures and mole faction profiles for the end-gas of an SI-engine. Top left: temperature profile, top right: HO**<sup>2</sup>** molefraction representing a radical profile, bottom: fuel profiles, the iso-octane and the n-heptane. Different reduced mechanisms are compared

Model Reduction Techniques for Chemical Mechanisms 105

and CH**<sup>3</sup>** that participate in chain reactions, is important since the formation of HO**2,** for example, is crucial for the formation of an initial radical pool (Westbrook et al., 1991). The plot at the top left in Figure (9) shows the profiles of HO**<sup>2</sup>** that result from employing the skeletal mechanisms as compared with the results from the detailed mechanism for inlet temperatures of 800K. The two lower plots show fuel decomposition as a function of CAD at the same inlet temperature. The plots reveal a close agreement between the reduced and the detailed mechanisms, showing clearly the positive effects on the ignition timing in the

The most obvious differences between the various reduced mechanisms in prediction of ignition delay times. Too early an onset of ignition results in both the decomposition of the fuel and the production of the radicals occurring earlier. However, since the errors in the ignition timing are within 1-2 CAD, the mechanisms are seen as performing well within the

In contrast to the previously discussed reduction schemes, in which chemical species only are selected if they are in steady state throughout the process, the *adaptive* method allows species to be selected at each operating point or domain separately, generating adaptive chemical kinetics. This is a dynamic reduction procedure that can be employed to ignition systems that are changing over time and to flame systems that change over the flame coordinate. As discussed for the LOI, in some cases the maximum accumulated value over the computational

with the detailed and skeletal mechanisms (Soyhan et al., 2000)

end-gas achieved by decreasing the inlet temperature.

**5. Application of adaptive kinetics**

specified range.

Fig. 8. The NO molefraction as a function of the equivalence ratio in the PSR. Results were obtained for reduced mechanism with increasing degrees of reduction. (Løvås et al., 2001)

#### **4.3 SI engine**

The degree of reduction depends strongly on the combustion system and on the degree of complexity involved. Ignition processes often require a more detailed mechanism than simple flame configurations. The on-set timing of the ignition is sensitive to the accuracy of species concentrations because of their dependency on the reaction rates.

In Figure (9), temperature and species profile plots obtained in an investigation of autoignition in the end-gas of an SI-engine, known as knock, as presented by Soyhan et al. (Soyhan et al., 2000), are shown. The calculations are obtained in employing a two-zone model (burnt and unburnt zones), the detailed mechanism for iso-octane and n-heptane mixtures being compiled from Chevalier (C**1**-C**4**) (Chavelier, 1993) and Muller (C**5**-C**8**) (Muller et al., 1992), consisting of 75 species and 510 reactions. In the work displayed in the figure, this mechanism was reduced with reaction flow and sensitivity analysis together with LOI. The results obtained with different reduced mechanisms are displayed in the figure, in which they are compared with the results of the detailed mechanism. The skeletal mechanism, obtained on the basis of the reaction flow and sensitivity analysis (denoted "skel" in the figure), was developed prior to the mechanisms from a reduction based on lifetime analysis (denoted "red" in the figure).

In the temperature profile for the end-gas as shown in Figure (9) one can note that at an inlet temperature of 1200K the end gas ignites at around -24 CAD. This is far ahead of the onset of the ignition by the spark plug, and even further ahead of the propagating flame front reaching the wall. Under such conditions, knocking occurs. The presented work concerns the performance of various reduced mechanisms and their performance as compared with the detailed mechanism. The comparison shows that over a large range of physical conditions the profiles of the chemical species and the temperature are the same for the various mechanism, except for changes in the ignition delay times. Thus, the basic characteristics of the detailed mechanism are preserved. The accuracy of the calculation of radicals such as the OH, HO**<sup>2</sup>**

26 Will-be-set-by-IN-TECH

Fig. 8. The NO molefraction as a function of the equivalence ratio in the PSR. Results were obtained for reduced mechanism with increasing degrees of reduction. (Løvås et al., 2001)

The degree of reduction depends strongly on the combustion system and on the degree of complexity involved. Ignition processes often require a more detailed mechanism than simple flame configurations. The on-set timing of the ignition is sensitive to the accuracy of species

In Figure (9), temperature and species profile plots obtained in an investigation of autoignition in the end-gas of an SI-engine, known as knock, as presented by Soyhan et al. (Soyhan et al., 2000), are shown. The calculations are obtained in employing a two-zone model (burnt and unburnt zones), the detailed mechanism for iso-octane and n-heptane mixtures being compiled from Chevalier (C**1**-C**4**) (Chavelier, 1993) and Muller (C**5**-C**8**) (Muller et al., 1992), consisting of 75 species and 510 reactions. In the work displayed in the figure, this mechanism was reduced with reaction flow and sensitivity analysis together with LOI. The results obtained with different reduced mechanisms are displayed in the figure, in which they are compared with the results of the detailed mechanism. The skeletal mechanism, obtained on the basis of the reaction flow and sensitivity analysis (denoted "skel" in the figure), was developed prior to the mechanisms from a reduction based on lifetime analysis (denoted "red"

In the temperature profile for the end-gas as shown in Figure (9) one can note that at an inlet temperature of 1200K the end gas ignites at around -24 CAD. This is far ahead of the onset of the ignition by the spark plug, and even further ahead of the propagating flame front reaching the wall. Under such conditions, knocking occurs. The presented work concerns the performance of various reduced mechanisms and their performance as compared with the detailed mechanism. The comparison shows that over a large range of physical conditions the profiles of the chemical species and the temperature are the same for the various mechanism, except for changes in the ignition delay times. Thus, the basic characteristics of the detailed mechanism are preserved. The accuracy of the calculation of radicals such as the OH, HO**<sup>2</sup>**

concentrations because of their dependency on the reaction rates.

**4.3 SI engine**

in the figure).

Fig. 9. Calculated temperatures and mole faction profiles for the end-gas of an SI-engine. Top left: temperature profile, top right: HO**<sup>2</sup>** molefraction representing a radical profile, bottom: fuel profiles, the iso-octane and the n-heptane. Different reduced mechanisms are compared with the detailed and skeletal mechanisms (Soyhan et al., 2000)

and CH**<sup>3</sup>** that participate in chain reactions, is important since the formation of HO**2,** for example, is crucial for the formation of an initial radical pool (Westbrook et al., 1991). The plot at the top left in Figure (9) shows the profiles of HO**<sup>2</sup>** that result from employing the skeletal mechanisms as compared with the results from the detailed mechanism for inlet temperatures of 800K. The two lower plots show fuel decomposition as a function of CAD at the same inlet temperature. The plots reveal a close agreement between the reduced and the detailed mechanisms, showing clearly the positive effects on the ignition timing in the end-gas achieved by decreasing the inlet temperature.

The most obvious differences between the various reduced mechanisms in prediction of ignition delay times. Too early an onset of ignition results in both the decomposition of the fuel and the production of the radicals occurring earlier. However, since the errors in the ignition timing are within 1-2 CAD, the mechanisms are seen as performing well within the specified range.

#### **5. Application of adaptive kinetics**

In contrast to the previously discussed reduction schemes, in which chemical species only are selected if they are in steady state throughout the process, the *adaptive* method allows species to be selected at each operating point or domain separately, generating adaptive chemical kinetics. This is a dynamic reduction procedure that can be employed to ignition systems that are changing over time and to flame systems that change over the flame coordinate. As discussed for the LOI, in some cases the maximum accumulated value over the computational

saved by applying the reduced mechanism is lost. Figure (10) shows temperature profiles resulting from employing this method of adaptive kinetics with varying threshold limits in selecting steady state species, which result in a greater degree of reduction. The profiles are compared with the corresponding profiles for the detailed mechanism ( Løvås et al., 2002a). It is clear that as the level of reduction is increased, the reduced models reproduce the temperature profiles with increasing error. However, it seems to be a very clear cut-off

Model Reduction Techniques for Chemical Mechanisms 107

Fig. 10. Temperature profiles of a HCCI engine cycle, employing adaptive kinetics. The upper plot is a result of applying the pure chemical lifetime. The middle plot is a result from applying mass fraction weighted lifetime. The lower plot results from an initial temperature

For 2D and 3D dynamic simulations such on-line reduction are very computationally costly. A less CPU time intensive approach is to pre-define the combustion domains or zones in which a certain sub-set of the detailed mechanism is used. This is often used for diffusion flame calculations where the domains can be defined by the fuel rich and the fuel lean zones. Hence, schemes that find the smallest chemical sub-set locally in time or space have been proposed e.g. by Schwer et al. (Schwer et al., 2003)). Here each computational cell was assigned a certain sub-mechanism based on a set of physical criteria such as temperature, pressure, species concentrations etc. For highly turbulent flames where these criteria can change rapidly and steeply from cell to cell, this method can be demanding. A single criterion was therefore proposed by Løvås et al. (Løvås et al., 2010), where the sub-mechanism was chosen based on *mixture fraction* alone. This approach will act as example of implementation of adaptive

356K +/- 2K applying mass fraction weighted lifetime. ( Løvås et al., 2002a)

when the reduced adaptive schemes can not reproduce ignition at all.

range was employed, but the values in the reaction zones were found to be a better choice of steady state candidates. However, this means that species that are of importance in the reaction zone are kept throughout the computation, also in regions in which the species may be in steady state. In order to improve the efficiency of the reduction procedure, it is desirable to only consider true steady state species at each point in the calculations. For this reason, methods of reducing the mechanism on-line or adaptively have been developed.

There are two main issues in developing an adaptive reduction procedure. One is the choice of a useful selection criterion for species or reactions to be removed, one that is accurate in selecting the correct species but does not require any considerable amount of CPU time as the analysis is repeated in the course of the computations. For true on-line reduction the selection of steady state species needs to be performed for each operating point or domain, allowing for the fact that species may move in and out of steady state in accordance with their lifetime under the conditions in hand. The second issue is that of implementation into flow codes, the question of how to implement the possibility of a mechanism changing in both size and matter during the simulation.

#### **5.1 Selection criteria for adaptive kinetics**

The first issue concerning adaptive kinetics is that of the selection criteria. This is rather straight forward for homogeneous ignition scenarios employing adaptive chemistry as only the time evolution of the chemistry needs to be considered. An on-the-fly reduction scheme was proposed by Liang et al. (Liang et al., 2009) for a homogeneous ignition scenario (HCCI), where the basis of the reduction was a modified version of the DRG procedure described earlier. Their procedure removes the locally redundant species from the detailed treatment, and "freezes" their mass fractions in the continuation of the computation. This is performed for each time step through the computation. The DRG analysis is sufficiently efficient that performing this on-line does not outperform the reduction compared to employing the detailed mechanism throughout.

A different approach to a similar problem can be to introduce a separation between vectors for *all* the species, containing specific species data that is kept available in module routines, and dynamic vectors for the *active* species, those being treated in detail. This is the approach proposed by Løvås et al. ( Løvås et al., 2002a). The vector of the active species changes during the run according to an array of logicals that hold the information concerning the species' status as steady state. Initially the logicals are assigned their value based on an a priori LOI analysis. The species set to steady state are assigned the logical true such that the species concentrations are to be calculated using algebraic equations instead of the original differential equations. This implies that algebraic equations for *all* the species need to be available in the code for when/if the species is selected to be in steady state. From the list of active species, the source terms are calculated and used to complete the calculations at this point. The combustion process proceeds at each point and the physical conditions change accordingly. This suggests that a new analysis of the species' lifetimes should be performed to allow for modifications in the mechanism so as to account for these changes. The code assesses the new lifetimes based on the inverse of the Jacobian matrix already available; species that has a considerable change in chemical lifetime on the basis of the new conditions being added or removed accordingly. A buffer for numerical inaccuracy needs to be included so as to prevent species with lifetimes very close to the specified threshold limit from systematically going in and out of steady state without adding any accuracy of the computations. Redefining the mechanism at almost every operating point becomes tedious and can mean that the CPU time 28 Will-be-set-by-IN-TECH

range was employed, but the values in the reaction zones were found to be a better choice of steady state candidates. However, this means that species that are of importance in the reaction zone are kept throughout the computation, also in regions in which the species may be in steady state. In order to improve the efficiency of the reduction procedure, it is desirable to only consider true steady state species at each point in the calculations. For this reason,

There are two main issues in developing an adaptive reduction procedure. One is the choice of a useful selection criterion for species or reactions to be removed, one that is accurate in selecting the correct species but does not require any considerable amount of CPU time as the analysis is repeated in the course of the computations. For true on-line reduction the selection of steady state species needs to be performed for each operating point or domain, allowing for the fact that species may move in and out of steady state in accordance with their lifetime under the conditions in hand. The second issue is that of implementation into flow codes, the question of how to implement the possibility of a mechanism changing in both size and

The first issue concerning adaptive kinetics is that of the selection criteria. This is rather straight forward for homogeneous ignition scenarios employing adaptive chemistry as only the time evolution of the chemistry needs to be considered. An on-the-fly reduction scheme was proposed by Liang et al. (Liang et al., 2009) for a homogeneous ignition scenario (HCCI), where the basis of the reduction was a modified version of the DRG procedure described earlier. Their procedure removes the locally redundant species from the detailed treatment, and "freezes" their mass fractions in the continuation of the computation. This is performed for each time step through the computation. The DRG analysis is sufficiently efficient that performing this on-line does not outperform the reduction compared to employing the

A different approach to a similar problem can be to introduce a separation between vectors for *all* the species, containing specific species data that is kept available in module routines, and dynamic vectors for the *active* species, those being treated in detail. This is the approach proposed by Løvås et al. ( Løvås et al., 2002a). The vector of the active species changes during the run according to an array of logicals that hold the information concerning the species' status as steady state. Initially the logicals are assigned their value based on an a priori LOI analysis. The species set to steady state are assigned the logical true such that the species concentrations are to be calculated using algebraic equations instead of the original differential equations. This implies that algebraic equations for *all* the species need to be available in the code for when/if the species is selected to be in steady state. From the list of active species, the source terms are calculated and used to complete the calculations at this point. The combustion process proceeds at each point and the physical conditions change accordingly. This suggests that a new analysis of the species' lifetimes should be performed to allow for modifications in the mechanism so as to account for these changes. The code assesses the new lifetimes based on the inverse of the Jacobian matrix already available; species that has a considerable change in chemical lifetime on the basis of the new conditions being added or removed accordingly. A buffer for numerical inaccuracy needs to be included so as to prevent species with lifetimes very close to the specified threshold limit from systematically going in and out of steady state without adding any accuracy of the computations. Redefining the mechanism at almost every operating point becomes tedious and can mean that the CPU time

methods of reducing the mechanism on-line or adaptively have been developed.

matter during the simulation.

detailed mechanism throughout.

**5.1 Selection criteria for adaptive kinetics**

saved by applying the reduced mechanism is lost. Figure (10) shows temperature profiles resulting from employing this method of adaptive kinetics with varying threshold limits in selecting steady state species, which result in a greater degree of reduction. The profiles are compared with the corresponding profiles for the detailed mechanism ( Løvås et al., 2002a). It is clear that as the level of reduction is increased, the reduced models reproduce the temperature profiles with increasing error. However, it seems to be a very clear cut-off when the reduced adaptive schemes can not reproduce ignition at all.

Fig. 10. Temperature profiles of a HCCI engine cycle, employing adaptive kinetics. The upper plot is a result of applying the pure chemical lifetime. The middle plot is a result from applying mass fraction weighted lifetime. The lower plot results from an initial temperature 356K +/- 2K applying mass fraction weighted lifetime. ( Løvås et al., 2002a)

For 2D and 3D dynamic simulations such on-line reduction are very computationally costly. A less CPU time intensive approach is to pre-define the combustion domains or zones in which a certain sub-set of the detailed mechanism is used. This is often used for diffusion flame calculations where the domains can be defined by the fuel rich and the fuel lean zones. Hence, schemes that find the smallest chemical sub-set locally in time or space have been proposed e.g. by Schwer et al. (Schwer et al., 2003)). Here each computational cell was assigned a certain sub-mechanism based on a set of physical criteria such as temperature, pressure, species concentrations etc. For highly turbulent flames where these criteria can change rapidly and steeply from cell to cell, this method can be demanding. A single criterion was therefore proposed by Løvås et al. (Løvås et al., 2010), where the sub-mechanism was chosen based on *mixture fraction* alone. This approach will act as example of implementation of adaptive

Fig. 11. The LOI profiles for selected species through the flame in mixture fraction space Z.

Model Reduction Techniques for Chemical Mechanisms 109

Fig. 12. Instantaneous contour plots of H2 in a turbulent ethylene diffusion flame resulting

gives excellent correlation to the full treatment. Since these are instantaneous plots, the flame is not identical. However, the averaged values indicated by the stapled lines show clearly the close correlation. It was demonstrated that when adaptive chemistry is employed the number of species that need to be treated in detail is substantially reduced, only true steady state

from adaptively reduced scheme with 15 species (left) in 4 sub-domains and direct integration of the full mechanism with 75 species (right). Average iso-lines are included to

The vertical lines indicate domain limits. (Løvås et al., 2010)

indicate the statistical similarities. (Løvås et al., 2010)

species being selected at each operating point.

chemistry into turbulent combustion modeling and will be described in more detailed in the following sub section.

#### **5.2 Implementation of adaptive kinetics**

When simulation a highly turbulent diffusion flame details regarding both the flow field and the chemical interactions are important. In the present discussion a turbulent ethylene flame is simulated suing large eddy simulations (LES). It is not the scope here to discuss the implementation of turbulence models in CFD. However, it can be noted that LES describe the turbulent flow field by resolving the large scales, but employing a sub-grid model for the small unresolved scales. This is a rather CPU intensive approach, and with detailed chemistry the computations becomes cumbersome.

Løvås et al. (Løvås et al., 2010) therefore proposed to combine a hierarchy of methods to simplify the treatment of the chemistry in such large simulations of turbulent combustion. At first it was recognized that the turbulent flow needed a much finer grid than the chemical system. Therefore a coarser "super grid" for the chemistry was imposed onto the spatial CFD grid for which mixing and transport is treated. However, due to the turbulent nature of the flame it was not practical to let the super grid follow the physical 2D or 3D co-ordinates. Instead, the chemistry was solved in the 1D mixture fraction space with the mixture fraction ranging from 0 (pure oxidant) to 1 (pure fuel). The interaction between the turbulent mixing and chemical kinetics are often modeled employing either the flamelet approach (Peters, 1984) or the conditional moment closure (CMC) (Klimenko & Bilger, 1999) method, where the latter was the choice of Løvås et al.

The LES/CMC approach for turbulent reactive flow modeling has already been developed without reduction (Navarro-Martinez et al., 2005). Løvås et al. extended this approach to also include an adaptive chemistry treatment. At first, distinct domains had to be defined using a domain splitting method. In this case the domains were identified in mixture fraction space and the analysis was therefore conveniently limited to one dimension. However, more rigorous automatic domain splitting methods using clustering techniques can be employed (Blurock et al., 2003). Since the choice of selection parameter was the LOI, the mixture fraction space was divided into 4 domains in which the species' LOI was relatively unchanged towards each other. This is illustrated in Figure (11). In this case the cut off limit was chosen such that the number of species was constant throughout the domains. However, the species changed between domains according to the highest ranked species.

Once the domains and the active species within each domain was identified, the RCCE approach described in the previous sub chapter was employed to treat the chemical system. In the LES-CMC code the reduction procedure is implemented on the CMC grid. The adaptive procedure is as follows: the local mixture fraction from the LES-CMC selects the relevant LOI-domain and therefore determines which species are locally leading. The RCCE uses the LOI information to select which species to constrain for a user-given number of constraints and provides a reduced mechanism. Equation (25) is then solved using an iterative Newton solver while non-constrained species are obtained directly from Equation (26). Since concentrations of the non-constrained species are known, the transitions between domains are smooth.

Figure (12) shows the resulting flame simulation based on the LES-CMC LOI-RCCE approach as described above. Employing 4 spatial sub-domains, each with their optimally reduced model for the given conditions (fuel rich, reaction zone, post-reaction zone, and fuel lean) 30 Will-be-set-by-IN-TECH

chemistry into turbulent combustion modeling and will be described in more detailed in the

When simulation a highly turbulent diffusion flame details regarding both the flow field and the chemical interactions are important. In the present discussion a turbulent ethylene flame is simulated suing large eddy simulations (LES). It is not the scope here to discuss the implementation of turbulence models in CFD. However, it can be noted that LES describe the turbulent flow field by resolving the large scales, but employing a sub-grid model for the small unresolved scales. This is a rather CPU intensive approach, and with detailed chemistry

Løvås et al. (Løvås et al., 2010) therefore proposed to combine a hierarchy of methods to simplify the treatment of the chemistry in such large simulations of turbulent combustion. At first it was recognized that the turbulent flow needed a much finer grid than the chemical system. Therefore a coarser "super grid" for the chemistry was imposed onto the spatial CFD grid for which mixing and transport is treated. However, due to the turbulent nature of the flame it was not practical to let the super grid follow the physical 2D or 3D co-ordinates. Instead, the chemistry was solved in the 1D mixture fraction space with the mixture fraction ranging from 0 (pure oxidant) to 1 (pure fuel). The interaction between the turbulent mixing and chemical kinetics are often modeled employing either the flamelet approach (Peters, 1984) or the conditional moment closure (CMC) (Klimenko & Bilger, 1999) method, where the latter

The LES/CMC approach for turbulent reactive flow modeling has already been developed without reduction (Navarro-Martinez et al., 2005). Løvås et al. extended this approach to also include an adaptive chemistry treatment. At first, distinct domains had to be defined using a domain splitting method. In this case the domains were identified in mixture fraction space and the analysis was therefore conveniently limited to one dimension. However, more rigorous automatic domain splitting methods using clustering techniques can be employed (Blurock et al., 2003). Since the choice of selection parameter was the LOI, the mixture fraction space was divided into 4 domains in which the species' LOI was relatively unchanged towards each other. This is illustrated in Figure (11). In this case the cut off limit was chosen such that the number of species was constant throughout the domains. However, the species changed

Once the domains and the active species within each domain was identified, the RCCE approach described in the previous sub chapter was employed to treat the chemical system. In the LES-CMC code the reduction procedure is implemented on the CMC grid. The adaptive procedure is as follows: the local mixture fraction from the LES-CMC selects the relevant LOI-domain and therefore determines which species are locally leading. The RCCE uses the LOI information to select which species to constrain for a user-given number of constraints and provides a reduced mechanism. Equation (25) is then solved using an iterative Newton solver while non-constrained species are obtained directly from Equation (26). Since concentrations of the non-constrained species are known, the transitions between domains are

Figure (12) shows the resulting flame simulation based on the LES-CMC LOI-RCCE approach as described above. Employing 4 spatial sub-domains, each with their optimally reduced model for the given conditions (fuel rich, reaction zone, post-reaction zone, and fuel lean)

following sub section.

**5.2 Implementation of adaptive kinetics**

the computations becomes cumbersome.

was the choice of Løvås et al.

smooth.

between domains according to the highest ranked species.

Fig. 11. The LOI profiles for selected species through the flame in mixture fraction space Z. The vertical lines indicate domain limits. (Løvås et al., 2010)

Fig. 12. Instantaneous contour plots of H2 in a turbulent ethylene diffusion flame resulting from adaptively reduced scheme with 15 species (left) in 4 sub-domains and direct integration of the full mechanism with 75 species (right). Average iso-lines are included to indicate the statistical similarities. (Løvås et al., 2010)

gives excellent correlation to the full treatment. Since these are instantaneous plots, the flame is not identical. However, the averaged values indicated by the stapled lines show clearly the close correlation. It was demonstrated that when adaptive chemistry is employed the number of species that need to be treated in detail is substantially reduced, only true steady state species being selected at each operating point.

Côme, G.M., Warth, V., Glaude, P.A., Fournet, R., Battin-Leclerc, & F. Scacchi, G. (1996).

Model Reduction Techniques for Chemical Mechanisms 111

Fieweger, K., Blumenthal, R., & Adomeit, G.,(1997). Self-Ignition of SI Engine Model Fuels: A Shock Tube Investigation at High Pressure, *Combustion and Flame*, 109:599-619 Green, W.H., Allen, J.W., Ashcraft, R.W., Beran, G.J., Class, C.A., Gao, C., Goldsmith, C.F.,

Goussis, D.A., (1996). On the Construction and Use of Reduced Chemical Kinetic Mechanisms

Jones, W.P. & Rigopoulos, S. (2005). Rate-controlled constrained equilibrium: Formulation

Jones, W.P. & Rigopoulos, S. (2007). Reduced chemistry for hydrogen and methanol premixed flames via RCCE, *Combustion Theory and Modelling* Vol. 11 (5) pp 755-780. Kantak, M.V., De Manrique, K.S., Aglave, R.H. & Hesketh, R.P. (1997). Methylamine Oxidation

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Klimenko, A. & Bilger R. (1999). Conditional Moment Closure for Turbulent Combustion,

Lam, S.H. (1993). Using CSP to understand complex chemical kinetics, *Combustion Science and*

Lam, S. H. & Gaussis, D.A., (1994). The CSP Method for Simplifying Kinetics, *International*

Lewis, B. & von Elbe, G. (1987). Combustion, Flames and Explosions of Gases, Academic Press

Li, J., Zhao, Z., Kazakov, A., & Dreyer, F.L.(2004). An updated comprehensive kinetic model

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Løvås, T., Nilsson, D. & Mauss, F. (1999). Development of Reduced Chemical Mechanisms

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reactive flow computations, *Proceedings of the Combustion Institute* Vol. 32, pp 527-534.

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Flames, *Combustion Science and Technology*, Vol. 124, pp.249-276.

v3.3 Home Page, http://rmg.sourceforge.net/

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Structure of Low-Pressure Premixed n-Heptane-O2-Ar and iso-Octane-O2-Ar

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and application to nonpremixed laminar flames, *Combustion and Flame* Vol. 142, pp

in a Flow Reactor: Mechanism and Modeling, *Combustion and Flame* Vol.108, pp.

#### **6. Conclusion**

The present chapter intended to outline the main model reduction techniques for chemical systems used in combustion modeling, with the end goal of presenting the recent developments of reducing chemical mechanisms through application of automatic and adaptive procedures. In the past, reduced mechanisms have been used extensively in simulations of combustion processes but they were carried out by hand, which is a tedious process. They were therefore often limited to very specific physical conditions and often represented very simple fuels. However, as chemical models of complex fuels have become very large and detailed, the need for automatic procedures and smart implementation into CFD is evident. These automatic procedures are however based on the ideas of the manual and analytical procedures developed in the past. Although the chapter emphasizes the classic fundamental reduction procedures, it was the goal here to demonstrate the usage of automatic reduction procedures suitable to a wide range of conditions and applicable to realistic multi-component fuels.

It is important to state that this field of research is under constant development, and there are still very promising and interesting techniques being proposed in literature which has not been discussed here. This is not a result of lack of importance, but rather the limitation of space. The reader is therefore encouraged to follow up from here, and based on the current review make a decision in his or her own mind for which approach is the most applicable to the problem in hand.

#### **7. References**


32 Will-be-set-by-IN-TECH

The present chapter intended to outline the main model reduction techniques for chemical systems used in combustion modeling, with the end goal of presenting the recent developments of reducing chemical mechanisms through application of automatic and adaptive procedures. In the past, reduced mechanisms have been used extensively in simulations of combustion processes but they were carried out by hand, which is a tedious process. They were therefore often limited to very specific physical conditions and often represented very simple fuels. However, as chemical models of complex fuels have become very large and detailed, the need for automatic procedures and smart implementation into CFD is evident. These automatic procedures are however based on the ideas of the manual and analytical procedures developed in the past. Although the chapter emphasizes the classic fundamental reduction procedures, it was the goal here to demonstrate the usage of automatic reduction procedures suitable to a wide range of conditions and applicable to

It is important to state that this field of research is under constant development, and there are still very promising and interesting techniques being proposed in literature which has not been discussed here. This is not a result of lack of importance, but rather the limitation of space. The reader is therefore encouraged to follow up from here, and based on the current review make a decision in his or her own mind for which approach is the most applicable to

Blurock, E.S. (1995). Reaction: System for Modelling Chemical Reactions, *J. Chem. Info. Comp.*

Blurock, E.S. (2000). Generation and Subsequent Reduction of Large Detailed Combustion Mechanisms, *Proceedings of the Combustion Institute*, WIP 4-D19, p 331. Blurock, E., Løvås, T. & Mauss, F. (2003). Steady State Reduced Mechanisms Based on

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**6. Conclusion**

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**1. Introduction**

non-equilibrium gas flows.

non-equilibrium reacting gas mixtures.

**0**

**5**

*Russia*

**Vibrational and Chemical Kinetics in**

In this chapter, we consider the vibrational and chemical kinetics in reacting gas flows under the conditions of strong deviations from thermodynamic equilibrium. Such conditions occur, for example, near surfaces of nonexpendable space vehicles in their reentry into the Earth and Mars atmospheres, in experiments carried out in high-enthalpy facilities, in supersonic gas flows in nozzles and jets, in chemical technology processes. In many cases, the characteristic times of vibrational relaxation and chemical reactions appear to be comparable with the characteristic time for the variation of basic gas-dynamic parameters of a flow. Therefore, the equations of gas dynamics and non-equilibrium kinetics should be considered jointly. Consequently, the set of governing equations for macroscopic parameters includes not only the conservation equations for the momentum and total energy, but also the equations for chemical reactions and vibrational energy relaxation. The latter equations contain the rates of energy transitions and chemical reactions which are needed in order to solve the equations of

Originally, non-equilibrium chemical reactions were studied in thermally equilibrium gas mixtures which were assumed to be spatially homogeneous Kondratiev & Nikitin (1974). Later on, different models for vibrational–chemical coupling were proposed on the basis of the kinetic theory methods. One of the first works in this area is that by I. Prigogine Prigogine & Xhrouet (1949), followed by studies Present (1960), and Ludwig & Heil (1960). The effect of non-equilibrium distributions on the chemical reaction rate coefficients was considered in Shizgal & Karplus (1970). Later on, this effect was studied using various distributions of reacting gas molecules over the internal energy (see, for instance, Refs. Belouaggadia & Brun (1997); Knab (1996))). Most of these models are based on thermally-equilibrium distributions or non-equilibrium Boltzmann distributions over the vibrational energy of the reagents. More rigorous models of non-equilibrium kinetics in a flow take into account the non-Boltzmann quasi-stationary distributions or the state-to-state vibrational and chemical kinetics Kustova et al. (1999); Nagnibeda & Kustova (2009). The influence of state-to-state and multi-temperature distributions on reaction rates in particular flows are studied in Kustova & Nagnibeda (2000); Kustova et al. (2003). Recently the comparison of kinetic models for transport properties in reacting gas flows has been discussed in Kustova & Nagnibeda (2011). In the present contribution, we propose mathematical description for the chemical kinetics in gas flows on the basis of the Chapman–Enskog method, generalized for strongly

**Non-Equilibrium Gas Flows**

E.V. Kustova and E.A. Nagnibeda *Saint Petersburg State University*


## **Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows**

E.V. Kustova and E.A. Nagnibeda *Saint Petersburg State University Russia*

#### **1. Introduction**

36 Will-be-set-by-IN-TECH

114 Chemical Kinetics

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Yetter, R.A., Dryer, F. L. & Rabitz, H., (1991). Flow Reactor Studies of Carbon Monoxide/Hydrogen/Oxygen Kinetics, *Combust. Sci. and Tech.* pp 79:129. Zeuch, T., Moerac, G., Ahmed, S.S. & Mauss, F. (2008). A comprehensive skeletal mechanism

Zheng, X.L., Lu, T.F., & Law, C.K. (2007). Experimental Counterflow Ignition Temperatures

INTERNATIONAL EDITIONS, Mechanical Engineering Series

Physics, New Series, m 15, Springer Verlag, pp. 76-101.

*Proceedings of the Combustion Institute* Vol. 18, pp. 369-384.

to n-Hexadecane, *Combustion and Flame* Vol. 156 (1) pp 181-199

2001-ICE-416.

Verlag.

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automatically reduced mechanism, ASME Fall Technical Conference, Technical Paper

in internal combustion engines using reduced chemistry, *Combustion Science and*

investigation and reduction of combustion mechanisms, in : M.J.Pilling (Ed.), Low-Temperature Combustion and Autoignition, Comprehensive Chemical

Mechanisms Based on Inner Iteration, in: Peters, N. and Rogg, B. (Eds.), Reduced Kinetic Mechanisms for Application in Combustion Systems, Lecture Notes in

Chemical Kinetic Reaction Mechanism for n-Alkane Hydrocarbons from n-Octane

Splitting Schemes in the Computational Implementation of Combustion Chemistry,

for the oxidation of n-heptane generated by chemistry-guided reduction , *Comb. and*

and Reaction Mechanisms of 1,3-Butadiene, *Proceedings of the Combustion Institute*,

In this chapter, we consider the vibrational and chemical kinetics in reacting gas flows under the conditions of strong deviations from thermodynamic equilibrium. Such conditions occur, for example, near surfaces of nonexpendable space vehicles in their reentry into the Earth and Mars atmospheres, in experiments carried out in high-enthalpy facilities, in supersonic gas flows in nozzles and jets, in chemical technology processes. In many cases, the characteristic times of vibrational relaxation and chemical reactions appear to be comparable with the characteristic time for the variation of basic gas-dynamic parameters of a flow. Therefore, the equations of gas dynamics and non-equilibrium kinetics should be considered jointly. Consequently, the set of governing equations for macroscopic parameters includes not only the conservation equations for the momentum and total energy, but also the equations for chemical reactions and vibrational energy relaxation. The latter equations contain the rates of energy transitions and chemical reactions which are needed in order to solve the equations of non-equilibrium gas flows.

Originally, non-equilibrium chemical reactions were studied in thermally equilibrium gas mixtures which were assumed to be spatially homogeneous Kondratiev & Nikitin (1974). Later on, different models for vibrational–chemical coupling were proposed on the basis of the kinetic theory methods. One of the first works in this area is that by I. Prigogine Prigogine & Xhrouet (1949), followed by studies Present (1960), and Ludwig & Heil (1960). The effect of non-equilibrium distributions on the chemical reaction rate coefficients was considered in Shizgal & Karplus (1970). Later on, this effect was studied using various distributions of reacting gas molecules over the internal energy (see, for instance, Refs. Belouaggadia & Brun (1997); Knab (1996))). Most of these models are based on thermally-equilibrium distributions or non-equilibrium Boltzmann distributions over the vibrational energy of the reagents. More rigorous models of non-equilibrium kinetics in a flow take into account the non-Boltzmann quasi-stationary distributions or the state-to-state vibrational and chemical kinetics Kustova et al. (1999); Nagnibeda & Kustova (2009). The influence of state-to-state and multi-temperature distributions on reaction rates in particular flows are studied in Kustova & Nagnibeda (2000); Kustova et al. (2003). Recently the comparison of kinetic models for transport properties in reacting gas flows has been discussed in Kustova & Nagnibeda (2011).

In the present contribution, we propose mathematical description for the chemical kinetics in gas flows on the basis of the Chapman–Enskog method, generalized for strongly non-equilibrium reacting gas mixtures.

Kušˇcer (1991); Ludwig & Heil (1960); Nagnibeda & Kustova (2009); Rydalevskaya (1977):

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 117

� *mcmd mc*�*md*�

�<sup>3</sup>

*cd*, *ijkl* is the differential cross section of the exchange reaction, and the

*mc mc*�*mf* �

distribution functions after the collision are denoted *fc*�*i*�*j*� = *fc*�*i*�*j*�(**r**, **u***c*� , *t*), *fd*�*k*�*l*� =

*fc*� = *fc*�(**r**, **u***c*� , *t*), *f <sup>f</sup>* � = *f <sup>f</sup>* �(**r**, **u***<sup>f</sup>* � , *t*) are the distribution functions of atomic dissociation

the statistical weight of the internal states *i* and *j* of a component *c*, *g* is the relative velocity, Ω

Expressions (6), (7) are written taking into account the principle of microscopic reversibility for reactive collisions considered in Alexeev et al. (1994); Ern & Giovangigli (1998); Kušˇcer

> , Ω) = *s c ijs d klm*<sup>2</sup> *cm*<sup>2</sup> *<sup>d</sup> <sup>g</sup>*2*σc*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* (**g**, Ω�

In the frame of the method proposed in Kustova & Nagnibeda (1998); Nagnibeda & Kustova (2009) for the solution of Eqs. (2), the distribution functions are expanded in a power series of the small parameter *ε*. The peculiarity of the modified Chapman-Enskog method is that the distribution functions and macroscopic parameters are determined by the collision invariants of the most frequent collisions. Under condition (1), the set of collision invariants contains the invariants of any collision (momentum and total energy) and the additional invariants of rapid processes. In our case, these additional invariants are any variables independent of the velocity and internal energy and depending arbitrary on chemical species *c* because chemical reactions are supposed to be frozen in rapid processes This set of collision invariants provides the following set of macroscopic parameters for a closed flow description: number densities

Closed set of equations of the flow are derived from the kinetic equations (2). Integrating these equations over velocities and summing over the internal energy levels we obtain equations of chemical kinetics in the flow. Multiplying kinetic equations by the collision invariants of any collision, integrating over the velocity and summing over the internal energy levels, we obtain the conservation equations for the momentum and total energy. Finally the set of governing

*c ijm*<sup>3</sup> *<sup>c</sup> <sup>g</sup>σdiss*

*<sup>d</sup>*, **u***c*, **u***d*) = *s*

�<sup>3</sup>

− *fcij fdkl*

*<sup>d</sup>*, *<sup>t</sup>*), *<sup>h</sup>* is the Plank constant, *mc* is the mass of a molecule *<sup>c</sup>*, *<sup>s</sup><sup>c</sup>*

<sup>−</sup> *fcij fdkl*�

⎤ <sup>⎦</sup> *<sup>g</sup>σdiss*

*gσc*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl <sup>d</sup>*2Ω*d***u***d*, (6)

*<sup>d</sup>*) is the formal cross section of dissociation,

*cij*, *<sup>d</sup>*(**u***c*, **u***d*, **u***c*� , **u***<sup>f</sup>* � , **u**�

*cij*, *<sup>d</sup>d***u***dd***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*<sup>d</sup>*. (7)

*ij* is

), (8)

*<sup>d</sup>*), (9)

*sc ijsd kl*

> *c ij* �

*sc*� *<sup>i</sup>*�*j*�*sd*� *k*�*l*�

*cij*, *<sup>d</sup>*(**u***c*, **u***d*, **u***c*� , **u***<sup>f</sup>* � , **u**�

*J ex cij* = ∑ *dc*�*d*� ∑ *ki*�*k*� ∑ *lj*�*l*�

*J diss cij* = ∑ *d* ∑ *k* ∑ *l*

In Eq. (6), *<sup>σ</sup>c*�

products; *f* �

*<sup>σ</sup>rec*, *cij*

*d*� , *i* � *j* � *k*� *l* �

*fd*�*k*�*l*�(**r**, **u***d*� , *t*); in Eq. (7), *σdiss*

*s c*� *<sup>i</sup>*�*j*�*s d*� *<sup>k</sup>*�*l*�*m*<sup>2</sup> *<sup>c</sup>*�*m*<sup>2</sup> *<sup>d</sup>*� *<sup>g</sup>*�2*<sup>σ</sup>*

*m*3 *<sup>c</sup>*�*m*<sup>3</sup> *f* � *<sup>h</sup>*<sup>3</sup> *<sup>σ</sup>rec*, *cij*

**2.2 Governing equations. Reaction rates**

*dkl* = *fdkl*(**r**, **u**�

� �

� <sup>⎡</sup> ⎣*f* �

*fc*�*i*�*j*� *fd*�*k*�*l*�

is the solid angle in which a molecule appear after a collision.

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* (**u***c*� , **u***<sup>f</sup>* � , **u**�

*cd*, *ijkl c*�*d*�

, *<sup>i</sup>*�*j*�*k*�*l*�(**g**�

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* is the probability density for a triple collision resulting in dissociation.

of species *nc*(**r**, *t*) (*c* = 1, ..., *L*), gas velocity **v**(**r**, *t*) and temperature *T*(**r**, *t*).

equations for macroscopic parameters *nc*(**r**, *t*), **v**(**r**, *t*), *T*(**r**, *t*) takes the form:

(1991); Ludwig & Heil (1960); Rydalevskaya (1977):

*dkl fc*� *<sup>f</sup> <sup>f</sup>* � *<sup>h</sup>*3*<sup>s</sup>*

First, we consider the one-temperature model for the non-equilibrium chemical kinetics in thermally equilibrium gas flows or deviating weakly from thermal equilibrium state. Then, the models for vibrational–chemical coupling in gas flows are derived from the kinetic theory taking into account state-to-state and multi-temperature vibrational distributions.

The influence of non-equilibrium distributions, gas compressibility and space inhomogeneity on the reaction rates for different processes is discussed.

#### **2. One-temperature model for non-equilibrium kinetics**

#### **2.1 Kinetic equations. Distribution functions**

We consider strong non-equilibrium chemical kinetics in a flow under the following conditions for relaxation times

$$
\tau\_{el} < \tau\_{int} \ll \tau\_{react} \sim \theta. \tag{1}
$$

Here *τel*, *τint*, *τreact* and *θ* are mean times for relaxation of translation and internal degrees of freedom, chemical reactions and gas dynamic parameters changing respectively. The kinetic equations for the distribution functions have the form Nagnibeda & Kustova (2009):

$$\frac{\partial f\_{\rm cij}}{\partial t} + \mathbf{u}\_{\rm c} \cdot \nabla f\_{\rm cij} = \frac{1}{\varepsilon} f\_{\rm cij}^{rap} + f\_{\rm cij}^{sl} \tag{2}$$

*ε* = *τrap*/*τsl* ∼ *τrap*/*θ* � 1 is the small parameter, *J rap cij* and *<sup>J</sup>sl cij* are the collision integral operators for rapid and slow processes, *c*, *i*, *j* denote chemical species, vibrational and rotational levels respectively, **r**, **u**, *t* are coordinates, molecular velocities and time. Under condition (2), integral operators of rapid processes describe elastic collisions and collisions with rotational and vibrational energies change and can be written in the form

$$J\_{\rm cij}^{rap} = J\_{\rm cij}^{el} + J\_{\rm cij}^{rot} + J\_{\rm cij}^{vibr} = J\_{\rm cij}^{el} + J\_{\rm cij}^{int} \tag{3}$$

The operator of slow processes *Jsl cij* <sup>=</sup> *<sup>J</sup>react cij* includes the integrals of reactive collisions and describes exchange reactions

$$A\_{\mathcal{C}}(\mathbf{u}\_{\mathcal{C}},\mathbf{i},\mathbf{j}) + A\_{d}(\mathbf{u}\_{d'}\mathbf{k},\mathbf{l}) \rightleftharpoons A\_{\mathcal{C}'}(\mathbf{u}\_{\mathcal{C}'},\mathbf{i}',\mathbf{j}') + A\_{d'}(\mathbf{u}\_{d'},\mathbf{k}',\mathbf{l}'),\tag{4}$$

and dissociation-recombination reactions

$$A\_{\mathcal{C}}(\mathbf{u}\_{\mathcal{C}}, i\_{\prime}) + A\_{d}(\mathbf{u}\_{d\prime}k\_{\prime}l) \rightleftharpoons A\_{\mathcal{C}}(\mathbf{u}\_{\mathcal{C}}) + A\_{f^{\prime}}(\mathbf{u}\_{f^{\prime}}) + A\_{d}(\mathbf{u}\_{d\prime}^{\prime}k\_{\prime}l),\tag{5}$$

*c*� , *f* � are the atomic species forming as reaction products; **u***c*� , **u***<sup>f</sup>* � , **u**� *<sup>d</sup>* are the particle velocities after the collision. For the simplicity we consider dissociation of only diatomic molecules, therefore products of dissociation are only atoms. In addition to this, it is commonly supposed that the dissociation cross section does not depend on the internal state of a partner in the reaction, and this state does not vary as a result of dissociation and recombination.

The collision operator *Jreact cij* represents the sum of two terms, *<sup>J</sup>ex cij* and *<sup>J</sup>diss cij* . Expressions for these operators are given, for instance, in Alexeev et al. (1994); Ern & Giovangigli (1998); 2 Will-be-set-by-IN-TECH

First, we consider the one-temperature model for the non-equilibrium chemical kinetics in thermally equilibrium gas flows or deviating weakly from thermal equilibrium state. Then, the models for vibrational–chemical coupling in gas flows are derived from the kinetic theory

The influence of non-equilibrium distributions, gas compressibility and space inhomogeneity

We consider strong non-equilibrium chemical kinetics in a flow under the following

Here *τel*, *τint*, *τreact* and *θ* are mean times for relaxation of translation and internal degrees of freedom, chemical reactions and gas dynamic parameters changing respectively. The kinetic

operators for rapid and slow processes, *c*, *i*, *j* denote chemical species, vibrational and rotational levels respectively, **r**, **u**, *t* are coordinates, molecular velocities and time. Under condition (2), integral operators of rapid processes describe elastic collisions and collisions

*ε J rap cij* + *J sl*

*vibr cij* = *J el cij* + *J int*

> � , *j* �

*rap cij* and *<sup>J</sup>sl*

equations for the distribution functions have the form Nagnibeda & Kustova (2009):

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> **<sup>u</sup>***<sup>c</sup>* · ∇ *fcij* <sup>=</sup> <sup>1</sup>

with rotational and vibrational energies change and can be written in the form

*cij* <sup>=</sup> *<sup>J</sup>react*

*Ac*(**u***c*, *i*, *j*) + *Ad*(**u***d*, *k*, *l*) *Ac*�(**u***c*�) + *Af* �(**u***<sup>f</sup>* �) + *Ad*(**u**�

reaction, and this state does not vary as a result of dissociation and recombination.

after the collision. For the simplicity we consider dissociation of only diatomic molecules, therefore products of dissociation are only atoms. In addition to this, it is commonly supposed that the dissociation cross section does not depend on the internal state of a partner in the

*cij* represents the sum of two terms, *<sup>J</sup>ex*

these operators are given, for instance, in Alexeev et al. (1994); Ern & Giovangigli (1998);

*Ac*(**u***c*, *i*, *j*) + *Ad*(**u***d*, *k*, *l*) *Ac*�(**u***c*� , *i*

, *f* � are the atomic species forming as reaction products; **u***c*� , **u***<sup>f</sup>* � , **u**�

*τel < τint* � *τreact* ∼ *θ*. (1)

*cij* includes the integrals of reactive collisions and

, *l* �

*cij* and *<sup>J</sup>diss*

) + *Ad*�(**u***d*� , *k*�

*cij*, (2)

*cij* are the collision integral

*cij* , (3)

), (4)

*<sup>d</sup>*, *k*, *l*), (5)

*cij* . Expressions for

*<sup>d</sup>* are the particle velocities

taking into account state-to-state and multi-temperature vibrational distributions.

on the reaction rates for different processes is discussed.

**2.1 Kinetic equations. Distribution functions**

conditions for relaxation times

The operator of slow processes *Jsl*

and dissociation-recombination reactions

describes exchange reactions

The collision operator *Jreact*

*c*�

**2. One-temperature model for non-equilibrium kinetics**

*∂ fcij*

*ε* = *τrap*/*τsl* ∼ *τrap*/*θ* � 1 is the small parameter, *J*

*J rap cij* = *J el cij* + *J rot cij* + *J* Kušˇcer (1991); Ludwig & Heil (1960); Nagnibeda & Kustova (2009); Rydalevskaya (1977):

$$f\_{\rm cij}^{\rm cx} = \sum\_{\rm dc'} \sum\_{\rm k'k'} \sum\_{\rm lj'l'} \int \left[ f\_{\rm c'i'j'} f\_{\rm d'k'l'} \frac{s\_{\rm ij}^{\rm c} s\_{\rm kl}^{\rm d}}{s\_{\rm i'j'}^{\rm c'} s\_{\rm k'l'}^{\rm d}} \left( \frac{m\_{\rm c} m\_{\rm d}}{m\_{\rm c'} m\_{\rm d'}} \right)^3 - f\_{\rm cij} f\_{\rm dkl} \right] g \frac{\varepsilon^{\rm c'd', i'j'} l'^{\rm l} l'}{c d\_{\rm c} i\_{\rm i} \rm jkl} d^2 \Omega d \mathbf{u}\_{\rm d'} \tag{6}$$

$$f\_{\rm cij}^{\rm diss} = \sum\_{d} \sum\_{k} \sum\_{l} \int \left[ f\_{\rm dkl}' f\_{\rm c'} f\_{\rm f'} h^3 s\_{\rm ij}^c \left( \frac{m\_c}{m\_{\rm c'} m\_{\rm f'}} \right)^3 - f\_{\rm cij} f\_{\rm dkl} \right] g \sigma\_{\rm cij,d}^{\rm diss} d\mathbf{u}\_d d\mathbf{u}\_{\rm c'} d\mathbf{u}\_{\rm f'} d\mathbf{u}\_d^\prime. \tag{7}$$

In Eq. (6), *<sup>σ</sup>c*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* is the differential cross section of the exchange reaction, and the distribution functions after the collision are denoted *fc*�*i*�*j*� = *fc*�*i*�*j*�(**r**, **u***c*� , *t*), *fd*�*k*�*l*� = *fd*�*k*�*l*�(**r**, **u***d*� , *t*); in Eq. (7), *σdiss cij*, *<sup>d</sup>*(**u***c*, **u***d*, **u***c*� , **u***<sup>f</sup>* � , **u**� *<sup>d</sup>*) is the formal cross section of dissociation, *fc*� = *fc*�(**r**, **u***c*� , *t*), *f <sup>f</sup>* � = *f <sup>f</sup>* �(**r**, **u***<sup>f</sup>* � , *t*) are the distribution functions of atomic dissociation products; *f* � *dkl* = *fdkl*(**r**, **u**� *<sup>d</sup>*, *<sup>t</sup>*), *<sup>h</sup>* is the Plank constant, *mc* is the mass of a molecule *<sup>c</sup>*, *<sup>s</sup><sup>c</sup> ij* is the statistical weight of the internal states *i* and *j* of a component *c*, *g* is the relative velocity, Ω is the solid angle in which a molecule appear after a collision.

Expressions (6), (7) are written taking into account the principle of microscopic reversibility for reactive collisions considered in Alexeev et al. (1994); Ern & Giovangigli (1998); Kušˇcer (1991); Ludwig & Heil (1960); Rydalevskaya (1977):

$$s\_{\mathbf{i'}\mathbf{j'}}^{c'}s\_{\mathbf{k'}l'}^{d'}m\_{\mathbf{c'}l}^2m\_{\mathbf{d'}}^2g^2\sigma\_{\mathbf{c'}d',\mathbf{i'}\mathbf{j'}k'l'}^{\text{cd,ijkl}}(\mathbf{g'},\Omega)=s\_{\mathbf{i'}\mathbf{j}}^{c}s\_{\mathbf{k}l}^{d}m\_{\mathbf{c}}^2m\_{\mathbf{d}\mathbf{g}}^2g^2\sigma\_{\mathbf{cd,ijkl}}^{c'd',\mathbf{i'}\mathbf{j'}k'l'}(\mathbf{g},\Omega'),\tag{8}$$

$$\frac{m\_{c'}^3 m\_{f'}^3}{h^3} \sigma\_{c'f'd}^{\text{rec.} \text{ci}\text{j}}(\mathbf{u}\_{c'}, \mathbf{u}\_{f'}, \mathbf{u}\_{d'}', \mathbf{u}\_{c'}, \mathbf{u}\_d) = s\_{\text{i}\text{j}}^c m\_{c\text{S}}^3 g \sigma\_{\text{cij},d}^{\text{diss}}(\mathbf{u}\_{c'}, \mathbf{u}\_{d'}, \mathbf{u}\_{f'}, \mathbf{u}\_{d}'), \tag{9}$$

*<sup>σ</sup>rec*, *cij <sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* is the probability density for a triple collision resulting in dissociation.

In the frame of the method proposed in Kustova & Nagnibeda (1998); Nagnibeda & Kustova (2009) for the solution of Eqs. (2), the distribution functions are expanded in a power series of the small parameter *ε*. The peculiarity of the modified Chapman-Enskog method is that the distribution functions and macroscopic parameters are determined by the collision invariants of the most frequent collisions. Under condition (1), the set of collision invariants contains the invariants of any collision (momentum and total energy) and the additional invariants of rapid processes. In our case, these additional invariants are any variables independent of the velocity and internal energy and depending arbitrary on chemical species *c* because chemical reactions are supposed to be frozen in rapid processes This set of collision invariants provides the following set of macroscopic parameters for a closed flow description: number densities of species *nc*(**r**, *t*) (*c* = 1, ..., *L*), gas velocity **v**(**r**, *t*) and temperature *T*(**r**, *t*).

#### **2.2 Governing equations. Reaction rates**

Closed set of equations of the flow are derived from the kinetic equations (2). Integrating these equations over velocities and summing over the internal energy levels we obtain equations of chemical kinetics in the flow. Multiplying kinetic equations by the collision invariants of any collision, integrating over the velocity and summing over the internal energy levels, we obtain the conservation equations for the momentum and total energy. Finally the set of governing equations for macroscopic parameters *nc*(**r**, *t*), **v**(**r**, *t*), *T*(**r**, *t*) takes the form:

*<sup>ν</sup>diss*, *<sup>c</sup> <sup>k</sup> <sup>f</sup>* , *diss* = −N*<sup>A</sup>* ∑

*<sup>ν</sup>diss*, *<sup>c</sup> kb*, *diss* <sup>=</sup> −N <sup>2</sup>

˙ *ξ<sup>r</sup>* = *k <sup>f</sup>* ,*<sup>r</sup>*

approximation of the Chapman–Enskog method.

**2.3 Zero-order reaction-rate coefficients**

*f* (0) *cij* =

with the internal partition function *Zint*

function for atomic species reads

 *mc* 2*πkT*

*Zint*

*f* (0) *<sup>c</sup>* <sup>=</sup>

Maxwell-Boltzmann distributions

relations:

for exchange reactions, and

conventional form:

Here *ε<sup>c</sup>*

*jl* ∑ *ik*

*<sup>A</sup>* ∑ *jl* ∑ *ik*

*<sup>k</sup> <sup>f</sup>* ,*<sup>r</sup>* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup> νrc kdd*�

*<sup>k</sup> <sup>f</sup>* ,*<sup>r</sup>* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup> νrc kd*

for dissociation and recombination reactions, N*<sup>A</sup>* is the Avogadro number.

*L* ∏*c*=1  *ρ<sup>c</sup> Mc*

3/2 *nc Zint <sup>c</sup>* (*T*) *s c ij* exp

*<sup>c</sup>* (*T*) = ∑

 *mc* 2*πkT*

The zero order transport terms in the flow equations take the form **V**(0)

*<sup>c</sup>* given by

*ij s c ij* exp

3/2

**q**(0) = 0, the pressure is *p* = *nkT*, *n* is the total number density. Thus, in the zero-order

*ij* is the internal energy of a molecule at the *i*th vibrational and *j*th rotational levels, *k* is the Boltzmann constant, **c***<sup>c</sup>* = **u***<sup>c</sup>* − **v** is the peculiar velocity. The zero-order distribution

*nc* exp

Using equations (16), (17)–(20), we can write the expression for the reaction rate ˙

*ν* (*r*) *rc*

One can notice that the general expressions for the rate coefficients depend on the cross-section of corresponding reactions as well as on the distribution functions, and, consequently, on the

The zero-order solution of Eqs. (2) for molecular species has the form of the

 *fcij fdkl ncnd*

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 119

*fc*� *f <sup>f</sup>* � *f* �

Rate coefficients (17)–(20) are connected with those appearing in equations (13), (14) by the

*dkl nc*�*nf* �*nd*

*cc*� , *kb*,*<sup>r</sup>* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*<sup>c</sup>*, *diss*, *kb*,*<sup>r</sup>* <sup>=</sup> <sup>−</sup> <sup>N</sup> <sup>2</sup>

− *kb*,*<sup>r</sup>*

*L* ∏*c*=1  *ρ<sup>c</sup> Mc*

<sup>−</sup> *mcc*<sup>2</sup> *c* <sup>2</sup>*kT* <sup>−</sup> *<sup>ε</sup><sup>c</sup> ij kT*

− *εc ij kT* .

> <sup>−</sup> *mcc*<sup>2</sup> *c* <sup>2</sup>*kT*

*ν* (*p*) *rc*

*g σdiss*

*<sup>σ</sup>rec*, *cij*

*cij*, *<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*νrc kd*� *d*

*A νrc kd* *<sup>d</sup>*, (19)

*ξ<sup>r</sup>* in the

*<sup>d</sup> d***u***<sup>c</sup> d***u***d*. (20)

*<sup>c</sup>*�*<sup>c</sup>* (21)

*rec*, *<sup>c</sup>* (22)

. (23)

, (24)

. (25)

*<sup>c</sup>* = 0, *P*(0) = *pI*,

$$\frac{dn\_{\mathcal{C}}}{dt} + n\_{\mathcal{C}} \nabla \cdot \mathbf{v} + \nabla \cdot (n\_{\mathcal{C}} \mathbf{V}\_{\mathcal{C}}) = R\_{\mathcal{C}}^{\text{react}}, \ \mathcal{c} = 1, \dots, L,\tag{10}$$

$$
\rho \frac{d\mathbf{v}}{dt} + \nabla \cdot \mathbf{P} = 0,\tag{11}
$$

$$
\rho \frac{d\mathbf{U}}{dt} + \nabla \cdot \mathbf{q} + \mathbf{P} : \nabla \mathbf{v} = 0. \tag{12}
$$

Here *U* is the total energy per unit mass which is the function of temperature and species number densities, *P* is the pressure tensor, **q** is the total heat flux, **V***<sup>i</sup>* are the diffusion velocities.

The production terms in Eqs. (10) take the form

$$R\_{\mathcal{C}}^{\varepsilon\chi} = \sum\_{d\mathbf{c'}d'} \left( n\_{\mathbf{c'}} n\_{d'} k\_{\mathbf{c'}c}^{d'd} - n\_{\mathbf{c}} n\_d k\_{\mathbf{c}c'}^{d d'} \right) \,, \tag{13}$$

$$R\_c^{\rm diss} = \sum\_d n\_d \left( n\_{c'} n\_{f'} k\_{\rm rec,c}^d - n\_c k\_{c, \rm diss}^d \right) . \tag{14}$$

Production terms (13) (14) contain the rate coefficients of exchange chemical reactions *kdd*� *cc*� , dissociation *k<sup>d</sup> <sup>c</sup>*, *diss*, and recombination *<sup>k</sup><sup>d</sup> rec*,*<sup>c</sup>* Nagnibeda & Kustova (2009).

For practical calculations it is more suitable to use component mass fractions *α<sup>c</sup>* = *ρc*/*ρ* instead of number densities *nc* (*ρ* is the total mass density, *ρ<sup>c</sup>* is the component *c* mass density). In this case, the set of macroscopic parameters includes *αc*(**r**, *t*) (*c* = 1, ..., *L*), **v**(**r**, *t*), *T*(**r**, *t*). The equations of chemical kinetics take the form:

$$\rho \frac{d\mathbf{u}\_{\mathcal{C}}}{dt} = -\nabla \cdot (\rho\_{\mathcal{C}} \mathbf{V}\_{\mathcal{C}}) + \sum\_{r} \dot{\xi}\_{r} \nu\_{\text{rc}} M\_{\mathcal{C}r} \quad \mathcal{c} = 1, \dots, L \tag{15}$$

here *Mc* is the component molar mass, ˙ *ξ<sup>r</sup>* is the chemical reaction rate for reaction *r* (*r* = 1, ..., *R*, *R* is the number of reactions in a mixture), *νrc* = *ν* (*p*) *rc* <sup>−</sup> *<sup>ν</sup>* (*r*) *rc* is global stoichiometric coefficient, *ν* (*r*) *rc* , *ν* (*p*) *rc* are the stoichiometric coefficients of reactants and products.

The source terms are defined by expressions:

$$
\sum\_{r} \dot{\xi}\_{r} \nu\_{\rm rc} M\_{\rm c} = m\_{\rm c} \sum\_{\rm ij} \int f\_{\rm cij}^{\rm sl} d\mathbf{u}\_{\rm c} = m\_{\rm c} \sum\_{\rm ij} \int (f\_{\rm cij}^{\rm ex} + f\_{\rm cij}^{\rm diss}) d\mathbf{u}\_{\rm c}.\tag{16}
$$

Let us introduce the rate coefficients of forward and backward reactions *k <sup>f</sup>* ,*r*, *kb*,*r*. For exchange reaction (4) (*r* = *ex*), and recombination-dissociation reaction (5) (*r* = *diss*) they have the form:

$$\nu\_{\rm ex,c} \, k\_{f,\rm ex} = -\mathcal{N}\_A \sum\_{\rm j\bar{j}\bar{j}\bar{l}\,\rm ik\bar{r}\,\rm k\bar{r}} \int \frac{f\_{\rm c\bar{j}} f\_{\rm d\bar{k}l}}{n\_{\rm c} n\_{\rm d}} \, g \, \sigma\_{\rm cd,j\bar{j}\bar{k}l}^{\rm c'\bar{d}\bar{l}\,\rm l\bar{r}\,\rm l\bar{l}\,\rm l}^{\prime} \, d^2 \Omega \, d\mathbf{u}\_{\rm d} \, d\mathbf{u}\_{\rm c\bar{c}} \tag{17}$$

$$\nu\_{\rm ex,c} k\_{b,\rm ex} = -\mathcal{N}\_A \sum\_{jl'l'} \sum\_{\rm ik'k'} \int \frac{f\_{c'l'j'} f\_{d'k'l'}}{n\_{c'l'} n\_{d'}} \, \text{g}' \, \sigma\_{c'd',i'j'k'l'}^{\rm cd,ijkl} \, d^2 \Omega \, d\mathbf{u}\_{d'} \, d\mathbf{u}\_{c'l} \tag{18}$$

$$\nu\_{\rm diss,c} k\_{f,\rm diss} = -\mathcal{N}\_A \sum\_{\rm jl} \sum\_{\rm ik} \int \frac{f\_{\rm cij} f\_{\rm dkl}}{n\_{\rm c} n\_{\rm d}} \, g \, \sigma\_{\rm cij,d}^{\rm diss} \, d\mathbf{u}\_{\rm c} \, d\mathbf{u}\_{\rm d} \, d\mathbf{u}\_{\rm c'} \, d\mathbf{u}\_{\rm f'} \, d\mathbf{u}\_{\rm d'} \tag{19}$$

$$\nu\_{\rm diss,c} \, k\_{b,\rm diss} = -\mathcal{N}\_A^2 \sum\_{\rm jl} \sum\_{\rm ik} \int \frac{f\_{c'} f\_{f'} f\_{\rm dkl}'}{n\_{c'} n\_{f'} n\_d} \sigma\_{c'f'd}^{\rm rec,cij} \, d\mathbf{u}\_{c'} \, d\mathbf{u}\_{f'} \, d\mathbf{u}\_d' \, d\mathbf{u}\_c \, d\mathbf{u}\_d. \tag{20}$$

Rate coefficients (17)–(20) are connected with those appearing in equations (13), (14) by the relations:

$$k\_{f,r} = -\frac{N\_A}{\nu\_{rc}} k\_{cc'}^{dd'} \qquad \qquad \qquad k\_{b,r} = -\frac{N\_A}{\nu\_{rc}} k\_{c'c}^{d'd} \tag{21}$$

for exchange reactions, and

4 Will-be-set-by-IN-TECH

Here *U* is the total energy per unit mass which is the function of temperature and species number densities, *P* is the pressure tensor, **q** is the total heat flux, **V***<sup>i</sup>* are the diffusion

> *nc*�*nd*� *kd*� *d*

Production terms (13) (14) contain the rate coefficients of exchange chemical reactions *kdd*�

For practical calculations it is more suitable to use component mass fractions *α<sup>c</sup>* = *ρc*/*ρ* instead of number densities *nc* (*ρ* is the total mass density, *ρ<sup>c</sup>* is the component *c* mass density). In this case, the set of macroscopic parameters includes *αc*(**r**, *t*) (*c* = 1, ..., *L*), **v**(**r**, *t*), *T*(**r**, *t*). The

˙

(*p*) *rc* are the stoichiometric coefficients of reactants and products.

*cijd***u***<sup>c</sup>* = *mc* ∑

Let us introduce the rate coefficients of forward and backward reactions *k <sup>f</sup>* ,*r*, *kb*,*r*. For exchange reaction (4) (*r* = *ex*), and recombination-dissociation reaction (5) (*r* = *diss*) they

> *fcij fdkl ncnd*

 *fc*�*i*�*j*� *fd*�*k*�*l*� *nc*�*nd*�

*ij*

*<sup>g</sup> <sup>σ</sup>c*� *d*� , *i* � *j* � *k*� *l* �

*g*� *σ cd*, *ijkl c*�*d*�

 (*J ex cij* + *J diss*

*dt* <sup>+</sup> ∇· *<sup>P</sup>* <sup>=</sup> 0, (11)

*dt* <sup>+</sup> ∇· **<sup>q</sup>** <sup>+</sup> *<sup>P</sup>* : <sup>∇</sup>**<sup>v</sup>** <sup>=</sup> 0. (12)

*<sup>c</sup>*�*<sup>c</sup>* <sup>−</sup> *ncndkdd*�

*rec*,*<sup>c</sup>* <sup>−</sup> *nck<sup>d</sup>*

*rec*,*<sup>c</sup>* Nagnibeda & Kustova (2009).

*cc*� 

*c*,*diss* 

*<sup>c</sup>* , *c* = 1, .., *L*, (10)

, (13)

. (14)

*rc* is global stoichiometric

*cij* )*d***u***c*. (16)

*cd*, *ijkl <sup>d</sup>*2<sup>Ω</sup> *<sup>d</sup>***u***<sup>d</sup> <sup>d</sup>***u***c*, (17)

, *<sup>i</sup>*�*j*�*k*�*l*� *<sup>d</sup>*2<sup>Ω</sup> *<sup>d</sup>***u***d*� *<sup>d</sup>***u***c*� , (18)

*ξrνrcMc*, *c* = 1, .., *L* (15)

*ξ<sup>r</sup>* is the chemical reaction rate for reaction *r* (*r* =

(*p*) *rc* <sup>−</sup> *<sup>ν</sup>* (*r*) *cc*� ,

*dt* <sup>+</sup> *nc*∇ · **<sup>v</sup>** <sup>+</sup> ∇ · (*nc***V***c*) = *<sup>R</sup>react*

*dt* <sup>=</sup> −∇ · (*ρc***V***c*) <sup>+</sup> <sup>∑</sup>*<sup>r</sup>*

*ij*

*jlj*�*l*� ∑ *iki*�*k*�

*jlj*�*l*� ∑ *iki*�*k*�

 *J sl*

*dnc*

*ρ d***v**

*ρ dU*

The production terms in Eqs. (10) take the form

*Rex <sup>c</sup>* = ∑ *dc*�*d*�

*Rdiss <sup>c</sup>* = ∑ *d nd nc*�*nf* � *k<sup>d</sup>*

1, ..., *R*, *R* is the number of reactions in a mixture), *νrc* = *ν*

*ξrνrcMc* = *mc* ∑

*<sup>ν</sup>ex*, *<sup>c</sup> <sup>k</sup> <sup>f</sup>* ,*ex* = −N*<sup>A</sup>* ∑

*<sup>ν</sup>ex*, *<sup>c</sup> kb*,*ex* = −N*<sup>A</sup>* ∑

*<sup>c</sup>*, *diss*, and recombination *<sup>k</sup><sup>d</sup>*

equations of chemical kinetics take the form:

here *Mc* is the component molar mass, ˙

The source terms are defined by expressions:

∑*r* ˙ *ρ dαc*

velocities.

dissociation *k<sup>d</sup>*

coefficient, *ν*

have the form:

(*r*) *rc* , *ν*

$$k\_{f,r} = -\frac{\mathcal{N}\_A}{\nu\_{rc}} k\_{c,\text{diss}}^d \qquad k\_{b,r} = -\frac{\mathcal{N}\_A^2}{\nu\_{rc}} k\_{\text{rec},c}^d \tag{22}$$

for dissociation and recombination reactions, N*<sup>A</sup>* is the Avogadro number.

Using equations (16), (17)–(20), we can write the expression for the reaction rate ˙ *ξ<sup>r</sup>* in the conventional form:

$$\dot{\xi}\_r = k\_{f,r} \prod\_{c=1}^{L} \left(\frac{\rho\_c}{M\_c}\right)^{\nu\_{\kappa}^{(\ell)}} - k\_{b,r} \prod\_{c=1}^{L} \left(\frac{\rho\_c}{M\_c}\right)^{\nu\_{\kappa}^{(\ell)}}.\tag{23}$$

One can notice that the general expressions for the rate coefficients depend on the cross-section of corresponding reactions as well as on the distribution functions, and, consequently, on the approximation of the Chapman–Enskog method.

#### **2.3 Zero-order reaction-rate coefficients**

The zero-order solution of Eqs. (2) for molecular species has the form of the Maxwell-Boltzmann distributions

$$f\_{\rm cij}^{(0)} = \left(\frac{m\_{\rm c}}{2\pi kT}\right)^{3/2} \frac{n\_{\rm c}}{Z\_{\rm c}^{int}(T)} s\_{ij}^{\rm c} \exp\left(-\frac{m\_{\rm c}c\_{\rm c}^2}{2kT} - \frac{\mathfrak{E}\_{ij}^{\rm c}}{kT}\right),\tag{24}$$

with the internal partition function *Zint <sup>c</sup>* given by

$$Z\_c^{int}(T) = \sum\_{ij} s\_{ij}^c \exp\left(-\frac{\varepsilon\_{ij}^c}{kT}\right).$$

Here *ε<sup>c</sup> ij* is the internal energy of a molecule at the *i*th vibrational and *j*th rotational levels, *k* is the Boltzmann constant, **c***<sup>c</sup>* = **u***<sup>c</sup>* − **v** is the peculiar velocity. The zero-order distribution function for atomic species reads

$$f\_c^{(0)} = \left(\frac{m\_c}{2\pi kT}\right)^{3/2} n\_c \exp\left(-\frac{m\_c c\_c^2}{2kT}\right). \tag{25}$$

The zero order transport terms in the flow equations take the form **V**(0) *<sup>c</sup>* = 0, *P*(0) = *pI*, **q**(0) = 0, the pressure is *p* = *nkT*, *n* is the total number density. Thus, in the zero-order

For the dissociation reaction (5), the zero-order forward-rate coefficient is obtained in the form

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 121

 *mcd* 2*πkT*

*cij*, *<sup>d</sup>*(*g*, **u***c*� , **u***<sup>f</sup>* � , **u**�

3/2 <sup>∑</sup> *ij*

exp

*<sup>d</sup>*) *d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

exp

*<sup>d</sup>*) *dg d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

<sup>−</sup> *mc*� *<sup>c</sup>*<sup>2</sup> *c*� <sup>2</sup>*kT* <sup>−</sup> *mf* � *<sup>c</sup>*<sup>2</sup>

<sup>−</sup> *mcd <sup>g</sup>*<sup>2</sup> <sup>2</sup>*kT*

> *f* � <sup>2</sup>*kT* <sup>−</sup> *mdc*�<sup>2</sup>

exp *Dc* <sup>+</sup> *Dd* <sup>−</sup> *Dc*� <sup>−</sup> *Dd*�

*<sup>c</sup>* (*T*) exp *Dc*

*Fcij*∇ · **<sup>v</sup>** <sup>−</sup> <sup>1</sup>

*n Gcij*

×

*<sup>d</sup>*. (34)

*d* <sup>2</sup>*kT*

*<sup>d</sup>*. (35)

*kT*

*kT*

×

, (36)

, (37)

. (38)

*νdiss*, *<sup>c</sup> k*

*νdiss*, *<sup>c</sup> k*

(0)

reaction rate coefficients:

*k* (0) *<sup>b</sup>*,*ex*(*T*)

*k* (0) *<sup>f</sup>* ,*ex*(*T*)

*K*(0) *diss*(*T*) =

(0) *cij* − 1 *n*

Functions **A***cij*, **B***cij*, **D***<sup>d</sup>*

stress tensor we obtain

*<sup>K</sup>*(0) *ex* (*T*) =

internal energy.

*f* (1) *cij* = *f* *s c ij* exp

*<sup>b</sup>*, *diss*(*T*) = −N <sup>2</sup>

(0)

*<sup>f</sup>* , *diss*(*T*) = <sup>−</sup> <sup>4</sup>*π*N*<sup>A</sup>*

− *εc ij kT*

*A*

*<sup>σ</sup>rec*, *cij*

models such as the Arrhenius one are commonly used.

 *mcmd mc*�*md*�

= N*<sup>A</sup>*

**<sup>A</sup>***cij* · ∇ ln *<sup>T</sup>* <sup>−</sup> <sup>1</sup>

 *mc mc*�*mf* �

**2.4 First order reaction rate coefficients. Chemical kinetics in viscous gases**

*<sup>n</sup>* ∑ *d* **D***d*

operators of elastic collisions and inelastic ones with internal energy transitions.

=

*k* (0) *b*, *diss k* (0) *f* , *diss* *Zint <sup>c</sup>* (*T*)

*g*3*σdiss*

The zero-order recombination (backward for dissociation) rate coefficient reads

3/2 *Zint*

*Zint <sup>c</sup>*� (*T*)*Zint*

3/2

9/2 ∑ *ij*

Thus, if the cross-sections of the corresponding reactions are known, the zero-order rate coefficients can easily be calculated. However, for practical applications, phenomenological

Using the detailed balance principle (8)–(9), one can obtain the ratios of forward and backward

*<sup>c</sup>* (*T*)*Zint*

3/2

*Dc* is the dissociation energy of molecule *c*, *Dc* + *Dd* − *Dc*� − *Dd*� is heat effect of an exchange reaction. Formulas (36), (37) express the chemical-equilibrium constants well known from thermodynamics and hold only for Maxwell-Boltzmann distributions over velocity and

First-order distribution functions are obtained in Nagnibeda & Kustova (2009) in the form

*cij* · **<sup>d</sup>***<sup>d</sup>* <sup>−</sup> <sup>1</sup>

Let us consider the first-order transport terms in equations Eqs. (10)–(12). For the viscous

Here, *prel* is the relaxation pressure, *η* and *ζ* are the coefficients of shear and bulk viscosity. In the one-temperature approach, the additional terms connected to the bulk viscosity and

*n*

**<sup>B</sup>***cij* : <sup>∇</sup>**<sup>v</sup>** <sup>−</sup> <sup>1</sup>

*P* = (*p* − *prel*)*I* − 2*ηS* − *ζ* ∇ ·**v***I*. (39)

*cij*, *Fcij* and *Gcij* satisfy the linear integral equations with the linearized

*n*

*<sup>d</sup>* (*T*)

*<sup>d</sup>*� (*T*)

*h*3(2*πkT*)−3/2*Zint*

*mc*�*mf* �*md*

(2*πkT*)

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* (**u***c*, **u***d*, **u***c*� , **u***<sup>f</sup>* � , **u**�

(Euler) approximation, the governing equations describe non-equilibrium chemical kinetics in a thermally equilibrium inviscid non-conducting gas mixture:

$$
\rho \frac{d\mathbf{a}\_{\mathcal{C}}}{dt} = \sum\_{r} \dot{\xi}\_{r}^{(0)} \nu\_{\text{rc}} M\_{\text{c}} \qquad \mathbf{c} = 1, \dots, L,\tag{26}
$$

$$
\rho \frac{d\mathbf{v}}{dt} = -\nabla p\_{\prime} \tag{27}
$$

$$
\rho \frac{d\mathbf{U}}{dt} = -p \,\nabla \cdot \mathbf{v}.\tag{28}
$$

The right-hand side in Eqs. (26) contain the zero-order reaction rates:

$$\dot{\xi}\_r^{(0)} = k\_{f,r}^{(0)} \prod\_{c=1}^{L} \left(\frac{\rho\_c}{M\_c}\right)^{\nu\_{\rm ref}^{(r)}} - k\_{b,r}^{(0)} \prod\_{c=1}^{L} \left(\frac{\rho\_c}{M\_c}\right)^{\nu\_{\rm ref}^{(p)}}.\tag{29}$$

Here *k* (0) *f* ,*r* , *k* (0) *<sup>b</sup>*,*<sup>r</sup>* are the thermal-equilibrium reaction-rate coefficients. For the exchange reaction (4), when the partner *Ad* is a molecule, *k* (0) *<sup>f</sup>* ,*ex*, *k* (0) *<sup>b</sup>*,*ex* are given by:

$$\nu\_{\varepsilon x,\varepsilon}k\_{f,\varepsilon x}^{(0)}(T) = -\frac{\mathcal{N}\_A}{Z\_c^{\rm int}(T)Z\_d^{\rm int}(T)} \left(\frac{m\_{cd}}{2\pi kT}\right)^{3/2} \sum\_{\vec{k}\vec{l}\vec{k}\prime} \sum\_{\vec{l}\vec{l}\prime\prime l} \int \exp\left(-\frac{m\_{cd}\xi^2}{2kT}\right) \times$$

$$s\_{\vec{l}\prime j}^{\varepsilon} s\_{\vec{k}\prime}^{d} \exp\left(-\frac{\varepsilon\_{\vec{l}\prime}^{\varepsilon} + \varepsilon\_{\vec{k}\prime}^{d}}{kT}\right) g^{3} \sigma\_{cd,\vec{j}\prime\prime l}^{\varepsilon'd\prime\prime l}(\mathcal{g},\Omega) \, dg \, d^2\Omega,\tag{30}$$

$$\nu\_{\varepsilon x,\varepsilon} \, k\_{b,\varepsilon x}^{(0)}(T) = -\frac{\mathcal{N}\_A}{Z\_{c'}^{\rm int}(T)Z\_{d'}^{\rm int}(T)} \left(\frac{m\_{c'd'}}{2\pi kT}\right)^{3/2} \sum\_{\vec{l}\vec{l}\prime\prime l'} \sum\_{\vec{l}\vec{l}\prime\prime l'l'} \int \exp\left(-\frac{m\_{c'd'}\xi^2}{2kT}\right) \times$$

$$s\_{\vec{l}\prime j}^{\varepsilon'} s\_{\vec{k}\prime\prime l}^{d\prime\prime} \exp\left(-\frac{\varepsilon\_{\vec{l}\prime\prime}^{\varepsilon} + \varepsilon\_{\vec{k}\prime\prime l}^{d\prime}}{kT}\right) g^{3} \sigma\_{c'd,\vec{j}\prime\prime l}^{d\prime\prime l\prime}(\mathcal{g}',\Omega) \, dg' \, d^2\Omega,\tag{31}$$

and if *Ad* is an atom,

$$\nu\_{cx,c}k\_{f,cx}^{(0)}(T) = -\frac{N\_A}{Z\_c^{int}(T)} \left(\frac{m\_{cd}}{2\pi kT}\right)^{3/2} \sum\_{\overrightarrow{ll'}} \sum\_{\overrightarrow{jj'}} \int \exp\left(-\frac{m\_{cd}g^2}{2kT}\right) \times$$

$$s\_{ij}^c \exp\left(-\frac{\xi\_{ij}^c}{kT}\right) g^3 \sigma\_{cd,ij}^{c'd',i'j'}(g,\Omega) \, dg \, d^2\Omega. \tag{32}$$

$$\nu\_{cx,c} \, k\_{b,cx}^{(0)}(T) = -\frac{N\_A}{Z\_{c'}^{int}(T)} \left(\frac{m\_{c'd'}}{2\pi kT}\right)^{3/2} \sum\_{\overrightarrow{ll'}} \sum\_{\overrightarrow{jj'}} \int \exp\left(-\frac{m\_{c'd'}g'^2}{2kT}\right) \times$$

$$s\_{i'j'}^{\mathcal{E}'} \exp\left(-\frac{\xi\_{i'j}^{\mathcal{E}'}}{kT}\right) g'^3 \sigma\_{c'd',i'j'}^{c,d,ij}(g',\Omega) \, dg' \, d^2\Omega. \tag{33}$$

6 Will-be-set-by-IN-TECH

(Euler) approximation, the governing equations describe non-equilibrium chemical kinetics

(0) *<sup>r</sup> <sup>ν</sup>rcMc*, *<sup>c</sup>* <sup>=</sup> 1, .., *<sup>L</sup>*, (26)

*dt* <sup>=</sup> −∇*<sup>p</sup>* , (27)

*dt* <sup>=</sup> <sup>−</sup>*<sup>p</sup>* ∇· **<sup>v</sup>**. (28)

� *ρ<sup>c</sup> Mc* �*ν* (*p*) *rc*

� exp �

� exp �

. (29)

� ×

> � ×

<sup>−</sup> *mcd <sup>g</sup>*<sup>2</sup> 2*kT*

<sup>−</sup> *mc*�*d*� *<sup>g</sup>*�<sup>2</sup> 2*kT*

, Ω) *dg*� *d*2Ω, (31)

� ×

> � ×

, Ω) *dg*� *d*2Ω. (33)

*cd*, *ijkl* (*g*, <sup>Ω</sup>) *dg d*2Ω, (30)

<sup>−</sup> *mcd <sup>g</sup>*<sup>2</sup> 2*kT*

*cd*, *ij* (*g*, <sup>Ω</sup>) *dg d*2Ω. (32)

<sup>−</sup> *mc*�*d*� *<sup>g</sup>*�<sup>2</sup> 2*kT*

˙ *ξ*

� *ρ<sup>c</sup> Mc*

> (0) *<sup>f</sup>* ,*ex*, *k* (0)

*<sup>d</sup>* (*T*)

*<sup>d</sup>*� (*T*)

*<sup>i</sup>*�*j*� <sup>+</sup> *<sup>ε</sup>d*� *k*�*l*� *kT*

�*ν* (*r*) *rc* − *k* (0) *b*,*r L* ∏*c*=1

� *mcd* 2*πkT*

� *g*3*σc*� *d*� , *i* � *j* � *k*� *l* �

� *mc*�*d*� 2*πkT*

> ⎞ <sup>⎠</sup> *<sup>g</sup>*�3*<sup>σ</sup>*

� *mcd* 2*πkT*

� *g*3*σc*� *d*� , *i* � *j* �

� *mc*�*d*� 2*πkT*

> ⎞ <sup>⎠</sup> *<sup>g</sup>*�3*<sup>σ</sup>*

*<sup>b</sup>*,*<sup>r</sup>* are the thermal-equilibrium reaction-rate coefficients. For the exchange reaction

�3/2

�3/2

*cd*, *ijkl c*�*d*�

�3/2 ∑ *ii*� ∑ *jj*�

�3/2 ∑ *ii*� ∑ *jj*�

*cd*, *ij c*�*d*� , *<sup>i</sup>*�*j*�(*g*�

*<sup>b</sup>*,*ex* are given by:

∑ *iki*�*k*� ∑ *jlj*�*l*�

∑ *iki*�*k*� ∑ *jlj*�*l*�

, *<sup>i</sup>*�*j*�*k*�*l*�(*g*�

� exp �

� exp �

in a thermally equilibrium inviscid non-conducting gas mixture:

*ρ dαc dt* <sup>=</sup> <sup>∑</sup>*<sup>r</sup>*

*ρ d***v**

*ρ dU*

˙ *ξ* (0) *<sup>r</sup>* <sup>=</sup> *<sup>k</sup>* (0) *f* ,*r L* ∏*c*=1

*<sup>f</sup>* ,*ex*(*T*) = <sup>−</sup> <sup>N</sup>*<sup>A</sup> Zint <sup>c</sup>* (*T*)*Zint*

> � −*εc ij* <sup>+</sup> *<sup>ε</sup><sup>d</sup> kl kT*

⎛ ⎝−*εc*�

*<sup>f</sup>* ,*ex*(*T*) = <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*s c ij* exp

*<sup>b</sup>*,*ex*(*T*) = <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*s c*� *<sup>i</sup>*�*j*� exp

*Zint <sup>c</sup>* (*T*)

> � − *εc ij kT*

*Zint <sup>c</sup>*� (*T*)

> ⎛ ⎝−*εc*� *i*�*j*� *kT*

*s c ijs d kl* exp

*s c*� *<sup>i</sup>*�*j*�*s d*� *<sup>k</sup>*�*l*� exp

*νex*, *c k* (0)

*νex*, *c k* (0)

*<sup>b</sup>*,*ex*(*T*) = <sup>−</sup> <sup>N</sup>*<sup>A</sup> Zint <sup>c</sup>*� (*T*)*Zint*

(4), when the partner *Ad* is a molecule, *k*

*νex*, *c k* (0)

*νex*, *c k* (0)

and if *Ad* is an atom,

Here *k* (0) *f* ,*r* , *k* (0)

The right-hand side in Eqs. (26) contain the zero-order reaction rates:

For the dissociation reaction (5), the zero-order forward-rate coefficient is obtained in the form

$$\nu\_{\rm diss,c}k\_{f,\rm diss}^{(0)}(T) = -\frac{4\pi N\_A}{Z\_c^{int}(T)} \left(\frac{m\_{cd}}{2\pi kT}\right)^{3/2} \sum\_{ij} \int \exp\left(-\frac{m\_{cd}g^2}{2kT}\right) \times$$

$$s\_{ij}^c \exp\left(-\frac{\varepsilon\_{ij}^c}{kT}\right) g^3 \sigma\_{cij,d}^{diss}(g, \mathbf{u}\_{\mathbf{c}'}, \mathbf{u}\_{f'}, \mathbf{u}\_d') \, dg \, d\mathbf{u}\_{\mathbf{c}'} \, d\mathbf{u}\_{f'} \, d\mathbf{u}\_d'.\tag{34}$$

The zero-order recombination (backward for dissociation) rate coefficient reads

$$\nu\_{\rm diss,c} k\_{b,\rm diss}^{(0)}(T) = -N\_A^2 \frac{\left(m\_{c'} m\_{f'} m\_d\right)^{3/2}}{\left(2\pi kT\right)^{9/2}} \sum\_{\vec{m}} \int \exp\left(-\frac{m\_{c'} c\_{c'}^2}{2kT} - \frac{m\_{f'} c\_{f'}^2}{2kT} - \frac{m\_d c\_d'^2}{2kT}\right) \times$$

$$\sigma\_{c'f'd}^{\rm rec,c\vec{m}}(\mathbf{u}\_{c'}, \mathbf{u}\_{d'}, \mathbf{u}\_{f'}, \mathbf{u}\_{f'}') \, d\mathbf{u}\_c \, d\mathbf{u}\_d \, d\mathbf{u}\_{d'} \, d\mathbf{u}\_{f'} \, d\mathbf{u}\_{d'}'.\tag{35}$$

Thus, if the cross-sections of the corresponding reactions are known, the zero-order rate coefficients can easily be calculated. However, for practical applications, phenomenological models such as the Arrhenius one are commonly used.

Using the detailed balance principle (8)–(9), one can obtain the ratios of forward and backward reaction rate coefficients:

$$K\_{\rm ex}^{(0)}(T) = \frac{k\_{b,\rm ex}^{(0)}(T)}{k\_{f,\rm ex}^{(0)}(T)} = \left(\frac{m\_{\rm c}m\_{\rm d}}{m\_{\rm c}m\_{\rm d'}}\right)^{3/2} \frac{Z\_{\rm c}^{\rm int}(T)Z\_{\rm d}^{\rm int}(T)}{Z\_{\rm c'}^{\rm int}(T)Z\_{\rm d'}^{\rm int}(T)} \exp\left(\frac{D\_{\rm c} + D\_{\rm d} - D\_{\rm c'} - D\_{\rm d'}}{kT}\right), \tag{36}$$

$$K\_{\rm diss}^{(0)}(T) = \frac{k\_{b,\rm diss}^{(0)}}{k\_{f,\rm diss}^{(0)}} = \mathcal{N}\_A \left(\frac{m\_c}{m\_{c'}m\_{f'}}\right)^{3/2} h^3 (2\pi kT)^{-3/2} Z\_c^{\rm int}(T) \exp\left(\frac{D\_c}{kT}\right),\tag{37}$$

*Dc* is the dissociation energy of molecule *c*, *Dc* + *Dd* − *Dc*� − *Dd*� is heat effect of an exchange reaction. Formulas (36), (37) express the chemical-equilibrium constants well known from thermodynamics and hold only for Maxwell-Boltzmann distributions over velocity and internal energy.

#### **2.4 First order reaction rate coefficients. Chemical kinetics in viscous gases**

First-order distribution functions are obtained in Nagnibeda & Kustova (2009) in the form

$$f\_{\vec{\alpha}\vec{j}}^{(1)} = f\_{\vec{\alpha}\vec{j}}^{(0)} \left( -\frac{1}{n} \mathbf{A}\_{\vec{\alpha}\vec{j}} \cdot \nabla \ln T - \frac{1}{n} \sum\_{d} \mathbf{D}\_{\vec{\alpha}\vec{j}}^{d} \cdot \mathbf{d}\_{d} - \frac{1}{n} \mathbf{B}\_{\vec{\alpha}\vec{j}} : \nabla \mathbf{v} - \frac{1}{n} F\_{\vec{\alpha}\vec{j}} \nabla \cdot \mathbf{v} - \frac{1}{n} G\_{\vec{\alpha}\vec{j}} \right). \tag{38}$$

Functions **A***cij*, **B***cij*, **D***<sup>d</sup> cij*, *Fcij* and *Gcij* satisfy the linear integral equations with the linearized operators of elastic collisions and inelastic ones with internal energy transitions.

Let us consider the first-order transport terms in equations Eqs. (10)–(12). For the viscous stress tensor we obtain

$$\mathbf{P} = (p - p\_{rel})\mathbf{I} - 2\eta \mathbf{S} - \zeta \nabla \cdot \mathbf{v} \mathbf{I}.\tag{39}$$

Here, *prel* is the relaxation pressure, *η* and *ζ* are the coefficients of shear and bulk viscosity. In the one-temperature approach, the additional terms connected to the bulk viscosity and

*νex*, *<sup>c</sup>* ˜ *k* (1)

*νex*, *<sup>c</sup>* ¯ *k* (1)

*<sup>ν</sup>diss*, *<sup>c</sup>* ¯ *k* (1)

*<sup>ν</sup>diss*, *<sup>c</sup>* ˜ *k* (1)

*<sup>ν</sup>diss*, *<sup>c</sup>* ¯ *k* (1)

*<sup>ν</sup>diss*, *<sup>c</sup>* ˜ *k* (1)

expanding flows.

*νex*, *<sup>c</sup>* ˜ *k* (1) *<sup>f</sup>* ,*ex* <sup>=</sup> −∇ · **<sup>v</sup>** <sup>N</sup>*<sup>A</sup>*

*<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

> *<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

*<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

> *<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

> > *f* (0) *<sup>c</sup>*� *f* (0) *<sup>f</sup>* � *<sup>f</sup>* �(0) *dkl*

tensor connected to the bulk viscosity and relaxation pressure.

*<sup>b</sup>*,*ex* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*<sup>b</sup>*,*ex* <sup>=</sup> −∇ · **<sup>v</sup>** <sup>N</sup>*<sup>A</sup>*

*<sup>f</sup>* , *diss* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*<sup>f</sup>* , *diss* <sup>=</sup> −∇ · **<sup>v</sup>** <sup>N</sup>*<sup>A</sup>*

*A <sup>n</sup>* ∑ *ik* ∑ *jl*

> *A <sup>n</sup>* ∑ *ik* ∑ *jl*

*<sup>b</sup>*, *diss* <sup>=</sup> <sup>−</sup> <sup>N</sup> <sup>2</sup>

*<sup>b</sup>*, *diss* <sup>=</sup> −∇· **<sup>v</sup>** <sup>N</sup> <sup>2</sup>

*<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

> *f* (0) *<sup>c</sup>*�*i*�*j*� *f* (0) *d*�*k*�*l*� *nc*�*nd*�

> > *f* (0) *<sup>c</sup>*�*i*�*j*� *f* (0) *d*�*k*�*l*� *nc*�*nd*�

 *f* (0) *cij f* (0) *dkl ncnd*

> *f* (0) *cij f* (0) *dkl ncnd*

*nc*�*nf* �*nd*

*nc*�*nf* �*nd*

 *f* (0) *<sup>c</sup>*� *f* (0) *<sup>f</sup>* � *<sup>f</sup>* �(0) *dkl*

 *f* (0) *cij f* (0) *dkl ncnd*

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 123

*Gc*�*i*�*j*� + *Gd*�*k*�*l*�

*Gcij* <sup>+</sup> *Gdkl*

*Fc*�*i*�*j*� + *Fd*�*k*�*l*�

*Fcij* <sup>+</sup> *Fdkl*

*Gc*� <sup>+</sup> *Gf* � <sup>+</sup> *Gdkl*

*Fc*� <sup>+</sup> *Ff* � <sup>+</sup> *Fdkl*

It can be noted that the first-order corrections for the reaction rate coefficients depend on the same functions *Fcij* and *Gcij* which define the additional diagonal elements of the pressure

Algorithms for the calculation of vector and tensor transport properties and first order corrections to reaction rate coefficients are described in details in Nagnibeda & Kustova (2009). In Ref. Alexeev & Grushin (1994), a procedure for the calculation of the first-order reaction rate coefficients has been developed for gases without internal degrees of freedom. In Ref. Kustova et al. (2008), the scalar functions *Fcij* and *Gcij* are considered, and transport linear systems for the calculation of bulk viscosity, chemical-reaction contribution to the normal mean stress and first-order reaction rate coefficients are derived taking into account internal energy of molecules. Numerical estimations of the first-order rate coefficients in reacting viscous gas flows remain an open question up to now. In the simulations of viscous flows, the first-order corrections to the rate coefficients are usually neglected as well as relaxation pressure and bulk viscosity. Some results in this field have been recently obtained in Ref. Kustova (2009) where numerical estimations of the normal mean stress and the first order corrections to the dissociation and recombination rates in the mixture N2/N have been performed. It is shown that whereas the first-order contribution to the normal mean stress remains small, the first-order corrections to the reaction rates are not negligible in both shock heated and

*Fcij* <sup>+</sup> *Fdkl*

*<sup>g</sup> <sup>σ</sup>c*� *d*� , *i* � *j* � *k*� *l* �

 *g*� *σ cd*, *ijkl c*�*d*�

> *g*� *σ cd*, *ijkl c*�*d*�

*g σdiss*

*g σdiss*

*<sup>σ</sup>rec*, *cij*

*<sup>σ</sup>rec*, *cij*

*cd*, *ijkl <sup>d</sup>*2Ω*d***u***dd***u***c*, (44)

, *<sup>i</sup>*�*j*�*k*�*l*� *<sup>d</sup>*2Ω*d***u***d*� *<sup>d</sup>***u***c*� , (45)

, *<sup>i</sup>*�*j*�*k*�*l*� *<sup>d</sup>*2Ω*d***u***d*� *<sup>d</sup>***u***c*� , (46)

*<sup>d</sup>*, (47)

*<sup>d</sup>*, (48)

*d*,

(49)

(50)

*d*.

*cij*, *<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*cij*, *<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

relaxation pressure appear in the diagonal terms of the stress tensor due to rapid inelastic internal energy transitions. The existence of the relaxation pressure is caused also by slow processes of chemical reactions proceeding on the gas-dynamic time scale.

The transport coefficients in the expression (39) can be written in terms of functions **B***cij*, *Fcij*, and *Gcij*:

$$
\eta = \frac{kT}{10} \left[ \mathbf{B} , \mathbf{B} \right], \qquad \zeta = kT \left[ F , F \right], \qquad p\_{rel} = kT \left[ F , G \right]. \tag{40}
$$

In these formulae, [*A*, *B*] (where *A*, *B* are arbitrary functions of molecular velocities) denotes a bilinear form depending on the linearized integral collision operator for rapid processes. In the kinetic theory, such bilinear forms are basically called bracket integrals. The bracket integrals in the expressions (40) are introduced in Nagnibeda & Kustova (2009) similarly to those defined in Ferziger & Kaper (1972) for a non-reacting gas mixture under the conditions of weak deviations from the equilibrium.

The diffusion velocity and the total energy flux in the considered approach are specified by the functions **D***<sup>d</sup> cij*, **<sup>A</sup>***<sup>d</sup> cij* also depending on the cross sections of rapid processes and are studied in Kustova & Nagnibeda (2011).

Thus, the governing equations (10)–(12) with the first order transport terms describe a flow of reacting mixture of viscous gases with strong non-equilibrium chemical reactions in the Navier-Stokes approximation. Transport properties in the one-temperature approach in reacting gas mixtures are considered in Ern & Giovangigli (1994); Kustova (2009); Kustova et al. (2008); Nagnibeda & Kustova (2009).

The chemical reaction rate coefficients contributing to the production terms *Rreact <sup>c</sup>* in the equations (10) or to the reaction rates in the equations (15) in the first-order approximation are defined by the first order distribution functions (38) and depend on the cross sections of reactive collisions.

The chemical-reaction rate in Eqs. (15) in the first-order approximation has the form (23) where

$$k\_{f,r} = k\_{f,r}^{(0)}\left(T\right) - \check{k}\_{f,r}^{(1)}\left(\mathfrak{a}\_{1}, \dots, \mathfrak{a}\_{L'}\rho\_{\prime}T\right) - \check{k}\_{f,r}^{(1)}\left(\mathfrak{a}\_{1}, \dots, \mathfrak{a}\_{L'}\rho\_{\prime}T\right),\tag{41}$$

$$k\_{b,r} = k\_{b,r}^{(0)}\left(T\right) - \tilde{k}\_{b,r}^{(1)}\left(a\_1, \dots, a\_{L,r}\rho\_r T\right) - \tilde{k}\_{b,r}^{(1)}\left(a\_1, \dots, a\_{L,r}\rho\_r T\right). \tag{42}$$

Quantities ¯ *k* (1) *f* ,*r* , ˜ *k* (1) *f* ,*r* , ¯ *k* (1) *<sup>b</sup>*,*r*, ˜ *k* (1) *<sup>b</sup>*,*<sup>r</sup>* express first-order corrections to the reaction-rate coefficients (17)–(20). The terms ¯ *k* (1) *f* ,*r* , ¯ *k* (1) *<sup>b</sup>*,*<sup>r</sup>* are due to deviations from Maxwell-Boltzmann distributions over velocities and internal energies whereas the terms ˜ *k* (1) *f* ,*r* , ˜ *k* (1) *<sup>b</sup>*,*<sup>r</sup>* are due to spatial non-homogeneity. If internal degrees of freedom are neglected, the coefficients ˜ *k* (1) *<sup>f</sup>* ,*<sup>r</sup>* and ˜ *k* (1) *b*,*r* vanish.

The first-order corrections to the reaction-rate coefficients are defined by the expressions:

$$\left(\nu\_{\rm ex,c}\,\vec{k}\_{f,\rm ex}^{(1)} = -\frac{\mathcal{N}\_A}{n}\sum\_{\rm ikl'k'}\sum\_{\rm l\tilde{j}\,\tilde{l}'\,\rm l}\int \frac{f\_{\rm c\tilde{j}\,j}^{(0)}f\_{\rm dkl}^{(0)}}{n\_\rm c\eta\_{\rm d}}\left(\mathcal{G}\_{\rm c\tilde{j}\,\rm l} + \mathcal{G}\_{\rm dkl}\right)\,\rm g\,\sigma\_{\rm cd,ijkl}^{\rm c'd',\tilde{i}'\tilde{j}\,\rm l'\ell'}\,d^2\Omega d\mathbf{u}\_{\rm d}d\mathbf{u}\_{\rm c\nu} \tag{43}$$

8 Will-be-set-by-IN-TECH

relaxation pressure appear in the diagonal terms of the stress tensor due to rapid inelastic internal energy transitions. The existence of the relaxation pressure is caused also by slow

The transport coefficients in the expression (39) can be written in terms of functions **B***cij*, *Fcij*,

In these formulae, [*A*, *B*] (where *A*, *B* are arbitrary functions of molecular velocities) denotes a bilinear form depending on the linearized integral collision operator for rapid processes. In the kinetic theory, such bilinear forms are basically called bracket integrals. The bracket integrals in the expressions (40) are introduced in Nagnibeda & Kustova (2009) similarly to those defined in Ferziger & Kaper (1972) for a non-reacting gas mixture under the conditions

The diffusion velocity and the total energy flux in the considered approach are specified by

Thus, the governing equations (10)–(12) with the first order transport terms describe a flow of reacting mixture of viscous gases with strong non-equilibrium chemical reactions in the Navier-Stokes approximation. Transport properties in the one-temperature approach in reacting gas mixtures are considered in Ern & Giovangigli (1994); Kustova (2009); Kustova

equations (10) or to the reaction rates in the equations (15) in the first-order approximation are defined by the first order distribution functions (38) and depend on the cross sections of

The chemical-reaction rate in Eqs. (15) in the first-order approximation has the form (23) where

*<sup>f</sup>* ,*<sup>r</sup>* (*α*1, ..., *<sup>α</sup>L*, *<sup>ρ</sup>*, *<sup>T</sup>*) <sup>−</sup> ˜

*<sup>b</sup>*,*<sup>r</sup>* (*α*1, ..., *<sup>α</sup>L*, *<sup>ρ</sup>*, *<sup>T</sup>*) <sup>−</sup> ˜

non-homogeneity. If internal degrees of freedom are neglected, the coefficients ˜

 *f* (0) *cij f* (0) *dkl ncnd*

The first-order corrections to the reaction-rate coefficients are defined by the expressions:

*Gcij* + *Gdkl*

The chemical reaction rate coefficients contributing to the production terms *Rreact*

<sup>10</sup> [**B**, **<sup>B</sup>**] , *<sup>ζ</sup>* <sup>=</sup> *kT* [*F*, *<sup>F</sup>*] , *prel* <sup>=</sup> *kT* [*F*, *<sup>G</sup>*] . (40)

*cij* also depending on the cross sections of rapid processes and are studied

*k* (1)

*k* (1)

*<sup>b</sup>*,*<sup>r</sup>* express first-order corrections to the reaction-rate coefficients

*<sup>b</sup>*,*<sup>r</sup>* are due to deviations from Maxwell-Boltzmann distributions

 *<sup>g</sup> <sup>σ</sup>c*� *d*� , *i* � *j* � *k*� *l* �

*k* (1) *f* ,*r* , ˜ *k* (1) *<sup>c</sup>* in the

*<sup>f</sup>* ,*<sup>r</sup>* (*α*1, ..., *αL*, *ρ*, *T*), (41)

*<sup>b</sup>*,*<sup>r</sup>* (*α*1, ..., *αL*, *ρ*, *T*). (42)

*<sup>b</sup>*,*<sup>r</sup>* are due to spatial

*k* (1) *<sup>f</sup>* ,*<sup>r</sup>* and ˜ *k* (1) *b*,*r*

*cd*, *ijkl <sup>d</sup>*2Ω*d***u***dd***u***c*, (43)

processes of chemical reactions proceeding on the gas-dynamic time scale.

*<sup>η</sup>* <sup>=</sup> *kT*

of weak deviations from the equilibrium.

et al. (2008); Nagnibeda & Kustova (2009).

*k <sup>f</sup>* ,*<sup>r</sup>* = *k*

*kb*,*<sup>r</sup>* = *k*

*k* (1) *f* ,*r* , ¯ *k* (1)

*<sup>f</sup>* ,*ex* <sup>=</sup> <sup>−</sup> <sup>N</sup>*<sup>A</sup>*

*<sup>n</sup>* ∑ *iki*�*k*� ∑ *jlj*�*l*�

(0) *<sup>f</sup>* ,*<sup>r</sup>* (*T*) <sup>−</sup> ¯ *k* (1)

(0) *<sup>b</sup>*,*<sup>r</sup>* (*T*) <sup>−</sup> ¯ *k* (1)

over velocities and internal energies whereas the terms ˜

*cij*, **<sup>A</sup>***<sup>d</sup>*

in Kustova & Nagnibeda (2011).

and *Gcij*:

the functions **D***<sup>d</sup>*

reactive collisions.

Quantities ¯

vanish.

*k* (1) *f* ,*r* , ˜ *k* (1) *f* ,*r* , ¯ *k* (1) *<sup>b</sup>*,*r*, ˜ *k* (1)

(17)–(20). The terms ¯

*νex*, *<sup>c</sup>* ¯ *k* (1)

$$\nu\_{\varepsilon x,\varepsilon} \dot{\mathcal{L}}\_{f,\varepsilon\mathbf{x}}^{(1)} = -\nabla \cdot \mathbf{v} \, \frac{\mathcal{N}\_A}{n} \sum\_{\text{i}\mathbf{i}\mathbf{i}'\mathbf{k}'} \sum\_{\text{i}\mathbf{j}'\mathbf{j}'l'} \int \frac{f\_{\text{cij}}^{(0)} f\_{\text{d}\mathbf{k}l}^{(0)}}{n\_{\text{c}} n\_{\text{d}}} \left(\mathcal{F}\_{\text{cij}} + \mathcal{F}\_{\text{d}\mathbf{k}l}\right) \, \text{g} \, \sigma\_{\text{cd,ij}\mathbf{k}l}^{\varepsilon'\mathbf{d}',\varepsilon'\mathbf{j}'k'l'} d^2 \Omega d\mathbf{u}\_{\text{d}} d\mathbf{u}\_{\text{c}} \tag{44}$$

$$\nu\_{\rm cx,c} \cdot \vec{k}\_{b,\rm cx}^{(1)} = -\frac{\mathcal{N}\_A}{n} \sum\_{\rm i\vec{k}\vec{r}\,\rm l\prime} \sum\_{\vec{j}\vec{j}\,\rm l\prime} \int \frac{f\_{c'\vec{i}\,\rm l\prime}^{(0)} f\_{d'\vec{k}\,\rm l\prime}^{(0)}}{n\_{c'}n\_{d'}} \left(\mathbb{G}\_{c'\vec{i}\,\rm l\prime} + \mathbb{G}\_{d'\vec{k}\,\rm l\prime}\right) \,\rm g\,\prime \,\sigma\_{c'd',\vec{i}\,\rm l\prime}^{\rm cl,ij\,\rm kl} d^2 \Omega d\mathbf{u}\_{d'} d\mathbf{u}\_{c'\prime} \tag{45}$$

$$\nu\_{\rm ex,c} \tilde{k}\_{b,\rm ex}^{(1)} = -\nabla \cdot \mathbf{v} \frac{\mathcal{N}\_A}{n} \sum\_{\rm ikl'k'} \sum\_{\rm jlj'l'} \int \frac{f\_{\rm c'l'j'}^{(0)} f\_{\rm d'k'l'}^{(0)}}{n\_{\rm c'} n\_{\rm d'}} \left( \mathbf{F}\_{\rm c'l'j'} + \mathbf{F}\_{\rm d'k'l'} \right) \mathbf{g'}\_{\rm c'd',i'j'k'l'}^{\rm cd,ijkl} d^2 \Omega d\mathbf{u}\_{\rm d'} d\mathbf{u}\_{\rm c'} \quad \text{(46)}$$

$$\nu\_{\rm diss,c} \vec{k}\_{f,\rm diss}^{(1)} = -\frac{\mathcal{N}\_A}{n} \sum\_{\rm ikr'k'} \sum\_{\rm jlj'l'} \int \frac{f\_{\rm cij}^{(0)} f\_{\rm dkl}^{(0)}}{n\_{\rm c} n\_{\rm d}} \left( \mathcal{G}\_{\rm cij} + \mathcal{G}\_{\rm dkl} \right) \, g \, \sigma\_{\rm cij,d}^{\rm diss} \, d\mathbf{u}\_{\rm c} \, d\mathbf{u}\_{\rm d} \, d\mathbf{u}\_{\rm c'} \, d\mathbf{u}\_{\rm f'} \, d\mathbf{u}\_{\rm d'}' \tag{47}$$

$$\mathbf{v}\_{\text{diss},\text{c}}\tilde{\mathbf{k}}\_{f,\text{diss}}^{(1)} = -\nabla \cdot \mathbf{v} \, \frac{\mathcal{N}\_A}{n} \sum\_{\text{ik}\mathbf{i}'\mathbf{k}'} \sum\_{\text{j}\mathbf{j}'\mathbf{j}'} \int \frac{f\_{\text{cij}}^{(0)} f\_{\text{d}\mathbf{k}\mathbf{i}}^{(0)}}{n\_{\text{c}}n\_{\text{d}}} \left(\mathbf{F}\_{\text{cij}} + \mathbf{F}\_{\text{d}\mathbf{k}}\right) \, \mathbf{g} \, \sigma\_{\text{cij},\text{d}}^{\text{diss}} \, d\mathbf{u}\_{\text{d}} \, d\mathbf{u}\_{\text{d}'} \, d\mathbf{u}\_{\text{f}'} \, d\mathbf{u}\_{\text{f}'} \, \text{(48)}$$

$$\mathcal{V}\_{\text{diss},c} \,\vec{k}\_{b,\text{diss}}^{(1)} = -\frac{N\_A^2}{n} \sum\_{\text{ik}} \sum\_{\text{jl}} \int \frac{f\_{c'}^{(0)} f\_{f'}^{(0)} f'\_{\text{dkl}}^{(0)}}{n\_{\text{c}'} n\_{f'} n\_{\text{d}}} \left(\text{G}\_{\text{c'}} + \text{G}\_{f'} + \text{G}\_{\text{dkl}}\right) \,\sigma\_{\text{c'}f'\text{d}}^{\text{rec,c\'ij}} \, d\mathbf{u}\_{\text{c}} \, d\mathbf{u}\_{\text{d}} \, d\mathbf{u}\_{\text{c'}} \, d\mathbf{u}\_{f'} \, d\mathbf{u}\_{\text{d'}} \, \tag{49}$$

*<sup>ν</sup>diss*, *<sup>c</sup>* ˜ *k* (1) *<sup>b</sup>*, *diss* <sup>=</sup> −∇· **<sup>v</sup>** <sup>N</sup> <sup>2</sup> *A <sup>n</sup>* ∑ *ik* ∑ *jl f* (0) *<sup>c</sup>*� *f* (0) *<sup>f</sup>* � *<sup>f</sup>* �(0) *dkl nc*�*nf* �*nd Fc*� <sup>+</sup> *Ff* � <sup>+</sup> *Fdkl <sup>σ</sup>rec*, *cij <sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup> d***u***<sup>c</sup> d***u***<sup>d</sup> d***u***c*� *d***u***<sup>f</sup>* � *d***u**� *d*. (50)

It can be noted that the first-order corrections for the reaction rate coefficients depend on the same functions *Fcij* and *Gcij* which define the additional diagonal elements of the pressure tensor connected to the bulk viscosity and relaxation pressure.

Algorithms for the calculation of vector and tensor transport properties and first order corrections to reaction rate coefficients are described in details in Nagnibeda & Kustova (2009). In Ref. Alexeev & Grushin (1994), a procedure for the calculation of the first-order reaction rate coefficients has been developed for gases without internal degrees of freedom. In Ref. Kustova et al. (2008), the scalar functions *Fcij* and *Gcij* are considered, and transport linear systems for the calculation of bulk viscosity, chemical-reaction contribution to the normal mean stress and first-order reaction rate coefficients are derived taking into account internal energy of molecules. Numerical estimations of the first-order rate coefficients in reacting viscous gas flows remain an open question up to now. In the simulations of viscous flows, the first-order corrections to the rate coefficients are usually neglected as well as relaxation pressure and bulk viscosity. Some results in this field have been recently obtained in Ref. Kustova (2009) where numerical estimations of the normal mean stress and the first order corrections to the dissociation and recombination rates in the mixture N2/N have been performed. It is shown that whereas the first-order contribution to the normal mean stress remains small, the first-order corrections to the reaction rates are not negligible in both shock heated and expanding flows.

(54), (55) are completely specified by the macroscopic gas parameters *nci*(**r**, *t*) (*c* = 1, ..., *L*, *i* = 0, 1, ..., *Lc*, *Lc* is the number of excited vibration levels of molecular species *c*), *T*(**r**, *t*), and

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 125

The set of equations for the macroscopic parameters *nci*(**r**, *t*), **v**(**r**, *t*), and *T*(**r**, *t*) follows from Eq. (2) with collision operators (52). This system includes equations of state-to-state vibrational and chemical kinetics in a flow Kustova & Nagnibeda (1998); Nagnibeda &

coupled to the conservation equations for the momentum and total energy which formally coincide with Eqs. (11)–(12). Here, **V***ci* is the diffusion velocity of molecules c at the vibrational state *i*. The total energy per unit mass *U* is specified by level populations *nci*(**r**, *t*), atomic

The source terms in equations (57) characterize the variation of the vibrational level populations and atomic number densities caused by different vibrational energy exchanges and chemical reactions and are expressed via the integral operators of slow processes:

*cij<sup>d</sup>* **<sup>u</sup>***<sup>c</sup>* <sup>=</sup> *<sup>R</sup>vibr*

The equations (57), (11), (12) provide a detailed description of vibrational and chemical kinetics and flow dynamics for weak deviations from the equilibrium distributions over the velocity and rotational energy levels and arbitrary deviations from the equilibrium for the vibrational degrees of freedom and chemical species. Let us emphasize that for such an approach, the vibrational level populations are included to the set of main macroscopic parameters, and particles of various chemical species in different vibrational states represent

*nci*�*ndk*� *<sup>k</sup>d*, *<sup>k</sup>*�

*nc*�*i*�*nd*�*k*� *<sup>k</sup>d*�

*nc*�*nf* � *kdk*

*k*

*ci* <sup>+</sup> *<sup>R</sup>diss ci* ,

*k*� , *dk c*�*i*�

*<sup>c</sup>*, *<sup>i</sup>*�*<sup>i</sup>* <sup>−</sup> *ncindkk<sup>d</sup>*,*kk*�

, *ci* <sup>−</sup> *ncindkkdk*, *<sup>d</sup>*�

*rec*, *ci* <sup>−</sup> *ncikdk*

*c*,*ii*� 

*ci*, *diss*

*Aci* + *Adk Aci*� + *Adk*� , (62)

*Aci* + *Adk Ac*�*i*� + *Ad*�*k*� (63)

*Aci* + *Adk Ac*� + *Af* � + *Adk*. (64)

*k*� *ci*, *c*�*i*� 

*dt* <sup>+</sup> *nci* ∇· **<sup>v</sup>** <sup>+</sup> ∇· (*nci***V***ci*) = *Rci*, *<sup>c</sup>* <sup>=</sup> 1, ..., *<sup>L</sup>*, *<sup>i</sup>* <sup>=</sup> 0, ..., *Lc*, (57)

*ci* <sup>+</sup> *<sup>R</sup>react*

*ci* . (58)

, (59)

, (60)

. (61)

**v**(**r**, *t*) and correspond to the set of the collision invariants of rapid processes.

Kustova (2009):

*dnci*

number densities *nc*(**r**, *t*) and gas temperature.

*Rci* = ∑ *j*

 *J sl*

the mixture components. The expressions (58) can be written in the form:

*Rreact ci* <sup>=</sup> *<sup>R</sup>ex*

*ndk*

*Rvibr ci* = ∑ *dki*�*k*�

*Rdiss ci* = ∑ *dk*

Here the rate coefficients are introduced for the energy exchange:

*Rex ci* = ∑ *dc*�*d*� ∑ *ki*�*k*�

exchange chemical reactions:

and dissociation-recombination reactions:

#### **3. State-to-state model for vibrational–chemical coupling**

#### **3.1 Distribution functions. Governing equations**

In this section, chemical kinetics in multi-component reacting gas mixture flows are studied under the conditions of strong vibrational and chemical non-equilibrium. Experimental results on the relaxation times of various processes demonstrate that, in a wide temperature range, the equilibration of translational and rotational degrees of freedom proceeds much faster compared to the vibrational relaxation and chemical reactions. The characteristic relaxation times satisfy the relation

$$
\pi\_{el} < \pi\_{rot} \ll \pi\_{vibr} \sim \pi\_{react} \sim \theta. \tag{51}
$$

where *τrot*, *τvibr* are relaxation times for rotational and vibrational degrees of freedom. Under such conditions, the integral operators in the kinetic equations (2) take the form:

$$J\_{\rm cij}^{rap} = J\_{\rm cij}^{el} + J\_{\rm cij}^{rot}, \qquad J\_{\rm cij}^{sl} = J\_{\rm cij}^{vibr} + J\_{\rm cij}^{react} \,. \tag{52}$$

In this case, the vibrational-chemical coupling in reacting flows becomes important.

The kinetic equations for the distribution functions in the zero-order Chapman-Enskog approximation have the form:

$$J\_{c\text{ij}}^{el(0)} + J\_{c\text{ij}}^{rot(0)} = 0.\tag{53}$$

In this case, the system of collision invariants for the most frequent collisions includes along with the momentum and a particle total energy, any value independent of the velocity and rotational level *j* and depending arbitrarily on the vibrational level *i* and chemical species *c*. This values are conserved at the most frequent collisions because, according to the condition (51), vibrational energy transitions and chemical reactions are forbidden in the rapid processes. Based on the above set of the collision invariants, the solution of Eqs. (53) takes the form

$$f\_{c\bar{c}j}^{(0)} = \left(\frac{m\_c}{2\pi kT}\right)^{3/2} s\_{\bar{j}}^{c\bar{i}} \frac{n\_{c\bar{i}}}{Z\_{c\bar{l}}^{\text{rot}}(T)} \exp\left(-\frac{m\_c c\_c^2}{2kT} - \frac{\varepsilon\_{\bar{j}}^{c\bar{i}}}{kT}\right) \tag{54}$$

for molecular species, and

$$f\_c^{(0)} = \left(\frac{m\_c}{2\pi kT}\right)^{3/2} n\_{c,a} \exp\left(-\frac{m\_c c\_c^2}{2kT}\right) \tag{55}$$

for atomic species. Here *nci* is the number density of molecules *c* at the *i*-th vibrational level, *nc*,*<sup>a</sup>* is the number density of atoms *c*, *Zrot ci* is the partition functions of rotational degrees of freedom:

$$Z\_{ci}^{rot}(T) = \sum\_{j} s\_j^{ci} \exp\left(-\frac{\varepsilon\_j^{ci}}{kT}\right) \tag{56}$$

*εci <sup>j</sup>* is the rotational energy of a molecule at the *<sup>i</sup>*th vibrational level, *<sup>s</sup>ci <sup>j</sup>* is the rotational statistical weight.

The solution (54) represents the local equilibrium Maxwell-Boltzmann distribution over the velocity and rotational energy levels with the temperature *T* and strongly non-equilibrium distribution over chemical species and vibrational energy levels. The distribution functions 10 Will-be-set-by-IN-TECH

In this section, chemical kinetics in multi-component reacting gas mixture flows are studied under the conditions of strong vibrational and chemical non-equilibrium. Experimental results on the relaxation times of various processes demonstrate that, in a wide temperature range, the equilibration of translational and rotational degrees of freedom proceeds much faster compared to the vibrational relaxation and chemical reactions. The characteristic

where *τrot*, *τvibr* are relaxation times for rotational and vibrational degrees of freedom. Under

The kinetic equations for the distribution functions in the zero-order Chapman-Enskog

In this case, the system of collision invariants for the most frequent collisions includes along with the momentum and a particle total energy, any value independent of the velocity and rotational level *j* and depending arbitrarily on the vibrational level *i* and chemical species *c*. This values are conserved at the most frequent collisions because, according to the condition (51), vibrational energy transitions and chemical reactions are forbidden in the rapid processes. Based on the above set of the collision invariants, the solution of Eqs. (53) takes the

> *nci Zrot ci* (*T*)

3/2

for atomic species. Here *nci* is the number density of molecules *c* at the *i*-th vibrational level,

The solution (54) represents the local equilibrium Maxwell-Boltzmann distribution over the velocity and rotational energy levels with the temperature *T* and strongly non-equilibrium distribution over chemical species and vibrational energy levels. The distribution functions

exp 

> <sup>−</sup> *mcc*<sup>2</sup> *c* 2*kT*

*nc*,*<sup>a</sup>* exp

 − *εci j kT*

<sup>−</sup> *mcc*<sup>2</sup> *c* <sup>2</sup>*kT* <sup>−</sup> *<sup>ε</sup>ci j kT*

*ci* is the partition functions of rotational degrees of

, (56)

*<sup>j</sup>* is the rotational

*rot*(0)

*sl cij* = *J*

*vibr cij* + *J*

such conditions, the integral operators in the kinetic equations (2) take the form:

In this case, the vibrational-chemical coupling in reacting flows becomes important.

*J el*(0) *cij* + *J*

3/2 *s ci j*

 *mc* 2*πkT*

*ci* (*T*) = ∑

*<sup>j</sup>* is the rotational energy of a molecule at the *<sup>i</sup>*th vibrational level, *<sup>s</sup>ci*

*j s ci <sup>j</sup>* exp

*τel < τrot* � *τvibr* ∼ *τreact* ∼ *θ*. (51)

*react*

*cij* = 0. (53)

(54)

(55)

*cij* . (52)

**3. State-to-state model for vibrational–chemical coupling**

**3.1 Distribution functions. Governing equations**

*J rap cij* = *J el cij* + *J rot cij* , *J*

*f* (0) *cij* =

*nc*,*<sup>a</sup>* is the number density of atoms *c*, *Zrot*

 *mc* 2*πkT*

*f* (0) *<sup>c</sup>* <sup>=</sup>

*Zrot*

relaxation times satisfy the relation

approximation have the form:

for molecular species, and

form

freedom:

statistical weight.

*εci*

(54), (55) are completely specified by the macroscopic gas parameters *nci*(**r**, *t*) (*c* = 1, ..., *L*, *i* = 0, 1, ..., *Lc*, *Lc* is the number of excited vibration levels of molecular species *c*), *T*(**r**, *t*), and **v**(**r**, *t*) and correspond to the set of the collision invariants of rapid processes.

The set of equations for the macroscopic parameters *nci*(**r**, *t*), **v**(**r**, *t*), and *T*(**r**, *t*) follows from Eq. (2) with collision operators (52). This system includes equations of state-to-state vibrational and chemical kinetics in a flow Kustova & Nagnibeda (1998); Nagnibeda & Kustova (2009):

$$\frac{dn\_{\rm ci}}{dt} + n\_{\rm ci} \nabla \cdot \mathbf{v} + \nabla \cdot (n\_{\rm ci} \mathbf{V}\_{\rm ci}) = R\_{\rm ci}, \qquad \mathbf{c} = 1, \ldots, L, \ \mathbf{i} = 0, \ldots, L\_{\rm c} \tag{57}$$

coupled to the conservation equations for the momentum and total energy which formally coincide with Eqs. (11)–(12). Here, **V***ci* is the diffusion velocity of molecules c at the vibrational state *i*. The total energy per unit mass *U* is specified by level populations *nci*(**r**, *t*), atomic number densities *nc*(**r**, *t*) and gas temperature.

The source terms in equations (57) characterize the variation of the vibrational level populations and atomic number densities caused by different vibrational energy exchanges and chemical reactions and are expressed via the integral operators of slow processes:

$$\mathcal{R}\_{\rm ci} = \sum\_{j} \int f\_{\rm cij}^{\rm sl} d\, \mathbf{u}\_{\rm c} = \mathcal{R}\_{\rm cj}^{\rm vibr} + \mathcal{R}\_{\rm ci}^{\rm react}.\tag{58}$$

The equations (57), (11), (12) provide a detailed description of vibrational and chemical kinetics and flow dynamics for weak deviations from the equilibrium distributions over the velocity and rotational energy levels and arbitrary deviations from the equilibrium for the vibrational degrees of freedom and chemical species. Let us emphasize that for such an approach, the vibrational level populations are included to the set of main macroscopic parameters, and particles of various chemical species in different vibrational states represent the mixture components. The expressions (58) can be written in the form:

$$R\_{c\bar{i}}^{vibr} = \sum\_{d k \bar{i}' k'} \left( n\_{c\bar{i}'} n\_{d k'} k\_{c\_r \bar{i}' i}^{d, k' k} - n\_{c\bar{i}} n\_{d k} k\_{c, \bar{i}\bar{i}'}^{d, k k'} \right), \tag{59}$$

$$R\_{ci}^{react} = R\_{ci}^{ex} + R\_{ci}^{diss}$$

$$R\_{ci}^{ex} = \sum\_{d$$

$$R\_{c\bar{i}}^{\rm diss} = \sum\_{d\mathbf{k}} n\_{d\mathbf{k}} \left( n\_{c'} n\_{f'} k\_{\rm rec,ci}^{\rm dk} - n\_{c\bar{i}} k\_{c\bar{i},\rm diss}^{\rm dk} \right) . \tag{61}$$

Here the rate coefficients are introduced for the energy exchange:

$$A\_{\rm cl} + A\_{\rm dk} \rightleftharpoons A\_{\rm cl'} + A\_{\rm dk' \prime} \tag{62}$$

exchange chemical reactions:

$$A\_{c\bar{1}} + A\_{d\bar{k}} \rightleftharpoons A\_{c'\bar{1}'} + A\_{d'\bar{k}'} \tag{63}$$

and dissociation-recombination reactions:

$$A\_{\rm ci} + A\_{dk} \rightleftharpoons A\_{\rm c'} + A\_{f'} + A\_{dk}.\tag{64}$$

exchange reactions have the form Nagnibeda & Kustova (2009):

× *s ci j s dk <sup>l</sup>* exp

*σ*˜ *c*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* (*g*) =

−

 *mcd* 2*πkT*

> *σ*˜ *diss cij*, *<sup>d</sup>* = *σdiss*

Since it is supposed that the cross sections of dissociation *σdiss*

*kdk ci*, *diss* <sup>=</sup>*k<sup>d</sup>*

, *ci* <sup>=</sup> *<sup>k</sup>d*�

*k*� , *dk c*�*i*� , *ci kdk*, *<sup>d</sup>*�*k*� *ci*, *c*�*i*�

*rec*, *ci* <sup>=</sup> (*mc*�*mf* �*md*)3/2

*mc*�*u*<sup>2</sup>

3/2 <sup>∑</sup> *j*

(2*πkT*)9/2 ∑

*<sup>c</sup>*� <sup>+</sup> *mf* �*u*<sup>2</sup>

not depend on the vibrational state *k* of the partner *Adk* in the reaction (64), then:

energy. Thus for the rate coefficients of forward and backward reactions we obtain

<sup>=</sup> *<sup>s</sup><sup>c</sup> isd k sc*� *<sup>i</sup>*� *<sup>s</sup>d*� *k*�

the integral dissociation reaction cross section is introduced by the formula

 *mcd* 2*πkT*

> −*εci <sup>j</sup>* <sup>+</sup> *<sup>ε</sup>dk l kT*

sections over solid angles in which relative velocity appear before and after collision:

It is commonly supposed that the cross section depends on the absolute value *g* of the relative

*σc*� *d*� , *i* � *j* � *k*� *l* �

The recombination rate coefficients in the zero-order approximation can be represented in the

*<sup>f</sup>* � <sup>+</sup> *mdu*�<sup>2</sup> *d* <sup>2</sup>*kT*

The zero-order dissociation rate coefficients take the form Kušˇcer (1991); Ludwig & Heil (1960)

exp

*cij*, *<sup>d</sup>*(*g*, **u***c*� , **u***<sup>f</sup>* � , **u**�

*ci*, *diss*, *<sup>k</sup>dk*

The relations connecting the rate coefficients of forward and backward collisional processes follow from the microscopic detailed balance relations for reactive collisions (8)-(9) after averaging them with the Maxwell–Boltzmann distribution over the velocity and rotational

> *mcmd mc*�*md*�

*<sup>σ</sup>rec*, *cij*

<sup>−</sup> *mcd <sup>g</sup>*<sup>2</sup> <sup>2</sup>*kT s ci <sup>j</sup>* exp

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* (**u***c*� , **u***<sup>f</sup>* � , **u**�

*d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

*<sup>d</sup>*)*d***u***c*� *d***u***<sup>f</sup>* � *d***u**�

3/2 *Zrot*

*Zrot <sup>c</sup>*�*i*�*Zrot d*�*k*� ×

*ci <sup>Z</sup>rot dk*

*rec*, *ci* <sup>=</sup> *<sup>k</sup><sup>d</sup>*

*j*

4*π σc*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* (**g**, <sup>Ω</sup>)*d*2Ω*d*2Ω�

3/2<sup>∑</sup> *jlj*�*l*�

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 127

*cd*, *ijkl* designating the integral cross section of a collision resulting in a bimolecular reaction. The integral cross section is obtained integrating the corresponding differential cross

*g*3*σ*˜ *c*� *d*� , *i* � *j* � *k*� *l* �

exp

<sup>−</sup> *mcd <sup>g</sup>*<sup>2</sup> <sup>2</sup>*kT*

×

*cd*, *ijkl* (*g*, <sup>Ω</sup>)*d*2Ω. (69)

*<sup>d</sup>*, **u***c*, **u***d*)×

− *εci j kT*

*cd*, *ijkl dg*, (67)

. (68)

*<sup>d</sup>d***u***cd***u***d*. (70)

*<sup>d</sup>*. (72)

*cij*, *<sup>d</sup>dg*, (71)

*<sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* do

*g*3*σ*˜ *diss*

*cij*, *<sup>d</sup>* and recombination *<sup>σ</sup>rec*, *cij*

*rec*, *ci* (73)

*kdk*, *<sup>d</sup>*� *k*� *ci*, *<sup>c</sup>*�*i*� <sup>=</sup> <sup>4</sup>*<sup>π</sup> Zrot ci <sup>Z</sup>rot dk*

> *σ*˜ *c*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* (*g*) = <sup>1</sup>

velocity rather than the vector **g**. Then

*kdk*

*kdk*

*ci*, *diss* <sup>=</sup> <sup>4</sup>*<sup>π</sup>*

*Zrot ci*

> *Kd*� *k*� , *dk c*�*i*�

<sup>×</sup> exp

with *σ*˜ *c*� *d*� , *i* � *j* � *k*� *l* �

form

The rate coefficients for the forward reactions (62)–(64) (for collisions of particles *Aci* and *Adk*) are introduced, respectively, as *kd*, *kk*� *<sup>c</sup>*, *ii*� , *<sup>k</sup>dk*, *<sup>d</sup>*� *k*� *ci*, *<sup>c</sup>*�*i*� , and *<sup>k</sup>dk ci*, *diss*, the recombination rate coefficient is denoted as *kdk rec*, *ci*. Note that if *k*� = *k*, then Eq. (62) describes VT(TV) transitions for a molecule *Aci* during the collision with a molecule *Adk* with the rate coefficient *kdk <sup>c</sup>*, *ii*� of the forward transition. If *d* is an atom, then the corresponding rate coefficient of the forward transition (62) is *k<sup>d</sup> <sup>c</sup>*, *ii*� .

If *k* �= *k*� then the reaction (62) describes either VV1 exchange of the vibrational energy between molecules of the same chemical species (for *c* = *d*) or VV2 transitions between molecules of different chemical species (for *c* �= *d*). Note that VV1 and VV2 transitions of the vibrational energy are almost always accompanied with the transfer of the part of vibrational energy into the translational or rotational modes. However, the probability of a simultaneous exchange between three and more energy modes during one collision is rather low, consequently, these exchanges are usually omitted in the production terms of the kinetic equations.

In the dissociation and recombination reactions (64), the particle *Adk* can also be either a molecule or an atom. Therefore, different dissociation rate coefficients should be introduced: *kdk rec*, *ci*, *<sup>k</sup><sup>d</sup> rec*, *ci*. The rate coefficients of the above processes depend on the order of the distribution function approximation.

The Chapman–Enskog method generalized for the conditions (51) gives the possibility to express, in any approximation, the transport and relaxation terms in Eqs. (57), (11), (12) as functions of the main macroscopic parameters *nci*(**r**, *t*), **v**(**r**, *t*), and *T*(**r**, *t*) and thus to close completely the set of governing equations.

In the zero-order approximation (54), (55),

$$\mathbf{P}^{(0)} = nkT\mathbf{I}, \qquad \mathbf{q}^{(0)} = 0, \qquad \mathbf{V}\_{\rm ci}^{(0)} = 0 \quad \forall \, \mathbf{c}, \mathbf{i}, \tag{65}$$

and the governing equations contain the equations of state-to-state kinetics

$$\frac{d n\_{\rm ci}}{dt} + n\_{\rm ci} \nabla \cdot \mathbf{v} = R\_{\rm ci}^{(0)}, \qquad \mathbf{c} = 1, \ldots, L, \ \mathbf{i} = \mathbf{0}, \ldots, L\_{\rm c} \tag{66}$$

coupled to the conservation equations in the form (27), (28).

The right hand sides of Eqs. (66) *R*(0) *ci* are specified by the zero-order distribution function. The expressions for *Rci* contain the microscopic rate coefficients for vibrational energy exchanges and chemical reactions. The equations (66) describe detailed state-to-state vibrational and chemical kinetics in an inviscid non-conductive gas mixture flow in the Euler approximation. In the first-order approximation state-dependent transport properties and reaction rates in reacting non-equilibrium flows are studied in Kustova & Nagnibeda (1998); Kustova et al. (1999); Nagnibeda & Kustova (2009).

#### **3.2 State dependent reaction rate coefficients**

Let us consider state dependent rate coefficients for chemical reactions appearing in Eqs. (59)–(61). In the zero-order Chapman-Enskog approximation rate coefficients for 12 Will-be-set-by-IN-TECH

The rate coefficients for the forward reactions (62)–(64) (for collisions of particles *Aci* and *Adk*)

transition. If *d* is an atom, then the corresponding rate coefficient of the forward transition

If *k* �= *k*� then the reaction (62) describes either VV1 exchange of the vibrational energy between molecules of the same chemical species (for *c* = *d*) or VV2 transitions between molecules of different chemical species (for *c* �= *d*). Note that VV1 and VV2 transitions of the vibrational energy are almost always accompanied with the transfer of the part of vibrational energy into the translational or rotational modes. However, the probability of a simultaneous exchange between three and more energy modes during one collision is rather low, consequently, these exchanges are usually omitted in the production terms of the kinetic

In the dissociation and recombination reactions (64), the particle *Adk* can also be either a molecule or an atom. Therefore, different dissociation rate coefficients should be introduced:

The Chapman–Enskog method generalized for the conditions (51) gives the possibility to express, in any approximation, the transport and relaxation terms in Eqs. (57), (11), (12) as functions of the main macroscopic parameters *nci*(**r**, *t*), **v**(**r**, *t*), and *T*(**r**, *t*) and thus to close

expressions for *Rci* contain the microscopic rate coefficients for vibrational energy exchanges and chemical reactions. The equations (66) describe detailed state-to-state vibrational and chemical kinetics in an inviscid non-conductive gas mixture flow in the Euler approximation. In the first-order approximation state-dependent transport properties and reaction rates in reacting non-equilibrium flows are studied in Kustova & Nagnibeda (1998); Kustova et al.

Let us consider state dependent rate coefficients for chemical reactions appearing in Eqs. (59)–(61). In the zero-order Chapman-Enskog approximation rate coefficients for

*P*(0) = *nkTI*, **q**(0) = 0, **V**(0)

and the governing equations contain the equations of state-to-state kinetics

*dt* <sup>+</sup> *nci* ∇· **<sup>v</sup>** <sup>=</sup> *<sup>R</sup>*(0)

coupled to the conservation equations in the form (27), (28).

*rec*, *ci*. The rate coefficients of the above processes depend on the order of the

*rec*, *ci*. Note that if *k*� = *k*, then Eq. (62) describes VT(TV) transitions for a molecule

*ci*, *diss*, the recombination rate coefficient is

*ci* = 0 ∀ *c*, *i*, (65)

*ci* , *c* = 1, ..., *L*, *i* = 0, ..., *Lc*, (66)

*ci* are specified by the zero-order distribution function. The

*<sup>c</sup>*, *ii*� of the forward

*k*� *ci*, *<sup>c</sup>*�*i*� , and *<sup>k</sup>dk*

*<sup>c</sup>*, *ii*� , *<sup>k</sup>dk*, *<sup>d</sup>*�

*Aci* during the collision with a molecule *Adk* with the rate coefficient *kdk*

are introduced, respectively, as *kd*, *kk*�

distribution function approximation.

completely the set of governing equations. In the zero-order approximation (54), (55),

*dnci*

The right hand sides of Eqs. (66) *R*(0)

(1999); Nagnibeda & Kustova (2009).

**3.2 State dependent reaction rate coefficients**

denoted as *kdk*

*<sup>c</sup>*, *ii*� .

(62) is *k<sup>d</sup>*

equations.

*kdk rec*, *ci*, *<sup>k</sup><sup>d</sup>* exchange reactions have the form Nagnibeda & Kustova (2009):

$$k\_{\rm ci,cf'i'}^{dk,d'k'} = \frac{4\pi}{Z\_{\rm ci}^{\rm rot} \Sigma\_{d\rm dk}^{\rm rot}} \left(\frac{m\_{\rm cd}}{2\pi kT}\right)^{3/2} \sum\_{jl'l'} \int \exp\left(-\frac{m\_{\rm cd}g^2}{2kT}\right) \times$$

$$\times s\_j^{\rm ci} s\_l^{dk} \exp\left(-\frac{\varepsilon\_j^{ci} + \varepsilon\_l^{dk}}{kT}\right) g^3 \tilde{\sigma}\_{cd,ijkl}^{\rm c'd',i'j'k'l'} dg\_{\rm s} \tag{67}$$

with *σ*˜ *c*� *d*� , *i* � *j* � *k*� *l* � *cd*, *ijkl* designating the integral cross section of a collision resulting in a bimolecular reaction. The integral cross section is obtained integrating the corresponding differential cross sections over solid angles in which relative velocity appear before and after collision:

$$
\sigma\_{\rm cd,ijkl}^{\varepsilon'd',i'j'k'l'}(\mathbf{g}) = \frac{1}{4\pi} \int \sigma\_{\rm cd,ijkl}^{\varepsilon'd',i'j'k'l'}(\mathbf{g},\boldsymbol{\Omega}) d^2\boldsymbol{\Omega} d^2\boldsymbol{\Omega}'.\tag{68}
$$

It is commonly supposed that the cross section depends on the absolute value *g* of the relative velocity rather than the vector **g**. Then

$$
\sigma\_{\rm cd,ijkl}^{\varepsilon'd',i'j'k'l'}(\mathbf{g}) = \int \sigma\_{\rm cd,ijkl}^{\varepsilon'd',i'j'k'l'}(\mathbf{g},\boldsymbol{\Omega})d^2\boldsymbol{\Omega}.\tag{69}
$$

The recombination rate coefficients in the zero-order approximation can be represented in the form

$$k\_{\rm rec,ci}^{dk} = \frac{(m\_{\rm c'} m\_{f'} m\_d)^{3/2}}{(2\pi kT)^{9/2}} \sum\_j \int \sigma\_{\rm c'f'd}^{\rm rec,cij} (\mathbf{u}\_{\rm c'}, \mathbf{u}\_{f'}, \mathbf{u}\_{d'}^{\prime}, \mathbf{u}\_{c'}, \mathbf{u}\_d) \times$$

$$\times \exp\left(-\frac{m\_{\rm c'} u\_{\rm c'}^2 + m\_{f'} u\_{f'}^2 + m\_d u\_d^{\prime 2}}{2kT}\right) d\mathbf{u}\_{\rm c'} d\mathbf{u}\_{f'} d\mathbf{u}\_d^{\prime} d\mathbf{u}\_c d\mathbf{u}\_d.\tag{70}$$

The zero-order dissociation rate coefficients take the form Kušˇcer (1991); Ludwig & Heil (1960)

$$k\_{\rm ci,diss}^{\rm kl} = \frac{4\pi}{Z\_{\rm ci}^{\rm rot}} \left(\frac{m\_{\rm cd}}{2\pi kT}\right)^{3/2} \sum\_{\dot{j}} \int \exp\left(-\frac{m\_{\rm cd}\xi^2}{2kT}\right) s\_{\dot{j}}^{\rm ci} \exp\left(-\frac{\varepsilon\_{\rm j}^{\rm ci}}{kT}\right) g^3 \tilde{\sigma}\_{\rm c\dot{j},\rm d}^{\rm diss} dg\_{\rm s} \tag{71}$$

the integral dissociation reaction cross section is introduced by the formula

$$
\sigma\_{\rm cij,d}^{\rm diss} = \int \sigma\_{\rm cij,d}^{\rm diss}(\mathbf{g}, \mathbf{u}\_{\mathbf{c}'}, \mathbf{u}\_{f'}, \mathbf{u}\_d') d\mathbf{u}\_{\mathbf{c}'} d\mathbf{u}\_{f'} d\mathbf{u}\_d'.\tag{72}
$$

Since it is supposed that the cross sections of dissociation *σdiss cij*, *<sup>d</sup>* and recombination *<sup>σ</sup>rec*, *cij <sup>c</sup>*� *<sup>f</sup>* �*<sup>d</sup>* do not depend on the vibrational state *k* of the partner *Adk* in the reaction (64), then:

$$k\_{\rm ci,diss}^{dk} = k\_{\rm ci,diss}^{d} \qquad k\_{\rm rec,ci}^{dk} = k\_{\rm rec,ci}^{d} \tag{73}$$

The relations connecting the rate coefficients of forward and backward collisional processes follow from the microscopic detailed balance relations for reactive collisions (8)-(9) after averaging them with the Maxwell–Boltzmann distribution over the velocity and rotational energy. Thus for the rate coefficients of forward and backward reactions we obtain

$$K\_{\mathbf{c'i'},\mathbf{ci}}^{d'k',dk} = \frac{k\_{\mathbf{c'i'},\mathbf{ci}}^{d'k',dk}}{k\_{\mathbf{c}i,\mathbf{c'i'}}^{dk,d'k'}} = \frac{s\_{\mathbf{i}}^{\mathbf{c}}s\_{\mathbf{k}}^{d}}{s\_{\mathbf{i'}}^{\mathbf{c'}}s\_{\mathbf{k'}}^{d'}} \left(\frac{m\_{\mathbf{c'}}m\_{d}}{m\_{\mathbf{c'}}m\_{d'}}\right)^{3/2} \frac{\mathbf{Z}\_{\mathbf{c}i}^{\mathbf{rot}}\mathbf{Z}\_{\mathbf{d}k}^{\mathbf{rot}}}{\mathbf{Z}\_{\mathbf{c'i'},\mathbf{i}}^{\mathbf{rot}}\mathbf{Z}\_{\mathbf{d}'k'}^{\mathbf{rot}}} \times 1$$

*U* is a parameter of the model.

Stupochenko et al. (1967).

rate coefficients *kN*<sup>2</sup>

calculate *k<sup>d</sup>*

*kd*

<sup>10</sup>-30 <sup>10</sup>-28 <sup>10</sup>-26 <sup>10</sup>-24 <sup>10</sup>-22 <sup>10</sup>-20 <sup>10</sup>-18 <sup>10</sup>-16

*ki*, diss, cm3

Thus, the state-dependent dissociation rate coefficient *k<sup>d</sup>*

diss, eq(*T*), the empirical Arrhenius law can be applied:

diss, eq <sup>=</sup> *AT<sup>n</sup>* exp

the coefficients *A* and *n* are generally obtained as a best fit to experimental data. The tables of the coefficients in the Arrhenius formula for various chemical reactions can be found in Refs. Gardiner (1984); Kondratiev & Nikitin (1974); Park (1990); Phys-Chem (2002);

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 129

In Ref. Esposito, Capitelli, Kustova & Nagnibeda (2000), the dissociation rate coefficients

*<sup>i</sup>*, *diss* calculated within the framework of the Treanor–Marrone model are compared with those obtained from trajectory calculations Esposito, Capitelli & Gorse (2000), some recommendations for the optimum choice for the parameter *U* for the specific reactions are given. Figure 1 presents the temperature dependence of the state-dependent dissociation

values of the parameter *U* for two vibrational quantum numbers: *i* = 0 and *i* = 20. The results of trajectory calculation for *ki*, *diss* taken from Ref. Esposito, Capitelli & Gorse (2000) are also plotted. We can see that for low vibrational levels, the choice for *U* = ∞ results in significant overestimation for *ki*, *diss*, which confirms the common assumption of the

With the increase in the vibrational quantum number, for *U* = ∞ we obtain more realistic values for *ki*, *diss*, and for *i >* 40, we have the best agreement with the results of accurate trajectory calculations. *U* = *D*/(6*k*) and *U* = 3*T* provide good consistency for *ki*, *diss* at intermediate levels (20 *< i <* 40). Furthermore, *U* = *D*/(6*k*) results in better consistency for low temperatures, whereas *U* = 3*T* is good in the high temperature range (*T >* 6000 K).

> *ki*, diss, cm3 /s

*T*, K

*i* = 20 (*b*). The curve *1* represents the results obtained in Ref. Esposito, Capitelli & Gorse

It should be emphasized that using the same value for the parameter *U* for any *i* and *T* can result in considerable errors in the calculation for the state-to-state dissociation rate

Fig. 1. The temperature dependence of the dissociation rate coefficient *k*

(2000), curves *2–4* correspond to *U* = *D*/(6*k*), *U* = 3*T*, and *U* = ∞.

<sup>10</sup>-25 <sup>10</sup>-23 <sup>10</sup>-21 <sup>10</sup>-19 <sup>10</sup>-17 <sup>10</sup>-15 <sup>10</sup>-13 <sup>10</sup>-11 − *D kT*

*<sup>i</sup>*, *diss* in an (N2, N) mixture. The coefficients are calculated for different

*kd*

averaged thermal equilibrium coefficient *k<sup>d</sup>*

preferential dissociation from high vibrational states.

/s (a)

*1 2*

*4*

*3*

2000 4000 6000 8000 10000

*<sup>i</sup>*, *diss* is expressed in terms of the

, (79)

*diss*,*eq*(*T*) and non-equilibrium factor (77). To

2000 4000 6000 8000 10000

N2

*<sup>i</sup>*, diss for *i* = 0 (*a*) and

*3*

*1*

*<sup>4</sup> <sup>2</sup>*

*T*, K

(b)

$$\times \exp\left(\frac{\varepsilon\_{\mathbf{r}'}^{\varepsilon'} + \varepsilon\_{k'}^{d'} - \varepsilon\_{i}^{c} - \varepsilon\_{k}^{d}}{kT}\right) \exp\left(\frac{D\_{c} + D\_{d} - D\_{c'} - D\_{d'}}{kT}\right),\tag{74}$$

$$K\_{\rm rec-diss,ci}^d = \frac{k\_{\rm rec,ci}^d}{k\_{\rm ci,diss}^d} = s\_i^c \left(\frac{m\_c}{m\_{c'}m\_{f'}}\right)^{3/2} h^3 \left(2\pi kT\right)^{-3/2} Z\_{\rm ci}^{\rm rot} \exp\left(-\frac{\varepsilon\_i^c - D\_\odot}{kT}\right). \tag{75}$$

In the last formula, *mc* = *mc*� + *mf* � .

For diatomic gases, the vibrational statistical weight *s<sup>c</sup> <sup>i</sup>* = 1. Moreover, for the rigid rotator model, the rotational partition function is independent of the vibrational state *Zrot ci* <sup>=</sup> *<sup>Z</sup>rot <sup>c</sup>* . In this case, the ratio of the backward and forward reaction rate coefficients takes the reduced form. The expression (74) can be simplified if the collision partner is an atom.

For the application of Egs. (66), (27), (28) to particular problems of non-equilibrium fluid dynamics, the analytical expressions for the dependence of the reaction rate coefficients on the vibrational states of molecules participating in the reactions are needed.

As for vibrational energy transitions, a number of theoretical and experimental estimates for rate coefficients of these transitions are available in the literature in different temperature intervals. These data can be found, for example, in Nagnibeda & Kustova (2009); Phys-Chem (2002; 2004). The comparison of rate coefficients for vibational energy transitions of N2 molecules obtained using different models is given in Nagnibeda & Kustova (2009).

The rate coefficients for dissociation from different vibrational levels have been studied much less widely than for vibrational energy transitions. Two models are commonly used for calculations: the ladder-climbing model assuming dissociation only from the last vibrational level (see, for instance, Armenise et al. (1996; 1995); Capitelli et al. (1997); Osipov (1966)), and that of Treanor and Marrone Marrone & Treanor (1963) allowing for dissociation from any vibrational state.

In the frame of ladder climbing model, the rate of dissociation is specified by the number of molecules occurring on the last vibrational level. Consequently, the dissociation rate is totally specified by the probabilities for the vibrational energy transitions to the last level. In the case when dissociation can occur from any vibrational level, the expression for the rate coefficient for dissociation of a molecule on the vibrational level *i* can be written in the form Nagnibeda & Kustova (2009):

$$k\_{i, \text{diss}}^d = Z\_i^d(T) k\_{\text{diss},eq}^d(T). \tag{76}$$

Here, *k<sup>d</sup> diss*,*eq*(*T*) is the thermal equilibrium dissociation rate coefficient obtained by averaging the state-dependent rate coefficient with the one-temperature Boltzmann distribution; *Z<sup>d</sup> <sup>i</sup>* is the non-equilibrium factor. Using the Treanor–Marrone model Marrone & Treanor (1963), the expression for *Z<sup>d</sup> <sup>i</sup>* (*T*) was obtained in Nagnibeda & Kustova (2009):

$$Z\_i^d(T) = Z\_i(T, \mathcal{U}) = \frac{Z^{\text{vibr}}(T)}{Z^{\text{vibr}}(-\mathcal{U})} \exp\left[\frac{\varepsilon\_i}{k}\left(\frac{1}{T} + \frac{1}{\mathcal{U}}\right)\right],\tag{77}$$

*Zvibr* is the equilibrium vibrational partition function

$$Z^{\rm vibr}(T) = \sum\_{\vec{l}} s\_{\vec{l}}^{\varepsilon} \exp\left(-\frac{\varepsilon\_{\vec{l}}}{kT}\right),\tag{78}$$

*U* is a parameter of the model.

14 Will-be-set-by-IN-TECH

 *Dc* <sup>+</sup> *Dd* <sup>−</sup> *Dc*� <sup>−</sup> *Dd*� *kT*

<sup>−</sup>3/2 *Zrot*

*ci* exp

*<sup>i</sup>* = 1. Moreover, for the rigid rotator

*diss*,*eq*(*T*). (76)

 −*εc <sup>i</sup>* − *Dc kT*

, (74)

*ci* <sup>=</sup> *<sup>Z</sup>rot*

. (75)

*<sup>c</sup>* . In

*<sup>i</sup>* is

, (77)

, (78)

 exp

3/2

this case, the ratio of the backward and forward reaction rate coefficients takes the reduced

For the application of Egs. (66), (27), (28) to particular problems of non-equilibrium fluid dynamics, the analytical expressions for the dependence of the reaction rate coefficients on

As for vibrational energy transitions, a number of theoretical and experimental estimates for rate coefficients of these transitions are available in the literature in different temperature intervals. These data can be found, for example, in Nagnibeda & Kustova (2009); Phys-Chem (2002; 2004). The comparison of rate coefficients for vibational energy transitions of N2

The rate coefficients for dissociation from different vibrational levels have been studied much less widely than for vibrational energy transitions. Two models are commonly used for calculations: the ladder-climbing model assuming dissociation only from the last vibrational level (see, for instance, Armenise et al. (1996; 1995); Capitelli et al. (1997); Osipov (1966)), and that of Treanor and Marrone Marrone & Treanor (1963) allowing for dissociation from any

In the frame of ladder climbing model, the rate of dissociation is specified by the number of molecules occurring on the last vibrational level. Consequently, the dissociation rate is totally specified by the probabilities for the vibrational energy transitions to the last level. In the case when dissociation can occur from any vibrational level, the expression for the rate coefficient for dissociation of a molecule on the vibrational level *i* can be written in the form Nagnibeda

*<sup>i</sup>* (*T*)*k<sup>d</sup>*

the state-dependent rate coefficient with the one-temperature Boltzmann distribution; *Z<sup>d</sup>*

the non-equilibrium factor. Using the Treanor–Marrone model Marrone & Treanor (1963), the

*<sup>Z</sup>*vibr(−*U*)

*i s c <sup>i</sup>* exp − *εi kT* 

*diss*,*eq*(*T*) is the thermal equilibrium dissociation rate coefficient obtained by averaging

exp *εi k* 1 *<sup>T</sup>* <sup>+</sup> 1 *U* 

*h*<sup>3</sup> (2*πkT*)

× exp

*rec*−*diss*, *ci* <sup>=</sup> *<sup>k</sup><sup>d</sup>*

In the last formula, *mc* = *mc*� + *mf* � .

*Kd*

vibrational state.

& Kustova (2009):

expression for *Z<sup>d</sup>*

*Zd*

*Zvibr* is the equilibrium vibrational partition function

Here, *k<sup>d</sup>*

*<sup>ε</sup>c*� *<sup>i</sup>*� <sup>+</sup> *<sup>ε</sup>d*�

*rec*, *ci kd ci*, *diss*

For diatomic gases, the vibrational statistical weight *s<sup>c</sup>*

= *s c i mc mc*�*mf* �

*<sup>k</sup>*� <sup>−</sup> *<sup>ε</sup><sup>c</sup>*

*kT*

*<sup>i</sup>* <sup>−</sup> *<sup>ε</sup><sup>d</sup> k*

model, the rotational partition function is independent of the vibrational state *Zrot*

molecules obtained using different models is given in Nagnibeda & Kustova (2009).

form. The expression (74) can be simplified if the collision partner is an atom.

the vibrational states of molecules participating in the reactions are needed.

*kd*

*<sup>i</sup>* (*T*) = *Zi*(*T*, *<sup>U</sup>*) = *<sup>Z</sup>*vibr(*T*)

*<sup>Z</sup>vibr*(*T*) = ∑

*<sup>i</sup>*, *diss* <sup>=</sup> *<sup>Z</sup><sup>d</sup>*

*<sup>i</sup>* (*T*) was obtained in Nagnibeda & Kustova (2009):

Thus, the state-dependent dissociation rate coefficient *k<sup>d</sup> <sup>i</sup>*, *diss* is expressed in terms of the averaged thermal equilibrium coefficient *k<sup>d</sup> diss*,*eq*(*T*) and non-equilibrium factor (77). To calculate *k<sup>d</sup>* diss, eq(*T*), the empirical Arrhenius law can be applied:

$$k\_{\rm diss,eq}^d = AT^n \exp\left(-\frac{D}{kT}\right),\tag{79}$$

the coefficients *A* and *n* are generally obtained as a best fit to experimental data. The tables of the coefficients in the Arrhenius formula for various chemical reactions can be found in Refs. Gardiner (1984); Kondratiev & Nikitin (1974); Park (1990); Phys-Chem (2002); Stupochenko et al. (1967).

In Ref. Esposito, Capitelli, Kustova & Nagnibeda (2000), the dissociation rate coefficients *kd <sup>i</sup>*, *diss* calculated within the framework of the Treanor–Marrone model are compared with those obtained from trajectory calculations Esposito, Capitelli & Gorse (2000), some recommendations for the optimum choice for the parameter *U* for the specific reactions are given. Figure 1 presents the temperature dependence of the state-dependent dissociation rate coefficients *kN*<sup>2</sup> *<sup>i</sup>*, *diss* in an (N2, N) mixture. The coefficients are calculated for different values of the parameter *U* for two vibrational quantum numbers: *i* = 0 and *i* = 20. The results of trajectory calculation for *ki*, *diss* taken from Ref. Esposito, Capitelli & Gorse (2000) are also plotted. We can see that for low vibrational levels, the choice for *U* = ∞ results in significant overestimation for *ki*, *diss*, which confirms the common assumption of the preferential dissociation from high vibrational states.

With the increase in the vibrational quantum number, for *U* = ∞ we obtain more realistic values for *ki*, *diss*, and for *i >* 40, we have the best agreement with the results of accurate trajectory calculations. *U* = *D*/(6*k*) and *U* = 3*T* provide good consistency for *ki*, *diss* at intermediate levels (20 *< i <* 40). Furthermore, *U* = *D*/(6*k*) results in better consistency for low temperatures, whereas *U* = 3*T* is good in the high temperature range (*T >* 6000 K).

Fig. 1. The temperature dependence of the dissociation rate coefficient *k* N2 *<sup>i</sup>*, diss for *i* = 0 (*a*) and *i* = 20 (*b*). The curve *1* represents the results obtained in Ref. Esposito, Capitelli & Gorse (2000), curves *2–4* correspond to *U* = *D*/(6*k*), *U* = 3*T*, and *U* = ∞.

It should be emphasized that using the same value for the parameter *U* for any *i* and *T* can result in considerable errors in the calculation for the state-to-state dissociation rate

few macroscopic parameters, consequently, non-equilibrium kinetics can be described by a

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 131

It is known from experiments Gordiets et al. (1988) that in a vibrationally excited gas, near-resonant vibrational energy exchanges between molecules of the same chemical species proceed much faster than non-resonant transitions between different molecules, as well as transfers of vibrational energy to other modes and chemical reactions. Therefore the following

*τVV*<sup>1</sup> is the mean time between the collisions of the same species; *τVV*<sup>2</sup> is the characteristic time of the vibrational energy exchange between different molecules. In the multi-component reacting gas mixture under the condition (80) the integral operators in the kinetic equations (2) includes the operator of VV1 vibrational energy transitions between molecules of the same species along with the operators of elastic collisions and collisions with rotational energy

between molecules of different species, the operator describing the transfer of vibrational

Governing equations of the reacting flows and distribution functions in the zero-order and first-order approximations, under condition (80) are studied in details in Nagnibeda & Kustova (2009). The distribution function is totally specified by the macroscopic parameters

which reflects the conservation of the number of vibrational quanta of each molecular species in rapid processes. The zero-order distribution functions in this case may be written in the

> −*εc <sup>i</sup>* <sup>−</sup> *<sup>i</sup>ε<sup>c</sup>* 1 *kT* <sup>−</sup> *<sup>i</sup>ε<sup>c</sup>*

−*εc <sup>i</sup>* <sup>−</sup> *<sup>i</sup>ε<sup>c</sup>* 1 *kT* <sup>−</sup> *<sup>i</sup>ε<sup>c</sup>*

<sup>1</sup> is the vibrational energy of a molecule *c* at the first level. Here the vibrational energy is

The expression (83) yields the non-equilibrium quasi-stationary Treanor distribution Treanor et al. (1968) generalized for a multi-component reacting gas mixture taking into account

*TRV cij* + *J*

*VV*<sup>2</sup> *cij* + *J*

*τel τrot < τVV*<sup>1</sup> � *τVV*<sup>2</sup> *< τTRV < τreact* ∼ *θ*, (80)

*VV*<sup>1</sup>

*react*

*cij* consists of the operator of VV2 vibrational transitions

<sup>1</sup> is the vibrational temperature of the first vibrational

<sup>1</sup> is associated to the additional collision invariant *ic*

1 *kT<sup>c</sup>* 1

1 *kT<sup>c</sup>* 1

*cij* . (81)

*cij* , as well as the operator of chemical

*cij* . (82)

. (83)

. (84)

considerably reduced set of governing equations.

exchanges:

reactions *Jreact*

*nc*, **v**, *T*, and *T<sup>c</sup>*

*εc*

*cij* :

The operator of slow processes *Jsl*

relation between the characteristic relaxation times is fulfilled:

*J rap cij* = *J el cij* + *J rot cij* + *J*

*J sl cij* = *J*

form (54) where level populations *nci* are described by the relation:

<sup>1</sup> ) = ∑ *i s c <sup>i</sup>* exp

anharmonic molecular vibrations and rapid exchange of vibrational quanta.

*nci* <sup>=</sup> *nc Zvibr <sup>c</sup>* (*T*, *T<sup>c</sup>* 1 ) *s c <sup>i</sup>* exp

*Zvibr <sup>c</sup>* (*T*, *<sup>T</sup><sup>c</sup>*

counted from the energy of the zeroth level.

energy into rotational and translational modes *JTRV*

<sup>1</sup> , where the parameter *<sup>T</sup><sup>c</sup>*

level of molecules *c*. The parameter *T<sup>c</sup>*

with vibrational partition function

coefficients. The choice for the parameter should be specified by the conditions of a particular problem (the temperature range, basic channels of dissociation, etc.). In some studies Armenise et al. (1996); Candler et al. (1997); Capitelli et al. (1997), a possibility for dissociation from any vibrational state is suggested within the framework of the ladder-climbing model. To this end, it is supposed that a transition to the continuum occurs as a result of multi-quantum vibrational energy transfers.

The rate coefficients for bimolecular exchange reactions depending on the vibrational states of reagents and products have been less thoroughly studied than those for dissociation processes. Theoretical and experimental studies for the influence of the vibrational excitation of reagents on reaction rates were started by J. Polanyi Polanyi (1972); some experimental results were also obtained in Birely & Lyman (1975). The accurate theoretical approach to this problem primarily requires a calculation for the state-dependent differential cross sections for collisions resulting in chemical reactions, and their subsequent averaging over the velocity distributions. In the recent years, the dynamics of atmospheric reactions has been studied, and quasi-classical trajectory calculations for the cross sections and state-dependent rate coefficients for the reactions N2(*i*) + O → NO + N and O2(*i*) + N → NO + O have been carried out by several authors Gilibert et al. (1992; 1993).

At present time, two kinds of analytical expressions are available in the literature. The first kind includes analytical approximations for numerical results obtained for particular reactions (see Refs. Bose & Candler (1996); Capitelli et al. (1997; 2000); Phys-Chem (2002)). These expressions are sufficiently accurate and convenient for practical use; however, their application is restricted by the considered temperature range. Another approach is based on the generalizations of the Treanor–Marrone model to exchange reactions suggested in Refs. Aliat (2008); Knab (1996); Knab et al. (1995); Seror et al. (1997). These models can be used for more general cases, but the theoretical expressions for the rate coefficients contain additional parameters, which should be validated using experimental data. A lack of the data for these parameters restricts the implementation of the above semi-empirical models.

Therefore, the development of justified theoretical models for cross sections of reactive collisions and state-dependent rate coefficients for exchange reactions remains a very important problem of the non-equilibrium kinetics.

#### **4. Multi-temperature models for vibrational-chemical kinetics**

#### **4.1 Governing equations**

The approach proposed in the previous section makes it possible to develop the most rigorous model of reacting gas mixtures, since it takes into account the detailed state-to-state vibrational and chemical kinetics in a flow. However, practical implementation of this method leads to serious difficulties. The first important problem encountered in the realization of the state-to-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. The second fundamental problem is that experimental and theoretical data on the state-specific rate coefficients and especially on the cross sections of inelastic processes are rather scanty. Due to the above problems, simpler models based on quasi-stationary vibrational distributions are rather attractive for practical applications. In quasi-stationary approaches, the vibrational level populations are expressed in terms of a 16 Will-be-set-by-IN-TECH

coefficients. The choice for the parameter should be specified by the conditions of a particular problem (the temperature range, basic channels of dissociation, etc.). In some studies Armenise et al. (1996); Candler et al. (1997); Capitelli et al. (1997), a possibility for dissociation from any vibrational state is suggested within the framework of the ladder-climbing model. To this end, it is supposed that a transition to the continuum occurs as a result of

The rate coefficients for bimolecular exchange reactions depending on the vibrational states of reagents and products have been less thoroughly studied than those for dissociation processes. Theoretical and experimental studies for the influence of the vibrational excitation of reagents on reaction rates were started by J. Polanyi Polanyi (1972); some experimental results were also obtained in Birely & Lyman (1975). The accurate theoretical approach to this problem primarily requires a calculation for the state-dependent differential cross sections for collisions resulting in chemical reactions, and their subsequent averaging over the velocity distributions. In the recent years, the dynamics of atmospheric reactions has been studied, and quasi-classical trajectory calculations for the cross sections and state-dependent rate coefficients for the reactions N2(*i*) + O → NO + N and O2(*i*) + N → NO + O have been

At present time, two kinds of analytical expressions are available in the literature. The first kind includes analytical approximations for numerical results obtained for particular reactions (see Refs. Bose & Candler (1996); Capitelli et al. (1997; 2000); Phys-Chem (2002)). These expressions are sufficiently accurate and convenient for practical use; however, their application is restricted by the considered temperature range. Another approach is based on the generalizations of the Treanor–Marrone model to exchange reactions suggested in Refs. Aliat (2008); Knab (1996); Knab et al. (1995); Seror et al. (1997). These models can be used for more general cases, but the theoretical expressions for the rate coefficients contain additional parameters, which should be validated using experimental data. A lack of the data for these parameters restricts the implementation of the above semi-empirical models.

Therefore, the development of justified theoretical models for cross sections of reactive collisions and state-dependent rate coefficients for exchange reactions remains a very

The approach proposed in the previous section makes it possible to develop the most rigorous model of reacting gas mixtures, since it takes into account the detailed state-to-state vibrational and chemical kinetics in a flow. However, practical implementation of this method leads to serious difficulties. The first important problem encountered in the realization of the state-to-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. The second fundamental problem is that experimental and theoretical data on the state-specific rate coefficients and especially on the cross sections of inelastic processes are rather scanty. Due to the above problems, simpler models based on quasi-stationary vibrational distributions are rather attractive for practical applications. In quasi-stationary approaches, the vibrational level populations are expressed in terms of a

multi-quantum vibrational energy transfers.

carried out by several authors Gilibert et al. (1992; 1993).

important problem of the non-equilibrium kinetics.

**4.1 Governing equations**

**4. Multi-temperature models for vibrational-chemical kinetics**

few macroscopic parameters, consequently, non-equilibrium kinetics can be described by a considerably reduced set of governing equations.

It is known from experiments Gordiets et al. (1988) that in a vibrationally excited gas, near-resonant vibrational energy exchanges between molecules of the same chemical species proceed much faster than non-resonant transitions between different molecules, as well as transfers of vibrational energy to other modes and chemical reactions. Therefore the following relation between the characteristic relaxation times is fulfilled:

$$
\tau\_{el} \lesssim \tau\_{rot} < \tau\_{VV\_1} \ll \tau\_{VV\_2} < \tau\_{TRV} < \tau\_{\text{react}} \sim \theta\_\prime \tag{80}
$$

*τVV*<sup>1</sup> is the mean time between the collisions of the same species; *τVV*<sup>2</sup> is the characteristic time of the vibrational energy exchange between different molecules. In the multi-component reacting gas mixture under the condition (80) the integral operators in the kinetic equations (2) includes the operator of VV1 vibrational energy transitions between molecules of the same species along with the operators of elastic collisions and collisions with rotational energy exchanges:

$$J\_{\rm cij}^{rap} = J\_{\rm cij}^{el} + J\_{\rm cij}^{rot} + J\_{\rm cij}^{VV\_1}.\tag{81}$$

The operator of slow processes *Jsl cij* consists of the operator of VV2 vibrational transitions between molecules of different species, the operator describing the transfer of vibrational energy into rotational and translational modes *JTRV cij* , as well as the operator of chemical reactions *Jreact cij* :

$$J\_{\rm cij}^{\rm sl} = J\_{\rm cij}^{VV\_2} + J\_{\rm cij}^{TRV} + J\_{\rm cij}^{react} \,. \tag{82}$$

Governing equations of the reacting flows and distribution functions in the zero-order and first-order approximations, under condition (80) are studied in details in Nagnibeda & Kustova (2009). The distribution function is totally specified by the macroscopic parameters *nc*, **v**, *T*, and *T<sup>c</sup>* <sup>1</sup> , where the parameter *<sup>T</sup><sup>c</sup>* <sup>1</sup> is the vibrational temperature of the first vibrational level of molecules *c*. The parameter *T<sup>c</sup>* <sup>1</sup> is associated to the additional collision invariant *ic* which reflects the conservation of the number of vibrational quanta of each molecular species in rapid processes. The zero-order distribution functions in this case may be written in the form (54) where level populations *nci* are described by the relation:

$$m\_{c\dot{\imath}} = \frac{n\_{\mathcal{C}}}{Z\_{\mathcal{C}}^{\text{vibr}}(T, T\_1^{\text{c}})} s\_{\dot{\imath}}^{\text{c}} \exp\left(-\frac{\varepsilon\_{\dot{\imath}}^{\text{c}} - i\varepsilon\_{1}^{\text{c}}}{kT} - \frac{i\varepsilon\_{1}^{\text{c}}}{kT\_1^{\text{c}}}\right). \tag{83}$$

with vibrational partition function

$$Z\_{\varepsilon}^{\text{vibr}}(T, T\_1^{\varepsilon}) = \sum\_{i} s\_i^{\varepsilon} \exp\left(-\frac{\varepsilon\_i^{\varepsilon} - i\varepsilon\_1^{\varepsilon}}{kT} - \frac{i\varepsilon\_1^{\varepsilon}}{kT\_1^{\varepsilon}}\right). \tag{84}$$

*εc* <sup>1</sup> is the vibrational energy of a molecule *c* at the first level. Here the vibrational energy is counted from the energy of the zeroth level.

The expression (83) yields the non-equilibrium quasi-stationary Treanor distribution Treanor et al. (1968) generalized for a multi-component reacting gas mixture taking into account anharmonic molecular vibrations and rapid exchange of vibrational quanta.

**4.2 Multi-temperature reaction rate coefficients**

multi-temperature rate coefficients *kdd*�

recombination rate coefficients *k<sup>d</sup>*

considered in section 3.2:

*k dd*� (0) *cc*� (*T*, *<sup>T</sup><sup>c</sup>*

*k dd*� (0) *cc*� (*T*, *<sup>T</sup><sup>c</sup>*

*k d* (0) *<sup>c</sup>*,*diss*(*T*, *<sup>T</sup><sup>c</sup>*

coefficients as follows

The production terms in Eqs. (89) may be written in the form (13), (14) similar to the equations obtained in the one-temperature approximation. However, the coefficients in these expressions differ from those in the one-temperature approach and contains the

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 133

In the zero-order approximation, the multi-temperature rate coefficients of exchange and dissociation reactions can be expressed in terms of the state-specific rate coefficients

*ncindk k*

*dk*,*d*� *k*� (0)

or a molecule and an atom A*d*), the two-temperature coefficients of dissociation *k<sup>d</sup>*

*ncnd* ∑ *iki*�*k*�

> *nc* ∑ *i nci k d* (0)

exchange reactions occurring as a result of collisions of two molecules take the form

<sup>1</sup> )*Zvibr*

1 *kT<sup>c</sup>* 1

as well as the dissociation rate coefficient, depends on two temperatures (*T* and *T<sup>c</sup>*

<sup>1</sup> ) <sup>∑</sup> *ii*� *s c <sup>i</sup>* exp

<sup>1</sup> ) <sup>∑</sup> *i s c <sup>i</sup>* exp

*k d* (0) *d* (0)

*i k d* (0)

and depends on the gas temperature *T* only. One should keep in mind that the superscript "0" in the notations for the state-to-state rate coefficients indicates that they are calculated by averaging the corresponding inelastic collision cross sections with the Maxwell–Boltzmann distribution over the velocity and rotational energy. The relations connecting the multi-temperature rate coefficients of forward and backward reactions can be

rec,*<sup>c</sup>* (*T*) = ∑

For the generalized Treanor distribution (83), the multi-temperature rate coefficients of

*<sup>d</sup>* (*T*, *<sup>T</sup><sup>d</sup>*

<sup>−</sup> *kdε<sup>d</sup>* 1 *kT<sup>d</sup>* 1

The rate coefficient for the exchange reaction in a collision of a diatomic molecule and an atom,

<sup>1</sup> ) <sup>∑</sup> *iki*�*k*� *s c is d <sup>k</sup>* exp

 *k dk*, *d*� *k*� (0)

−*εc <sup>i</sup>* <sup>−</sup> *icε<sup>c</sup>* 1 *kT* <sup>−</sup> *icε<sup>c</sup>*

−*εc <sup>i</sup>* <sup>−</sup> *icε<sup>c</sup>* 1 *kT* <sup>−</sup> *icε<sup>c</sup>*

*rec*,*c*.

*k d* (0) *<sup>c</sup>*,*diss* <sup>=</sup> <sup>1</sup>

where *nci* denotes some non-equilibrium quasi-stationary distribution.

<sup>1</sup> ) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T*, *T<sup>c</sup>*

> *<sup>k</sup>* <sup>−</sup> *kdε<sup>d</sup>* 1 *kT* <sup>−</sup> *icε<sup>c</sup>*

*k dd*� (0) *cc*� <sup>=</sup> <sup>1</sup>

<sup>1</sup> , *<sup>T</sup><sup>d</sup>*

−*εd*

<sup>1</sup> ) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T*, *T<sup>c</sup>*

<sup>1</sup> ) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T*, *T<sup>c</sup>*

The total recombination rate coefficient *k*

*cc*� of the reaction (63) (during a collision of two molecules

*ci*,*c*�*i*� (*T*), (91)

*ci*, *diss*(*T*), (92)

−*εc <sup>i</sup>* <sup>−</sup> *icε<sup>c</sup>* 1 *kT* −

1 *kT<sup>c</sup>* 1

1 *kT<sup>c</sup>* 1

rec,*<sup>c</sup>* is defined in terms of the state-specific rate

rec,*ci*(*T*) (96)

 *k dd*� (0)

 *k d* (0)

*ci*, *<sup>c</sup>*�*i*� (*T*). (93)

1 ):

*ci*, *<sup>c</sup>*�*i*� (*T*), (94)

*ci*, *diss*(*T*). (95)

*<sup>c</sup>*, *diss*, and

The closed set of governing equations for the macroscopic parameters *nc*(**r**, *t*), **v**(**r**, *t*), *T*(**r**, *t*), and *T<sup>c</sup>* <sup>1</sup> (**r**, *t*) derived in Chikhaoui et al. (2000; 1997) includes the equations of the multi-temperature chemical kinetics for the species number densities

$$\frac{dn\_{\mathcal{L}}}{dt} + n\_{\mathcal{c}} \nabla \cdot \mathbf{v} + \nabla \cdot (n\_{\mathcal{c}} \mathbf{V}\_{\mathcal{c}}) = R\_{\mathcal{c}}^{\text{react}}, \ \mathcal{c} = 1, \dots, L,\tag{85}$$

relaxation equations for the specific numbers of vibrational quanta *Wc* in each molecular species c:

$$
\rho\_{\rm c} \frac{d\mathbf{W}\_{\rm c}}{dt} + \nabla \cdot \mathbf{q}\_{\rm w,c} = \mathbf{R}\_{\rm c}^{w} - \mathbf{W}\_{\rm c} m\_{\rm c} R\_{\rm c}^{\rm react} + \mathbf{W}\_{\rm c} \nabla \cdot (\rho\_{\rm c} \mathbf{V}\_{\rm c}) \,, \quad \mathbf{c} = 1 \,, \dots \, \mathbf{L}\_{\rm m} . \tag{86}
$$

along with the conservation equations for the momentum and the total energy, *Lm* is the number of molecular species in the mixture. The latter equations formally coincide with the corresponding equations (11) and (12) obtained in the two previous approaches. One should however bear in mind that in the multi-temperature approach, the total energy is a function of *T*, *T<sup>c</sup>* <sup>1</sup> , and *nc*, and the transport terms are expressed as functions of the same set of macroscopic parameters *T*, *T<sup>c</sup>* <sup>1</sup> , and *nc*.

The source terms in Eqs. (85) are determined by the collision operator of chemical reactions

$$\mathcal{R}\_{\mathbf{c}}^{react} = \sum\_{ij} \int J\_{cij}^{react} d\mathbf{u}\_{\mathbf{c}}.\tag{87}$$

The production terms in the relaxation equations (86) are expressed as functions of collision operators of all slow processes: VV2 and TRV vibrational energy transfers and chemical reactions,

$$\boldsymbol{R}\_{\mathcal{c}}^{w} = \sum\_{ij} \boldsymbol{i} \int f\_{\mathrm{cij}}^{\mathrm{sl}} \boldsymbol{d} \, \mathbf{u}\_{\mathcal{c}} = \boldsymbol{R}\_{\mathcal{c}}^{w, VV} + \boldsymbol{R}\_{\mathcal{c}}^{w, TRV} + \boldsymbol{R}\_{\mathcal{c}}^{w, reactant} \,. \tag{88}$$

The value **q***w*,*<sup>c</sup>* in Eq. (86) has the physical meaning of the vibrational quanta flux of *c* molecular species and is introduced on the basis of the additional collision invariant of the most frequent collisions *ic*:

$$\mathbf{q}\_{w,c} = \sum\_{ij} \mathbf{i} \int \mathbf{c}\_c f\_{cij} d\mathbf{u}\_c.$$

It is obvious that the system of governing equations in the multi-temperature approach is considerably simpler than the corresponding system in the state-to-state approach, since it contains much fewer equations. In the zero-order approximation of the Chapman-Enskog method, the system of governing equations takes the form typical for inviscid non-conductive flows. In this case equations (85), (86) read:

$$\frac{dn\_{\mathcal{L}}}{dt} + n\_{\mathcal{c}} \nabla \cdot \mathbf{v} = \mathcal{R}\_{\mathcal{c}}^{react(0)}, \qquad \mathbf{c} = \mathbf{1}, \ldots, \mathbf{L}, \tag{89}$$

$$
\rho\_{\mathcal{E}} \frac{dW\_{\mathcal{E}}}{dt} = R\_{\mathcal{E}}^{w(0)} - m\_{\mathcal{E}} \mathcal{W}\_{\mathcal{E}} R\_{\mathcal{E}}^{\text{react}(0)}, \ \ c = 1, \dots, L\_{\mathfrak{m}}.\tag{90}
$$

#### **4.2 Multi-temperature reaction rate coefficients**

18 Will-be-set-by-IN-TECH

The closed set of governing equations for the macroscopic parameters *nc*(**r**, *t*), **v**(**r**, *t*),

relaxation equations for the specific numbers of vibrational quanta *Wc* in each molecular

along with the conservation equations for the momentum and the total energy, *Lm* is the number of molecular species in the mixture. The latter equations formally coincide with the corresponding equations (11) and (12) obtained in the two previous approaches. One should however bear in mind that in the multi-temperature approach, the total energy is a function of

<sup>1</sup> , and *nc*, and the transport terms are expressed as functions of the same set of macroscopic

The source terms in Eqs. (85) are determined by the collision operator of chemical reactions

 *J react*

The production terms in the relaxation equations (86) are expressed as functions of collision operators of all slow processes: VV2 and TRV vibrational energy transfers and chemical

The value **q***w*,*<sup>c</sup>* in Eq. (86) has the physical meaning of the vibrational quanta flux of *c* molecular species and is introduced on the basis of the additional collision invariant of the

It is obvious that the system of governing equations in the multi-temperature approach is considerably simpler than the corresponding system in the state-to-state approach, since it contains much fewer equations. In the zero-order approximation of the Chapman-Enskog method, the system of governing equations takes the form typical for inviscid non-conductive

**c***<sup>c</sup> fcijd***u***c*.

*dt* <sup>+</sup> *nc* ∇· **<sup>v</sup>** <sup>=</sup> *<sup>R</sup>react*(0) *<sup>c</sup>* , *<sup>c</sup>* <sup>=</sup> 1, ..., *<sup>L</sup>*, (89)

*dt* <sup>=</sup> *<sup>R</sup>w*(0) *<sup>c</sup>* <sup>−</sup> *mcWcRreact*(0) *<sup>c</sup>* , *<sup>c</sup>* <sup>=</sup> 1, ..., *<sup>L</sup>*m. (90)

**q***w*,*<sup>c</sup>* = ∑ *ij i* 

*Rreact <sup>c</sup>* = ∑ *ij*

multi-temperature chemical kinetics for the species number densities

*dt* <sup>+</sup> *nc*∇ · **<sup>v</sup>** <sup>+</sup> ∇ · (*nc***V***c*) = *<sup>R</sup>react*

*<sup>c</sup>* <sup>−</sup> *WcmcRreact*

*dnc*

*dt* <sup>+</sup> ∇ · **<sup>q</sup>***w*,*<sup>c</sup>* <sup>=</sup> *<sup>R</sup><sup>w</sup>*

<sup>1</sup> , and *nc*.

*Rw <sup>c</sup>* = ∑ *ij i J sl*

flows. In this case equations (85), (86) read:

*dnc*

*ρc dWc*

<sup>1</sup> (**r**, *t*) derived in Chikhaoui et al. (2000; 1997) includes the equations of the

*<sup>c</sup>* , *c* = 1, ..., *L*, (85)

*<sup>c</sup>* + *Wc*∇ · (*ρc***V***c*), *c* = 1, ..., *L*m. (86)

*cij d***u***c*. (87)

*cij<sup>d</sup>* **<sup>u</sup>***<sup>c</sup>* <sup>=</sup> *<sup>R</sup>w*, *VV*<sup>2</sup> *<sup>c</sup>* <sup>+</sup> *<sup>R</sup>w*, *TRV <sup>c</sup>* <sup>+</sup> *<sup>R</sup>w*,*react <sup>c</sup>* . (88)

*T*(**r**, *t*), and *T<sup>c</sup>*

species c:

*T*, *T<sup>c</sup>*

reactions,

most frequent collisions *ic*:

parameters *T*, *T<sup>c</sup>*

*ρc dWc* The production terms in Eqs. (89) may be written in the form (13), (14) similar to the equations obtained in the one-temperature approximation. However, the coefficients in these expressions differ from those in the one-temperature approach and contains the multi-temperature rate coefficients *kdd*� *cc*� of the reaction (63) (during a collision of two molecules or a molecule and an atom A*d*), the two-temperature coefficients of dissociation *k<sup>d</sup> <sup>c</sup>*, *diss*, and recombination rate coefficients *k<sup>d</sup> rec*,*c*.

In the zero-order approximation, the multi-temperature rate coefficients of exchange and dissociation reactions can be expressed in terms of the state-specific rate coefficients considered in section 3.2:

$$k\_{c c'}^{dd'}(0) = \frac{1}{n\_c n\_d} \sum\_{ikl'k'} n\_{ci} n\_{dk} \, k\_{ci, c'l'}^{dk, d'k'}(0) \,\prime \tag{91}$$

$$\boldsymbol{k}\_{c,diss}^{d(0)} = \frac{1}{n\_c} \sum\_{i} n\_{ci} \, k\_{ci,diss}^{d(0)}(T) \,\tag{92}$$

where *nci* denotes some non-equilibrium quasi-stationary distribution.

For the generalized Treanor distribution (83), the multi-temperature rate coefficients of exchange reactions occurring as a result of collisions of two molecules take the form

$$k\_{\varepsilon\varepsilon'}^{dd'}(0^{\circ},T\_1^{\varepsilon},T\_1^d) = \frac{1}{Z\_{\varepsilon}^{\mathrm{vibr}}(T,T\_1^{\varepsilon})Z\_d^{\mathrm{vibr}}(T,T\_1^d)}\sum\_{ikl'k'}s\_l^{\varepsilon}s\_k^d \exp\left(-\frac{\varepsilon\_l^{\varepsilon}-i\_{\varepsilon}\varepsilon\_1^{\varepsilon}}{kT}-\frac{\varepsilon\_l^{\varepsilon}-i\_{\varepsilon}\varepsilon\_1^{\varepsilon}}{kT\_1^{\varepsilon}-kT\_1^d}\right)$$

$$-\frac{\varepsilon\_k^d-k\_d\varepsilon\_1^d}{kT}-\frac{i\_{\varepsilon}\varepsilon\_1^{\varepsilon}}{kT\_1^{\varepsilon}}-\frac{k\_d\varepsilon\_1^d}{kT\_1^d}\Big)k\_{\varepsilon i\_{\varepsilon}\varepsilon'l'}^{dk,d'k'}(T). \tag{93}$$

The rate coefficient for the exchange reaction in a collision of a diatomic molecule and an atom, as well as the dissociation rate coefficient, depends on two temperatures (*T* and *T<sup>c</sup>* 1 ):

$$k\_{\varepsilon\varepsilon'}^{dd'(0)}(T, T\_1^{\varepsilon}) = \frac{1}{Z\_{\varepsilon}^{\text{vibr}}(T, T\_1^{\varepsilon})} \sum\_{\vec{i}\vec{i}'} s\_{\vec{i}}^{\varepsilon} \exp\left(-\frac{\varepsilon\_{\vec{i}}^{\varepsilon} - \dot{t}\_{\vec{\epsilon}} \varepsilon\_{1}^{\varepsilon}}{kT} - \frac{\dot{t}\_{\vec{\epsilon}} \varepsilon\_{1}^{\varepsilon}}{kT\_1^{\varepsilon}}\right) k\_{\vec{c}\vec{i}, \vec{\epsilon}'\vec{i}'}^{dd'(0)}(T),\tag{94}$$

$$k\_{c, \text{diss}}^{d(0)}(T, T\_1^{\varepsilon}) = \frac{1}{Z\_{\varepsilon}^{\text{vibr}}(T, T\_1^{\varepsilon})} \sum\_{i} s\_i^{\varepsilon} \exp\left(-\frac{\varepsilon\_i^{\varepsilon} - i\_{\varepsilon} \varepsilon\_1^{\varepsilon}}{kT} - \frac{i\_{\varepsilon} \varepsilon\_1^{\varepsilon}}{kT\_1^{\varepsilon}}\right) k\_{c\bar{i}, \text{diss}}^{d(0)}(T). \tag{95}$$

The total recombination rate coefficient *k d* (0) rec,*<sup>c</sup>* is defined in terms of the state-specific rate coefficients as follows

$$k\_{\text{rec},\mathcal{c}}^{d}(T) = \sum\_{i} k\_{\text{rec},\mathcal{c}i}^{d}(T) \tag{96}$$

and depends on the gas temperature *T* only. One should keep in mind that the superscript "0" in the notations for the state-to-state rate coefficients indicates that they are calculated by averaging the corresponding inelastic collision cross sections with the Maxwell–Boltzmann distribution over the velocity and rotational energy. The relations connecting the multi-temperature rate coefficients of forward and backward reactions can be

Using the Treanor distribution (83) for *ni*, the factor *Z* is given by the relation

*<sup>Z</sup>vibr*(−*U*)*Zvibr*(*T*, *<sup>T</sup>*1)<sup>∑</sup>

*Z*(*T*, *Tv*, *U*) =

*Tf* =

 1 *Tv* − 1 *<sup>T</sup>* <sup>−</sup> <sup>1</sup> *U*

*i*

For the harmonic oscillator model, the non-equilibrium factor is specified by the vibrational

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 135

Figure 2 presents the temperature dependence of the non-equilibrium factor *Z*(*T*, *T*1, *U*) in nitrogen for fixed vibrational temperature values. The non-equilibrium factor is calculated for both anharmonic (104) and harmonic (105) oscillator models. We can see that for minor deviations from the equilibrium (*T*1/*T* ∼ 1), both models yield similar results, whereas for the ratio *T*1/*T* essentially different from unity, the values of *Z* for harmonic and anharmonic oscillators differ considerably. In particular, for the selected dissociation model, the non-equilibrium factor and hence the dissociation rate coefficient of harmonic oscillators

oscillators. For *T*1/*T <* 1, the use of the harmonic oscillator model yields lower *Z* and *k<sup>d</sup>*

1000 2000 3000 4000 5000 6000 7000

temperatures *T*<sup>1</sup> and *U* = *D*/(6*k*). The solid lines represent anharmonic oscillators, dashed — harmonic oscillators. The curves *1, 1*� — *T*<sup>1</sup> = 3000; *2, 2*� — *T*<sup>1</sup> = 5000; *3, 3*� — *T*<sup>1</sup> = 7000 K.

*3'*

Fig. 2. The non-equilibrium factor *Z* in N2 as a function of temperature *T* for fixed

*1'*

*si* exp

*Zvibr*(*T*)*Zvibr*(*Tf*) *<sup>Z</sup>vibr*(−*U*)*Zvibr*(*Tv*)

> −<sup>1</sup> ,

*iε*1 *k*

 1 *<sup>T</sup>* <sup>−</sup> <sup>1</sup> *T*1 + *εi kU*

*diss*, respectively, when calculated for anharmonic

*T*, K

. (104)

*diss*

, (105)

*<sup>Z</sup>*(*T*, *<sup>T</sup>*1, *<sup>U</sup>*) = *<sup>Z</sup>vibr*(*T*)

where the effective temperature *Tf* is defined as

at *T*1/*T >* 1 significantly exceed *Z* and *k<sup>d</sup>*

<sup>10</sup>-8 1

10<sup>8</sup>

10<sup>16</sup>

10<sup>24</sup>

10<sup>32</sup>

10<sup>40</sup>

10<sup>48</sup> *Z*

than those obtained taking into account anharmonic effects.

*2'*

*1*

*3*

*2*

temperature *Tv* and can be calculated using the expression:

obtained applying the microscopic detailed balance relations for the collision cross sections. For bimolecular reactions we find

$$k\_{c'c}^{d'd}(^0(T, T\_1^{c'}, T\_1^{d'}) = \frac{1}{Z\_{c'}^{vibr}(T, T\_1^{c'}) Z\_{d'}^{vibr}(T, T\_1^{d'})} \sum\_{ikl'k'} s\_{l'}^{c'} s\_{k'}^{d'} \exp\left(-\frac{i\_{c'}^{t} \varepsilon\_1^{c'}}{kT\_1^{t'}} - \frac{1}{kT\_1^{d'}}\right)$$

$$-\frac{k\_{d'}^{t} \varepsilon\_1^{d'}}{kT\_1^{d'}} - \frac{\varepsilon\_{l'}^{c'} - i\_{c'}^{t} \varepsilon\_1^{c'}}{kT} - \frac{\varepsilon\_{k'}^{d'} - k\_{d'}^{t} \varepsilon\_1^{d'}}{kT} \right) k\_{c i\_c' d' i'}^{dk, d'k'}(0) \quad \text{ $\mathbf{K}\_{c'l, c1}^{d'k}$  ( $T$ )\_{\prime}} = 1$$

whereas for dissociation-recombination we can write

$$k\_{\rm rec,c}^{d}(0) = \sum\_{i} k\_{\rm rec,ci}^{d}(T) = \sum\_{i} k\_{\rm ci,diss}^{d}(T) K\_{\rm rec-diss,ci}^{d}(T). \tag{98}$$

In these formulae, the ratios for the state-to-state rate coefficients *Kd*� *k*� , *dk* (0) *c*�*i*� , *ci* (*T*), *<sup>K</sup><sup>d</sup>* (0) *rec*−*diss*,*ci*(*T*) are defined in (74), (75).

In the zero-order approximation, *<sup>R</sup>w*(0) *<sup>c</sup>* includes the vibrational distributions (83) and the state-to-state rate coefficients of VV2, VT vibrational energy transitions and chemical reactions.

If anharmonic effects are neglected then the Boltzmann vibrational distributions with the vibrational temperature *T<sup>c</sup> <sup>v</sup>* are valid, and the multi-temperature rate coefficients of the reactions (93)–(95) take the form

$$k\_{cc'}^{dd'\ (0)}(T, T\_{v\prime}^c, T\_{\overline{v}}^d) = \frac{1}{Z\_c^{vibr}(T\_{\overline{v}}^c) Z\_d^{vibr}(T\_{\overline{v}}^d)} \times$$

$$\times \sum\_{ikl'k'} s\_l^c s\_k^d \exp\left(-\frac{\varepsilon\_l^c}{kT\_{\overline{v}}^c} - \frac{\varepsilon\_k^d}{kT\_{\overline{v}}^d}\right) k\_{ci, \epsilon'i'}^{dk, d'k' \ (0)}(T) \,\,, \tag{99}$$

if *d* is a molecule,

$$k\_{\rm c\prime}^{dd'\prime(0)}(T, T\_{\rm v}^{\rm c}) = \frac{1}{Z\_{\rm c}^{\rm vibr}(T\_{\rm v}^{\rm c})} \sum\_{\rm i\prime} s\_{\rm i}^{\rm c} \exp\left(-\frac{\varepsilon\_{\rm i}^{\rm c}}{kT\_{\rm v}^{\rm c}}\right) k\_{\rm c i, \rm c'i\prime}^{dd'\prime(0)}(T),\tag{100}$$

if *d* is an atom, and

$$k\_{\rm c,diss}^{d}(T, T\_{\rm v}^{\rm c}) = \frac{1}{Z\_{\rm c}^{\rm vibr}(T\_{\rm v}^{\rm c})} \sum\_{\rm i} s\_{\rm i}^{\rm c} \exp\left(-\frac{\mathfrak{E}\_{\rm i}^{\rm c}}{kT\_{\rm v}^{\rm c}}\right) k\_{\rm c,diss}^{d}(T). \tag{101}$$

For the calculation of two-temperature dissociation rate coefficients in the most studies (see Marrone & Treanor (1963); Phys-Chem (2002))), the two-temperature dissociation rate coefficient *k<sup>d</sup> diss* is associated with the equilibrium averaged coefficient *<sup>k</sup><sup>d</sup> diss*,*eq* by introducing the two-temperature non-equilibrium factor *Z*(*T*, *T*1, *U*) rather than the state-to-state factor *Zi*(*T*, *U*):

$$k\_{\rm diss}^d = Z(T, T\_1, \mathcal{U}) k\_{\rm diss, eq}^d(T). \tag{102}$$

where

$$Z(T, T\_1, U) = \frac{1}{n\_{\rm m}} \sum\_{i} n\_i Z\_i(T, U). \tag{103}$$

20 Will-be-set-by-IN-TECH

obtained applying the microscopic detailed balance relations for the collision cross sections.

<sup>1</sup> )*Zvibr*

*<sup>k</sup>*�− *k*� *<sup>d</sup>*�*εd*� 1 *kT*

*rec*,*ci*(*T*) = ∑

*i k d* (0)

In the zero-order approximation, *<sup>R</sup>w*(0) *<sup>c</sup>* includes the vibrational distributions (83) and the state-to-state rate coefficients of VV2, VT vibrational energy transitions and chemical

If anharmonic effects are neglected then the Boltzmann vibrational distributions with the

*<sup>v</sup>* ) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T<sup>c</sup>*

> *<sup>v</sup>* ) <sup>∑</sup> *ii*� *s c <sup>i</sup>* exp − *εc i kT<sup>c</sup>* v *k dd*� (0)

*<sup>v</sup>* ) <sup>∑</sup> *i s c <sup>i</sup>* exp − *εc i kT<sup>c</sup>* v *k d* (0)

*diss* is associated with the equilibrium averaged coefficient *<sup>k</sup><sup>d</sup>*

*diss* <sup>=</sup> *<sup>Z</sup>*(*T*, *<sup>T</sup>*1, *<sup>U</sup>*)*k<sup>d</sup>*

*<sup>Z</sup>*(*T*, *<sup>T</sup>*1, *<sup>U</sup>*) = <sup>1</sup>

For the calculation of two-temperature dissociation rate coefficients in the most studies (see Marrone & Treanor (1963); Phys-Chem (2002))), the two-temperature dissociation rate

the two-temperature non-equilibrium factor *Z*(*T*, *T*1, *U*) rather than the state-to-state factor

*<sup>n</sup>*<sup>m</sup> ∑ *i*

*<sup>v</sup>* )*Zvibr <sup>d</sup>* (*T<sup>d</sup> v* ) ×

> *k dk*,*d*� *k*� (0)

− *εd k kT<sup>d</sup>* v

*<sup>d</sup>*� (*T*, *<sup>T</sup>d*�

 *k dk*, *d*� *k*� (0) *ci*, *<sup>c</sup>*�*i*� *<sup>K</sup>d*�

<sup>1</sup> ) <sup>∑</sup> *iki*�*k*� *s c*� *i*� *s d*� *<sup>k</sup>*� exp −*i* � *<sup>c</sup>*�*εc*� 1 *kTc*� 1 −

*ci*,*diss*(*T*)*K<sup>d</sup>* (0)

*<sup>v</sup>* are valid, and the multi-temperature rate coefficients of the

*k*� , *dk* (0)

> *k*� , *dk* (0)

*c*�*i*�

, *ci* (*T*), (97)

*rec*−*diss*,*ci*(*T*). (98)

, *ci* (*T*), *<sup>K</sup><sup>d</sup>* (0)

*ci*,*c*�*i*� (*T*), (99)

*ci*,*c*�*i*� (*T*), (100)

*ci*,diss(*T*). (101)

*diss*,*eq*(*T*). (102)

*niZi*(*T*, *U*). (103)

*diss*,*eq* by introducing

*rec*−*diss*,*ci*(*T*)

*c*�*i*�

For bimolecular reactions we find

−*k*� *<sup>d</sup>*�*εd*� 1 *kTd*� 1

*k*

<sup>1</sup> , *<sup>T</sup>d*�

− *εc*� *<sup>i</sup>*� − *i* � *<sup>c</sup>*�*εc*� 1 *kT* <sup>−</sup> *<sup>ε</sup>d*�

whereas for dissociation-recombination we can write

*<sup>d</sup>* (0) *rec*,*<sup>c</sup>* (*T*) = ∑

*i k d* (0)

In these formulae, the ratios for the state-to-state rate coefficients *Kd*�

*<sup>v</sup>*, *<sup>T</sup><sup>d</sup>*

<sup>v</sup>) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T<sup>c</sup>*

*<sup>v</sup>* ) = <sup>1</sup> *Zvibr <sup>c</sup>* (*T<sup>c</sup>*

*kd*

 − *εc i kT<sup>c</sup>* v

<sup>1</sup> ) = <sup>1</sup> *Zvibr <sup>c</sup>*� (*T*, *<sup>T</sup>c*�

*k d*� *d* (0) *<sup>c</sup>*�*<sup>c</sup>* (*T*, *<sup>T</sup>c*�

are defined in (74), (75).

vibrational temperature *T<sup>c</sup>*

reactions (93)–(95) take the form

*k dd*� (0) *cc*� (*T*, *<sup>T</sup><sup>c</sup>*

*k dd*� (0) *cc*� (*T*, *<sup>T</sup><sup>c</sup>*

*kd*

*<sup>c</sup>*,diss(*T*, *<sup>T</sup><sup>c</sup>*

× ∑ *iki*�*k*� *s c is d <sup>k</sup>* exp

reactions.

if *d* is a molecule,

if *d* is an atom, and

coefficient *k<sup>d</sup>*

*Zi*(*T*, *U*):

where

Using the Treanor distribution (83) for *ni*, the factor *Z* is given by the relation

$$Z(T, T\_1, U) = \frac{Z^{\text{vibr}}(T)}{Z^{\text{vibr}}(-U)Z^{\text{vibr}}(T, T\_1)} \sum\_{i} s\_i \exp\left[\frac{i\varepsilon\_1}{k}\left(\frac{1}{T} - \frac{1}{T\_1}\right) + \frac{\varepsilon\_i}{kLI}\right]. \tag{104}$$

For the harmonic oscillator model, the non-equilibrium factor is specified by the vibrational temperature *Tv* and can be calculated using the expression:

$$Z(T, T\_{\upsilon}, \mathcal{U}) = \frac{Z^{\upsilon \text{ibr}}(T) Z^{\upsilon \text{ibr}}(T\_f)}{Z^{\upsilon \text{ibr}}(-\mathcal{U}) Z^{\upsilon \text{ibr}}(T\_\upsilon)},\tag{105}$$

where the effective temperature *Tf* is defined as

$$T\_f = \left(\frac{1}{T\_v} - \frac{1}{T} - \frac{1}{U}\right)^{-1} J$$

Figure 2 presents the temperature dependence of the non-equilibrium factor *Z*(*T*, *T*1, *U*) in nitrogen for fixed vibrational temperature values. The non-equilibrium factor is calculated for both anharmonic (104) and harmonic (105) oscillator models. We can see that for minor deviations from the equilibrium (*T*1/*T* ∼ 1), both models yield similar results, whereas for the ratio *T*1/*T* essentially different from unity, the values of *Z* for harmonic and anharmonic oscillators differ considerably. In particular, for the selected dissociation model, the non-equilibrium factor and hence the dissociation rate coefficient of harmonic oscillators at *T*1/*T >* 1 significantly exceed *Z* and *k<sup>d</sup> diss*, respectively, when calculated for anharmonic oscillators. For *T*1/*T <* 1, the use of the harmonic oscillator model yields lower *Z* and *k<sup>d</sup> diss* than those obtained taking into account anharmonic effects.

Fig. 2. The non-equilibrium factor *Z* in N2 as a function of temperature *T* for fixed temperatures *T*<sup>1</sup> and *U* = *D*/(6*k*). The solid lines represent anharmonic oscillators, dashed — harmonic oscillators. The curves *1, 1*� — *T*<sup>1</sup> = 3000; *2, 2*� — *T*<sup>1</sup> = 5000; *3, 3*� — *T*<sup>1</sup> = 7000 K.

0 10 20 30 40 50

x/R

Fig. 4. Averaged dissociation rate coefficient *k*

oscillator model; 4: one-temperature model.

O2 /O

a)

10-50 10-48 10-46 10-44 1x10-42 10-40 10-38 10-36 10-34 1x10-32 10-30 10-28 10-26 1x10-24 10-22

dissociation rate coefficient

Vibrational and Chemical Kinetics in Non-Equilibrium Gas Flows 137

(*mol*)

O2/O, *T*<sup>∗</sup> = 4000 K, *p*<sup>∗</sup> = 1 atm; **(b)** N2/N, *T*<sup>∗</sup> = 7000 K, *p*<sup>∗</sup> = 1 atm. Curves 1: state-to-state model; 2: two-temperature anharmonic oscillator model; 3: two-temperature harmonic

oscillators and a complex distribution for anharmonic oscillators studied in Kustova et al. (2003). One can see a quite strong influence of the kinetic model on the averaged dissociation

obtained in the rigorous state-to-state approximation, the same effect is obtained for *kat*

In this Chapter, the theoretical models for non-equilibrium chemical kinetics in multi-component reacting gas flows are proposed on the basis of three approaches of the kinetic theory. In the frame of the one-temperature approximation the chemical kinetics in thermal equilibrium flows or deviating weakly from thermal equilibrium is studied. The coupling of chemical kinetics and fluid dynamics equations is considered in the Euler and Navier-Stokes approximations. Chemical kinetics in vibrationally non-equilibrium flows is considered on the basis of the state-to-state and multi-temperature approaches. Different models for vibrational-chemical coupling in the flows of multi-component mixtures are derived. The influence of non-equilibrium distributions on reaction rates in the flows behind

We are grateful to the Russian Foundation for Basic Research (Grant 11-01-00408) and Ministry of Education and Science of RF (Contract 13.G25.31.0076) for the support of this study.

Alexeev, B., Chikhaoui, A. & Grushin, I. (1994). Application of the generalized

Chapman-Enskog method to the transport-coefficients calculation in a reacting gas

rate coefficients, all quasi-stationary models give the values of *kmol*

*diss* is higher than *<sup>k</sup>mol*

shock waves and in nozzle expansion is demonstrated.

mixture, *Phys. Review E* 49: 2809.

4

0 10 20 30 40 50

x/R

*diss* (m3/s) versus x/R in a conic nozzle. **(a)**

*diss*, i.e., atoms are more efficient as partners in the

2

1x10-20 N2

3

*diss* rather far from those

1

/N

b)

*diss*.

3

2 1

10-48 10-46 10-44 1x10-42 10-40 10-38 10-36 10-34 1x10-32 10-30 10-28 10-26 1x10-24 10-22

dissociation rate coefficient

4

In all considered cases, *kat*

**6. Acknowledgements**

**7. References**

dissociation process.

**5. Conclusions**

Note in addition that the ratio of the dissociation and recombination rate coefficients *K*rec–diss under the non-equilibrium conditions can also be expressed in terms of the averaged non-equilibrium factor:

$$K\_{\rm rec-diss} = \frac{k\_{\rm rec}^d}{k\_{\rm diss}^d} = \frac{1}{Z} K\_{\rm rec-diss}^{eq}(T) \tag{106}$$

where *Keq rec*−*diss*(*T*) is the ratio of the dissociation and recombination rate coefficients in a thermal equilibrium gas. Calculating *Z* from the state-to-state or quasi-stationary vibrational distributions *ni*/*<sup>n</sup>* for various dissociation models, we can find *Krec*−*diss* under thermal non-equilibrium conditions and estimate its deviation from the equilibrium constant.

The equations of non-equilibrium reacting flows derived in the state-to-state, multi-temperature and one-temperature approaches were applied for calculations of distributions and macroscopic parameters in particular flows of air components behind shock waves, in nozzles, in non-equilibrium boundary layer (see Nagnibeda & Kustova (2009) and references in this book). On the basis of obtained distributions, global reaction rates (92) were calculated in relaxation zone behind the shock wave Kustova & Nagnibeda (2000) and in nozzle expansion Kustova et al. (2003) in different approaches. The results obtained for the relaxation zone behind the shock wave at the following free stream conditions: *T*<sup>0</sup> = 293 K, *p*<sup>0</sup> = 100 Pa, *M*<sup>0</sup> = 15 are presented in Fig. 3.

Fig. 3. The averaged dissociation rate coefficient *kN*<sup>2</sup> *diss* as a function of *x*. Curves *1, 2, 3* are, respectively, for the state-to-state, two-temperature, and one-temperature approaches.

It is seen that the one-temperature model describes the behavior of the dissociation rate coefficient inadequately, particularly close to the shock front. The two-temperature approach provides more realistic values for the dissociation rate coefficient, overestimating however *kN*<sup>2</sup> *diss* in comparison to the state-to-state approximation at *x <* 0.5 cm.

The averaged dissociation rate coefficient *kmol diss* calculated for O2/O and N2/N mixtures in a conic nozzle in four approaches using state-to-state, multi-temperature and one-temperature distributions, is presented in Fig. 4. The following conditions in the throat are considered: for O2/O mixture, *T*<sup>∗</sup> = 4000 K, *p*<sup>∗</sup> = 1 atm; for N2/N mixture, *T*<sup>∗</sup> = 7000 K, *p*<sup>∗</sup> = 1 atm. Two kinds of multi-temperature distributions are applied: the Boltzmann distribution for harmonic

Fig. 4. Averaged dissociation rate coefficient *k* (*mol*) *diss* (m3/s) versus x/R in a conic nozzle. **(a)** O2/O, *T*<sup>∗</sup> = 4000 K, *p*<sup>∗</sup> = 1 atm; **(b)** N2/N, *T*<sup>∗</sup> = 7000 K, *p*<sup>∗</sup> = 1 atm. Curves 1: state-to-state model; 2: two-temperature anharmonic oscillator model; 3: two-temperature harmonic oscillator model; 4: one-temperature model.

oscillators and a complex distribution for anharmonic oscillators studied in Kustova et al. (2003). One can see a quite strong influence of the kinetic model on the averaged dissociation rate coefficients, all quasi-stationary models give the values of *kmol diss* rather far from those obtained in the rigorous state-to-state approximation, the same effect is obtained for *kat diss*. In all considered cases, *kat diss* is higher than *<sup>k</sup>mol diss*, i.e., atoms are more efficient as partners in the dissociation process.

#### **5. Conclusions**

22 Will-be-set-by-IN-TECH

Note in addition that the ratio of the dissociation and recombination rate coefficients *K*rec–diss under the non-equilibrium conditions can also be expressed in terms of the averaged

thermal equilibrium gas. Calculating *Z* from the state-to-state or quasi-stationary vibrational distributions *ni*/*<sup>n</sup>* for various dissociation models, we can find *Krec*−*diss* under thermal non-equilibrium conditions and estimate its deviation from the equilibrium constant.

The equations of non-equilibrium reacting flows derived in the state-to-state, multi-temperature and one-temperature approaches were applied for calculations of distributions and macroscopic parameters in particular flows of air components behind shock waves, in nozzles, in non-equilibrium boundary layer (see Nagnibeda & Kustova (2009) and references in this book). On the basis of obtained distributions, global reaction rates (92) were calculated in relaxation zone behind the shock wave Kustova & Nagnibeda (2000) and in nozzle expansion Kustova et al. (2003) in different approaches. The results obtained for the relaxation zone behind the shock wave at the following free stream conditions: *T*<sup>0</sup> = 293 K,

0 0.5 1.0 1.5 2.0

respectively, for the state-to-state, two-temperature, and one-temperature approaches.

It is seen that the one-temperature model describes the behavior of the dissociation rate coefficient inadequately, particularly close to the shock front. The two-temperature approach provides more realistic values for the dissociation rate coefficient, overestimating however

conic nozzle in four approaches using state-to-state, multi-temperature and one-temperature distributions, is presented in Fig. 4. The following conditions in the throat are considered: for O2/O mixture, *T*<sup>∗</sup> = 4000 K, *p*<sup>∗</sup> = 1 atm; for N2/N mixture, *T*<sup>∗</sup> = 7000 K, *p*<sup>∗</sup> = 1 atm. Two kinds of multi-temperature distributions are applied: the Boltzmann distribution for harmonic

1

<sup>=</sup> <sup>1</sup> *<sup>Z</sup> <sup>K</sup>eq*

*rec*−*diss*(*T*) is the ratio of the dissociation and recombination rate coefficients in a

*rec*−*diss*(*T*), (106)

x, cm

*diss* as a function of *x*. Curves *1, 2, 3* are,

*diss* calculated for O2/O and N2/N mixtures in a

*rec kd diss*

*Krec*−*diss* <sup>=</sup> *<sup>k</sup><sup>d</sup>*

non-equilibrium factor:

*p*<sup>0</sup> = 100 Pa, *M*<sup>0</sup> = 15 are presented in Fig. 3.

<sup>k</sup>diss·1020, m3

/s

2

4

3

Fig. 3. The averaged dissociation rate coefficient *kN*<sup>2</sup>

The averaged dissociation rate coefficient *kmol*

2

*diss* in comparison to the state-to-state approximation at *x <* 0.5 cm.

6

8

10

where *Keq*

*kN*<sup>2</sup>

In this Chapter, the theoretical models for non-equilibrium chemical kinetics in multi-component reacting gas flows are proposed on the basis of three approaches of the kinetic theory. In the frame of the one-temperature approximation the chemical kinetics in thermal equilibrium flows or deviating weakly from thermal equilibrium is studied. The coupling of chemical kinetics and fluid dynamics equations is considered in the Euler and Navier-Stokes approximations. Chemical kinetics in vibrationally non-equilibrium flows is considered on the basis of the state-to-state and multi-temperature approaches. Different models for vibrational-chemical coupling in the flows of multi-component mixtures are derived. The influence of non-equilibrium distributions on reaction rates in the flows behind shock waves and in nozzle expansion is demonstrated.

#### **6. Acknowledgements**

We are grateful to the Russian Foundation for Basic Research (Grant 11-01-00408) and Ministry of Education and Science of RF (Contract 13.G25.31.0076) for the support of this study.

#### **7. References**

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260: 49–72.


**6** 

**Numerical Analysis of the Effect of** 

 **Rise Rate in HCCI Engine by Using** 

 **Multi-Zone Chemical Kinetics** 

Ock Taeck Lim1 and Norimasa Iida2

*1University of Ulsan, 2Keio University, 1Republic of Korea* 

*2Japan* 

**Inhomogeneous Pre-Mixture on Pressure** 

In HCCI, a premixed air/fuel mixture is inhaled to the combustion chamber and ignited by adiabatic compression. The HCCI engine, which is using the bulk combustion, is promising concept of the high efficient and low emission engine. However, power produced in HCCI engine is limited by knocking cause of excessive heat release and pressure rise at local area in condition of high load. Therefore, the main investigation of the HCCI is avoiding knocking problem to be able to operate at a high load. Delaying the combustion angle and stratifying temperature or concentration of fuel resulting in dispersion of combustion timing is the well-known effective method. In this study, effect of the pressure increase rate in combustion chamber by stratified temperature and fuel concentration of pre mixture according to fuel characteristic was investigated. Numerical analysis in chemical reaction

In this paper, DME(Di-Methyle Ether) and Methane were used as a fuel. The Numerical analysis was conducted for the purpose of investigating that fuel or temperature stratification effect on pressure rise rate and emission using the multi-zone code with detailed chemical kinetics. DME has two stages of heat reaction which is called HTR(High Temperature Reaction) and LTR(Low Temperature Reaction). In case of DME, proportion of LTR is more than those of HTR. It is known that the more DME concentration make heat release increase during the LTR. Due to those facts, existing of DME fuel stratification in combustion chamber is expected to contribute reduction of gas pressure rise rate because heat release difference in local area during LTR causes stratification of temperature before the HTR which makes combustion duration extend. On the other hand, Methane has only 1 stage heat release and self-ignition temperature is high. The detailed specifications of the test engine are presented in Table 1. The hermetic gas is used for calculation duration from

was conducted for investigating an emission property.

**1. Introduction** 

**2. Experimental method** 

