# Combustion kinetics I¶

- Equilibrium is
*approached*in combustion, but often kinetics are limiting and required for an accurate representation of temperature and composition. - Kinetics are important for soot and NOx
- Flame extinction and ignition phenomena
**Combustion chemistry is complex:**hundreds (thousands) of species and reactions.

## Elementary reactions¶

- Bimolecular reaction $$A + B \rightarrow C + D$$

$$\frac{d[A]}{dt} = -k[A][B]$$

- In general: $$ aA + bB \rightleftharpoons cC + dD$$ $$\frac{d[A]}{dt} = -ak_f[A]^a[B]^b + ak_r[C]^c[D]^d$$
- For elementary reactions the powers on [A], [B], etc. are integers, and correspond to the coefficients in the reaction.

$\alpha$ is a pre-exponential rate factor.

$\beta$ is a temperature coefficient.

$E_a$ is the activation energy.

Unimolecular reaction: $A\rightarrow B$

Termolecular reaction: $A+B+M\rightarrow \text{products}$

In combustion, we often see species M, which is some generic species.

- Normally occurs in radical reactions.

## Global reactions¶

- Lump many elementary steps into one (or more) overall reaction. $$\text{Fuel} + a\text{Oxidizer} \rightarrow b\text{Products}$$
- For methane $$\frac{d[CH_4]}{dt} = -k[CH_4]^n[O_2]^m$$
- Generally $n$, and $m$ are not integers.
- Can be greater than or less than 1.

- The reaction rate is
**empirical**and only valid within the range of conditions used to fit the data. **Use with caution**- Can be very convenient.
- See Turns Table 5.1
- See also Westbrook 1981 simplified reaction mechanisms

## Compact notation¶

- To do combustion kinetics requires computer codes.
- Need a
*framework*to express reactions and rates on a consistent basis.

### Reactions¶

- Consider reaction $i$, made up of
**all**species $j$ $$\sum_{j=1}^{N_{sp}} \nu_{i,j}^\prime X_j\rightleftharpoons \sum_{j=1}^{N_{sp}} \nu_{i,j}^{\prime\prime}X_j$$ - $\nu_{i,j}^{\prime}$ is the
**reactant**coefficient of species $j$ in reaction $i$. - $\nu_{i,j}^{\prime\prime}$ is the
**product**coefficient of species $j$ in reaction $i$. - $X_j$ simplly denotes species $j$ in the reaction.
- Most of the $\nu$ are zero, since most reactions only involve a very few species.

#### Matrix form¶

$$[V^{\prime}]\mathbf{X} \rightleftharpoons [V^{\prime\prime}]\mathbf{X}$$- $[V^{\prime}]$ has $N_{rxns}$ rows, and $N_{sp}$ columns. $[V^{\prime\prime}]$ also.
- $\mathbf{X}$ is the vector of all species.

#### Example: Hydrogen:¶

- Species: $O_2$, $H_2$, $H_2O$, $HO_2$, $O$, $H$, $OH$, $M$
- Reactions: $$H_2 + O_2 \rightleftharpoons HO_2 + H$$ $$H + O_2 \rightleftharpoons OH + O$$ $$OH + H_2 \rightleftharpoons H_2O + H$$ $$H + O_2 + M \rightleftharpoons HO_2 + M$$

$[V^{\prime}] = $

Rxn | $O_2$ | $H_2$ | $H_2O$ | $HO_2$ | $O$ | $H$ | $OH$ | $M$ |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

3 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

4 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |

$[V^{\prime\prime}] = $

Rxn | $O_2$ | $H_2$ | $H_2O$ | $HO_2$ | $O$ | $H$ | $OH$ | $M$ |
---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |

3 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |

4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |

### Reaction rates¶

- Reaction rates are (forward rate) - (reverse rate)

For the reaction $$ aA + bB \rightleftharpoons cC + dD,$$ we have $$R_f = k_f[A]^a[B]^b[C]^0[D]^0 = k_f[A]^a[B]^b,$$ $$R_r = k_r[A]^0[B]^0[C]^c[D]^d = k_r[C]^c[D]^d.$$

The net rate for each reaction $i$ is called the

**rate of progress variable**, and is given by $$q_i = R_{f,i} - R_{r,i}$$The net reaction rate for species $j$ is then $$\dot{\omega}_j = \sum_{i=1}^{N_{rxns}}\underbrace{(\nu_{i,j}^{\prime\prime}-\nu_{i,j}^{\prime})}_{\nu_{i,j}}q_i$$ Or,

where the elements of $[V]$ are $\nu_{i,j}^{\prime\prime}-\nu_{i,j}^{\prime}$.

## Reverse reaction rate constants¶

- Reaction rate constants are independent of composition.
- Hence, for a reaction at equilibrium, $q_i=R_{f,i}-R_{r,i}=0$.
- This gives $R_{f,i}=R_{r,i}$, or $$k_{f,i}\prod_j[X_j]^{\nu_{i,j}^\prime} = k_{r,i}\prod_j[X_j]^{\nu_{i,j}^{\prime\prime}}$$ $$\frac{k_{f,i}}{k_{r,i}} = \prod_j[X_j]^{\nu_{i,j}} = \prod_j\left[\frac{P_i}{RT}\right]^{\nu_{i,j}} = \prod_j\left[\frac{P_i}{P_o}\right]^{\nu_{i,j}}\left[\frac{P_o}{RT}\right]^{\nu_{i,j}} = K_{eq}\prod_j\left[\frac{P_o}{RT}\right]^{\nu_{i,j}} = K_c$$

## Quasi-steady-state approximation (QSSA)¶

- Fast reacting species, like radical intermediates may be formed and destroyed very fast relative to other species.
- Assume that the fast species are in steady state relative to the slower species.

### Zeldovich Mechanism¶

$$O + N_2 \rightarrow NO + N\phantom{xxxxx}\text{slow}$$ $$N + O_2 \rightarrow NO + O\phantom{xxxxx}\text{fast}$$

$$\frac{d[N]}{dt} = k_1[O][N_2] - k_2[N][O_2] = 0$$ Solve this for $[N]$

$$[N]_{ss} = \frac{k_1}{k_2}\frac{[O][N_2]}{[O_2]}.$$- We have replaced an ODE with an algebraic expression for $[N]$ in terms of other species.
- $[N]_{ss}$ is a function of time though, through the time dependence of $[O]$, $[N_2]$, and $[O_2]$.

## Partial equilibrium¶

- We can also treat fast reactions as if they were in equilibrium (see Turns).
- This again leads to algebraic expressions.

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