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MAE 5310: COMBUSTION FUNDAMENTALS

Chemical Kinetics:

Steady-State Approximation and Chain Reactions

October 1, 2012

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

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STEADY-STATE APPROXIMATION

• What are we talking about:

– In many chemical combustion systems, many highly reactive intermediate species (radicals) are formed

• Physical explanation:

– Rapid initial build-up of radical concentration

– Then radicals are destroyed as quickly as they are being created

• Implies that rate of radical formation = rate of radical destruction

– Situation typically occurs when the reaction forming the radicals is slow and the reaction destroying the radicals is fast

– This implies that the concentration of radicals is small in comparison with those of the reactants and products

• The radical species can thus be assumed to be at steady-state

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MECHANISM FOR UNIMOLECULAR REACTIONS

1

1

*

*

***

*

*

*

Mkk

Mkk

productsA

Mkk

MAk

kMk

MAkkAk

dt

prodd

kMk

MAkA

AkMAkMAkdt

Ad

productsA

MAMA

MAMA

uni

de

eapp

k

uni

de

e

unide

euniuni

unide

e

unidee

k

k

k

app

uni

de

e

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CHAIN AND CHAIN-BRANCHING REACTIONS• Chain reactions involve production of a radical species that subsequently reacts to produce another

radical. This radical in turn, reacts to produce yet another radical

• Sequence of events, called chain reaction, continues until a reaction involving the formation of a stable species from two radicals breaks the chain

• Consider a hypothetical reaction, represented by a global mechanism:

MABMBA

AABAB

BABBA

MAAMA

ABBA

k

k

k

k

4

3

2

1

2

2

2

22 2

Chain-initiation reaction

Chain-propagating reactions

Chain-terminating reaction

Global model

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CHAIN AND CHAIN-BRANCHING REACTIONS: EXAMPLE

21

4

32121

22

21

22

41

322

12

21

42

31

21

2

22

1

22

41

32

2

2

2

1

42322

4232221

42322

222

23212

411

2

411

2

0

02

k

kkkBA

M

B

kk

kkAM

k

dt

Bd

kk

kk

B

A

M

B

kk

kk

B

AM

k

kA

MBAkABkBAk

MBAkABkBAkMAk

MBAkABkBAkdt

ABd

ABkdt

Bd

BAkMAkdt

Ad

Steady-state approximationfor radical concentration

In the early stages of reaction, the concentrationof the product AB is small, as are the concentrationsof A and B throughout the course of the reactiontherefore, reverse reactions may be neglected atthis reaction stage

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CONCLUSIONS

• Term in brackets [ ] >> 1 because the rate coefficients for the radical concentrations, k2 and k3 are much larger than k1 and k4

• Can write approximate expressions for [A] and d[B2]/dt

• Radical concentration depends on the square root ratio of k1 to k4

– The greater the initiation rate, the greater the radical concentration

– The greater the termination rate, the lesser the radical concentration

– Rate coefficients of chain-propagating steps are likely to have little effect upon radical concentration because k2 and k3 appear as a ratio, and their influence on the radical concentration would be small for rate coefficients of similar magnitude.

• Increasing k2 and k3 results in an increased disappearance of [B2]

• Note that these scalings break down at pressure sufficiently high to cause 4k2k3[B2]/(k1k4[M]2) >> 1

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COMMENTS ON CHAIN-BRANCHING

• Chain Branching

– Chain branching reactions involve formation of two radical species from a reaction that consumes only one radical

• Example: O + H2O → OH + OH

– Existence of a chain-branching step in a chain reaction mechanism can have an explosive effect

• Example: Explosions in H2 and O2 mixtures, that we will examine in our study of detailed mechanisms, are a direct result of chain-branching steps

• Example: Chain-branching reactions are responsible for a flame being self-propagating

– In systems with chain branching, it is possible for the concentration of a radical species to build up geometrically, causing the rapid formation of products

– Unlike the previous hypothetical example, the rate of chain-initiation step does not control the overall reaction rate

– With chain-branching, the rates of radical reactions dominate

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CONCLUSIONS AND COMMENTS ON CHAIN-BRANCHING• Conclusions:

– First term in both equations dominates at low pressures– Concentration of A depends directly on the ratio of the initiation-step rate coefficient, k1, to the

first propagation step, k2, rate coefficient– The rate at which B2 disappears is governed by the initiation step– Increasing k2 and k3 has virtually no effect on the production rates of the products– The termolecular rate coefficient, k4, has virtually no effect on either the radical concentration or

the overall reaction rate, however at higher pressures it does have an influence in the 2nd terms

• Chain Branching– Chain branching reactions involve formation of two radical species from a reaction that consumes

only one radical• Example: O + H2O → OH + OH

– Existence of a chain-branching step in a chain reaction mechanism can have an explosive effect• Example: Explosions in H2 and O2 mixtures, that we will examine in our study of detailed

mechanisms, are a direct result of chain-branching steps• Example: Chain-branching reactions are responsible for a flame being self-propagating

– In systems with chain branching, it is possible for the concentration of a radical species to build up geometrically, causing the rapid formation of products

– Unlike the previous hypothetical example, the rate of chain-initiation step does not control the overall reaction rate

– With chain-branching, the rates of radical reactions dominate

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EXAMPLE: ZELDOVICH MECHANISM FOR NO• A famous mechanism for the formation of nitric oxide from atmospheric nitrogen is called the

Zeldovich (thermal) mechanism, given by:

• Because the second reaction is much fast than the first, the steady-state approximation can be used to evaluate the N-atom concentration. Furthermore, in high-temperature systems, the NO formation reaction is typically much slower than other reactions involving O2 and O. This, the O2 and O can be assumed to be in equilibrium, i.e., O2 ↔ 2O.

• Construct a global mechanism:

• Determine kglobal, m, and n using the elementary rate coefficients, etc., from the detailed mechanism• Using these results and the heating of air to 2500 K and 3 atmospheres, determine:

1. The initial NO formation rate in ppm/s2. The amount of NO formed (in ppm) in the 0.25 ms– The rate coefficient, k1f, is k1f=1.82x1014exp(-38,370/T)

ONOON

NNOONf

f

k

k

2

1

2

2

nmglobal

k

ONkdt

NOd

NOONglobal

22

22 2