1 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

2 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

3 MECHANISM FOR UNIMOLECULAR REACTIONS

4 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: Chain-initiation reaction Chain-propagating reactions Chain-terminating reaction Global model

5 CHAIN AND CHAIN-BRANCHING REACTIONS: EXAMPLE Steady-state approximation for radical concentration In the early stages of reaction, the concentration of the product AB is small, as are the concentrations of A and B throughout the course of the reaction therefore, reverse reactions may be neglected at this reaction stage

6 CONCLUSIONS Term in brackets [ ] >> 1 because the rate coefficients for the radical concentrations, k 2 and k 3 are much larger than k 1 and k 4 Can write approximate expressions for [A] and d[B 2 ]/dt Radical concentration depends on the square root ratio of k 1 to k 4 –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 k 2 and k 3 appear as a ratio, and their influence on the radical concentration would be small for rate coefficients of similar magnitude. Increasing k 2 and k 3 results in an increased disappearance of [B 2 ] Note that these scalings break down at pressure sufficiently high to cause 4k 2 k 3 [B 2 ]/(k 1 k 4 [M] 2 ) >> 1

7 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 + H 2 O → OH + OH –Existence of a chain-branching step in a chain reaction mechanism can have an explosive effect Example: Explosions in H 2 and O 2 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

8 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, k 1, to the first propagation step, k 2, rate coefficient –The rate at which B 2 disappears is governed by the initiation step –Increasing k 2 and k 3 has virtually no effect on the production rates of the products –The termolecular rate coefficient, k 4, 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 2 nd terms Chain Branching –Chain branching reactions involve formation of two radical species from a reaction that consumes only one radical Example: O + H 2 O → OH + OH –Existence of a chain-branching step in a chain reaction mechanism can have an explosive effect Example: Explosions in H 2 and O 2 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

9 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 O 2 and O. This, the O 2 and O can be assumed to be in equilibrium, i.e., O 2 ↔ 2O. Construct a global mechanism: Determine k global, 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/s 2.The amount of NO formed (in ppm) in the 0.25 ms –The rate coefficient, k 1f, is k 1f =1.82x10 14 exp(-38,370/T)