MAE 5310: COMBUSTION FUNDAMENTALS

1. MAE 5310: COMBUSTION FUNDAMENTALS Chemical Kinetics Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. CHEMICAL KINETICS OVERVIEW • In many combustion processes, chemical reaction rates control rate of combustion • Chemical reaction rates determine pollutant formation and destruction • Ignition and flame extinction are dependent on rate processes • Overall reaction of a mole of fuel, F, with a moles of oxidizer, O, to form b moles of products, P, can be expressed by a global reaction mechanism as: • From experimental measurements, rate at which the fuel is consumed expressed as: • [ci] is molar concentration of ith species in mixture • Equation states that rate of disappearance of fuel is proportional to each of reactants raised to a power • Constant of proportionality, kglobal, is called global rate coefficient, which is a strong function of temperature and minus sign indicates that fuel concentration decreases with time • Exponents n and m relate to reaction order • Reaction is nth order with respect to fuel • Reaction is mth order with respect to oxidizer • Reaction is (n+m)th order overall

3. EXAMPLE OF INTERMEDIATE SPECIES • Consider global reaction of conversion of hydrogen and oxygen to water • The following elementary reactions are important: • First reaction produces hydroperoxy, HO2 and a hydrogen atom, H • HO2 and H are called radicals • Radicals, or free radicals, are reactive molecules, or atoms, that have unpaired electrons • To have a complete picture of hydrogen and oxygen combustion over 20 elementary reactions are necessary • Collection of elementary reactions necessary to describe an overall reaction is called a mechanism

4. MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS • Molecular collision theory an be used to provide insight into form of bimolecular reaction rates and to suggest the temperature dependence of the bimolecular rate coefficient • Consider a single molecule of diameter s traveling at constant speed v and experiencing collisions with identical, but stationary, molecules • If distance between traveled between collisions (mean free path, l) is large then moving molecule sweeps out a cylindrical volume in which collisions are possible = vps2Dt in a time interval Dt. • At ambient conditions for gases: • Time between collisions ~ O(10-9 s) • Duration of collisions ~ O(10-12 – 10-13 s) • If stationary molecules distributed randomly and have a number density, n/V, the number of collisions experienced by the traveling molecule per unit time is: Z = collisions per unit time = (n/V)vps2 • In actual gas all molecules are moving • Assuming a Maxwellian distribution for all molecules, the collision frequency, Zc, is given by:

5. MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS • So far theory applies to identical molecules • Extend analysis to collisions between unlike molecules have diameters sA and sB. Diameter of collision volume is then given as sAB=(sA+ sB)/2 • This is an expression for the frequency of collision of a single A molecule with all B molecules • Ultimately we want collision frequency associated with all A and B molecules • Total number of collisions per unit volume and per unit time is obtained by multiplying collision frequency of a single A molecule by the number of A molecules per unit volume and using the appropriate mean molecular speed (RMS) • ZAB/V = Number of collisions between all A and all B / Unit volume Unit time

6. MOLECULAR KINETIC AND COLLISION THEORY OVERVIEW:BIMOLECULAR REACTIONS • NAvogadro = 6.022x1023 molecules/mol or 6.022x1026 molecules/kmol • Probability, P, that a collision leads to reaction can be expressed as product of two terms • Energy factor, exp[-EA/RT] • Expresses the fraction of collisions that occur with an energy above the threshold level necessary for reaction, EA, or activation energy • Geometrical or steric factor, p • Takes into account the geometry of collisions between A and B More common curve fit A, n and EA are empirical parameters

7. EXAMPLE: H2 OXIDATION AND NET PRODUCTION RATES Global reaction Partial mechanism Find: d[O2]/dt, d[H]/dt, etc. System of 1st order, ordinary differential equations Initial conditions for each participating species

8. GENERAL NOTATION

9. EXAMPLE Determine the collision-theory steric factor for the reaction O + H2→ OH + H at T=2000 K give the sphere diameters, sO=3.050 and sH2=2.827 Å using the data in Appendix 2 of Glassman Comments • Pay attention to units: • kb=1.381x10-23 J/K = 1.381x10-16 g cm2/s2 K

10. RELATION BETWEEN RATE COEFFICIENTS AND EQUILIBRIUM CONSTANTS Equilibrium condition d[A]dt→0 Relationship between concentration and partial pressure [M] ↑ when P ↑ at constant cM and T True for all reactions True for all bimolecular reactions

11. 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

12. 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

13. MECHANISM FOR UNIMOLECULAR REACTIONS

14. 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: Global model Chain-initiation reaction Chain-propagating reactions Chain-terminating reaction

15. CHAIN AND CHAIN-BRANCHING REACTIONS: EXAMPLE 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 Steady-state approximation for radical concentration

16. 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

17. INCORRECT CONCLUSIONS FROM 2nd EDITION • Governing Diff EQ’s are WRONG • Conclusions are wrong: • 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

18. COMMENTS ON 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 chain-branching step in reaction mechanism can have 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 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

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

20. Insight gained from knowledge of chemical time scale, tchem What is approximate magnitude of time scale? Characteristic time is defined as time required for concentration of A to fall from its initial value to a value equal to 1/e times the initial value [A](t=tchem)/[A]0=1/e RC circuit analog CHEMICAL TIME SCALES Unimolecular reactions Bimolecular reactions Bimolecular reactions case where [B]0 >> [A]0 Termolecular reactions Termolecular reactions case where [B]0 >> [A]0

21. EXAMPLE: CHEMICAL TIME SCALE CALCULATIONS 1 • Reaction 1 is an important step in oxidation of CH4 • Reaction 2 is key step in CO oxidation • Reaction 3 is a rate-limiting step in prompt-NO mechanism • Reaction 4 is a typical radical recombination reaction • Estimate tchem associated with least abundant reactant in each reaction for following 2 conditions • Assume that each of 4 reactions is uncoupled from all others and that third-body collision partner concentration is the sum of the N2 and H2O concentrations) k=1.0x108T1.6exp(-1570/T) k=4.76x107T1.23exp(-35.2/T) k=2.86x108T1.1exp(-10,267/T) k=2.2x1022T-2.0 2 3 4 Condition II: High Temperature T=2199 K, P=1 atm cCH4=3.773x10-6 cN2=0.7077 cCO=1.106x10-2 cOH=3.678x10-3 cH=6.634x10-4 cCH=9.148x10-9 cH2O=0.1815 Condition I: Low Temperature T=1344 K, P=1 atm cCH4=2.012x10-4 cN2=0.7125 cCO=4.083x10-3 cOH=1.818x10-4 cH=1.418x10-4 cCH=2.082x10-9 cH2O=0.1864

22. Turns Example Solution • Increasing T shortens chemical time scales for all species, but most dramatically for CH4 + OH and CH + N2 reactions • For the CH4 + OH reaction, the dominant factor is the decrease in the given CH4 concentration • For the CH + N2 reaction, the large increase in the rate coefficient dominates • Characteristic times at either condition vary widely among the various reactions • Notice long characteristic times associated with recombination reaction H + OH + M → H2O + M, compared with the bimolecular reactions. • That recombination reactions are relatively slow plays a key role in using partial equilibrium assumptions to simplify complex chemical mechanisms (see next slide) • Note that tchem=1/[B]0kbimolec could have been used with good accuracy for reaction 3 at both low and high temperature conditions, as well as reaction 1 at high temperature because in these cases one of reactants was much more abundant than other

23. PARTIAL EQUILIBRIUM • Many combustion process simultaneously involve both fast and slow reactions such that the fast reactions are rapid in both the forward and reverse directions • These fast reactions are usually chain-propagating or chain-branching steps • The slow reactions are usually the termolecular recombination reactions • Treating fast reactions as if they were equilibrated simplifies chemical kinetics by eliminating need to write rate equations for radical species involved, which is called the partial-equilibrium approximation • Consider the following hypothetical mechanism:

24. PARTIAL EQUILIBRIUM VS. STEADY-STATE • Result of invoking either partial equilibrium assumption or steady-state approximation is same: • A radical concentration is determined by an algebraic equation rather than through integration of an ordinary differential equation • Keep in mind that physically the two approximations are different: • Partial equilibrium approximationforcing a reaction, or sets of reactions, to be essentially equilibrated • Steady-state approximationforces the individual net production rate of one or more species to be essentially zero • There are many examples in combustion literature where partial equilibrium approximation is invoked to simplify a problem • Examples: • Calculation of CO concentrations during the expansion stroke of a spark-ignition engine • Calculation of NO emissions in turbulent jet flames • In both examples, slow recombination reactions cause radical concentrations to build up to levels in excess of their values for full equilibrium