AME 513 Principles of Combustion Lecture 5 Chemical kinetics II – Multistep mechanisms

2 AME Fall Lecture 5 - Chemical Kinetics II Outline  Analytical solution for 1-step reaction  Irreversible  Reversible  Steady-state approximation  Pressure effects – Lindemann mechanism  Spreadsheet-level modeling  Partial equilibrium approximation  Online chemical kinetics calculator

3 AME Fall Lecture 5 - Chemical Kinetics II 1-step irreversible reaction  First consider 1 single (irreversible) reaction: A + B  C + D (forward reaction, rate constant k f ) But [A] and [B] are not independent, any decrease in [A] results in an identical decrease in [B] and increase in [C] & [D]; for a stoichiometric excess of [B], i.e. [B] o > [A] o

4 AME Fall Lecture 5 - Chemical Kinetics II 1-step reversible reaction  Now consider 1 single (but reversible) reaction: A + B  C + D (forward reaction, rate constant k f ) C + D  A + B (reverse reaction, rate constant k b )

5  Initial condition: t = 0, [A]= [A] o  In general the algebra horrendous but consider special case [A] o = [B] o, [C] o = [D] o = 0 AME Fall Lecture 5 - Chemical Kinetics II 1-step reversible reaction

6 AME Fall Lecture 5 - Chemical Kinetics II 1-step reversible reaction

7 AME Fall Lecture 5 - Chemical Kinetics II 1-step reversible reaction k f = 10, k b = 1, [A] o = 1 [A] eq = k f = 1, k b = 10, [A] o = 10 [A] eq = 7.60

8 AME Fall Lecture 5 - Chemical Kinetics II Steady-state approximation - concept  These calculations are tedious, need some simplifications to handle more complex reaction mechanisms!  Consider a two-step reaction with very reactive radical N O + N 2  NO + N (rate constant k 1 ) N + O 2  NO + O (rate constant k 2 )  Net production of N:  The steady-state approximation assumes d[N]/dt ≈ 0, i.e. in which case  Does NOT imply [N] = constant, only that [N] varies slowly compared to its production & consumption rates individually  When valid? Typically when a rapidly-reacting intermediate (N in this case) is produced by a slow reaction (O + N 2  NO + N)

9 AME Fall Lecture 5 - Chemical Kinetics II Lindemann (1922) - P effects on decomposition  Unimolecular decomposition of A (e.g. H 2 O 2 + M  2 OH) A + M  A * + M (rate constant k 1f ; A* = activated state of A) A * + M  A + M (rate constant k 1b ; k 1f /k 1b = K equilibrium = K A /K A* ) A *  B (rate constant k 2 )  Steady state assumption for A * yields  “Apparent” overall reaction rate: find k eff such that d[A]/dt = -d[B]/dt (i.e. rate of consumption of reactant A = rate of production of product B):  Note at low P ([M] small): k eff  k 1f [M] ~ P 1 ; high P: k eff  k 1f k 2 /k 1b ~ P 0 (falloff in rate at high P)

10 AME Fall Lecture 5 - Chemical Kinetics II Steady-state approx. – chain branching  Overall reaction A 2 + B 2  2AB A 2 + M  A + A + M (rate constant k 1, chain initiation) A + B 2  AB + B (rate constant k 2, chain branching) A 2 + B  AB + A (rate constant k 3, chain branching) A + B + M  AB + M (rate constant k 4, chain termination)  Steady state assumption for A and B:

11 AME Fall Lecture 5 - Chemical Kinetics II Steady-state approx. – chain branching  Text states that k 1 & k 4 should be much smaller than k 2 & k 3 since k 2 & k 3 are radical-molecule reactions - what’s wrong with that statement?  Anyway if the 2 nd term inside the square root is >> 1 then  Rate of [B 2 ] consumption increases with k 1, k 2, k 3 but decreases with k 4 due to loss of A and B radicals  Note that the requirement always breaks down at sufficiently high P since [B 2 ] ~ P but [M] 2 ~ P 2, thus steady-state assumption for this reaction mechanism fails at high enough P

12 AME Fall Lecture 5 - Chemical Kinetics II Excel spreadsheet model  OK is this for real? Create Excel spreadsheet model and compare “exact” solutions using Euler’s method to advance solution from time step t to t+  t: f(t+  t) ≈ f(t) + {df(t)/dt}  t, e.g. Euler’s method very simple not very stable or accurate – use e.g. 4 th order Runge-Kutta when Euler fails (actually Euler’s method gives the first term in the 4 th order Runge-Kutta)  Spreadsheet inputs: k 1 – k 4 ; initial concentrations of A 2, B 2, A, B, AB, M;  t  Outputs: concentrations vs. time; time to reach 90% completion of reaction to AB; time to peak [A]; time to peak [B]; validity of steady-state approximation (rates of formation & destruction of A > 10x net rate of change of [A])

13 AME Fall Lecture 5 - Chemical Kinetics II Excel spreadsheet model – baseline case  k 1 = 0.01, k 2 = 2, k 3 = 1, k 4 = 1  Time = 0: [A 2 ] = 1, [B 2 ] = 1, [M] = 10, [A] = [B] = [AB] = 0

14 AME Fall Lecture 5 - Chemical Kinetics II Validity of steady-state approximation  Approximation invalid initially when no radical pool exists  Also invalid when reactants A 2 & B 2 have been depleted significantly thus k 2 [A][B 2 ] & k 3 [B][A 2 ] are not large

15 AME Fall Lecture 5 - Chemical Kinetics II Excel spreadsheet model - results  Radical (A, B) concentrations are much lower than major (stable) species (A 2, B 2, AB)  Radical pool builds quickly to peak, reaches steady-state, then decreases slowly as reactants are consumed  Product AB follows mirror-image of reactants A 2 & B 2, showing that steady-state exists, radical concentrations shift quickly as reactant concentrations shift  How do changes in input parameters affect reaction time (90% of AB formed)? For 10% increase in parameter (in case of [A], [B], [AB], added 0.1 to initial mixture), % change in reaction time: k 1, k 2, k 3 increases overall reaction rate, k 4 decreases rate due to loss of radicals; adding [A] decreases rate more than adding [B] due to bottleneck of initiation A 2 + M  2A + M Propertyk1k1 k2k2 k3k3 k4k4 [M][A 2 ][B 2 ][A][B] % change

16 AME Fall Lecture 5 - Chemical Kinetics II Partial equilibrium approximation  Analogous to steady-state approximation, but applies to a single (reversible) reaction rather than a single species  Requires both forward and reverse rates to be fast compared to net rate, e.g. for A + B 2  AB + B  Example: for reactants A 2, B 2, product A 2 B, radicals A, B, AB A + B 2  AB + BRate constant k 1f AB + B  A + B 2 Rate constant k 1b B + A 2  AB + ARate constant k 2f AB + A  B + A 2 Rate constant k 2b AB + A 2  A 2 B + ARate constant k 3f A 2 B + A  AB + A 2 Rate constant k 3b A + AB + M  A 2 B + MRate constant k 4 (NOT in partial equil.)

17 AME Fall Lecture 5 - Chemical Kinetics II Partial equilibrium approximation  Combining these results to solve for radical species A, B, AB  Then the product formation rate is  Obviously cannot apply at t = 0 since [A 2 B] = 0, but early on, before [A 2 ] and [B 2 ] depletion is significant, t 1/2 behavior t 1/2 behavior

18 AME Fall Lecture 5 - Chemical Kinetics II Online chemical kinetics calculator  What if the mechanism is more complex or less stable?  Chemical mechanisms for H 2 -O 2, CH 4 -O 2 and a few others  Input page simple except for need to choose by trial & error:  Total integration time (long enough to see fuel consumption & product formation)  Time interval (  t) (small enough to resolve transients)  Output format hard to feed into to Excel!

19 AME Fall Lecture 5 - Chemical Kinetics II Online chemical kinetics calculator  Typical example - stoich. H 2 -O 2, T = 850K, P = 1 atm, near 3 rd explosion limit, above 2 nd limit so mole fractions of H, O, OH very small, ≈  Similar to schematic A 2 + B 2 results

20 AME Fall Lecture 5 - Chemical Kinetics II Summary  Chemical kinetic systems for realistic fuels are comprised of many individual reactions, resulting in coupled Ordinary Differential Equations in terms of the concentrations of reactants [A], [B], [C], [D], … of the form  Analytical solution of these equations is impossible for all but the simplest cases  Techniques exist for simplifying these large sets of equations  Steady-state (for one species)  Partial equilibrium (for one reversible reaction)  Many others…  Simplifying methods exploit the fact that some reactions (e.g. radical + stable species) are typically faster than others (e.g. chain initiation, 3-body recombination, radical + radical) and so adjust to changing concentrations on a shorter time scale