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Hydrocarbon combustion around droplets and in sprays J.F. Griffiths

Hydrocarbon combustion around droplets and in sprays J.F. Griffiths School of Chemistry, The University, Leeds, LS2 9JT, UK.

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Hydrocarbon combustion around droplets and in sprays J.F. Griffiths

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  1. Hydrocarbon combustion around droplets and in sprays J.F. Griffiths School of Chemistry, The University, Leeds, LS2 9JT, UK

  2. Burning n-heptane droplet (1.3 mm dia) in a turbulent system(√q = 30 cm s-1). Sequence a – l, elapsed time not given(M. Birouk et al, Proc. Comb. Inst. 28, 1015, 2000)

  3. AUTOIGNITION Ignition delay for n-decane suspended droplets (O.7 mm dia) as a function of ambient T at different pressures (Moriue et al, Proc. Comb. Inst. 28, 969, 2000) The ignition delay includes both physics and chemistry. Physical processes might be dominant at certain conditions If this is so, the necessary chemistry detail might be subsidiary. Nevertheless, these results reflect very significant changes in chemistry over the range of ambient temperature Increase p

  4. Autoignition temperature of alkane vapours(e.g. IEC 60079-4, ASTM 659-78 and BS 4056 tests) Ignition delays are very long at the limiting temperature. The minimum autoignition temperature occurs in very rich mixtures Vapour pressures (approx.) 0.3 atm 0.15 atm From Zabetakis US Bureau of Mines Bull. 627 (1966)

  5. High temperature hydrocarbon combustion chemistry(after Warnatz and others) At T > 900 K, large fragments readily break down to the C1 and C2 species, especially in fuel rich conditions, as indicated by the heavy arrows

  6. Low temperature hydrocarbon combustion chemistry(as illustrated by n-butane) At T < 900 K, the predominant routes involve oxidation of the primary carbon structure leading to complex competitive sequences of reactions RH RO2 / R QOOH O2QOOH

  7. The need for mechanism reduction • Comprehensive chemical mechanisms can contain hundreds of chemical species and thousands of reactions, making their solution in CFD codes virtually impossible. • Owing to the range of time-scales present in kinetic systems, the resulting chemical rate equations are “stiff”. • It is possible to identify redundant species and reactions using formal methods, without major loss of kinetic detail. This reduces computational cost. • The reduced schemes are still ordinary chemical reaction mechanisms, i.e. sets of elementary chemical reactions that constitute a minimal set required to reproduce the behaviour of the full scheme.

  8. Basis of simulation for model reduction– zero-dimensional closed vessel calculations Mechanisms in CHEMKIN format, including list of species, NASA polynomials, reaction stoichiometry and Arrhenius parameters: Set up and solve system of ODEs Rate of change of concentration is given by where vij is the stoichiometric coefficient of the species i in the reaction j andRj is the jthreaction rate. (e.g. A + B -> C + D gives R = k cA.cB , with k = A Tn e(-E/RT))

  9. Zero-dimensional (i.e. spatially uniform) closed vessel calculations (continued) • Rate of change of temperature calculated by • where Cv is heat capacity, ΔUj is the internal energy of reaction j, V the • volume, A the reactor surface area, U the heat transfer coefficient • and Ta is the ambient temperature. • System of ODEs is solved numerically using a stiff integration solver (In Leeds we tend to use SPRINT).

  10. Validation tests – based on homogeneous gas-phase chemistry(over wide ranges of temperature, pressure and composition) • Ignition delay • Isothermal chemistry • p – Ta ignition diagrams An ignition diagram generated from a comprehensive scheme (358 species, 2411 reactions) for n-heptane in air at f = 1. Validation of the full scheme constitutes the comparison between simulation and experiment.

  11. Reduction methodologies • Sensitivity based methods to eliminate species and reactions, whilst retaining the original kinetic structure. • Quasi steady state approximation used to represent fast species as algebraic functions of other species further combined with reaction lumping to give reduced kinetic scheme with lumped reaction rates. Both species and reactions are removed as result. • Further time scale analysis such as computational singular perturbation and intrinsic low dimensional manifold to establish a repro-model– i.e. fitted models or look-up tables. Effective in CFD but difficult to interpret kinetically. Large time investment required to set up.

  12. Sensitivity: Identification of Necessary Species Necessary species are identified using sensitivity analysis to measure the effect of a change of concentration of a species on the rate of production of an N-member group of important species. where fn is the rate of production of the N-member group The necessary species with the highest Bi valuejoin the N-member group after each iteration until the iteration converges. An amalgamation of necessary species is taken over all the time points and redundant species are then removed.

  13. The basis of model reduction is sensitivity analysis of the temperature – time profile (e.g. two-stage ignition of n-C4H10) 30 22 34 Time points are selected to reflect important stages – e.g. initial conditions, inflexions, temperature range, maximum rates of temperature change 23 14 15 35 12 20 The numbers indicate the numbers of necessary species identified as being important at that point – which are then amalgamated.

  14. n-heptane low temperature combustion as a testcase - starting at 358 species in 2411 reactions Comparisons of T – t profiles after redundant species and reaction removal Point 2 Point 1 Point 3 • full scheme (358 / 2411) • intermediate scheme (257 / 1256) • skeleton scheme (236 / 810)

  15. How well is the n-heptane p – Ta ignition diagram reproduced?

  16. Further mechanism reduction: lumping techniques • Designed to further reduce reaction mechanisms by lumping species and/or reactions. • Advantage of reducing yet further the mechanism size and, with it, the computational effort. • Disadvantage that the scheme that is produced starts to • lose identifiable individual elementary chemical reactions. • May end up as a purely abstract set of mathematical equations. An illustration by application of the Quasi-Steady State Approximation (QSSA) shows some difficulties. (e.g. The application is not amenable to automation, and the resulting mechanisms are less flexible in their use.)

  17. k1 k2 A B C k-1 d[B] dt k1 k-1+ k2 = 0, hence k1[A]=[B](k-1+ k2), and [B]=[A] k1k2 k-1+ k2 k’ where k’ = A C Generalised QSSA Reduction Example • Formal mathematical techniques allow identification of most, if not all, radicals.

  18. QSSA Example: n-heptane low temperature oxidation RH C7H16 8 radical abstractions alkene + HO2 +O2 R (4 isomers) other isomers C7H15 6 product channels alkene + R· +O2 +HO2 RO2 (4 isomers) C7H15O2 C7H15OOH + O2 31 product channels C7H14OOH other isomers QOOH (25 isomers) +O2 O2QOOH (25 isomers) O2C7H14OOH 2 product channels (chain branching) Aim to remove R, RO2, QOOH and O2QOOH by systematic application of QSSA

  19. Reduction Example – n-heptane C7H16 8 radical abstractions alkene + HO2 +O2 6 product channels other isomers C7H15 (4 isomers) alkene + R· +O2 +HO2 C7H15OOH + O2 C7H16O2 (4 isomers) 31 product channels C7H15OOH (25 isomers) other isomers +O2 (25 isomers) O2C7H15OOH OH + products (2 product channels)

  20. Reduction Example – n-heptane C7H16 alkene + HO2 8 radical abstractions (6 product channels) C7H15O2 (4 isomers) alkene + R· 33 product channels • Nevertheless, the kinetic consequence of all QSSA species is retained in the complex functions of the form k’ (which can be derived by computation, e.g. Maple). • Each product channel has a unique effective rate coefficient expressed in terms of all of the other rate coefficients that were removed by applying the QSSA. • In deriving the functions the procedure is to work backwards.

  21. Reduction Example – n-heptane C7H16 8 radical abstractions alkene + HO2 +O2 6 product channels other isomers C7H15 (4 isomers) alkene + R· +O2 +HO2 C7H15OOH + O2 C7H16O2 (4 isomers) 31 product channels C7H15OOH (25 isomers) other isomers +O2 (25 isomers) O2C7H15OOH OH + product

  22. Reduction Example – n-heptane C7H16 8 radical abstractions alkene + HO2 +O2 6 product channels other isomers C7H15 (4 isomers) alkene + R· +O2 +HO2 C7H15OOH + O2 C7H16O2 (4 isomers) 31 product channels C7H15OOH (25 isomers) other isomers +O2 (25 isomers) O2C7H15OOH OH + product

  23. Reduction Example – n-heptane C7H16 8 radical abstractions alkene + HO2 +O2 6 product channels other isomers C7H15 (4 isomers) alkene + R· +O2 +HO2 C7H15OOH + O2 C7H16O2 (4 isomers) 31 product channels C7H15OOH (25 isomers) other isomers +O2 (25 isomers) O2C7H15OOH OH + product

  24. Reduction Example – n-heptane ─full (358 species, 2411 reactions). ─species & reactions removed (236 species, 810 reactions). ─QSSA reduction of the reduced scheme (118 species, 530 reactions). Point 2 Point 1 Point 3

  25. Reduction Example – n-heptane ─ full (358 species, 2411 reactions), 140s ─ species & reactions removed (236 species, 810 reactions), 58s ─ QSSA reduction of the reduced scheme (118 species, 530 reactions), 19s ─ further normal reduction of the QSSA scheme (110 species, 452 reactions), <9s ─ replacement of “product only” species with a dummy species (83 species, 452 reactions) Point 2 Point 1 Point 3

  26. Reduction Example – n-heptane

  27. Second Reduction Example - Cyclohexane

  28. Reduction Example - Cyclohexane Computing times: Full - ~1200s Final - ~6s

  29. Conclusions • Methods of applying sensitivity analysis have been automated. • Allows mechanisms to be reduced efficiently with minimal user intervention. • Methods allow redundant species and reactions to be identified and removed. • Lumping methods, e.g “reaction lumping” as implemented by applying the QSSA, allow significant further reductions with minimal loss of accuracy. • Demonstrated here with respect to n-heptane and cyclohexane.

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