hydrocarbon combustion around droplets and in sprays · ambient temperature autoignition increase p...

Hydrocarbon combustion around droplets and in sprays

J.F. Griffiths

School of Chemistry, The University, Leeds, LS2 9JT, UK

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)

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

AUTOIGNITION

Increase p

Autoignition temperature of alkane vapours (e.g. IEC 60079-4, ASTM 659-78 and BS 4056 tests)

Vapour pressures (approx.)

0.15 atm

0.3 atm

From Zabetakis US Bureau of Mines Bull. 627 (1966)

Ignition delays are very long at the limiting temperature. The minimum autoignition temperature occurs in very rich mixtures

High temperature hydrocarbon combustion chemistry (after Warnatz and others)

CH4 C2H6

CH3

CH2O

CHO

CO

CO2

C2H5

C2H4

C2H3

C2H2

C2H

C4H2 C3H4

CH2

CHCO

CH2CO

CO

CH

CO2

CO2

CH3CHO CH3CO CH3

CH3, CH2O, CHO

CH3, CH2O, CHO

H H, O,OH

O, OH

H, O, OH

H, O, OH

M, O2

H, O, OH

O, O2

CH3

CH3

CH3

H

H

H, OH

OH

O, OH

O M

HM, H,O2

H, O, OH

C2H2 C2H2

O, O2

O, OH

H, O, OH

O

O

O

H

O, O2

O2

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

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

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.

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 and Rj is the jth reaction rate.

(e.g. A + B -> C + D gives R = k cA.cB , with k = A Tn e(-E/RT))

[ ]j

jij

i Rvdtcd

∑=

•  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).

Zero-dimensional (i.e. spatially uniform) closed vessel calculations (continued)

( ) )( ajj

ojv TT

VUARU

dtdTC −−Δ=∑

Validation tests – based on homogeneous gas-phase chemistry (over wide ranges of temperature, pressure and composition)

•  Ignition delay •  Isothermal chemistry •  p – Ta ignition diagrams

500 550 600 650 700 750 800 8500.0

0.5

1.0

1.5

2.0

1 2

3

Ignition Boundary Cool Flame Boundary Mechanism Reduction Initial Conditions

Slowreaction

Cool flamesP

/atm

T/K

Ignition An ignition diagram generated from a comprehensive scheme (358 species, 2411 reactions) for n-heptane in air at φ = 1.

Validation of the full scheme constitutes the comparison between simulation and experiment.

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.

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 value join 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.

( )2

1ln/ln∑

=

∂∂=N

nini cfB

0.0 0.1 0.2 0.3 0.4 0.5

600

800

1000

1200

T / K

t / s

12 20

35 15

23 14

34 22

30

The basis of model reduction is sensitivity analysis of the temperature – time profile (e.g. two-stage ignition of n-C4H10)

The numbers indicate the numbers of necessary species identified as being important at that point – which are then amalgamated.

Time points are selected to reflect important stages – e.g. initial conditions, inflexions, temperature range, maximum rates of temperature change

0.0 0.2 0.4 0.6650

700

750

800

T/K

t/s

0.0 0.1 0.2 0.3 0.4 0.5700

800

900

1000

1100

1200

T/K

t/s0 1 5.0 5.5 6.0

600

800

1000

1200

T/K

t/s

Point 1

Point 2

Point 3

Comparisons of T – t profiles after redundant species and reaction removal

•  full scheme (358 / 2411) •  intermediate scheme (257 / 1256) •  skeleton scheme (236 / 810)

n-heptane low temperature combustion as a test case - starting at 358 species in 2411 reactions

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

500 550 600 650 700 750 800 8500.0

0.5

1.0

1.5

2.0

1 2

3

Full Scheme (358 species, 2411 reactions) Skeleton Scheme (236 species, 810 reactions) Mechanism Reduction Initial Conditions

P/a

tm

T/K

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.

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.)

•  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.

Generalised QSSA Reduction Example

Formal mathematical techniques allow identification of most, if not all, radicals.

A B C k1

k-1

k2

A C k’

d[B] dt = 0, hence k1[A]=[B](k-1+ k2), and [B]=[A] k1

k-1+ k2

where k’ = k1 k2 k-1+ k2

QSSA Example: n-heptane low temperature oxidation

C7H16

O2C7H14OOH

2 product channels (chain branching)

+O2

C7H15O2

C7H15

+O2 alkene + HO2

alkene + R·

C7H14OOH

+O2

+HO2 C7H15OOH + O2

31 product channels other isomers

6 product channels

8 radical abstractions

other isomers

RO2 (4 isomers)

RH

R (4 isomers)

QOOH (25 isomers)

O2QOOH (25 isomers)

Aim to remove R, RO2, QOOH and O2QOOH by systematic application of QSSA

other isomers

Reduction Example – n-heptane C7H16

O2C7H15OOH

OH + products

+O2

C7H16O2

C7H15

+O2 alkene + HO2

alkene + R· (4 isomers)

(4 isomers)

C7H15OOH

+O2

(25 isomers)

(25 isomers)

+HO2 C7H15OOH + O2


6 product channels


(2 product channels)


33 product channels

C7H15O2

alkene + HO2


(6 product channels) 8 radical abstractions

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.


O2C7H15OOH

OH + product

+O2

C7H16O2

C7H15

+O2 alkene + HO2


(4 isomers)

C7H15OOH

+O2

(25 isomers)

(25 isomers)

+HO2 C7H15OOH + O2


6 product channels


other isomers

Reduction Example – n-heptane

Point 1 Point 2

Point 3

─ full (358 species, 2411 reactions). ─ species & reactions removed (236 species, 810 reactions). ─ QSSA reduction of the reduced scheme (118 species, 530 reactions).

0.0 0.2 0.4 0.6650

700

750

800

T/K

t/s

0 1 2 3 4 5 6

600

800

1000

1200

T/K

t/s0.0 0.1 0.2 0.3 0.4 0.5 0.6

700

800

900

1000

1100

1200

T/K

t/s


Point 1 Point 2

Point 3

─ 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)

0 1 2 3 4 5 6

600

800

1000

1200

T/K

t/s0.0 0.1 0.2 0.3 0.4 0.5 0.6

700

800

900

1000

1100

1200

T/K

t/s

0.0 0.2 0.4 0.6650

700

750

800

T/K

t/s


500 550 600 650 700 750 800 8500

200

400

600

800

1000

1200

1400

1600

Full mechanism (358 species 2411 reactions) Skeleton mechanism (236 species 810 reactions) QSSA reduced mechanism (83 species 452 reactions)

Slowreaction

Cool flames

P/to

rr

T/K

Ignition

Second Reduction Example - Cyclohexane

500 520 540 560 580 6000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Full Scheme (497 species) Reduced Scheme (77 species) QSSA reduced scheme (45 species) QSSA reduced scheme & dummy product (35 species)

P /

atm

Ta / K

Slow reaction Cool flames

Ignition

Reduction Example - Cyclohexane

0 5 10 15 180 185 190 195 200 205500

600

700

800

900

1000

1100

1200

1300

1400

Full Scheme (497 species) Reduced Scheme (77 species) QSSA reduced scheme (45 species) QSSA reduced scheme, with dummy product (35 species)

T/K

t/s

T = 510K, P = 2300 torr

Computing times: Full - ~1200s Final - ~6s

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.

hydrocarbon combustion around droplets and in sprays · ambient temperature autoignition increase p...

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