COMPUTATIONALLY EFFICIENT MODELING OF HYDROCARBON OXIDATION CHEMISTRY

AND FLAMES USING CONSTITUENTS AND SPECIES

K. G. Harstad* and J. Bellan*,**

*Jet Propulsion Laboratory, California Institute of Technology,Pasadena, CA 90019-8099

**Mechanical Engineering Dept., California Institute of Technology,Pasadena, CA 91125

MACCCR Fuels SummitArgonne National Laboratories

September 20-23, 2011

Sponsored by the Army Research Office under the direction of Dr. Ralph Anthenien

Outline• Background

• Concept description for chemical kinetics in a constant-volume perfectly stirred reactor

• Fuel pyrolysis in the presence of nitrogen (non-premixed flames)

• Flame model• Premixed• Non-premixed

• Results

• Summary and conclusions

• Future work

Background

• Necessity to reduce the computational cost of chemistry in simulations of turbulent combustion

• None of the existing chemistry reduction schemes is considered the ultimate solution to the problem of chemical kinetic reduction

• Novel methods of reduction are still sought

Chemical kinetics: The concept of a base

Species list

Light (we do not decompose) Heavy (we decompose)

Air, final combustionproducts

Light radicals/molecules(e.g. CH3, CH4, H2O2)

Radicals Stable

Constituents

The base = Light species + Constituents(26) (13)

For general alkane or alkene with air:

Remove NOx, C, C2, and N at this preliminary stage, and then,

The base = Light species + Constituents + N2(20) (13) (null rates)

Kinetic model summary, 1 of 2

∙ n-heptane: 13 constituents, CH2, CH3, CH, C2H3, C2H2, C2, HC2, CO (keto), HCO, HO, HO2, OO, O

N c ≡ ∑ k113 N k

∙ iso-octane, PRF fuels, mixtures of iso-octane and n-pentane or iso-hexane: 14 constituents (C atom added)

N c ≡ ∑ k114 N k

∙ other aspects of the modelN ∗ ≡ Nn2

Nref

≡ T−T0T r,N∗

T r ≡ 2065N ∗0.06 w

w 1.51.3110.711.1 2

Kinetic model summary, 2 of 2

R i ≡ dNi

dt reac dNi

dt heavies dN i

dt lights

dNi

dt heavies N cKG i − X iKL i

N cC p ,h ∑ i∈lights C p ,iN idTdt

−∑ i∈lights h iR i N cRu T refK h

C p ,h ≡∑ l∈heavies Cp,lN l

Nc

K h ≡ − ∑ l∈heavies h lR l1

R u TrefNc

Findings from the kinetic model1. Nc exhibits a self-similarity with respect to θ, modulo T0 (600-1200 K), for a wide range of p0 (5-50 bar) and (1/4 to 4)

2. In the cold ignition regime (600-900 K), the self-similarity is independent of T0

3. The molar densities of O2 and H2O exhibit a quasi-linear behavior versus θ

4. Eleven species progress variables: the unsteady light species

5. The capability of the model was assessed from the viewpoints of predicting species, the temperature evolution and the ignition time.

- The species prediction was excellent in all cases, with the exception of the multivalued regions occurring for very rich situations.

- The temperature values were excellently to very well predicted, except for the temperature decrease from the maximum observed for rich mixtures.

- The ignition times were reproduced within percentages of those computed with the full mechanism.

6. The above evaluation was valid for all single fuels and mixtures considered.

Fuel pyrolysis in counterflow diffusion flames

Non-premixed model- air jet counterflow to fuel/nitrogen jet- need estimate of fuel pyrolysis before mixing with air occurs- calculations performed using LLNL rates with Chemkin II

Results- rates minimal for T ≤ 900 K (≤ 1 s-1)- only 3 light species (H2, CH4, C2H4) are produced having mole fractions

as large as O(10-2); all other mole fractions are much smaller- a single rate form describes the rates of these species

N i N c K i, K i ai 1012 ′0.37 exp−Tp /T

• Tp= 2.9×104 K is the characteristic pyrolysis temperature• ’ is an effective equivalence ratio (air replaced by N2)• aH2,CH4 ≐ 0.35 + 4.25 woct – 2.7 (woct)2 0 ≤ woct ≤ 1 is the octane fraction in the PRF• aC2H4 ≐ 0.78 + 3.9 woct – 4.6 (woct)2

Interpolation procedure for the rate functions

The reduced rate functions have the general form

where Tm and m are the temperature and at the initial fuel/air mixing; K is in s-1.

-Tables are generated numerically at fixed m and Tm.-The K rates are found by numerical interpolation. Detailed comparisons with the LLNL rates shows that this local interpolation isaccurate using

where s and T* are interpolation parameters.e.g.: s = (ln K2 – ln K1)/(ln m2 – ln m1 )

K f,m , T m , T − T m /T r, T r T rm , N N2

K Kf ms exp−T∗/T

Flame model, 1 of 3

v r∙

McontinuityR 3

dRdx

momentum ≐ 1 − 1 − L 0

x0 x≠x a

dx / 0

L0 x≠x a

dx x ≡ R0 /Rx

species Ji Ji,light Ji,heavy

J i,light −n X iDT,id ln T

dx∑

j

D ijdXj

dx

J i,heavy −n∑l

D ildX l

dx≡ JH,i

k ∈ ensemble C

l ∈ ensemble H

i, j ∈ ensemble L

JH,i ≐ −nDH,id ln C ave

dx

DH,i ≡ 1n ∑

l

D ilN l ∑l

D ilX l X HD iH

C ave ≡ Ncn ∑

k,l

C klX l

A.

B. 2 1 − k 0x x ′ x

′0

x ′dx ′k determined by BCs; =1 outside the flame; is a cst, model parameter inside the flame chosen to minimize the residue to the momentum eqn.

Flame model, 2 of 3

species Ji −n ∑j

D ijdX j

dx X HD iH

d ln Cave

dx X iDT,i

d ln Tdx

i and j to the unsteady light species and q to the quasi-steady light species

X u 1 − X H − ∑q

X q m X u

∑i

Yim i

D ia ≡ 1

X u∑

j

D ijX j .

Ji −n ∑j

D ij − D ia m

m j

dY j

dx BT,i

e dTdx

BT,ie ≡ X i

D T,i

T ∑q

D iq − Dia X q

T X HD iHln Cave

T − D ia X H

T

dY i

dx A

∙M

m iR i − ddx

A∙

Mm iJi

R i R i | lights R i |heavies R i |heavies N cK net,i

Flame model, 3 of 3

energy q ≐ − dTdx− ∑

i

m im H

hH − hi Ji hH 1/X H∑ l X lh l

Cpm

dTdx

ddx

A∙

M dT

dx− A

∙M

dTdx ∑

i

Cp,im i− Cp,H

m Hm iJi ∑

i

h iR i − RuTrefN cK H

equation of state

p Ru Tv PR − bm ix

− am ix

v PR2 2bm ixv PR − bm ix

2

am ix ∑ p ∑ n X p X n apn T bm ix ∑ p X p bp

apn 1 − k ′ pp nn bp 0. 077796

Ru T C,ppC,p

pp T ≡ 0. 457236Ru T C,p 2 1 cp 1 − T red ,p 2/pC,p

cp 0. 37464 1. 54226p − 0. 26992p2

Computation of Tm and m for non-premixed flames

• The original reduced rate model is based on premixed flow situations where the initial temperature and correspond to the fuel/air being completely mixed.

• For non-premixed situations, mixing occurs continuously through diffusion along fluid paths: the temperature and at which mixing occurs vary according to location.

• An assumption is made that mixing occurs continuously (no ‘puffs’) along the fluid path. This is valid for many, if not most, situations.

• Based on this view, the local mean ‘mixing temperature and ’ are estimated through

T m T T 0 /2

where T0 is the temperature at the first location where either there is non-negligible fuel in the air or non negligible air in the fuel. T0 is a constant for either side of the flow.

• At the location having temperature T the rates correspond to those computed from the tables using the local values of m, θ computed using Tm and m, and Tm.

Lean : m /2; Rich : m 2

Transport coefficients

• Mass diffusion– need the full matrix of pairwise diffusion coefficients– base the pairwise diffusion coefficients on binary diffusion

coefficient models (Harstad & Bellan 2004a) and all-pressure mixing rules (Harstad & Bellan 2004b)

• Thermal conductivity for the mixture– model from Reid et al.

• Thermal diffusion ratios– based on the all-pressure model of Harstad & Bellan 2004b

• Inviscid fluid: – in principle the viscosity is not needed– in fact it is needed for the modeling of the thermal

conductivity

Premixed and diffusion flames

• Premixed flames– Version A of momentum equation, large Pé (valid

outside but not throughout the flame); study the influence of diffusion for constant area flow

– Version B of momentum equation, general Pévalues (needed to compute through the flame peak)

• Counterflow diffusion flame– Version B of momentum equation, general Pé

values

Premixed flames, large Pé

• Defined two parameters

• Based on a large Péclet number approach, i.e. convection dominates diffusion

• The equations are solved using a stiff integrator

i ≡ −m iJi/u0

i′ ≡ − i/Yi uD,i/u0

| i′ | 1

Initial conditions

• 90% iso-octane and 10% n-heptane• 50% iso-octane and 50% n-heptane• = 1• (ρu)0=0.8 g/(cm2 s)• T0 prescribed• p0 prescribed• Constant area situation

Premixed flame, large Pé; 1 of 5

90/10; p0 = 30 bar, T0 = 800 K; no diffusion

x (cm)

Nc

(mol

/m3 )

0 0.1 0.2 0.3 0.4 0.5 0.605

1015202530354045505560

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

x (cm)

T(K

)

0 0.1 0.2 0.3 0.4 0.5 0.6

800

1200

1600

2000

2400

2800

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

x (cm)

Noh

(mol

/m3 )

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

x (cm)

No2

(mol

/m3 )

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

70

80

90

100

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4x (cm)

0

5

10

15

20

25

Nh2

o(m

ol/m

)3

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

x (cm)

Nco

2(m

ol/m

3 )

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

12

14

16

18

20

22

p0 = 30 barT0 = 800 Kφ = 1

(Rates only)

Premixed flame, large Pé; 2 of 590/10; p0 = 30 bar, T0 = 800 K; with diffusion

x (cm)

Nc

(mol

/m3 )

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

p0 = 30 barT0 = 800 Kφ = 1

x (cm)

T(K

)

0 0.1 0.2 0.3 0.4600

800

1000

1200

1400

1600

1800

2000p0 = 30 barT0 = 800 Kφ = 1

x (cm)

Noh

(mol

/m)3

0 0.1 0.2 0.3 0.40

0.02

0.04

p0 = 30 barT0 = 800 Kφ = 1

x (cm)

No2

(mol

/m3 )

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

80

90

100

p0 = 30 barT0 = 800 Kφ = 1

x (cm)

Nh2

o(m

ol/m

3 )

0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7

8

9

10

p0 = 30 barT0 = 800 Kφ = 1

x (cm)

Nco

2(m

ol/m

3 )

0 0.1 0.2 0.3 0.40

0.5

1

1.5

2

2.5

p0 = 30 barT0 = 800 Kφ = 1

Premixed flame, large Pé; 3 of 5

600 700 800 900T0 (K)

0

2

4

6

8

10

12

x fs(c

m)

p0 = 20p0 = 30p0 = 40

p0 in bar, T0 in K

90/10xfs is that at T0+900K

600 700 800 900T0 (K)

100

101

x fs(c

m)

p0 = 20p0 = 30p0 = 40

Premixed flame, large Pé; 4 of 5

600 700 800 900T0 (K)

50

60

70

80

90

100

110

120

130

140

150U

fs(c

m/s

)p0 = 20p0 = 30p0 = 40

90/10=1

Premixed flame, large Pé; 5 of 5

50/50

600 700 800 900T0 (K)

10-1

100

x fs(c

m)

p0 = 30

Heptane/Octane=50%/50%

600 700 800 900T0 (K)

60

70

80

90

100

110

120

Ufs

(cm

/s)

p0 = 30

Heptane/Octane=50%/50%

=1

Premixed flame; general Pé

0 0.1 0.2 0.3 0.4 0.5 0.6X (cm)

30040050060070080090010001100120013001400150016001700180019002000

T(K

)

0

20

40

60

80

100

120

140

160

180

200

220

240

U(c

m/s

)

Ji. C. et al., “Propagation and extinction of premixed C5-C12 n-alkane flames”, Combust. Flame, 157, 277-287, 2010.

The experimental profiles were stated to be for C5-C8 n-alkanes (unburned temperature: 353K), and n-heptane was used .

Counterflow flame: computation feasibility

0.5 0.75 1 1.25x/L

0

200

400

600

800

1000

1200

1400

1600

1800

2000

T(K

)

0.5 0.75 1 1.25x/L

0

0.05

0.1

0.15

0.2

0.25

Y

Oxygen

Fuel

p = 1 bar, Tjet = 350 K, (ρu)air-jet = 4.8 ×10-2 g/cm2, (ρu)fuel-jet = 4.85 ×10-2 g/cm2, L = 2 cm(ρuYF)fuel-jet / (ρuYO2)air-jet = s

u (@max R) = 56 cm/s

Summary and conclusions, 1 of 2

• Developed a model of steady quasi-1D flame with variable area, including– Full mass-diffusion coefficient matrix– Thermal diffusion factors– Mixture thermal conductivity– Real-gas EOS

• Preliminary results obtained for premixed flames and constant area flow show that the predictions are qualitatively correct– Ufs varies with the inverse of a power of pressure – The position of the flame initiation depends on T0

– The position of the flame initiation depends of the initial pressure over the entire T0 regime

– Increasing the % heptane in the PRF results in larger Ufs

Summary and conclusions, 2 of 2

• Limited results were obtained for premixed flow with variable area– Results compare well with experimental data

• The model has been extended to non-premixed flow– Preliminary results are qualitatively correct.– The numerics must be improved as stability problems are still

preventing routine computations and comparisons with experiments.

Future work

• Solidify the numerical method to enable routine computations

• Investigate whether the same constituents-and-species concept holds for higher alkanes and cycloalkanes