Accuracy and Flexibility of Simplified Kinetic Models for CFD

Accuracy and Flexibility of Simplified Kinetic Models for CFD applications A. Cuoci, A. Frassoldati, T. Faravelli, E. Ranzi Dipartimento di Chimica, Materiali, Ingegneria Chimica – Politecnico di Milano - ITALY

1. Introduction During the 80s there was a particular attention on the development of reliable global mechanisms for the combustion of hydrocarbons [1-3]. The reason of such attention was found in the fact that successful combustion modelling needs to take into account all the fundamental aspects and not only detailed chemistry. Turbulence, heat and mass transfer strongly affect simulation results. However, it was not possible at that time to couple detailed fluid dynamics with more than a few reactions. Despite the increasing power and memory of modern PC, there is still nowadays the need of simplified kinetic mechanisms to be coupled with CFD codes. Several reasons explain this need. From one side, the increasing detail of fluid dynamic description. RANS approaches are replaced by more accurate Large Eddy Simulation, computationally very expensive. In the next future, the scientists are sighting the Direct Numerical Simulations. On the other side, fuels different from simple methane are considered. Liquid fuels, like diesel or jet fuels, are investigated because of their importance in several applications like engines. In this last contest, a further push to revise actual available global mechanisms is the presence of mixtures, where the synergistic effect of the most reactive component has to be taken into account. Literature about single or multi-step oxidation mechanisms refers the evaluation of kinetic parameters and reaction orders to some ‘regression approaches’. Results carried out in Plug Flow Reactors or flame speed measures are used to adjust the kinetic parameters. Starting points are chemical analyses of the reaction pathways. More recently data from non adiabatic PSR were also used to identify the rate constants [4]. This paper presents new estimations of the parameters of multistep oxidation mechanisms. These evaluations are based on a regression of data with a very effective numerical algorithm [5]. The main novelty is the definition of the set of comparison measures. In order to take into account the quite broad temperature and stoichiometry ranges of a turbulent diffusive flame, the regression analysis is performed over data obtained by a detailed kinetic scheme in laminar diffusive counter-flow flames. These flames are assumed as a good representation of the real flame [6], still maintaining a simple solution, compatible with the numerical effort required by the regression algorithm. 2. Detailed and multistep kinetic models The source of data to be regressed comes from a detailed kinetic mechanism (already developed and validated for hydrocarbons up to 16 C atoms [7]), freely available on the web: www.chem.polim.it/creckmodeling. The model showed to be accurate in predicting the pyrolysis, oxidation and combustion of pure components and mixtures in wide ranges of conditions (in terms of pressures, temperatures, stoichiometries and residence times). On these bases, the results coming from this kinetic model are assumed as ‘correct measures’. As mentioned, several global mechanisms are available in the literature. In principle, the II-6, 1

Combustion Colloquia 2009

32nd Meeting on Combustion Reaction

1 2 3

CO + 0.5O2 ⎯⎯ → CO 2

Reaction rate r1 = 2.24 ⋅ 1012 e

CO 2 ⎯⎯ → CO + 0.5O2 H 2 + 0.5O2 ⎯⎯ → H 2O

−

r2 = 5 ⋅ 108 e

40700 RT

[CO ][ H 2O ]

40700 − RT

r3 = 9.87 ⋅108 e

7400 − RT

[CO2 ] [ H 2 ][O2 ]

Units of reaction parameters are: cal, mol, l, s.

Table 1. Dryer-Westbrook mechanism (WD). Reaction 1

Reaction rate

⎯⎯ → CO 2 + H 2 CO + H 2 O ←⎯ ⎯

2

H 2 + 0.5O2 ⎯⎯ → H 2O

3

H 2 O ⎯⎯ → H 2 + 0.5O2

r1 = 2.75 ⋅ 109 e

−

20000 RT

[CO ][ H 2O ]

40000 − −1 RT

r2 = 6.80 ⋅1015 T e r3 = 1.25 ⋅1017 T −0.877 e

−

98000 RT

[ H 2 ]0.25 [O2 ]1.50

[ H 2O ][ H 2 ]−0.75 [O2 ]


Table 2. Jones-Lindstedt mechanism (JL).

present approach allows to consider any of them or to propose new schemes. In the present paper, as an example of the whole strategy, the combustion of syngas (40% CO, 30% H2, 30% N2 by volume) is analyzed. The starting point is represented by the global mechanisms of Westbrook and Dryer [2] and Jones and Lindstedt [3, 8], reported in Table 1 and Table 2 respectively. The reverse rate constants of non elementary reactions have to be carefully derived and in particular the reaction orders (see for example reaction WD1). Generally speaking, assuming the following non elementary equilibrium reaction: NS

⎯⎯ → cC+dD rf = k f ( T ) ∏ Ciυ f ,i aA + bB ←⎯ ⎯

(1)

i =1

where υ f ,i is the order in the forward reaction of each of NS species and Ci its concentration, the reverse reaction expression is: kf rb = NS − ∑ ni K eq ⋅ ( RT ) i =1

NS

∏C i =1

υb ,i

(2)

i

where ni is the stoichiometric coefficient ( a, b, c, d ) of each species ( A, B, C , D ) and υb,i its order in the reverse reaction. υb,i can be simply derived from the order of the forward reaction and from the stoichiometric coefficient: υb,i = υ f ,i + ni . Non elementary reactions with orders lower than 1 might cause numerical problems, because of possible negative values of the concentration. A solution is a linearization of the rate expression when the concentration of the reactants becomes lower than a certain value. A simple example can show this approach. The species A is consumed with a reaction rate r = kC αA , where α is less than one. The mass balance can be then written as 1 (1−α )

, being C A0 the initial A dC A dt = − kC αA , whose solution is C A = ⎡⎣C 1A−0α − (1 − α ) kt ⎤⎦ concentration. When the time is higher than t = C 1A−0α ⎡⎣(1 − α ) k ⎤⎦ , C A becomes lower than II-6, 2


32nd Meeting on Combustion

zero with several problems arising if a numerical solution is adopted. To overcome these difficulties it is possible identify a small threshold value ( C AT ) of C A , below which an order % α , where k% is estimated making equal the two reaction rates one reaction is assumed: r = kC A

α −1 % % for C A = C AT : kC AT = kC AT ⇒ k = kC AT . The transition between the two reaction rates is obtained through an expression, able to avoid discontinuities in the function and in its derivatives. The final rate constant expression covering the whole time range is then: r = ξ kC αA + (1 − ξ ) kC αAT−1C A , where ξ is a proper function based on hyperbolic tangent, which α

allow the continuous transition: 1⎡ 2⎣

⎞ ⎤ CA − τ ⎟ + 1⎥ ⎝ C AT ⎠ ⎦ ⎛

ξ = ⎢ tanh ⎜ σ

(3)

where σ and τ are two constants. Figure 1 shows the impact of this approach on the solution. The two results are very similar and only zooming at very low concentrations, it is possible to observe the correction introduced.

0.08

Figure 1. Concentration of A with (blue line) and without (red line) the linearization (CA0=0.09 mol/l; =-0.3; k=10 mol0.3/l0.3/s). Small figure zooms the zone where the functions approach zero. The constants and are chosen equal to 23 and 17 respectively.

8e-9

cA

0.06 4e-9

0.04

0.0

0.02 0.00 0.000

0.002 t (s)

0.004

3. Tuning procedure The proposed procedure to determine the kinetic parameters of a global mechanism and the reaction orders is based on the ‘experimental measures’ generated by the detailed kinetic model [7]. Turbulent diffusive flames can be thought as the results of laminar flamelets. In this light, it is clear that selection of laminar counter-flow diffusion flames results particularly convenient as regression device. Furthermore, a sensitivity analysis showed that the best operative conditions where the temperature and composition profiles are affected by the rate parameters are those close to the flame extinction. It is quite obvious that the chemistry plays a fundamental role when the residence times are short, i.e. at high strain rates. The use of data obtained also for lower strain rates did not significantly modify either the values of the estimated parameters or the accuracy of the global model coming from the regression of just the flame close to extension. On the contrary, different feed compositions allowed to improve the performances. In particular, partially premixed flames were adopted. The introduction of the oxidizer in the fuel feed allows to enlarge the reaction zone and to better characterize the chemical phenomena in the rich side close to the flame front. It has to be noted that typical kinetic regressions are challenging problems. As a matter of facts, kinetic models are typically strongly nonlinear and, consequently, algorithms adopted for parameter estimation must be both robust and flexible enough to deal with constraints that are not always analytically definable. In this case the function evaluations are quite time consuming and then the algorithm has also to be effective. This work applies a numerical algorithm specifically conceived for kinetic regression: ‘BzzNonLinearRegression’ class [5], which is a free software for non commercial use. II-6, 3



4. Results Several counter-flow diffusion flames, at different Flame I Flame II strain rates (from 10 s-1 up to the extinction) and 100 500 strain rate (s-1) with different fuel compositions, were adopted to 29.87 149.3 vfuel (cm/s) carry out the regression. In particular, Table 3 reports two of these flames (called Flame I and 25 125 vair (cm/s) Flame II), for which some numerical results will be Table 3. Operative conditions of the proposed. ‘experimental flames’. Figure 2 shows the calculated peak temperature of counter flow diffusion flames at different strain rates by using the detailed kinetic mechanism and the simplified mechanisms in their original formulation (see Table 1 and Table 2). It is evident that the temperature is largely overestimated by both the global mechanisms (more than 300K), especially when the strain rate is low. Moreover, while the extinction strain rate predicted by the WD mechanisms is much larger (~6500 s-1) than the corresponding value predicted by the detailed scheme (~4500 s-1), the JL mechanisms predicts a smaller value (~3500 s-1). Figure 3 shows the predicted numerical profiles of temperature and carbon monoxide for Flame II. These curves mirror the results reported in Figure 2, confirming that the simplified mechanisms are not able to correctly describe the main reaction zone. It is reasonable to expect a similar trend also in a turbulent flame. The large differences in the predicted temperature profiles justify the need and the possibility to revise the original simplified mechanisms to obtain a better agreement with the detailed kinetic scheme. The first step in the ‘optimization’ procedure consists in finding the kinetic parameters which are more suitable for the non linear regression. In other words, the regression cannot be performed using all the available kinetic parameters, because their number is usually too large. It is better to choose only the parameters to which the flame is more sensitive. These parameters are chosen following the indications suggested by a sensitivity analysis, with respect to every kinetic parameter in the global kinetic scheme. Following this simple procedure, it is possible to select a relatively small number of parameters (usually less than 10), in order to reduce the computational time and the dimensions of the overall regression problem can be greatly reduced. The most significant parameters for the problem under investigation were found using this approach and are reported in Table 4 and Table 5. The regression was performed on different sets of ‘experimental data’, obtained through different combinations of counter flow diffusion flames. As previously mentioned, the numerical results suggest that it is better to use flames with large strain rates, because in these conditions the effects of the chemistry are more important and therefore the regression is Detailed

Detailed

2200

JL

2100

Flame II

WD

WD

temperature [K]

flame temperature [K]

2300

1900 1700 1500

JL

1800 1400 1000 600

1300

200 0

1000

2000

3000

4000

5000

6000

7000

0.35

strain rate [s-1]

0.40

0.45

0.50

0.55

0.60

0.65

distance from fuel nozzle [cm]

Figure 2. Flame temperature: comparison between kinetic schemes.

Figure 3. Temperature profiles in Flame II: comparison between kinetic schemes.

II-6, 4

Combustion Colloquia 2009 Reaction

Parameter

1 1 2 2 3 3

A Ea A Ea A Ea

3

υ f ,H

3

υ f ,O


Original Value 2.24·1012 40700 5·108 40700 9.87·108 7400

Optimized Value 2.30·1011 31700 4.45·109 41300 1.35·108 6900

1

0.87

2

1

2

Reaction

Parameter

1 1 2 2 3 3

A Ea A Ea A Ea

1.10

Original Value 2.75·109 20000 6.70·1015 40000 1.25·1017 98000

Optimized Value 2.63·1010 18500 6.80·1014 37700 1.07·1018 94000

2

υ f ,H

2

0.25

0.33

2

υ f ,O

2

1.50

1.40



Table 4. Optimized WD mechanism.

Table 5. Optimized JL mechanism.

performed on experimental data more sensitive to the kinetic parameters. For example, this means that the parameters obtained from the regression on the single Flame II (large strain rate) work pretty well also for Flame I (low strain rate), but the opposite is not true. Therefore, if it is necessary to reduce the number of ‘experimental flames’ for reasons related to the excessive computational time, it is more convenient to work on flames with a large strain rate. The kinetic parameters obtained from the regression are summarized in Table 4 and Table 5. These parameters describe pretty well the counter flow diffusion flames in a large range of strain rates. As an example, Figure 4 compares the temperature profiles of Flames I and II obtained using the optimized global kinetic schemes with the detailed kinetic scheme. The agreement is largely improved if compared with the original result (Figure 3). In particular, the peak temperature and the flame front are correctly predicted. The optimized kinetic schemes were then applied to a turbulent flame, in order to check if a real improvement with respect to the original mechanisms can be obtained under turbulent conditions. A simple jet flame, fed with syngas (with the same composition used for the counter flow diffusion flames), in a coflow of air, was chosen as a test case [9]. This is a relatively simple system, which can be simulated using a 2D grid and applying the same detailed kinetic model adopted for the regression. The simulations are performed on a structured grid with ~16000 cells, by using a RANS approach and the modified - model (C1 =1.60). The EDC model was chosen for describing the interactions between chemistry and turbulence. Figure 5 shows the temperature profiles along the axis of this flame, as obtained by using the detailed kinetic scheme and the original simplified mechanisms (Table 1 and Table 2) and the comparison with the experimental measurements. The detailed mechanism predicts very well the temperature profile, while both the global mechanisms lead to a large overestimation of the temperature, which confirms the results obtained in the 2200

2200 Detailed WD

1800

1400

1000

600

Flame II

WD

1800

JL

temperature [K]

temperature [K]

Detailed

Flame I

JL

1400

1000

600

200

200 0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.35


0.40

0.45

0.50

0.55


Figure 4. Temperature profiles in Flame I and II: optimized kinetic schemes.

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0.60

0.65


32nd Meeting on Combustion 2200 WD

Detailed

2200

Exp.

JL

1800

JL

1800

temperature [K]

temperature [K]

WD

Exp.

1400 1000

1400

1000

600

600 200

200 0.00

100.00

200.00

300.00

400.00

500.00

0.00

axial coordinate [mm]

100.00

200.00

300.00

400.00

500.00

axial coordinate [mm]

Figure 5. Axial profiles of temperature in the turbulent flame: original kinetic schemes.

Figure 6. Axial profiles of temperature in the turbulent flame: optimized kinetic schemes.

counter flow diffusion flames. This over-estimation (more than 250K) is not small and can lead to negative consequences on the prediction of pollutant species, heat transfer calculations, etc. On the contrary, the optimized kinetic schemes (Table 4 and Table 5) strongly improve the numerical predictions, as reported in Figure 5. The revised WD and JL mechanisms show a satisfactory agreement with the experimental measurements, by greatly reducing the original temperature over-estimation, even if they tend to move the peak temperature closer to the inlet. 5. Conclusions In this work the need and the possibility to improve the existing simplified mechanisms describing the combustion of syngas were demonstrated. A procedure based on non linear regression was used to improve two simplified mechanisms, available in the literature and largely adopted by the combustion community. The novelty of the proposed approach is represented by the choice of ‘experimental data’, which correspond to an appropriate set of counter flow diffusion flames. The ‘optimized mechanisms’ were applied to a turbulent jet flame, showing a strong improvement with respect the original mechanisms. Further investigations and improvements are needed to confirm the feasibility of this procedure. In particular, the choice of the counter flow diffusion flames to adopt for the optimization must be better defined. However, the results reported in this work appear promising, especially for the formulation of new kinetic schemes for fuel mixtures. 6. Acknowledgments The authors acknowledge the financial support of CNR/DET program “Gas naturale”. 7. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

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