2.3 Premixed Combustion
2.3.1 Unstretched Laminar Flames
2.3.1.2 Unstretched Laminar Burning Velocity
One-dimensional free flame calculations in Cantera 2.2[67] are one method to obtain the unstretched laminar burning velocitySl,0. The laminar burning velocity is equal to the resulting inlet flow velocity of the final solution. Alter-natively, Peters [66] suggests the following correlation including temperature
0.3 0.5 0.7 0.9 1.1 1.3 1100
1400 1700 2000 2300 2600
φ[–]
Tad[K]
Tu=293 K Tu=473 K Tu=673 K
(a) Effect ofTu(p=1 bar). Symbols: GRI-Mech 3.0 [69], lines:
Eq. (2.34) [66].
0.3 0.5 0.7 0.9 1.1 1.3
1100 1400 1700 2000 2300 2600
φ[–]
Tad[K]
Tu=293 K Tu=473 K Tu=673 K
(b) Effect ofpcomputed with GRI-Mech 3.0 [69]: Filled sym-bols:p=1 bar, empty symbols:p=20 bar.
Figure 2.7:Preheating temperature and pressure effect on the adiabatic flame temperature of methane-air mixtures.
2.3 Premixed Combustion
and pressure dependencies based on the work of Göttgens et al. [70]:
Sl,0=b1Yfb2e−
Assuming that the inner layer temperature TIL does not depend on equiva-lence ratio, it can be calculated from
p=b5e−
b6
TIL. (2.36)
The coefficients b1–b6 are fuel dependent. The adiabatic flame temperature can be obtained from Eq. (2.34) or from equilibrium kinetics computations.
If the unstretched laminar burning velocity at a reference temperatureTrefand a reference pressure pref is known, a third method to determine Sl,0 can be applied:
This method was introduced by Metghalchi and Keck [71, 72] for propane-air mixtures and was applied by Dahoe [73] to hydrogen-air flames. The power law exponentsc1 andc2 depend on fuel and equivalence ratio. The presented methods to determine Sl,0 will be discussed for hydrogen- and methane-air mixtures in the following paragraphs.
Hydrogen-Air Mixtures: Measured unstretched laminar burning velocities of hydrogen-air mixtures atTu=293 K andp=1 bar presented by Konnov [74]
are summarized in Fig. 2.8. The figure also includes a fourth order polynomial fit
Sl,0[m/s]=2.1600φ4−9.1537φ3+12.4930φ2−3.7952φ+0.3972 . (2.38) It will be used throughout this work to represent experimentally determined unstretched laminar burning velocities of hydrogen-air mixtures at ambient conditions. Figure 2.8 shows thatSl,0increases with equivalence ratio. In con-trast to the adiabatic flame temperature, the maximum is not found around φ=1.1 but aroundφ=1.5 [73] which is beyond the equivalence ratio range of interest in this work.
0.1 0.3 0.5 0.7 0.9 1.1 1.3 0
0.5 1 1.5 2 2.5 3
φ[–]
Sl,0[m/s]
[75, 76] [77]
[78] [79]
[80] [81]
Eq. (2.38)
Figure 2.8:Measured unstretched laminar burning velocitySl,0of hyrogen-air mixtures at ambient conditions compared to Eq. (2.38).
Table 2.2:Coefficients of unstretched laminar burning velocity correlation by Peters and Göttgens et al. [66, 70] (Eq. (2.35)) for hydrogen-air and methane-air mixtures.
Coefficient H2 CH4 Unit
b1 12928.8 0.22176 m/s
b2 1.08721 0.565 −
b3 2057.56 −6444.27 K
b4 3.535 2.516 −
b5 30044.1 3.1557×108 bar
b6 10200.8 23873 K
Figure 2.9 compares the unstretched laminar burning velocity at ambient con-ditions computed withCantera 2.2[67], the correlation by Peters and Göttgens et al. [66, 70] in Eq. (2.35) and the experimental data fit from Eq. (2.38). The filled symbols represent the Cantera 2.2 [67] data computed with the reac-tion mechanism of Ó Conaire et al. [68]. The results from Eq. (2.35) are shown as a dashed line. The coefficients are given in Tab. 2.2. Compared to the ex-perimental data shown as black line, the simulations underestimate Sl,0 at
2.3 Premixed Combustion
Figure 2.9:Computed unstretched laminar burning velocity Sl,0 of hyrogen-air mixtures at ambient conditions compared to Eq. (2.38).
lean conditions and overestimate it around stoichiometry. The presented re-sults from the correlation by Peters and Göttgens et al. [66, 70] are similar to theCantera 2.2 [67] data in the lean region but strongly overestimate the un-stretched laminar burning velocity around stoichiometry.
In the following, the influence of preheating temperature will be analyzed.
Apart from Cantera 2.2 [67] computations and the correlation by Peters and Göttgens et al. [66, 70], the power law method presented in Eq. (2.37) can be used for this purpose. Different methods to determine the power law expo-nent c1 are depicted in Fig. 2.10. Gelfand [82] derives values betweenc1 =1.6 around stoichiometry andc1 =2.5 at φ=0.35 from computations of spheri-cal flame propagation. Alternatively, the power law exponent can be extracted from one-dimensional free flame (FF) computations at 293 K and 473 K as de-scribed in [58]. If only the temperature dependence in Eq. (2.37) is considered, the temperature power law exponent is obtained from:
c1=
The exponents obtained with the reaction mechanism of Ó Conaire et al. [68]
0.3 0.5 0.7 0.9 1.1 1.3 1.5
2 2.5 3 3.5 4
φ[–]
c1[–]
[82]
Eq. (2.39) [83]
Eq. (2.40)
Figure 2.10:Power law exponentc1 for hydrogen-air mixtures.
are presented in Fig. 2.10 and range from c1 =1.6 at φ=1 to c1 = 3.6 at φ= 0.35. A third method to determinec1 is based on experimentally determined unstretched laminar burning velocities by Heimel [83] and is introduced in [35]. The power law exponents extracted from Heimel’s data are included in Fig. 2.10 as empty circles. They cover the range from φ = 0.9 to φ= 1.3. To extend the data to lean conditions a quadratic fit
c1=1.6429φ2−4.1907φ+4.2441 (2.40) is used for extrapolation. It has to be noted that such extrapolation holds a considerable uncertainty. The obtained power law exponents range fromc1= 1.7 atφ=1 toc1=3.0 atφ=0.35.
Figure 2.10 shows that all three presented methods deliver similar exponents c1 around stoichiometry. However, at lean conditions large deviations are ob-served leading to high uncertainties in the definition ofSl,0 at preheated con-ditions.
Figure 2.11 compares the unstretched laminar burning velocities at preheated conditions computed in laminar free flames with those obtained from the cor-relation by Peters and Göttgens et al. [66, 70] in Eq. (2.35). Filled symbols rep-resent data at Tu =473 K and empty symbols data at Tu =673 K. As expected
2.3 Premixed Combustion
0.3 0.5 0.7 0.9 1.1 1.3
0 3 6 9 12 15
φ[–]
Sl,0[m/s]
Eq. (2.38) Ó Conaire [68]
Eq. (2.35)
Figure 2.11:Unstretched laminar burning velocity at different preheating temperatures for hydrogen-air mixtures. Filled symbols:Tu = 473 K, empty symbols:Tu=673 K.
from the ambient temperature case presented in Fig. 2.9, the correlation by Peters and Göttgens widely overestimates the unstretched laminar burning velocity around stoichiometry. Therefore, it does not seem suitable to repre-sentSl,0for hydrogen-air mixtures in this work. For that reason, the free flame method is preferred in this work to defineSl,0at preheated conditions.
In order to obtain a correlation-based representation ofSl,0, free flame simula-tions at different preheating temperatures are performed and used to generate third order polynomials of the form
Sl,0(Tu)=b7Tu3+b8Tu2+b9Tu+b10. (2.41) The coefficients b7–b10 depend on equivalence ratio and are presented in Tab. C.1 for atmospheric pressure. An example polynomial forφ=1 is shown in Fig. 2.12. The unstretched laminar burning velocity at ambient temperature is calculated from Eq. (2.38) to include the available experimental data. Equa-tion (2.41) in combinaEqua-tion with the coefficients presented in App. C will be used throughout this work to calculateSl,0. It should be kept in mind that at low burning velocities, i.e. low equivalence ratios, and preheated conditions
250 360 470 580 690 800 2
4 6 8 10 12
Tu[K]
Sl,0[m/s]
Ó Conaire [68]
Eq. (2.38) Eqn. (2.41)
Figure 2.12:Effect of preheating temperature on unstretched laminar burning velocity at atmospheric pressure (φ=1).
Sl,0might be underestimated (cf. Fig. 2.9).
Similar to the dependence on temperature, little experimental data is avail-able to determine the influence of pressure on unstretched laminar burning velocity. Dahoe [73] introduces the power law exponentc2=0.194 (Eq. (2.37)) based on the work of Iijima and Takeno [84] for a pressure range of p =0.5–
5.5 bar and the equivalence ratio φ=1. This exponent reflects an increase of Sl,0 with pressure. Konnov [74] presents experimental data at ambient tem-perature by Aung et al. [76] for four different equivalence ratios ranging from φ= 0.75 to φ=3. The data in the lean and stoichiometric region are shown in Fig. 2.13 in comparison to one-dimensional free flame computations with the reaction mechanism of Ó Conaire et al. [68]. Around stoichiometry an ini-tial increase ofSl,0with pressure is observed up to atmospheric pressure. Be-yond that, the unstretched laminar burning velocity decreases contradicting the observations of Iijima and Takeno [84]. With decreasing equivalence ratio, the maximum unstretched laminar burning velocity is shifted to lower pres-sures. Comparing ambient temperature free flame simulations at p = 1 bar
2.3 Premixed Combustion
Figure 2.13:Pressure influence on Sl,0 of hydrogen-air mixtures Tu = 293 K.
Filled symbols: experimental data [74], empty symbols: one-dimensional free flame (Ó Conaire et al. [68]).
andp=20 bar, the power law exponent for Eq. (2.37) can be defined by c2=
The resulting power law exponents are depicted in Fig. 2.14. In contrast to the value presented by Dahoe [73],c2 is negative for equivalence ratios up toφ= 1.3. The absolute value decreases with increasing equivalence ratio indicating a reduced influence of pressure. However, it has to be noted that as shown in Fig. 2.15 the power law exponents are not suitable to represent the pressure influence over the whole displayed pressure range.
Alternatively, the unstretched laminar burning velocity at elevated pressure can be computed with the correlation of Peters and Göttgens et al. [66, 70]
in Eq. (2.35). In Fig. 2.16, the result is compared to one-dimensional free flame simulations with the reaction mechanism of Ó Conaire et al. [68] at p = 20 bar and different preheating temperatures. At lean conditions, both methods show similar results whereas around stoichiometry the correlation of Peters and Göttgens et al. [66, 70] underestimatesSl,0compared to the free
0.3 0.5 0.7 0.9 1.1 1.3
−1.5
−1.2
−0.9
−0.6
−0.3 0
φ[–]
c2[–]
Figure 2.14:Power law exponentsc2for hydrogen-air mixtures obtained from free flame simulations at ambient temperature (Ó Conaire et al. [68], Eq. (2.42)).
0 4 8 12 16 20
0 0.5 1 1.5 2 2.5
p [bar]
Sl,0[m/s]
φ=1.05 φ=0.75 φ=0.40
Figure 2.15:Pressure influence on Sl,0 of hydrogen-air mixtures at ambient temperature. Filled symbols: power law (Eq. (2.37)) with expo-nents from Eq. (2.42) (Sl,0(Tref,pref) from Eq. (2.38)), empty sym-bols: free flame simulation (Ó Conaire et al. [68]).
2.3 Premixed Combustion
0.3 0.5 0.7 0.9 1.1 1.3
0 2 4 6 8 10
φ[–]
Sl,0[m/s]
Tu=293 K Tu=473 K Tu=673 K
Figure 2.16:Sl,0 of hydrogen-air mixtures at p=20 bar and different preheat-ing. Filled symbols: Eq. (2.35) [66, 70], empty symbols: free flame simulation (Ó Conaire et al. [68]).
flame simulation. Compared to the values in Fig. 2.11 it can be observed that the unstretched laminar burning velocity is strongly reduced by pressure at lean conditions and only slightly reduced around stoichiometry as indicated by the computed power law exponents in Fig. 2.14.
Due to the lack of experimental data, the values computed from one-dimensional free flame simulations with the reaction mechanism of Ó Conaire et al. [68] presented as empty symbols in Fig. 2.16 will be used in this work to represent the unstretched laminar burning velocity at elevated pressure and different preheating temperatures. Similar to the atmospheric pressure case, third order polynomials in the form of Eq. (2.41) are extracted for 3, 5, 7 and 20 bar resulting in the coefficients presented in App. C.
Methane-Air Mixtures: Experimental data of several authors for the un-stretched laminar burning velocity of methane-air mixtures at ambient tem-perature and pressure were summarized by Chen [85]. They are shown in Fig. 2.17 compared to the results of one-dimensional free flame simulations
0.3 0.5 0.7 0.9 1.1 1.3 0
0.1 0.2 0.3 0.4 0.5
φ[–]
Sl,0[m/s]
[86] [87]
[88] [89]
[90] [91]
[92] GRI [69]
Figure 2.17:Measured unstretched laminar burning velocitySl,0 of methane-air mixtures at ambient conditions compared to laminar free flame computation with GRI-Mech 3.0 [69].
with the reaction mechanism GRI-Mech 3.0 [69]. As for hydrogen-air mixtures, the unstretched laminar burning velocity increases with equivalence ratio un-til a maximum is reached at slightly rich conditions. For methane-air mixtures this maximum is located close to stoichiometry around φ= 1.05. In general, Sl,0is lower compared to hydrogen-air mixtures (approximately by a factor of 10 at the maximum). Fig. 2.17 shows that the reaction mechanism GRI-Mech 3.0 [69] well represents the experimental data over a broad equivalence ratio range. Only slight deviations are observed around φ=0.5. For that reason, it is assumed that GRI-Mech 3.0 [69] can also be applied to investigate the effect of preheating temperature and pressure onSl,0.
Fig. 2.18 shows the unstretched laminar burning velocity of methane-air mix-tures for different preheating temperamix-tures at 1 bar and 20 bar computed with GRI-Mech 3.0 [69] in comparison to the correlation of Peters and Göttgens et al. [66, 70] in Eq. (2.35). The required parameters for Eq. (2.35) can be found in Tab. 2.2. The adiabatic flame temperature is taken from chemical equilib-rium calculations presented in Fig. 2.7(b). For methane-air mixtures, preheat-ing leads to an increase of Sl,0 whereas increasing pressure leads to aSl,0
re-2.3 Premixed Combustion
0.3 0.5 0.7 0.9 1.1 1.3
0 0.4 0.8 1.2 1.6 2
φ[–]
Sl,0[m/s]
Tu=293 K Tu=473 K Tu=673 K
(a) p=1 bar
0.3 0.5 0.7 0.9 1.1 1.3
0 0.15 0.3 0.45 0.6 0.75
φ[–]
Sl,0[m/s]
Tu=293 K Tu=473 K Tu=673 K
(b) p=20 bar
Figure 2.18:Sl,0 for methane-air mixtures at different preheating tempera-tures. Empty symbols: laminar free flame (Cantera 2.2 [67], GRI-Mech 3.0 [69]), filled symbols: correlation of Peters and Göttgens et al. [66, 70] (Eq. (2.35)).Tadis taken from Fig. 2.7(b).
duction. The correlation of Peters and Göttgens et al. [66, 70] fits the laminar free flame results at lean conditions and ambient pressure. High deviations are observed in the rich region and at elevated pressure. As Rozenchan et al. [89]
compared Sl,0 obtained from GRI-Mech 3.0 [69] with experimental data and found good agreement up to p =20 bar, the one-dimensional free flame re-sults will be used to representSl,0of methane-air mixtures in this work.
In order to obtain a correlation-based representation ofSl,0 at different pres-sures for varying preheating temperature similar to the hydrogen-air mixture case, third order polynomials in the form of Eq. (2.41) are extracted for 1 and 20 bar. The resulting coefficients are presented in Tab. C.6 and C.7.