Toward Predictive Modeling of Petroleum and Biobased Fuel Stability

Article pubs.acs.org/EF

Toward Predictive Modeling of Petroleum and Biobased Fuel Stability: Kinetics of Methyl Oleate/n‑Dodecane Autoxidation Arij Ben Amara,* André Nicolle, Maira Alves-Fortunato, and Nicolas Jeuland IFP Energies nouvelles, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France S Supporting Information *

ABSTRACT: Because of the recent changes in the formulation and handling of middle-distillate fuels, oxidation stability is becoming an increasingly important issue. However, liquid-phase oxidation kinetics of middle-distillate fuels remains poorly understood. The purpose of this study was to gain an in-depth understanding of the impact of fatty acid methyl ester (FAME) addition on autoxidation kinetics. A detailed kinetic mechanism for the autoxidation of a n-dodecane/methyl oleate (MO) surrogate mixture was generated and validated against original well-controlled accelerated oxidation experiments. Results emphasize the nonlinear oxidation promoting effect of MO on n-dodecane autoxidation. Pathway analyses reveal that HO2 and OH propagation steps as well as the duration of initiation and propagation phases strongly affected sensitivity analysis by MO addition. On the basis of these analyses and the detailed mechanism, an analytical model was derived and validated against experiments on binary surrogate mixtures as well as blends of conventional commercial fuels and FAME. These results open up the use of bottom-up liquid-phase oxidation modeling strategies for the in silico formulation of alternative fuels and the design of innovative injection fuel systems.

1. INTRODUCTION As a consequence of the diversification of energy ressources, the complexity of fuel logistics, and the increasing severeness of operating conditions, oxidation stability of middle-distillate fuels is becoming more and more of an issue for automotive and aerospace industries.1,2 Currently, liquid-phase oxidation kinetics of these fuels and the impact of additives or pollutants remain poorly understood. Although several semi-detailed liquid-phase oxidation mechanisms were proposed,3 very few detailed kinetic models focusing on the autoxidation of singlecomponent fuels have been developed.4 The ability of existing certification tests to assess the impact of fatty acid methyl ester (FAME) addition on fuel stability has been questioned recently.5,6 There is still a need for experimental and computational tools allowing for the specification of safe and economic solutions for alternative fuel formulation and use. The field of application of the Rancimat test (EN 15751), initially dedicated to the characterization of fats and oils,7,8 has widened over time.9 This test has been increasingly used to probe the stability of FAME/conventional diesel blends (EN 590). Using this experimental technique, Dunn10 and Farhoosh et al.11 put forward linear correlations between the logarithm of the stability index (OSI) and the inverse of the operating temperature for pure and commercial methyl esters. Classically, co-oxidation of hydrocarbon mixtures is described by reactivity ratios involving self- and cross-propagation rate constants as well as individual oxidizibility ratios,12 namely, the ratio of the chain propagation to the square root of the termination rate constant, kp/(kt1/2). Typically, the propagation rate constant kp accounts only for propagation reactions involving the reactants and alkyl peroxy radicals, therefore neglecting the contribution of alternative consumption pathways (hydrogen abstraction by hydroxyl or hydroperoxyl radicals, peresterification, hydrolysis, or reverse transesterifica© 2013 American Chemical Society

tion), which have been found to contribute to the autoxidation process.13,14 To more accurately predict fuel stability, these alternate pathways competing with the classical alkylperoxy route15 need to be accounted for in the oxidizibility ratio. The present detailed chemical kinetic study aims at understanding the impact of FAME addition on alternative fuel autoxidation kinetics and formulating a microkinetic-based analytical expression for the induction period (IP). Commercial FAMEs are typically composed of mono- and polyunsaturated compounds.16,17 They may also contain antioxidants,18,19 metal contaminants,20 impurities, such as glycerol,21 and microorganisms,18 which have been shown to affect autoxidation. Polyunsaturated esters, however, represent a huge challenge for kinetic modeling because of the large number of carbon atoms and the asymmetry of the molecules.22 Because it is present in most commercial biofuels, methyl oleate (MO) was chosen in this work as a good trade-off between complexity and representativity of FAME reactivity, thus representing a first step toward realistic biofuel surrogates. n-Dodecane was chosen to represent the reactivity of middle-distillate fuels,11,23 although it is established that other chemical classes besides n-alkanes can affect the reactivity and selectivity of the autoxidation process.24

2. MATERIALS AND METHODS 2.1. Experimental Setup. The experiments were conducted using a Rancimat 743 apparatus from Metrohm, Ltd. (Figure 1). Air was bubbled through fuel samples kept in a glass test tube held stationary within a thermostat-controlled block heater at a flow rate ranging from 10 to 25 standard liters per hour (SLPH). The operating temperature ranged from 130 to 160 °C with a maximum standard deviation of Received: July 17, 2013 Revised: September 16, 2013 Published: September 16, 2013 6125

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Table 1. Experimental Conditions in Our Rancimat Experiments experiment number

n-C12H26 n-C12H26 n-C12H26 n-C12H26 95% n-C12H26 95% n-C12H26 95% n-C12H26 90% n-C12H26 90% n-C12H26 90% n-C12H26 70% n-C12H26 70% n-C12H26 70% n-C12H26

1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 1. Schematic of the Rancimat apparatus.25

fuel composition (%, v/v)

+ + + + + + + + +

5% MO 5% MO 5% MO 10% MO 10% MO 10% MO 30% MO 30% MO 30% MO

temperature (°C)

volumetric flow rate (SLPH)

130 140 150 160 130 140 150 130 140 150 130 140 150

10 10 10a 10 10 10 10a 10 10 10 10 10 10

a

For this condition, an additional test was performed using a 20 SLPH air flow rate to check that gas−liquid mass transfer is not rate-limiting (see section 2.1).

±0.3 °C. The injected air passed through the sample carrying volatile fuel fractions and oxidation products, such as short-chain carboxylic acids, into a measuring vessel containing an absorption solution of distilled water, where the electrical conductivity is measured continuously. The electrical conductivity of the distilled water is 1.84 MΩ cm. Because the conductivity of a solution is roughly proportional to the concentration of species,11 its variation is a global indicator of the composition change. The maximum of the second temporal derivative of conductivity characterizes the onset of the oxidation regime, and the corresponding time is considered to be the IP (Figure 2). The measurement precision (IPR) depends upon the IP

1 presents the operating conditions explored in the experimental part of the study. 2.2. Chemical Kinetic Modeling. A detailed chemical kinetic scheme describing n-dodecane/MO blend autoxidation at atmospheric pressure was generated using the RMG code,27 involving 174 species and 3275 reactions. The mechanism and the associated kinetic and thermochemical files are provided in the Supporting Information. Thermochemistry estimates are based on Benson’s group additivity method and corrected for solvation using a gas/solvent partition coefficient estimated from Abraham’s model.28 The solvent considered in this study is n-dodecane. Table 2 gives the effective rate constant of the ith reaction

⎛ E ⎞ ki = k int , iF = Ai T ni exp⎜− i ⎟ ⎝ RT ⎠

(1)

where Ai and Ei correspond to the pre-exponential factor and the activation energy of the ith reaction, respectively, T is the temperature, ni is the temperature exponent, and R is the universal gas constant. This effective rate constant is evaluated from the corresponding intrinsic rate constant kint,i and from the diffusion factor F = (1 + (kint,i/ 4πRD))−1, accounting for the impact of diffusion into the solvent cage on the effective rate constant. R and D represent the sum of the radii and diffusivities of reacting species in the ith reaction, respectively. The individual diffusivities are calculated from viscosity using the Stokes− Einstein equation.29 The Senkin code from the Chemkin package30 was modified to account for liquid-phase reactions under isothermal and isobaric conditions at a nearly constant dissolved oxygen concentration. Except for O2, whose net sink term is set to 0, the mass balance equation for the jth species is

Figure 2. Recorded conductivity temporal profile during autoxidation at 0% (v/v) MO and 150 °C. The IP determination method used in this study is compared to the tangent intersection method previously proposed.26

d[Bj ] duration via the operating temperature. According to our experimental data, it scales linearly with IP, according to IPR (h) = 0.15IP − 0.37, in agreement with the precision of ca. 0.6 h previously reported in standard conditions.9 In our test conditions, IP precision ranges from 0.97 h at 130 °C to 0.02 h at 160 °C. The reproducibility was checked by performing replicate analyses. Additionally, experiments were performed with several air flow rates from 10 to 20 SLPH (Table 1), leading to a difference in the induction period below 0.6 h for both pure ndodecane and n-dodecane/MO blends. This difference remains insignificant according to the above-mentioned IP precision and confirms that gas−liquid mass transfer is not rate-limiting in the range of air flow rates studied. In the framework of this study, two fuels were used: n-dodecane (nC12H26) and MO, from Sigma-Aldrich, having a purity of >99%. Table

dt

=

NR

Ns

i=1

k=1

∑ vijki ∏ [Bk ]vki

= ωj

(2)

where νij denotes the stoichiometric coefficient of the jth species in the ith monodirectional reaction. The initial liquid density was computed from single-component densities found in the National Institute of Standards and Technology (NIST) database. On the basis of Henry’s constant of n-dodecane,31 the dissolved O2 mole fraction was fixed to 300 ppm by mole for the simulations. As noticed from Table 2, most reactions can be considered not to be diffusion-limited in the present operating conditions, which is however not the case for reactions R2 and R7. This is not surprising because oxygen addition to alkyl radicals is known to occur near the diffusion-controlled regime under ambient conditions.32 The reaction barrier generated by RMG for reaction R1 is similar to that recommended by Pfaendtner and Broadbelt33 Further, the activation 6126

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Table 2. Kinetic Parameters of the Most Important Reaction Families Ai (cm3 mol−1 s−1)

reactiona

R1H + O2 → R1 + HO2 R1 + O2 → R1O2

(R2) (R4)

R1O2 + H 2O2 → R1O2 H + HO2 R 2H + O2 → R 2 + HO2 R 2 + O2 → R 2O2

(R5)

0.899

7.54 × 1012

0.00

0.0

0.563

3

8.42 × 100 3.46 × 10 7.54 × 10

(R7)

R 2H + HO2 → R 2 + H 2O2 R1O2 + R1O2 → R1O4 R1

5.08 × 10

(R9)

5.08 × 10

(R11)

R1O2 H + R1O2 → Q 1O2 H + R1O2 H

(R14) (R15)

−4

−4

1.81 × 1012

(R12) c

12

5.08 × 10−4

(R10)

R 2H + R1O2 → R 2 + R1O2 H

1

8.42 × 100

(R8)

R 2H + R 2O2 → R 2 + R 2O2 H

R1H + OH → R1 + H 2O

47.4b

1.07 × 1014

(R6)

R1H + R 2O2 → R1 + R 2O2 H

Q 1O2 H → Q 1O + OH

F

−0.86

5.07 × 10

(R3)

R1H + R1O2 → R1 + R1O2 H

Ei (kcal mol−1)

5.95 × 10

(R1)

R1H + HO2 → R1 + H 2O2

ni

18

(R13)

6.09 × 10

0

1.09 × 10

22

1.58 × 107

2.63

12.3

b

1.000

3.49

11.3b

1.000

3.40

−1.14

0.989

−0.02

33.9b

0.904

0.00

0.0

0.577

3.49

11.3b

1.000

4.59

7.16

1.000

4.59

7.16

1.000

4.59

7.16

1.000

0.00

0.0

0.824

9.52

b

−2.30

1.84

b

1.90

0.16

3.40

1.000 1.000 0.898

a

R1H and R2H refer to n-dodecane and MO, respectively. bThe value was computed from the backward rate constant by MECHMOD39 using thermochemical data. cQ1O2H refers to alkyl hydroperoxy radicals obtained from R1O2H by hydrogen abstraction.

Figure 3. Comparison of induction times obtained using the detailed kinetic model (lines) to experimental results (symbols) from the literature40,41 and obtained in the present work (filled circles): (a) temperature-dependent profiles and (b) FAME-volume-fraction-dependent profiles. energies of reactions R2 and R4 are in excellent agreement with those previously proposed by Pfaendtner and Broadbelt33 and Kuprowicz et al.34 These kinetic parameters are also in line with the data compilation by Denisov and Afanas’ev.35 The pre-exponential factor used in this study for reaction R5 was calculated from the transition-state theory36 and is much lower than that by Zabarnick,37 which was based on average collision frequency.38 Note that the ester hydrolysis pathway evidenced by Fang et al.14 is not considered in the present model, although it can be anticipated that the acidity (hence, the hydrolysis rate) remains low during the induction period.

[R1H]IP + [R 2H]IP = 0.95 [R1H]0 + [R 2H]0

(3)

As seen in Figure 3, the model correctly captures the nonlinear promoting effect of MO addition on the IP. The correlation coefficient between our experimental and modeling data is best at 5% MO (0.989) and 30% MO (0.995) and worst at 0% MO (0.947) and 10% MO (0.968), even if the qualitative agreement between the model and experiment appears to be worst at intermediate MO concentrations. It is interesting to note that MO addition does not only affect the overall pre-exponential factor but also tends to induce a significant curvature in the Arrhenius plots obtained. We note a slight change in the slope of predicted IP at high temperatures (Figure 3a), which could be attributed to the increasing contribution of n-dodecyl polyperoxy radical propagation reactions at the expense of the n-dodecyl peroxy radical reaction R4. The specific analysis of

3. RESULTS AND DISCUSSION 3.1. Validation of the Detailed Kinetic Model. The detailed chemical kinetic model described in section 2.2 was validated against experimental IP data from the Rancimat apparatus described in section 2.1 and from the literature.40,41 In line with previous studies,42,43 the IP can be assumed to correspond to a fuel conversion of ca. 5%, namely 6127

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Figure 4. Impact of the MO content in the fuel (0 versus 10%, v/v) on (a and b) main n-dodecane consumption pathways and (c) main MO consumption pathways [10% (v/v) MO in the fuel] at half induction period (IP/2). The calculations were carried out using the detailed kinetic scheme.

peroxides (R1O2H). In the absence of MO (Figure 4a), five n-dodecyl radicals are formed by hydrogen abstraction (reaction R1). These radicals react quickly with the dissolved oxygen,4 yielding five peroxy radicals, which abstract ndodecane hydrogen to form hydroperoxides. Our analysis reveals a 12% higher net production rate for species 6 (see the Appendix) with respect to all of the other n-dodecyl radicals. This may be explained by the C−H bond dissociation energy difference according to carbon position. Sumathi et al.45 have reported this dependency for linear alkanes having 1−5 carbon atoms. The addition of 10% (v/v) MO significantly impacts ndodecane consumption pathways with a 17-fold increase of the R1O2 formation rate. As shown on Figure 4b, the rates of

reactions, including polyperoxy radicals, lies outside the scope of this work and would deserve a dedicated study. 3.2. Reaction Pathway Analysis. Because the detailed kinetic scheme was found to perform quite well, it was used with some confidence to conduct rate-of-production (ROP) analyses. Using the same notations given in eq 2, the normalized rate of the ith monodirectional reaction for the production of the jth species is given by44 (ROP)ij = ((max(vij,0)ki∏k =Ns 1[Bk]vki)/(∑i =NR1max(vij,0)ki∏k =Ns 1[Bk]vki)). Figure 4a illustrates the consumption pathways of n-dodecane and its blends with MO during oxidation involving the formation of n-dodecyl radicals (hereafter referred to as R1), n-dodecyl peroxy radicals (R1O2), and n-dodecyl hydro6128

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Figure 5. Impact of the MO content on the history of the main n-dodecane consumption pathways. Calculations were carried out using the detailed kinetic scheme. R1H corresponds to n-dodecane, while R2H represents MO. The colors represent the three phases described in section 3.2. The first phase corresponds to t ≤ τHO2; second phase corresponds to τHO2 ≤ t ≤ τ12; and third phase corresponds to τ12 ≤ t ≤ IP (see the text for notations).

radicals can be considered in steady state provided that 10−6 ≤ t/IP ≤ 10−2. On the basis of the ROP analyses and reaction subset {R1, ..., R12}, we can derive analytical expressions for R1O2 and HO2. Applying QSSA to R2, we obtain d[R 2O2 ] = ω7 − ω8 − ω9 = ω6 − ω8 + ω10 + ω11 dt (4)

Our rate analyses reveal that ω10 remains negligible with respect to ω6 and ω8, provided [R1H]0 ≠ 0. Application of QSSA to R1 gives the alkylperoxy radical concentration: d[R1O2]/dt = ω2 − ω4 − ω5 − ω11 − ω12 = 2ω1 + ω6 + ω8 − ω11 − k12[R1O2]2. According to our rate analysis, ω8 and ω11 nearly compensate each other and a good estimate of the R1O2 concentration may be written as [R1O2] = (2α/k12)1/2, with α = ω1 + ω6. ω1 and ω6 can be considered to be constant during the pre-induction phase because of the negligible depletion of reactants. On the basis of this concentration estimate, the integration of eq 4 yields

Figure 6. Assessment of the validity of the quasi-steady-state approximation for different radicals. The detailed kinetic scheme was used at 10% (v/v) MO and 150 °C.

ω6 + k11[R 2H]0 [R 2O2 ] =

secondary reactions involving peroxides are enhanced by MO addition. According to the detailed model, hydroperoxy (HO2) radicals are mainly produced by hydrogen abstraction from reactants (R1H and R2H) and n-dodecyl hydroperoxides (R1O2H). In line with previous findings,46 as the oxidation proceeds, the peroxy radicals tend to react more rapidly with the oxygenated species (e.g., R1O2H and MO) than with the unreacted n-dodecane (R1H), because the former frequently contains more reactive carbon−hydrogen bonds. Figure 4c illustrates the main MO consumption pathways in the case of 10% (v/v) MO in the fuel. According to the model, autoxidation of MO involves hydrogen abstraction on allylic carbons, yielding two radicals (species 34 and 37). The subsequent oxygen addition on these delocalized radicals results in the production of four isomeric R2O2-type peroxides (species 111, 113, 182, and 184) and hydroperoxides. Interestingly, the contribution of each corresponding pathway is similar, which is in line with existing chromatographic and nuclear magnetic resonance (NMR) measurements.47 Figure 6 illustrates that the quasi-steady-state approximation (QSSA) can be safely applied to R1 and R2 over the entire pre-induction period. Hydroperoxy

k 8[R1H]0

2α k12

[1 − exp( −k 8[R1H]0 t )] (5)

Similarly, applying QSSA to HO2, we obtain d[H2O2]/dt = ω3 + ω10 − ω5 = α. Applying QSSA to R1 and using eq 5, the differential equation for the alkylperoxy radical concentration is given by d[R1O2]/dt = 2α − k12[R1O2]2, which, according to the initial condition ([R1O2]0 = 0), solves

( )

tanh [R1O2 ] =

t τ12

k12τ12

(6)

with τ12 = (2αk12)−1/2. Equation 6 is similar to that of previous studies,35 and its asymptotic value is identical to the estimate obtained previously. However, this asymptotic value tends to underpredict R1O2 concentrations at t ≥ IP/1000. Neglecting the impact of termination (reaction R12), we obtain a linear temporal profile (of slope 2α) for R1O2, which is in better agreement with the detailed mechanism results. The hydroperoxy concentration is then deduced from the QSSA equation 6129

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Figure 7. Comparison of R1H, R1O2, R2O2, and HO2 temporal concentration profiles obtained using the analytical model [eqs 5, 6 (linearized form; see the text), 7, and 8] and calculations using the full detailed kinetic scheme.

termination on the HO2 concentration becomes significant. To obtain a more complete picture of the autoxidation process, n-dodecane consumption pathways were studied as a function of time (Figure 5) for different MO amounts in the binary fuel blend. Our analysis suggests that the IP can be broken down into three distinct phases, namely, initiation, propagation, and hydroxyl phases. In the initiation phase, reactions R1 and R6 are the main contributors to n-dodecane consumption. During this phase of the order of τHO2, the contributions of reactions R3 and R4 are very similar because HO2 and R1O2 are both produced by initiation (reaction R1) and fast O2 addition (reaction R2). In the presence of MO (90:10 blend), the contribution of reaction R3 increases as HO2 is produced by both n-dodecane (reaction R1) and MO (reaction R6) initiations. In the 90:10 blend condition, the contribution of reaction R6 supersedes that of reaction R1; therefore, MO peroxide contribution (reaction R8) is much higher than that of n-dodecane peroxide (reaction R4) for propagation. During the propagation phase, the contribution of R1O2, R2O2, HO2, and OH radicals becomes significant. Reaction R4 dominates R1H consumption, albeit less markedly in the presence of MO. The HO2 pathway (reaction R3) contribution reaches a maximum and then decreases, whereas the contribution of the OH route (reaction R14) increases during the entire phase, regardless of the initial MO blending ratio. In the presence of MO, the contribution of reaction R8 reaches a plateau during the second phase until it decays during the following phase because of the higher conversion of MO than that of n-dodecane. The hydroxyl phase is characterized by a sharp increase of OH route (reaction R14). ROP analyses reveal that, in the

Figure 8. Comparison of analytical IP (eq 10) with experimental data from this study (n-dodecane/MO blends) and literature for pure compounds2,10 and middle-distilate/FAME blends.4950 Binary mixtures of n-dodecane and MO are considered as surrogates for biofuels.

⎛ ⎛ t ⎞⎞ k5 [HO2 ] = ατHO2⎜⎜1 + tanh⎜ ⎟t ⎟⎟ k12τ12 ⎝ τ12 ⎠ ⎠ ⎝

(7)

with τHO2 = (k3[R1H]0 + k10[R2H]0)−1. Looking at Figure 7, we note that the analytical expression of the HO2 concentration appears valid for 10−6 ≤ t/IP ≤ 10−2, in line with the results from Figure 6. After a rapid buildup phase (time scale of τHO2), the HO2 concentration remains remarkably constant up to times of the order of τ12, at which the impact of R1O2 6130

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Figure 9. Impact of the MO content in the fuel on normalized sensitivity coefficients defined in the form of logarithmic derivatives.30 Calculations were carried out using the detailed kinetic scheme. R1H stands for n-dodecane, while R2H represents MO.

encompasses not only the classical oxidizability factor through γ but also the contribution of HO2 to fuel autoxidation. The analytical model was further tested against a range of literature experimental data for blends of FAMEs and several middle-distillate hydrocarbons, such as jet fuel or diesel fuel. From the results of Figure 8, it can be concluded that the analytical model of the binary surrogate mixture was able to correctly represent the promoting effect of FAME on induction over a wide range of FAME contents and operating temperatures. Furthermore, the predicted increase of zero concentration slope (d(IP)/d[FAME]0)[FAME]0 = 0 with the operating temperature is in agreement with previous findings for the impact of soybean methyl ester on JP8 jet fuel.51 However, it is important to keep in mind that the simple surrogate used in this study involving only one monounsaturated methyl ester is not able, in its current formulation, to reproduce the impact of the FAME composition (e.g., amount of polyunsaturated FAMEs) on the FAME-promoting effect. 3.4. Sensitivity Analysis. Sensitivity analyses of the IP allow for deeper insight into the impact of MO addition on the most important rate parameters. Figure 9 provides evidence of the need for accurate density and oxygen concentration estimates. In the absence of MO, the IP is mostly sensitive to n-dodecane initiation reaction R1. Adding MO results in an increase of sensitivity with respect to the R6 initiation reaction at the expense of reaction R1. Within the current set of kinetic parameters, IP is not sensitive to the parameters related to the R3 or R15 hydrogen-abstraction reactions. Although the sensitivities obtained from eq 10 are remarkably similar to those given by the detailed model, opposite trends are observed regarding the impact of MO addition on sensitivity with respect to the R4 propagation reaction. The kinetics of other propagation reactions is found to have a negligible impact on the IP under the present operating conditions.

absence of MO, OH radicals are mainly formed from hydroperoxides through reactions R12 and R13. However, in the presence of MO, R2O2H conversion to OH contributes to hydroxyl radical production as well. Note that, according to the detailed kinetic mechanism, the hydrogen peroxide decomposition pathway, namely, H2O2 → 2OH, remains negligible (albeit existing) compared to the former route (reactions R12 and R13). At the end of the induction period, the relative contribution of the OH route to n-dodecane consumption represents up to 20% (Figure 5). 3.3. Analytical IP Model. To overcome the shortcomings of empirical IP correlations, it is desirable to derive an IP analytical expression based on the previously validated detailed kinetic model. This approach has already been applied for gasphase reactive systems48 but, to our knowledge, not yet for liquid-phase oxidation kinetics. From these concentrations, the temporal evolution of R1H can be deduced, neglecting the direct contribution of initiation [R1H] = [R1H]0 exp( −(β + γt )t )

(8)

with β = k3ατHO2 + ((ω6 + k11[R2H]0((2α/k12)1/2))/[R1H]0]) and γ = k4α + (k3k5ατHO2/2k12τ12). Similarly, for R2H [R 2H] = [R 2H]0 exp( −(δ + εt )t )

(9)

with δ = k10ατHO2 + k9((ω6 + k11[R2H]0((2α/k12)1/2))/ k8[R1H]0]) and ε = k11α. An analytical IP equation is finally obtained by replacing the exponential terms of eqs 8 and 9 by their Taylor expansions up to an order of 2 and solving eq 3 for time

IP =

b+

b2 − 4ac 2a

(10)

where a = 1/2([R1H]0(β − 2γ) + (δ − 2ε)[R2H]0), b = β[R1H]0 + δ[R2H]0, and c + 0.05([R1H]0 + [R2H]0). On the basis of only the rate parameters of Table 2 generated by RMG, eq 10 is able to reproduce fairly well the impact of the MO content in the fuel as well as the effect of the operating temperature on the IP (see Figure 8). This analytical IP expression obtained from a detailed kinetic mechanism 2

2

4. CONCLUSION The purpose of this study was to gain an in-depth understanding of the impact of FAME addition on autoxidation kinetics. A detailed kinetic mechanism for the autoxidation of a n-dodecane/MO surrogate mixture was generated and validated 6131

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against original well-controlled accelerated oxidation experiments. Our pathway analyses reveal that HO2 and OH propagation steps as well as the duration of initiation and propagation phases are strongly affected by MO addition. On the basis of these analyses and the detailed mechanism, an analytical IP model was derived and validated against experimental data involving single-component and commercial biofuels. These results open up the use of bottom-up liquidphase oxidation modeling strategies for the in silico formulation of alternative fuels and the design of innovative fuel injection systems. However, the contribution of hydroxyl radical pathway would still need to be included in the analytical model to improve its predictivity, particularly at a high FAME content. Work on more realistic surrogates of FAME (i.e., mixtures of methyl esters present in commercial FAME and naturally occurring antioxidants) is ongoing by our team.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work benefited from fruitful discussions with Prof. William Green and Dr. Richard West on the RMG code, Dr. Laurie Starck on fuel oxidation stability, and Dr. Anthony Velghe on the Senkin reactor code. The authors also thank the French National Institute for Industrial Environment and Risks (INERIS) for lending the Rancimat apparatus as well as Romain Cherblanc and Pascal Hayrault from IFP Energies nouvelles (IFPEN) for conducting the experiments.

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APPENDIX The nomenclature and standard enthalpies of formation of main species involved in the detailed kinetic mechanism are provided in Table A1. Table A1. Nomenclature and Standard Enthalpies of Formation of Main Species Involved in the Detailed Kinetic Mechanism

NOMENCLATURE FAME = fatty acid methyl ester IP = induction period (h) MO = methyl oleate RMG = reaction mechanism generator code R1H = n-dodecane R2H = MO SLPH = standard liters per hour kint,i = intrinsic rate constant of the ith reaction, where n represents the reaction order [cm3(n−1) mol−1(n−1) s−1] (see section 2.2) F = diffusion factor (see section 2.2) Ai = pre-exponential factor of the ith reaction, where n represents the reaction order [cm3(n−1) mol−1(n−1) s−1] (see section 2.2) Ei = activation energy of the ith reaction (kcal/mol−1) (see section 2.2) R = universal gas constant (J mol−1 K−1) ROP = normalized rate of production (see section 3.2) a = analytical model factor (mol cm−3 s−2) (see section 3.3) b = analytical model factor (mol cm−3 s−1) (see section 3.3) c = analytical model factor (mol cm−3) (see section 3.3)

Greek Letters

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ASSOCIATED CONTENT

S Supporting Information *

νij = stoichiometric coefficient of the jth species in the ith monodirectional reaction ωi = reaction rate of of the ith reaction (mol cm−3 mol−1 s−1) (see section 3.2) α = sum of the rates of reactions 1 and 6 (mol cm−3 s−1) (see section 3.2) τ12 = characteristic time for R1O2 termination (s) (see section 3.2) τHO2 = characteristic time for hydrogen abstraction by HO2 (s) (see section 3.2) β = analytical model factor in [R1H] expression (s−1) (see section 3.3) γ = analytical model factor (s−2) (see section 3.3) δ = analytical model factor (s−1) (see section 3.3) ε = analytical model factor (s−2) (see section 3.3)

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