Understanding the Relationship between Cetane Number and the

Article pubs.acs.org/EF

Understanding the Relationship between Cetane Number and the Ignition Delay in Shock Tubes for Different Fuels Based on a Skeletal Primary Reference Fuel (n‑Hexadecane/Iso-cetane) Mechanism Weiwei Fan, Ming Jia,* Yachao Chang, and Maozhao Xie School of Energy and Power Engineering, Dalian University of Technology, Dalian, P.R. China S Supporting Information *

ABSTRACT: A new skeletal oxidation mechanism for the primary reference fuel (PRF) was established with a decoupling methodology. The mechanism is composed of n-hexadecane and iso-cetane submechanisms, containing 44 species and 139 reactions. Using the present mechanism, the relationship between cetane number and the ignition delay in shock tubes was investigated. First, based on the ignition delay data in shock tubes, the cetane number of various fuels was estimated using the present PRF mechanism and a weighted least-squares method. The prediction of cetane number investigated in this study primarily focused on the operating conditions of practical diesel engines (i.e., the equivalence ratio of 1.0 and pressures from 19− 80 atm), which encompass the cetane number from 15 to 100. Under the test operating conditions, the mean absolute deviation of the predicted cetane number is within 3.327. Furthermore, according the cetane number of different fuels, the ignition delays in shock tubes were reproduced by the present mechanism focusing on a wide range of equivalence ratios (0.5−3.0) and pressures (20−50 atm). The results indicated that the predicted IDs of alkanes were more accurate than those of other types of fuels and blended fuels because of the consistent molecular structure of the n-hexadecane/iso-cetane used in the present mechanism. Because of the compact size of the skeletal mechanism, its application can considerably reduce the computational time for 3D combustion simulations, especially for practical fuels with complicated compositions. alternative methods was proposed by Ryan et al.,17,18 which was further standardized as ASTM D689019 and is commonly referred to as the IQT method. The CN measured by this method is defined as the derived cetane number (DCN). In addition, there are several other methods to estimate CN, such as the blending cetane number method.20 In this method, the CN of the test fuel is determined by a linear interpolation of the CN between two fuels, where the CN of one fuel is known. However, there are still some uncertainties in the linear relationship of the CNs for different blending fuels.21,22 As an auxiliary method to experimental measurement, theoretical and numerical prediction methods of CN have attracted increasing attention. Ladommatos and Goacher23 employed a series of formulas to predict CN according to the physical properties of fuels. Yang et al.24 established a neural network to estimate the CNs of different fuels. Moreover, Ghosh and Jaffe16 used a simple composition-based model to predict the CNs of various diesel fuels. Recently, Creton et al.25 applied a quantitative structure property relationship method and a genetic algorithm to predict the CNs of various hydrocarbons with consideration of the molecular structure of different fuels. Meanwhile, the computational coupling of the threedimensional (3D) computational fluid dynamics (CFD) with the fuel mechanism is extremely important for the research and development of engines with advanced combustion strategies.26,27 To more deeply understand the oxidation character-

1. INTRODUCTION With increasing attention on energy and the environment, research on improving engine combustion efficiency and reducing exhaust emissions is continuously being explored. The combustion in internal combustion engines is an extremely complex process, which is affected by many factors, such as fuel/air mixing, in-cylinder pressure and heat release rate, the ignition characteristics of fuels, etc. One of the most important parameters for evaluating the ignition characteristics of fuels is the ignition delay (ID), which considerably impacts engine performance and emissions, especially for advanced combustion strategies.1 ID for different fuels has been investigated in shock tubes (ST),2,3 rapid compression machines (RCM),4 ignition quality testers (IQT),5−8 and practical engines.9,10 As an important indicator of ID, cetane number (CN) is directly related to the magnitude of ID.11−13 With the enhancement of computing capability, the development and application of chemical kinetic mechanisms are becoming an indispensable approach for ID studies. At present, the CNs of different fuels are mainly obtained from experimental measurements. Boerlage and Broeze14 proposed a method for CN measurement that involves using 1-hexadecene (cetene) and α-methyl naphthalene as reference fuels. This method rapidly evolved into the standard test method ASTM D61315 for the measurement of CN, in which n-hexadecane and 2,2,4,4,6,8,8-heptamethylnonane (iso-cetane) were used as reference fuels. Additional methods for the measurement of CN have been proposed in recent years because of the disadvantages of the ASTM D613 method,15 which include sample consumption, its time-consuming nature, and poor reproducibility.16 One of the most prominent © 2015 American Chemical Society

Received: December 16, 2014 Revised: April 2, 2015 Published: April 6, 2015 3413

DOI: 10.1021/ef5028185 Energy Fuels 2015, 29, 3413−3427

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Energy & Fuels

the following studies. Haylett et al.28 compared the ignition delay of different diesel fuels with the detailed n-hexadecane/ iso-cetane PRF mechanism of Westbrook et al.,40,44,45 and the overall trend of fuels with a lower aromatic fraction can be reproduced well. Moreover, Kawanabe and Ishiyama 46 established a reduced n-hexadecane/iso-cetane mechanism for application in CFD simulations. For the prediction of CN using theoretical or numerical models, most of the previous studies are based on mathematical algorithms16,23,24 and rarely involve ID or a kinetic mechanism. Thus, these methods become nearly independent systems because of a lack of association with the combustion characteristics of fuels. That said, because of the one-to-one correspondence between chemical kinetic mechanisms and fuels for ID simulation, a large database composed of several mechanisms is required to reproduce the ignition behaviors of various fuels. Although the global model proposed by Gowdagiri et al.30,31 can predict the ID characteristics of different fuels, it is essentially a fuel-specific model that adjusts the reaction rate coefficients to best fit the measurements for each fuel. Moreover, the investigation on the ID of practical fuels is very limited because of their complex composition and low vapor pressure. Until recently, Haylett et al.28 studied the ID of diesel fuels using a detailed n-hexadecane/iso-cetane PRF mechanism40,44,45 with a special emphasis on the prediction of ID. In this study, a new skeletal n-hexadecane/iso-cetane PRF mechanism is established using a decoupling methodology,42,47−49 which is capable of reproducing the ignition characteristics reasonably well. Then, the PRF mechanism is used to predict the CNs of different fuels based on the ID data in shock tubes in a wide range of operating conditions. In addition, according to the CNs of various fuels, this mechanism is applied to describe the ID behaviors of various fuels over an extensive range of temperature, pressure, and equivalence ratio. The purpose of this study is to understand the correlation between the ID data obtained from shock tubes and the CN for different fuels based on the n-hexadecane/iso-cetane PRF mechanism, as demonstrated in Figure 1. For the cases with

istics of various fuels, the establishment of chemical kinetic mechanism is becoming an active area of research because such knowledge can provide detailed information on ignition delay, species evolution, flame propagation, and extinction characteristics. Many methods have been proposed to construct chemical kinetic mechanism. Because ID is usually utilized as a benchmark for the development of chemical kinetic mechanisms,28 the ID characteristics of fuels can be well-understood using chemical kinetic mechanisms. Researchers at the Lawrence Livermore National Laboratory (LLNL)29 established a series of detailed chemical kinetic mechanisms to investigate the ignition and oxidation processes of various fuels over a wide range of operating conditions. However, investigations into the detailed mechanisms require a substantial amount of computing resources and time. Therefore, it is difficult to integrate detailed chemical mechanisms with the CFD code for multidimensional combustion simulations. To solve this problem, further simplifications to the detailed mechanisms are required. Meanwhile, it is still very difficult to construct a detailed or reduced mechanism for practical fuels because of their complicated compositions. Recently, Gowdagiri et al.30,31 improved a global reaction model32 to describe the ID characteristics of various practical fuels, and a thorough analysis of the ignition delay characteristics was conducted. As an important surrogate fuel for practical fuels, the primary reference fuel (PRF) has attracted progressively more attention. In general, the n-heptane/iso-octane PRF is employed to evaluate the octane number (ON) of fuels, whereas the nhexadecane/iso-cetane PRF is used to estimate the fuel CN. The previously suggested mechanisms for n-hexadecane and iso-cetane are listed in Table 1. A detailed n-hexadecane/isocetane PRF mechanism was established by Westbrook et al.40,44,45 based on a hierarchical approach. The detailed nhexadecane/iso-cetane PRF mechanism served as the basis for Table 1. Previous Mechanisms for n-Hexadecane or isoCetane mechanism

scale

authors and literatures

species

reaction

nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane nhexadecane iso-cetane

detailed

Chevalier et al.33

1200

7000

simple kinetic model reduced

Khorasheh and Gray34 Habik et al.35

−

−

13

22

51

282

iso-cetane

semidetailed

36

semidetailed

Marchese et al.

detailed

Fournet et al.37

265

1787

detailed

Dagaut38

242

2801

semidetailed

Ranzi et al.39

231

5591

2116

8130

40

detailed

Westbrook et al.

detailed

Basevich et al.41

−

−

skeletal

Chang et al.42

36

128

reduced

Poon et al.43

49

97

detailed

Oehlschlaeger et al.44 Ranzi et al.39

614

4487

231

5591

Figure 1. Relationship between ID and CN.

available ID data obtained from shock tubes, the CN of the fuel can be predicted. In contrast, because the CN of a fuel is known, the ID characteristics of the fuel can be reproduced with the present n-hexadecane/iso-cetane PRF mechanism by adjusting the proportion of n-hexadecane and iso-cetane.

2. SKELETAL N-HEXADECANE/ISO-CETANE PRF MECHANISM 2.1. Development of the Mechanism. The PRF mechanism developed in this study is composed of nhexadecane and iso-cetane submechanisms. Although the CN of α-methyl naphthalene is 0, iso-cetane is still selected as a reference fuel to estimate CN because of its low toxicity compared to that of α-methyl naphthalene in practical 3414


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Figure 2. Ignition delay time of an iso-cetane/air mixture in a shock tube at (a) ϕ = 0.5, (b) ϕ = 1.0, and (c) ϕ = 1.5 (symbols are experimental data from Oehlschaeger et al.;44 solid lines are the computational data of the present mechanism; dashed lines are the computational data of the detailed mechanism44).

applications.21 Iso-cetane is used in the standard test methods of ASTM D61315 and ASTM D6890,19 and so the use of isocetane in this study can lead to consistent predictions of CN. The n-hexadecane/iso-cetane mechanism is established based on a decoupling methodology. The detailed description of the decoupling methodology has been reported in our previous works.42,47−49 For this reason, only a brief introduction is presented herein. In accordance with the decoupling methodology, the skeletal n-hexadecane/iso-cetane PRF mechanism can be divided into three parts. Taking iso-cetane as an example, the submechanism of small molecules H2/CO/C1 and the submechanism of large molecules are constructed in different ways. The submechanism of H2/CO/C150,51 is considered in detail in the decoupling methodology. The submechanism of large molecules is constructed in an extremely simple way for describing the low-temperature ignition and high-temperature decomposition pathways of fuel molecules. As a transition submechanism, a semidetailed C2−C3 submechanism52 is employed. The final iso-cetane mechanism contains 44 species and 139 reactions. For the submechanism of C4−C16, only the representative species are retained to reproduce the ignition characteristics of iso-cetane, which can considerably reduce the number of species in the final mechanism. In contrast, a detailed submechanism of H2/CO/C1 is introduced to accurately predict the heat release rate, flame speed, and emissions characteristics. Thus, the final mechanism is capable of reproducing the ignition and combustion characteristics in various reactors under a wide range of operating conditions.

The reactions involving large-molecules in the iso-cetane (iC16H34) mechanism are established as follows: iC16H34 + O2 = iC16H33 + HO2

(R1)

iC16H33 + O2 = iC16H32O2

(R2)

iC16H33O2 = iC16H32OOH

(R3)

iC16H32OOH + O2 = iO2 C16H32OOH

(R4)

iO2 C16H32OOH → iC16KET + OH

(R5)

iC16KET → C6H13CO + CH 2O + OH + 4C2H4

(R6)

C6H13CO + O2 → C3H 7 + C3H5 + CO + HO2

(R7)

iC16H34 + OH → iC16H33 + H 2O

(R8)

iC16H33 + O2 = iC16H32 + HO2

(R9)

iC16H32 + O2 → 2C3H 7 + C3H6 + C2H4 + C3H5 + CH 2O + HCO

(R10)

iC16H34 + HO2 → iC16H33 + H 2O2

(R11)

iC16H33 → 2C3H 7 + C3H6 + C3H5 + 2C2H4

(R12)

In the initial stage, the fuel is mainly consumed by the H atom abstraction reactions R1, R8, and R11. At low temperatures, subsequent ketohydroperoxide formation reactions occur, including two oxygen addition reactions, an isomerization reaction, and an OH releasing reaction (i.e., 3415


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Energy & Fuels reactions R2, R3, R4, and R5, respectively). The ketohydroperoxide formed in reaction R5 is decomposed by reactions R6 and R7. The high-temperature reactions are also considered in the present mechanism, such as alkyl radical decompositions, and are lumped into reaction R12. Olefin decompositions and the addition of radical species to olefins are lumped into reaction R10, and the alkyl radical isomerization reactions are lumped into reaction R9. At the core of the iso-cetane mechanism, the H2/CO/C1 submechanism is based on a detailed methanol mechanism50,51 as used in previous studies.42,47−49 The skeletal n-hexadecane mechanism established by Chang et al.42 is chosen in this study, in which the overall structure is consistent with the present isocetane mechanism. To better reproduce the experimental data in shock tubes (ST) and jet-stirred reactors (JSR), the reaction rate constants of the large-molecule reactions have been modified in the present study. 2.2. Validation of the Mechanism. CHEMKIN-PRO is used for the simulations in this and subsequent sections if not specifically mentioned. Because n-hexadecane has been wellverified by Chang et al.,42 only a brief comparison of the present mechanism and the mechanism of Chang et al.42 are presented herein. 2.2.1. Validations in a Shock Tube. Oehlschaeger et al.44 measured the ignition delay time of an iso-cetane/air mixture with temperatures (T) between 878 and 1397 K at pressures (p) between 10 and 40 atm and equivalence ratios (ϕ) from 0.5 to 1.5. Comparisons between the experimental data and the simulation results are shown in Figure 2. As seen, the present mechanism accurately reproduces the ignition characteristics of iso-cetane over the investigated conditions, except for slightly overpredicting the ignition delay time for temperatures below 1000 K at p = 40 atm and ϕ = 1.5 (Figure 2c). Moreover, the increase of pressure leading to a decreased ignition delay time is also well-captured by the present mechanism. Compared to the detailed mechanism, the present mechanism illustrates less pronounced negative temperature coefficient (NTC) behavior, especially at low equivalence ratios and low pressures, as shown in Figure 2a. When the equivalence ratio is higher than 0.5, the difference in the ignition delay in the NTC region between the two mechanisms becomes smaller. It can also be seen from Figure 2 that at high temperatures, the predicted ignition delays from the present mechanism are in better agreement with the measurements than those from the detailed mechanism. However, because of the lack of experimental data in the low-temperature and NTC regions, further validation of the isocetane mechanism is still needed in future work. Assad et al.53 measured the ignition delay time of an nhexadecane/air mixture with temperatures between 1238 and 1792 K at a pressure of 4 atm and an equivalence ratio of 1.0. As seen in Figure 3, the predictions from the present nhexadecane mechanism is consistent with those of the Chang et al. mechanism42 at high temperatures. Both mechanisms show good agreement with the experimental data, except for slightly underestimating the ignition delay at temperatures below 1275 K. The predicted ignition behavior in the NTC region is different between the two mechanisms because the H atom abstraction reactions are slightly modified in the present mechanism to better reproduce the ignition characteristics of various fuels according to their CNs. For all the cases tested in this section, the definition of ignition delay is based on the evolution of OH concentration, which is consistent with that used in the experiments.44,53 In

Figure 3. Ignition delay time of an n-hexadecane/air mixture in a shock tube at ϕ = 1.0 and p = 4 atm (symbols are experimental data from Assad et al.;53 solid lines are the computational data of the present mechanism; dashed lines are the computational data of the Chang et al. mechanism42).

fact, it is found that the different definitions based on the histories of temperature, pressure, and OH concentrations are identical, as shown in Figure 4.

Figure 4. An example to illustrate the definition of ignition time.

2.2.2. Validations in a Jet-Stirred Reactor. The concentrations of the reactants and final products for the oxidation of iso-cetane were studied by Dagaut and Hadj-Ali54 in a JSR. The experiment was conducted at p = 10 atm, ϕ = 0.5−2.0, T = 770−1070 K, and with a residence time of 1 s. The concentrations of iso-cetane, CO2, CO, and H2O from the simulations and measurements are compared in Figure 5. As seen, the consumption of iso-cetane (HMN) is well-reproduced for all of the operating conditions. The formation of CO is also satisfactorily predicted at ϕ = 0.5 and 1.0. However, the concentrations of CO are overpredicted at ϕ = 2.0, which may be caused by the lack of the initial fuel decomposition reactions in the present mechanism; however, the overall trend is captured reasonably well. The final product, CO2, is wellreproduced at ϕ = 1.0 and 2.0 and only a slight underestimation can be observed at ϕ = 0.5 (Figure 5a). For H2O, the mechanism slightly overpredicts its concentration at T > 900 K, although it is well-modeled at T < 900 K. It can also be seen from Figure 5 that the predicted concentrations of the major species using the present mechanism are nearly consistent with those of the detailed mechanism.44 In particular, the predicted CO and H2O concentrations at low temperatures using the present mechanism agree better with the measurements than 3416


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Figure 5. Concentrations of major species of iso-cetane/O2/N2 in a jet-stirred reactor at (a) ϕ = 0.5, (b) ϕ = 1.0, and (c) ϕ = 2.0 (symbols are experimental data;54 solid lines are the computational data of the present mechanism; dashed lines are the computational data of the detailed chemical kinetic mechanism44).

those using the detailed mechanism. This could be attributed to the modification of the rate constants of the low-temperature reactions in this study. Overall, from the above validations in both the shock tube and jet-stirred reactor, it can be concluded that the nhexadecane and iso-cetane mechanism developed in this study is capable of satisfactorily reproducing the ignition and oxidation behaviors of n-hexadecane and iso-cetane for a wide range of temperatures, pressures, and equivalence ratios. Thus, the present n-hexadecane/iso-cetane mechanism is further utilized to predict the CNs of different fuels in the following section.

Moreover, based on the available measured ID data in shock tubes, the present method of using the skeletal n-hexadecane/ iso-cetane PRF mechanism is similar to the standard test methods of ASTM D61315 or ASTM D689019 for the determination of CN. In all of these methods, the fuel CN is measured by comparing the ID data of the test fuel with the ID data of the PRF. Because the CNs of various fuels are primarily measured in engines or IQT with liquid fuel injection, and because the autoignition of a fuel/air mixture first occurs at stoichiometric conditions,56,57 the determination of CN in this study is primarily based on the ignition data in shock tubes at ϕ = 1.0. 3.1. Computational Methodology. The methodology for the quantitative prediction of CN used in this section is outlined in the following steps. First, CHEMKIN-PRO is used to calculate the ID of various proportions of n-hexadecane/isocetane over a wide temperature range, and the operating conditions are kept consistent with those used in the experiment for the measurement of the ID in shock tubes. Then, a weighted least-squares algorithm is used to determine the degree of similarity between the measured ID and the predicted ID of different n-hexadecane/iso-cetane mixtures based on a Mathworks Matlab program. Finally, the nhexadecane/iso-cetane mixture with the highest degree of similarity is used to define the predicted CN for the test fuel according to eq 1,21,58 which is also employed in ASTM D613.15 The highest degree of similarity refers to the smallest global discrepancy between the measured ignition delays and

3. PREDICTION OF CETANE NUMBER From their experiments in a constant volume vessel, Rabl et al.55 found a close association between ID and CN at a constant ambient pressure using a mixture with different proportions of n-hexadecane and iso-cetane. Recently, Gowdagiri and Oehlschlaeger30 satisfactorily reproduced the variation of the ID in shock tubes according to fuel CN for various aliphatic compounds using a global reduced model. On the basis of the above findings, this study attempts to apply the skeletal nhexadecane/iso-cetane PRF mechanism to determine the CN of various fuels when the ID data in shock tubes are available and to reproduce the ID characteristics under different operating conditions for fuels with known CNs. In the present method, the predicted CN directly relates to the ID characteristics, which makes CN a simple and intuitive criterion to evaluate the ignition behaviors of different fuels. 3417


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deviation62,63 of the ID is used in this study, which can effectively decrease the deviation caused by the uneven distribution of the ID data. 3.2. Procedure for the Prediction of CN. The experimental data of ignition delay in shock tubes for Jet-A at ϕ = 1 and p = 20 atm60 is chosen as an example to illustrate the procedure for the prediction of CN. These test data are shown in Table 2. First, the IDs of n-hexadecane/iso-cetane/air

the predictions from the mixture by sweeping the CN of the mixture from 15 to 100 with a step size of 1. CN = vol%n ‐ hexadecane + 0.15 × vol%iso ‐ cetane

(1)

where vol % is the volume fraction. In this study, the CN is estimated based on the ignition data over a wide temperature range because the ignition behaviors in both the low-temperature and NTC regions can affect the measured CN.59,60 Moreover, the ignition behaviors in the lowand high-temperature regions are closely associated with the ignition characteristics of fuels in the NTC region. For some fuels tested in this study, very limited ignition data over a small temperature range are available. Therefore, to apply the present method to estimate the CNs of these fuels, the ignition data over a relatively large temperature range are considered. In this section, the ID data for different temperatures were divided into three regimes, as suggested by Gowdagiri and Oehlschlaeger:30 low-temperature, NTC, and high-temperature. By comparing the predicted ID in shock tubes for the fuels with different CNs, Westbrook et al.40 found that the NTC regime was primarily located in the temperature range of 750−900 K, which is similar to the experimental results of Fieweger et al.59 An equivalent definition of the temperature range for NTC was used by Bogin et al.61 and Wang and Oehlschlaeger.60 Thus, a unified NTC boundary of 750−900 K is employed in this study. Moreover, Wang and Oehlschlaeger60 concluded that the largest difference of the ID among various fuels was in the lowtemperature and NTC regimes. Therefore, different weighted coefficients for the low-temperature, NTC, and high-temperature regimes are used in this study to more accurately predict the CNs of various fuels. When the temperature is higher than 900 K or less than 750 K, the weight coefficient is 0.1. For the temperatures 750−900 K, the weighted coefficient is designated as 0.8. Equation 2 is the weighted least-squares formula used in this study. It must be noted that the weighting coefficients are empirical, which were determined based on the accuracy of the present method for different fuels over a wide range of operating conditions. n ⎧ ∑ 1 (τ m − τi p)2 ⎪ Nj ,1 = 0.1 × 1 i (Ti > 900 K) ⎪ n1 ⎪ n ⎪ ∑1 2 (τim − τi p)2 = × N 0.8 ⎪ j ,2 n2 ⎪ ⎪ (750 K < Ti < 900 K) ⎪ n3 ⎨ m p 2 ⎪ N = 0.1 × ∑1 (τi − τi ) (Ti < 750 K) ⎪ j ,3 n3 ⎪ ⎪ ⎪ φj = ∑ (Nj ,1 + Nj ,2 + Nj ,3) ⎪ (j = 1, 2, ⋯ , r ) ⎪ ⎪ φ = min(φ ) (j = 1, 2, ⋯ , r ) j ⎩ min

Table 2. Test Data of Jet-A/air60 at ϕ = 1 and p = 20 atm τp (μs)

temperature (K)

τm (μs)

n-hexadecane/ iso-cetane (15.0/85.0 vol %, CN = 28)

671.41 673.01 686.07 704.88 721.97 745.76 770.13 809.82 816.83 822.77 832.46 836.15 855.11 883.14 904.32 928.05 941.96 949.87 959.53 991.49 1002.02 1010.98 1029.37 1040.74 1062.23 1088.80 1108.00 1132.40 1153.17 1167.45 1202.19 1209.90 1212.49 1220.34 1228.29

8136.69 7492.50 4859.23 2901.93 2219.57 1733.03 1882.03 1769.14 1733.03 1733.03 2174.27 1733.03 2043.84 2002.13 1921.24 1769.14 1439.49 1271.97 1500.10 1101.01 1013.84 895.86 744.12 569.15 472.74 384.66 319.50 229.72 179.36 155.25 83.63 88.97 94.65 68.05 70.92

11509.82 11173.43 8941.44 6869.68 5706.06 4733.54 4173.58 3681.93 3613.53 3554.96 3454.90 3414.54 3181.73 2762.26 2409.95 2003.52 1766.22 1633.42 1474.90 1005.12 875.46 776.41 604.23 517.29 387.08 274.40 216.95 163.92 131.26 113.62 82.14 76.79 75.10 70.27 65.81



7572.58 7326.26 5678.30 4131.11 3251.00 2509.11 2085.21 1768.97 1742.10 1724.20 1703.45 1698.06 1689.27 1722.91 1766.85 1779.10 1728.06 1674.09 1587.30 1217.22 1091.04 988.39 797.11 694.19 531.60 381.96 302.24 226.59 179.16 153.38 107.44 99.69 97.25 90.31 83.91

5707.94 5512.25 4198.09 2958.49 2255.07 1671.45 1352.59 1152.58 1143.64 1140.96 1145.69 1150.36 1198.54 1342.51 1493.59 1622.61 1623.50 1595.31 1537.53 1240.30 1129.09 1035.71 854.95 754.04 589.29 431.66 345.14 261.28 207.72 178.29 125.17 116.13 113.27 105.13 97.62

mixtures in a shock tube are calculated at given volume fractions of n-hexadecane and iso-cetane using CHEMKINPRO under the operating conditions of ϕ = 1 and p = 20 atm for each temperature measured in the experiment. Second, the predicted IDs are compared with the experimental data of JetA60 based on eq 2 for a specific n-hexadecane/iso-cetane mixture. The n-hexadecane/iso-cetane mixture with the minimum total deviations (i.e., φmin in eq 2) is chosen as the final surrogate fuel to determine the CN of Jet-A. Figure 6 shows an example of the prediction process. As seen, the prediction of CN48 (i.e., 38.8/61.2 vol % nhexadecane/iso-cetane) agrees better with the experimental

(2)

where ni represents the total number of experimental data in each temperature regime, Nj,i is the mean square deviation for the jth n-hexadecane/iso-cetane mixture in the ith temperature regime, τmi is the measured ignition delay, τpi is the predicted ignition delay, Ti is temperature, and φj is the sum of the Nj,i. φmin in eq 2 is the minimum of φj, which represents the deviation of the ID between the measured and predicted values. Instead of the sum of the variance, the sum of the mean square 3418


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data than either CN68 or CN28 over the entire temperature range, especially in the NTC regime. On the basis of eq 2, the deviation (φj) of CN48 is much smaller than that of CN68 and CN28, and the predicted ID using CN48 is closest to the experimental data. Therefore, the final predicted CN of Jet-A is 48, which is very close to the measured CN of 47.1.64,65 3.3. Extensive Validations. Using the above methods, the predicted CNs of various pure fuels, mixtures, surrogate fuels, and practical fuels are extensively validated in this section. Detailed information on the test data are listed in Table 3, in which the average predicted CNs are calculated at ϕ = 1.0 and various pressures. In Table 3, the measured CNs of the pure fuels are mainly taken from the experimental results of Puckett and Caudle,66 Hum and Smith,67 and Smagala et al.68 The CNs of the blends obtained from experimental data are calculated using eq 3 based on a linear interpolation of the CN of each individual fuel in the blend.58,90,91

Figure 6. Comparison between the predicted and measured ID for JetA/air at ϕ = 1 and p = 20 atm (symbols are experimental data;60 lines are computational data of the present mechanism).

Table 3. Detailed Data for the Predicted and Measured CNs prediction ϕ

p (atm)

T (K)

data source

predicted CN

average CNa

measured CN and data sources

n-heptane/air

1.0

13

666−1297

69,70

48

56.25

5320,71/ 53.772/ 53.819,21/ 54.173/ 5666,74

n-decane/air

1.0 1.0 1.0 0.67

19 30 40 50

744−1002 736−1066 749−1139 733−916

47,70 47,70 59,70 69

57 56 64 98

n-dodecane/air

1.0 1.0 1.0 2.0 2.0 0.5

13 50 80 13 50 20

697−1300 654−948 791−1005 708−1293 707−912 746−1179

69 69 79 69 69 80

67 82 82 64 76 89

methylcyclohexane/air

1.0 1.0

20 50

726−1137 816−1157

80 82

98 19

1.0 1.0

20 20

688−1249 686−1213

84 84

56 49

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

40 40 40 30 40c 50 40 40 55

727−1093 735−1059 739−1109 710−1097 709−1097 709−1088 847−1016 711−1182 858−1137

59 59 59 85 85 85 86 86 87

32 20 16 19 16 16 44 56 16

0.5 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0

20 11 20 20 20 20 20 20 20

706−1305 743−1311 673−1233 661−1188 681−1326 653−1250 671−1042 665−1009 651−1024

60 60 60 60 60 89 89 89 89

74 38 48 40 33 44 46 34 48

fuel and composition

2,6,10-trimethyl dodecane/air n-hexadecane/iso-cetane/air 45.9/54.1 mol % ON60/air ON80/air ON90/air n-heptane/toluene/air 35/65 vol %

60/40 vol % 90/10 vol % iso-octane/toluene/n-heptane/air 63/20/17 vol % Jet-A/air

Sasol IPK/air JP-5/air JP-8/air LPA-142/air LPA-210/air

6564,75/ 65.521/ 67.276/ 7677,78/ 76.967/ 7874 77

72.921/ 7471/ 7875,81/ 8066,77/ 87.667 2066,67,77/ 2283/ 23.519/ 2420/ 24.421 5868/ 59.184 54.02 29.6 20.8 16.4 16.35

31.6 49.9 16.08 47.164,65 43

31.388 42.330/ CI∼39−4889 49.365/ CI∼42−4789 40.689 50.589

absolute deviationb −8 1 0 8 21.1 −9.9 5.1 5.1 −12.9 −0.9 1.4 10.4 −1 −2 −5.02 2.4 −0.8 −0.4 2.65 −0.35 −0.35 12.4 6.1 −0.08 26.9 −9.1 0.9 −7.1 1.7 1.7 −3.3 −6.6 −2.5

On the basis of only the predictions at ϕ = 1.0. bCalculated using eq 4 based on the measured CNs from the experiment.66−68 cID data are scaled from p = 10, 30, and 50 atm by a pressure scaling of p−1.059, according to the work of Herzler et al.85 a

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Energy & Fuels CNblend =

∑ (Xi × CNi)

and pressures of 19−80 atm, except for the blend composed of 60/40 vol % n-heptane/toluene (Figure 8). It should be noted that the predicted CN by the present method is closely associated with the measured ID data in shock tubes. The larger deviation for the above n-heptane/toluene blend may be caused by an insufficient quantity of data because only three ID data at approximately 900 K are available for this blend. In addition, the uncertainty in the linear approximation of the CN for blends (i.e., eq 3) is also a possible reason for the discrepancy. To evaluate the overall deviation of the prediction, the mean absolute deviation is defined as

(3)

where CNblend is the CN of the blend, Xi is the volume fraction of the ith pure fuel in the mixture, and CNi is the CN of the ith individual fuel obtained from the experiment. We note that eq 3 is essentially identical to eq 1.58,90 As seen from Table 3, the predicted average CNs are very close to the measured values for various test fuels. It can also be found that the predicted CNs are affected considerably by the operating conditions (i.e., equivalence ratio and pressure). In particular, the predicted CNs based on the IDs at ϕ = 1.0 are better than those at lean or rich conditions. Additionally, the predicted CNs agree better with the measurements when the pressure is higher than 20 atm. This may be because the CNs of different fuels are mainly measured under engine-relevant conditions with high ambient pressures and stoichiometric fuel/air mixtures. Thus, for the predictions of CN based on the measurements of ID in shock tubes, special attention should be focused on ϕ = 1.0 and high pressure conditions. To evaluate the prediction error of the present method, the absolute deviation is calculated by eq 4 as

AD = CNP − CNM

MAD =

∑ |AD| × 100 n

(5)

where MAD is the mean absolute deviation, and ∑|AD| is the sum of the absolute deviation over n test cases. The mean absolute deviation at ϕ = 1.0 and pressures of 19− 80 atm is 3.327. This value is much lower than that for the entire test operating conditions, where a mean absolute deviation of 5.180 is determined. This finding further confirms the validity of using the ID data under the conditions of ϕ = 1.0 and p = 19−80 atm to estimate CN. It is worth noting that the deviation of the predicted CN is affected by many factors, such as the molecular structure of the fuel,58 the reliability of the measured ID data in shock tubes, the nonlinear relationship between ID and CN,12 and the uncertainty of the derived CN for blends using eq 3.58,90,91 In addition, Santana et al.92 found that the measured CN from different sources can dramatically affect the absolute deviation of the predicted CN. Overall, as an intuitive approach to establish the relationship between CN and the ID in shock tubes, the deviations of the present method to estimate the CN of various fuels can be considered to be satisfactory.

(4)

where CNP and CNM are the predicted and measured CN, respectively, and AD is the absolute deviation. It must be noted that the uncertainties in the measured IDs in shock tubes could also affect the final predicted CNs. For example, if the uncertainties in the measured IDs of methylcyclohexane82 are considered, the absolute deviation of the predicted CN will increase to ±4. Therefore, if the uncertainties of the experimental data are reduced effectively, the accuracy of the present method can be improved further. As seen in Figure 7, the absolute deviations of the predicted CN are mostly within ±10, and the deviation slightly increases

4. PREDICTION OF IGNITION DELAY The detailed data tested in this section are listed in Table 4, in which the CN of the blend fuel is calculated by eq 3. It has been widely recognized the IDs of a fuel in shock tubes are closely associated with its CN.55,58 In this section, the nhexadecane/iso-cetane PRF mechanism is applied to reproduce the ID behaviors of different fuels under a wide range of operating conditions. First, the n-hexadecane and iso-cetane blend is used as a surrogate for the test fuel, and the composition of the two components are determined according to the CN of the fuel based on eq 1. Then, by application of the n-hexadecane/iso-cetane PRF mechanism developed in this study, the IDs of the test fuel are estimated and compared with the measured values in shock tubes. Various fuels have been studied, including pure fuels, blend fuels, and practical fuels. 4.1. Pure Fuel. On the basis of the present n-hexadecane and iso-cetane mechanism, the computational IDs of normal alkanes, including n-heptane, n-decane, and n-dodecane, in air are compared with the experimental data in shock tubes under various pressures and equivalence ratios (Figure 9). Overall, good agreement between the predictions and measurements can be observed for the three test fuels under a wide range of operating conditions. The increase of equivalence ratio and pressure leading to a decrease of the ID is satisfactorily reproduced by the present mechanism. However, the model overpredicts the ID of n-heptane at T < 1000 K and ϕ = 0.5 (Figure 9b), and a small discrepancy exists for n-decane at T < 900 K and low equivalence ratios (Figure

Figure 7. Comparisons of the predicted and measured CNs.

with increased CN. Further illustration of the absolute deviations under various pressures and equivalence ratios are presented in Figure 8. In Figure 8, the top x axis presents three intervals of equivalence ratio, and the bottom x axis represents the variation of pressure in each interval of equivalence ratio. As seen, the deviation significantly increases when the equivalence ratio is less than 1.0. When the equivalence ratio is greater than 1.0, the deviation is very unstable. Moreover, for all of the cases with pressures less than 19 atm, the absolute deviation is clearly larger than that at higher pressures. Overall, satisfactory absolute deviations are obtained for various fuels at ϕ = 1.0 3420


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Energy & Fuels

Figure 8. Data analysis of the absolute deviation of the present method.

Table 4. Detailed Data Used for the Prediction of ID operating conditions for ID measurements in shock tubes fuel and composition n-heptane/air

5666,74

n-decane/air

76.967

n-dodecane/air

87.667

methylcyclohexane/air dimethyl ether/air 2,6,10-trimethyl dodecane/air ON60/air ON80/air ON90/air n-heptane/toluene/air 35/65 vol % 60/40 vol % 90/10 vol % Jet-A/air

Sasol IPK/air JP-5/air JP-8/air LPA-142/air LPA-210/air a

measured CN and sources

2066,67,77 7893 5868 29.6 20.8 16.4 16.35 31.6 49.9 47.164,65

31.388 42.330 49.365 40.689 50.589

ϕ

p (atm)

T (K)

data source

0.5 0.5 1.0 1.0 1.0 1.0 2.0 3.0 0.5 0.67 1.0 1.0 1.0 2.0 2.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0

13 40 13 19 30 40 40 13 13 50 13 50 80 13 50 20 20 50 13 40 20 40 40 40 40a 40 40 20 8 11 20 20 20 20 20 20 20

697−1187 778−1098 666−1297 744−1002 736−1066 749−1139 877−1048 697−1281 689−1302 733−916 697−1300 654−948 791−1005 708−1293 707−912 746−1179 726−1137 816−1157 656−1281 664−1161 688−1249 727−1093 735−1059 739−1109 709−1141 847−1016 711−1182 706−1305 847−1327 743−1311 673−1233 661−1188 681−1326 653−1250 671−1042 665−1009 651−1024

70 70 69,70 47,70 47,70 59,70 70 70 69 69 69 69 79 69 69 80 80 82 69 69 84 59 59 59 85 86 86 60 60 60 60 60 60 89 89 89 89

ID data are scaled by τ ∝ p−1.059, and the pressure exponent −1.059 is taken from the work of Herzler et al.85

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Energy & Fuels

Figure 9. Predicted and measured IDs in shock tubes at different operating conditions for (a) n-heptane/air, (b) n-heptane/air, (c) n-heptane/air, (d) n-decane/air, and (e) n-dodecane/air (symbols are experimental data;69,70,79,80 lines are computational data of the present mechanism).

hexane, and dimethyl ether in air, which represent branched alkane, cycloalkane, and ether, respectively. As seen, the present mechanism reproduces the ID data of methylcyclohexane and dimethyl ether very well. This is mainly because the ignition behaviors of MCH94 and DME69 are similar to alkanes. For 2,6,10-trimethyl dodecane, however, there are certain discrepancies between the predictions and measurements in the NTC regime. Similar simulation results have also been obtained by Won et al.84 using a detailed mechanism. One of the possible reasons for this behavior is that the ignition behavior of 2,6,10trimethyl dodecane is different from other alkanes. Therefore, additional experimental and theoretical studies should be performed in the future. 4.2. Blend Fuel. The predicted IDs using the present mechanism are compared with the experimental data of nheptane/iso-octane/air and n-heptane/toluene/air with various

9d). The development and validation of the present mechanism primarily focus on engine-relevant conditions (i.e., high pressure and low-to-high temperature). Thus, the deficiency of the present mechanism for conditions of low pressure and low equivalence ratio may originate from this feature. The discrepancies between the measurements and predictions for low equivalence ratios and low pressures could be caused by the simplified reaction pathways in the C4−C16 submechanism, which were implemented to maintain the compact size of the present mechanism. In addition, the ignition delay at low pressures (

Understanding the Relationship between Cetane Number and the

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