Letter pubs.acs.org/JPCL
Quantum Tunneling Affects Engine Performance Sibendu Som,† Wei Liu,‡ Dingyu D. Y. Zhou,§ Gina M. Magnotti,†,∥ Raghu Sivaramakrishnan,‡ Douglas E. Longman,† Rex T. Skodje,§ and Michael J. Davis*,‡ †
Energy Systems Division and ‡Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States § Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309, United States ∥ Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States S Supporting Information *
ABSTRACT: We study the role of individual reaction rates on engine performance, with an emphasis on the contribution of quantum tunneling. It is demonstrated that the effect of quantum tunneling corrections for the reaction HO2 + HO2 = H2O2 + O2 can have a noticeable impact on the performance of a high-fidelity model of a compression-ignition (e.g., diesel) engine, and that an accurate prediction of ignition delay time for the engine model requires an accurate estimation of the tunneling correction for this reaction. The threedimensional model includes detailed descriptions of the chemistry of a surrogate for a biodiesel fuel, as well as all the features of the engine, such as the liquid fuel spray and turbulence. This study is part of a larger investigation of how the features of the dynamics and potential energy surfaces of key reactions, as well as their reaction rate uncertainties, affect engine performance, and results in these directions are also presented here. SECTION: Kinetics and Dynamics
I
reactions whose reaction-rate uncertainties are liable to have an impact on engine performance and emissions, with a further focus on another set of reactions where tunneling is liable to be important in the same temperature range studied here, where significant chemistry starts between 800 and 1200 K. Rate coefficients for all the key reactions influence the outcome of the engine simulations here. Depending on the type of reaction, uncertainties for the rate coefficients can be due to numerous sources. However, the focus of the present work deals with the subtleties involved in the theoretical prediction of a rate coefficient for one particular reaction, the HO2 + HO2 reaction, that has a big influence on the engine simulations (see Figure 4 below). The uncertainty in the experimental estimate for this reaction has driven a state-of-the-art theoretical kinetics treatment of this system.7 Among the various dynamical features, tunneling was shown to play an important role in the theoretical predictions for the rate coefficient for this reaction. Given the importance of this reaction in the engine simulations, the first question that arose in our minds was what would be the effect of including/excluding tunneling within the theoretical calculations and how this seemingly esoteric concept of tunneling impacts the outcome of a realistic simulation of a practical device, an engine. As noted above, at the end of this Letter we study the effect of several other reactions that are
t is now possible to include complex chemistry in the modeling of compression-ignition engines.1−3 Reaction mechanisms as large as 2000 reactions with 200 species can be accommodated relatively routinely.4 A recent investigation by several of us studied a mechanism for a biodiesel surrogate that included 145 species and 869 reactions, which was a reduced version of a mechanism for a blend of methyl butanoate (MB) and n-heptane that had 661 species and 3019 reactions.4 In the course of that study, we became interested in how the ignition process would be affected by changes in the rate coefficients, which are uncertain by as much as an order of magnitude (for example, ref 5). This led to a further investigation of how features of the chemical dynamics and potential energy surface would affect the ignition process in the engine, and these sets of studies are summarized in the Letter. As part of the model building process in ref 4, quantum mechanical rate coefficients of several reactions in the MB system were calculated, demonstrating that with modern day computational resources one could move seamlessly from quantum chemistry to engine modeling. The calculations described in this Letter are meant to demonstrate in more detail how the features of the calculations of reaction rates translate directly into engine performance. In ref 4, one of the reactions studied was MB + H, and quantum tunneling corrections were used in the computation of its rate. This suggested to us that it might be interesting to first look at the effect of tunneling on engine performance, and we lead this Letter with our study of tunneling for one of the important reactions. Later we expand our calculations to include other © XXXX American Chemical Society
Received: April 24, 2013 Accepted: May 29, 2013 2021
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Figure 1. Schematic of the potential features of the reaction HO2 + HO2 = H2O2 + O2 along the reaction coordinate, with an accurate portrayal of the molecular structures, all adapted from ref 7. Different barrier heights are included to demonstrate the various features used in the engine simulations in the paper. The schematic barrier heights represent barrier heights of 4.57 kcal/mol, 5.57 kcal/mol, and 6.57 kcal/mol.
shown schematically with the broken arrow in Figure 1. The tunneling corrections were made with the method described in Section V of ref 7. The first column of Table 1 shows a set of conditions that were used to generate a sequence of rate coefficients. The rate
liable to influence engine performance. For a number of these reactions, there are important dynamical features, such as hindered rotations, that will influence engine performance, and we expect that studies of how these features influence engine performance would also be interesting. The role of tunneling in reaction rate theory is well established.6 As long as the temperature is not too high, it is expected to play a significant role in hydrogen abstraction reactions, particularly when the imaginary frequency used to describe the tunneling is large as it is for the reaction studied here (2547 cm−1), based on a recent paper.7 Here we show that tunneling plays a noticeable role in the ignition process in a compression-ignition engine, with ignition delay times varying by up to 6% whether tunneling is included or not for a single reaction, the self-reaction of HO2. Because reactions involving hydrogen abstraction by HO2 are important for accurate descriptions of the ignition process, as they are key chain branching reactions,8 it seems likely that tunneling can play a role in engine performance under many circumstances. Over the past few years, several of the coauthors have studied the role of reaction rate uncertainties on ignition9−12 and speciation12 in combustion processes under simple physical conditions. That work featured applications of global sensitivity analysis (GSA),13 which has a long history in the physical chemistry literature.14−17 GSA highlights those reactions whose accurate calculations are most likely to improve a chemical model.9,10 It became apparent in that work that the reaction HO2 + HO2 = H2O2 + O2 played a significant role in many combustion processes involving oxidation, leading to a detailed examination.7 It was demonstrated that a rather sophisticated treatment of quantum tunneling was necessary for an accurate rate coefficient for the reaction.7 Figure 1 shows features of the HO2 self-reaction relevant for the calculation of the rate coefficient,7 which had been studied many times previously (for example, refs 18 and 19). Figure 1 is schematic in regards to the quantitative values of the barrier height (whose values are listed accurately in the figure caption), but has an accurate shape and correctly depicts the molecular structures along the reaction path. In what follows, we will study how the height of the barrier affects ignition properties and the role of the tunneling correction through the barrier
Table 1. Characteristics of HO2 + HO2 Reaction Rate Coefficient reaction characteristica original no tunneling tunneling +1 kcal/mol − no tunneling +1 kcal/mol − Tunneling −1 kcal/mol − no tunneling −1 kcal/mol− tunneling
ignition delay: HRR based (CA°)b,c
ignition delay: temp. based (CA°)b,d
11.28 11.08 11.31 10.7
10.6 10.4 10.6 9.75
7.92 6.43 8.40 2.02
11.05
10.4
5.32 × 1011
11.4
10.56
1.01 × 1012
11.45
10.75
1.31 × 1012
k (T = 1000 K)e × × × ×
1011 1011 1011 1011
a
Original is from ref 19. All others refer to variations of the rate coefficients in ref 7. bThis is crank angle in degrees (CA°). One CA° is 0.1 ms for an engine operating at 1600 rpm. cIgnition delay time is based on heat release rate (HRR). Specifically, the crank angle (CA) location of 5% of the total heat release is determined for every simulation. dIgnition delay time calculated at point where there is a 400 K increase in temperature, which is 1400 K for the starting temperature of 1000 K used in the simulations. ecm3 mol−1 s−1.
coefficient labeled “original” is the one due to Troe and coworkers.19 The rate coefficient labeled “tunneling” is the recommended rate coefficient from ref 7, which also calculated a rate coefficient that did not include tunneling, and this is labeled as “no tunneling”. The rest of the rate coefficients in the table were calculated with higher or lower barrier heights, with and without tunneling corrections. Compression ignition or diesel engine simulations were performed using the Eulerian−Lagrangian approach in the computational fluid dynamics software CONVERGE.1,20 It incorporates state-of-the-art models for spray injection, atomization and breakup, turbulence, droplet collision, and 2022
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coalescence. The gas-phase flow field was described by using the Favre-averaged Navier−Stokes equations in conjunction with the RNG k-ε-based turbulence model, which included source terms for the effects of dispersed phase on gas-phase turbulence. These equations were solved using a finite volume solver. The details of these models can be found in previous publications.2 The Kelvin−Helmholtz (KH) and Rayleigh− Taylor (RT) models were used to predict the droplet breakup.21,22 A droplet evaporation model based on the Frossling correlation was used. Also used is a dynamic drag model based on the postulation that the drag coefficient depends on the shape of the droplet, which can vary between a sphere and a disk. The effects of turbulence on the droplet were accounted for using a turbulent dispersion model. Detailed kinetic modeling was performed using the SAGE chemical kinetic solver23 directly coupled with the gas-phase calculations using a well-stirred reactor model. This modeling approach has been extensively validated by Som et al.3,24 The results of the simulations are presented in column 2 of Table 1 and Figures 2 and 3. Table 1 demonstrates that the
Figure 3. Spatial distribution of the temperature for the two cases in the bottom panel of Figure 2 at CA = 2°. The top plot is for the case with no tunneling and the bottom for the tunneling case. The plots show a cut plane, as described in the text.
by 1 kcal/mol and includes tunneling, whereas the blue curve has a barrier height that is 1 kcal/mol higher than the barrier height calculated in ref 7. The bottom panel of Figure 2 compares two cases where there is and is not a tunneling correction and it demonstrates that tunneling alone can change the ignition delay time by about 6%. The HO2 self-reaction inhibits ignition,7 which explains the trends observed. For example, raising the barrier and turning off tunneling in the top panel reduces the rate of the reaction, causing ignition to occur more rapidly. Spatial views of the changes in the temperature contours inside the engine combustion chamber are presented in Figure 3 for the two cases in the bottom panel of Figure 2. The engine simulations were performed using the methodology described above. A single cylinder engine is simulated with the injector consisting of six orifices. In order to reduce wall-clock times, a 60° sector simulation was performed assuming axi-symmetry. The liquid fuel is injected into the combustion chamber near the end of the compression stroke (at −9 CA° in these cases), the total duration of injection being 21 CA°. Following injection, the fuel undergoes atomization and vaporization processes, followed by fuel-air mixing, ignition, and combustion inside the combustion chamber. The ignition delay is typically about 10−13 CA° i.e., 1−4 CA° after top-dead center (ATDC). The condition simulated mimics a typical midload engine operating condition. At an engine speed of 1600 revolutions per minute (rpm), this is an ignition delay of about 1 ms. The plots show a cut-plane through the center of the fuel jet during the time of ignition, i.e., 2 CA° ATDC, hence the local hot spot where the temperature is above 2200 K in the top panel. Comparisons of the top and bottom plots again demonstrate that tunneling corrections for the HO2 self-reaction changes ignition, this time showing how the spatial variations of the temperature changes. Because the cut plane used in Figure 3 is not in the spatial location where the temperature is a maximum, the crank angle where there is a significant difference in this spatial view is not the same as in the plots in the bottom of Figure 2. The study of the role of individual rate coefficients was extended to a series of reactions for the biodiesel blend, summarized in Figure 4. The rate coefficients were varied based
Figure 2. Profiles of maximum temperature in the cylinder as a function of crank angle (1 CA° ≈ 0.1 ms) are shown. The top panel compares cases where barrier heights and tunneling rates are different, and the bottom compares two cases with the same barrier height with and without tunneling corrections. The crank angle at start of injection was −9. Dashed lines show results with tunneling, and the solid lines show results without tunneling.
tunneling corrections and barrier heights can have a noticeable impact on the ignition delay time in the engine model, with a range of about 10% from the minimum to the maximum ignition delay time in the table. Figure 2 shows plots of the maximum temperature in the engine versus time. The top panel compares ignition for the two extreme cases. The red, dashed curve shows a case where the barrier height has been lowered 2023
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here is HO2 + C7H16 (there are four reactions of this type, depending on which hydrogen atom is abstracted from nheptane), and Figure 5 shows how the maximum temperature
Figure 4. Ignition delays for a series of engine simulations where rate coefficients of select reactions have been increased and decreased by their perturbation factors are presented here. Ignition delay time calculated based on heat release (column 2 of Table 1).
on the estimated uncertainties in the literature,5 as well as expert elicitation. There are four sets of reactions studied in Figure 4 that involve abstraction of a hydrogen atom. There are four distinct ways of doing this, and for three sets of reactions, the designation “(1−4)” indicates that all four abstractions are varied at the same time by the same factor. In one case (an OH abstraction), only one of the rate coefficients was varied, and the designation “(3)” indicates that a hydrogen atom is abstracted from the C-atom third from the end. The reactions were chosen based on their global sensitivity coefficients in homogeneous constant-volume ignition studies, summarized in the Supporting Information. The first three reactions in the figure legend had high sensitivity in the homogeneous case. The hydrogen abstraction reactions involving OH (there are four, as noted above) were chosen because they are important reactions for the breakdown of the fuel,25 but ignition is often not sensitive to their rate coefficients (see ref 12 and the Supporting Information). The other reactions were chosen as a control sample, as we did not expect ignition to be particularly sensitive to them. All engine simulations were run with only one type of reaction rate coefficient increased by the perturbation factor f or decreased by 1/f. Some additional simulations were run for the HO2 self-reaction and H-abstraction from n-heptane by HO2, where the rate coefficient was perturbed by a factor between these two extremes. The results in Figure 4 generally tracked with results of homogeneous ignition, with those reactions that showed large sensitivity coefficients for the homogeneous case showing significant sensitivity in the engine simulations. An interesting and important exception is the H-abstraction reaction via OH, which had relatively small sensitivity coefficients in the homogeneous case, but shows the highest sensitivity in the engine simulations. We ran one pair of simulations for the following H-abstraction: OH + C7H16 = H 2O + 3‐C7H15
Figure 5. Temperature profile for the two extreme cases of n-C7H16 + HO2.
profile changes within the uncertainty range when all four reactions are varied at the same time. As demonstrated in Figure 4, this set of reactions shows an approximately 2 CA° difference in ignition delay time, when using the heat release criterion (Table 1), which can lead to a misfire if the engine is operating near the stability limit.26 Although we would expect that each individual reaction in this set would have somewhat smaller tunneling corrections than the HO2 self-reaction, the combined effect for all four is liable to be higher than for the self-reaction, and a complete description of this set of reactions is very important for an accurate description of the ignition delay time. In this Letter we have demonstrated that a detailed understanding of the chemical dynamics of elementary reactions, including quantum tunneling, is necessary for an accurate description of the performance of compressionignition engine models, and that something as fundamental as quantum tunneling corrections could make a noticeable difference on such a complex model. We also demonstrated that the accuracy of several important chemical reactions could alter the modeling of these engines. In these simulations, the temperature at which chemistry began was approximately 1000 K, the temperature when the fuel was sprayed in during the compression stroke. The fuel could be sprayed in somewhat earlier, and from our experience with compression-ignition engine modeling, the initial temperature at which significant chemistry begins could be as low as 800 K, leading to engine performance that would show even stronger effects from tunneling corrections and barrier height changes than what was observed here. We finally note that hindered rotations should play an important role in the accurate calculation of many of the abstraction reactions27 important in our engine simulations, and these may have an even larger impact on engine performance than the tunneling corrections and barrier-height variations studied here.
(1)
because the ignition delay time was least sensitive for this reaction of all the OH abstraction reactions under constantvolume conditions. Figure 4 demonstrates that ignition delay times are more sensitive to this reaction than many of the other reactions studied, all of which had higher sensitivity coefficients for the constant-volume ignition cases we studied previously (see ref 4 and the Supporting Information for this Letter). Besides the HO2 self-reaction, a set of reactions that are liable to be sensitive to tunneling in the temperature range studied
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ASSOCIATED CONTENT
S Supporting Information *
More information about the engine simulations is provided in the Supporting Information, including the two definitions of ignition delay used in the Letter. Additional information is 2024
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Deconstruction of the Probability Density Function. J. Phys. Chem. A 2011, 115, 1556−1578. (12) Zhou, D. D. Y.; Davis, M. J.; Skodje, R. T. Multi-Target Global Sensitivity Analysis of n-Butanol Combustion. J. Phys. Chem. A 2013, 117, 3569−3584. (13) Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Sasana, M.; Tarantola, S. Global Sensitivity Analysis. The Primer; John Wiley & Sons: Hoboken, NJ, 2008. (14) Cukier, R. I.; Levine, H. B.; Shuler, K. E. Nonlinear sensitivity analysis of multiparameter model systems. J. Comput. Phys. 1978, 26, 1−42 and references cited therein.. (15) Li, G.; Rabitz, H.; Yelvington, P.; Oluwole, O.; Bacon, F.; Kolb, C.; Schoendorf, J. Global Sensitivity Analysis for Systems with Independent and/or Correlated Inputs. J. Phys. Chem. A 2010, 114, 6022−6032 and references cited therein.. (16) Tomlin, A. S.; Ziehn, T. The Use of Global Sensitivity Methods for the Analysis, Evaluation and Improvement of Complex Modelling Systems. In Coping with Complexity: Model Reduction and Data Analysis; Gorban, A. N., Roose, D., Eds.; Springer: Berlin, 2010; pp 9− 36, and references cited therein. (17) Zador, J.; Zseley, I. G.; Turanyi, T. Local and Global Uncertainty Analysis of Complex Chemical Kinetic Systems. Reliab. Eng. Syst. Saf. 2006, 91, 1232−1240. (18) Mozurkewich, M.; Benson, S. W. Self-Reaction of HO2 and DO2: Negative Temperature Dependence and Pressure Effects. Int. J. Chem. Kinet. 1985, 17, 787−807. (19) Kappel, C.; Luther, K.; Troe, J. Shock Wave Study of the Unimolecular Dissociation of H2O2 in Its Fall-Off Range and of Its Secondary Reactions. Phys. Chem. Chem. Phys. 2002, 4, 4392−4398. (20) Richards, K. J.; Senecal, P. K.; Pomraning, E. CONVERGETM (Version 1.2) Manualp; Convergent Science, Inc.: Middleton, WI, 2008. (21) Reitz, R. D. Modeling Atomization Processes in High Pressure Vaporizing Sprays. Atomization Spray Technol. 1987, 3, 309−337. (22) Patterson, M. A.; Reitz, R. D. Modeling the Effects of Fuel Spray Characteristics on Diesel Engine Combustion and Emissions; SAE International: Warrendale, PA, 1998; SAE Report No. 980131. (23) Senecal, P. K.; Pomraning, E.; Richards, K. J. Multi-Dimensional Modeling of Direct-Injection Diesel Spray Liquid Length and Flame Lift-Off Length using CFD and Parallel Detailed Chemistry. Proceedings of the Society of Automotive Engineers (SAE) World Congress and Exhibition; Detroit, MI, March 2003; SAE Paper 2003-01-1043. (24) Som, S.; Ramirez, A. I.; Longman, D. E.; Aggarwal, S. K. Effect of Nozzle Orifice Geometry on Spray, Combustion, and Emission Characteristics under Diesel Engines Conditions. Fuel 2011, 90, 1267−1276. (25) Glassman, I. Combustion, 3rd ed.; Academic Press: San Diego, CA, 1996; p 94. (26) Heywood, J. B. Internal Combustion Engine Fundamentals, McGraw-Hill, Inc.: New York, 1998. (27) See, for example, Seal, P.; Papajak, E.; Truhlar, D. G. Kinetics of the Hydrogen Abstraction from Carbon-3 of 1-Butanol by Hydroperoxyl Radical: Multi-Structural Variational Transition-State Calculations of a Reaction with 262 Conformations of the Transition State. J. Phys. Chem. Lett. 2012, 3, 264−271.
provided about the rate coefficients used for Table 1 and Figures 3 and 4, including expressions for all the rate coefficients. Information is also provided about the choice of reactions studied in Figures 4 and 5, including global sensitivity indices for 0-D simulations. This information is available free of charge via the Internet at http://pubs.acs.org/.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under Contract No. DE-AC02-06CH11357. This research was also funded by the DOE’s Office of Vehicle Technologies, Office of Energy Efficiency and Renewable Energy, under contract No. DEAC02-06CH11357. The authors wish to thank Wade Sisk and Gupreet Singh, program managers at the DOE, for their support. We gratefully acknowledge the computing resources provided on “Fusion,” a 320-node computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory.
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