Available online at www.sciencedirect.com Proceedings of the Combustion Institute 37 (2019) 807–815 www.elsevier.com/locate/proci Evolution of sensitivity directions during autoignition Weiqi Ji a, Zhuyin Ren a,∗, Chung K. Law a,b a Center for Combustion Energy, Tsinghua University, Beijing 100084, China b Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Received 1 December 2017; accepted 3 July 2018 Available online 17 July 2018 Abstract Sensitivity analysis of the ignition delay time and species profiles to kinetic parameters has been widely used to identify the rate-limiting steps during the autoignition process, providing insights for the optimization of the reaction mechanism. This work studies the time evolution of the sensitivity directions of the temper- ature and species concentration during autoignition. The direction is represented by a unit vector along the gradient of the simulation output to the kinetic parameters, and the alignment between the two directions are measured by the inner product between the corresponding unit vectors. We use evolution of the sensitivity directions to reveal changes in the rate-limiting steps and the correlation among species at different phases of the ignition delay period. It is found that the sensitivity directions of temperature and the concentrations of the major intermediate species are similar to each other during the entire ignition delay period. In partic- ular, they converge to the same direction when approaching the ignition state, and the direction is the same as the one for the ignition delay time. The correlation is validated for various fuels across a wide range of pressures and temperatures, and works for both single-stage ignition and two-stage ignition. Consequently the sensitivity of the ignition delay time can be efficiently evaluated based on the temperature sensitivity at the ignition point, for which a single run of the simulation can produce the sensitivity to all parameters, as otherwise the sensitivity of the ignition delay time has to be evaluated through finite difference, in which the number of runs equals to the number of parameters. It can also significantly reduce the computation cost of gradient-based algorithms for the purpose of mechanism optimization and uncertainty quantification. © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Ignition delay time; Sensitivity analysis; Sensitivity direction; Similarity; Gradient 1. Introduction ment and validation of kinetics mechanisms. The ignition delay time (IDT), which is the time for the Autoignition of hydrocarbon fuels has been ex- induction period before rapid heat release, is of pri- tensively studied for engine design and the develop- mary importance and is readily measurable using rapid compression machines and shock tubes. It has also been routinely simulated with detailed re- ∗ Corresponding author. action mechanisms, in which the kinetic parame- ters could have pronounced effect on the predicted E-mail address: zhuyinren@mail.tsinghua.edu.cn (Z. IDT. The sensitivity of IDT to the rate constants Ren). https://doi.org/10.1016/j.proci.2018.07.005 1540-7489 © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved. 808 W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 can reveal the rate-limiting steps controlling the Sin (t)/Sim (t) does not change with t within the inter- ignition process and provide valuable insights into val (t1 , t2 ). Similar patterns were also observed in the mechanism optimization. laminar flames and in models of biological prob- While sensitivity analysis of IDT has become lems [8]. a standard procedure for mechanism development, This work also focuses on the patterns among insufficient attention has been directed to the evolu- the sensitivity vectors during auto-ignition. While tion of the sensitivities of composition (e.g., species previous work has studied the similarity for hy- concentrations and temperature) with respect to ki- drogen and methane with single-stage ignition, this netic parameters during autoignition. In this work, work shall extend the study to large hydrocarbon we study the sensitivities to the pre-exponential fuels with two-stage ignition. More importantly, “A-factors” in the Arrhenius reaction rate expres- while the correlation between the sensitivity vector sions. The governing ordinary differential equa- of laminar flame speed and that of the local com- tions (ODEs) for composition evolution during the position sensitivity has been studied in [3–7], it is auto-ignition process are of the general form unclear how the sensitivity vector of the IDT is re- dϕ lated to the local composition sensitivity. Here, we = F (ϕ, t; k ), (1) shall show that the sensitivity vector of IDT and dt the local temperature sensitivity at the instance of where ϕ(t) is the composition vector consisting of ignition are parallel, which provides the possibil- species mass fractions and temperature, and k is the ity to evaluate the sensitivity of IDT based on the vector of reaction rate constants. The first-order composition sensitivity. The composition sensitiv- sensitivity coefficients are defined as ities are usually obtained by solving the govern- ∂ ϕi ing equations for the sensitivity matrix during the Si j = , (2) autoignition integration. They can be computed ∂kj via very efficient methods [7], such as Decoupled with the evolution equations being Direct Method (DDM) which is implemented in Chemkin [9]. DDM requires very little additional dSij ∂Fi ∂Fi computational cost for the calculation of the local = Slj + , (3) dt ∂ϕi ∂k j sensitivity vectors after the solution of Eq. (1). In addition, the computational cost might be further The corresponding sensitivity matrix S(t), of reduced via the adjoint method [10]. Then the sen- sizes nc × nr , evolves with time and depends on the sitivity of any other quantities defined as the inte- local composition, with nc being the dimension of gral of the solution variables can be readily com- the composition and nr the number of kinetic pa- puted by matrix manipulation. However, since IDT rameters. The kinetic parameters here refer to the is defined as the time instant when the temperature pre-exponential “A-factors” of the nr elementary profile reaches its inflection point, its sensitivity is reactions, and the ith row of S(t) represents the usually evaluated through brute force finite differ- sensitivity vector of the ith composition variable in ence approach, for which the kinetic parameters are the nr -dimensional parameter space, with its mag- perturbed one by one. Consequently the number of nitude and direction evolving with time. The rel- autoignition integrations required is equal to nr and ative importance of the kinetic parameters is re- the computational cost becomes very expensive for vealed by the individual components of the sensi- large hydrocarbon fuel with thousands to tens of tivity vector. thousands of reactions. Various interpretations of the time-evolving To alleviate the computational cost, we propose sensitivity vectors have been proposed, such as an efficient approach for computing the sensitiv- measuring the averaged sensitivity over a certain ity of IDT by exploiting the correlations between time range [1, 2]. For example, Vajda et al. [2] em- the sensitivities of IDT and composition. By do- ployed principal component analysis (PCA) on S(t) ing so, the computation of the sensitivity of IDT over a time interval, i.e., eigen decomposition of only requires two runs, and the cost can be com- ∫tt21 S T S dt, to reveal the sensitive reactions and the parable to that for the autoignition integration us- coupling between the reactions. The principal com- ing the adjoint method [10]. Then the computa- ponents, referred to the eigenvectors with large tional cost for statistical calibration of the reaction eigenvalues, are linear combinations of the kinetic mechanism [11,12], construction of the response parameters, along which the predicted composition surface for the kinetic parameters [11,13], and profiles mostly change. global sensitivity analysis [14] can be significantly Turányi and co-workers [3–7] have systemati- reduced by taking advantage of various gradient- cally studied three types of similarity among the based algorithms, such as those of steepest de- sensitivity vectors during autoignition, i.e., (i) Lo- scent method, active subspace [13,15,16] and so on. cal similarity: the ratio Sim /Sjm is the same for any For example, in [13], with the sensitivities of IDT, parameter km . (ii) Scaling relation: the ratio Sim /Sjm the active subspace method is employed to iden- ∂ϕ is equal to ∂∂tϕi / ∂tj . (iii) Global similarity: the ratio tify one-dimensional active subspaces in the 33 and W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 809 Table 1 Chemical mechanisms and conditions for the autoignition processes. No. No. Fuel/mech species reactions φ p [atm] T [K] Methane (CH4 ) 53 325 0.5, 1, 2 1, 20 1000–1500 [18] DME 55 290 0.5, 1, 2 1, 20 1000–1500 (CH3 OCH3 ) [19] n-heptane 88 729 0.5, 1, 2 1, 20 600–1500 (n-C7 H16 ) [20] iso-octane 143 1178 0.5, 1, 2 1, 20 600–1500 (i-C8 H18 ) [21] 217-dimensional uncertain kinetic parameter space Fig. 1. Time is normalized by the ignition delay for the IDT of hydrogen/air and methane/air mix- time and is denoted as NT. As expected, the magni- tures, respectively. Then the cost of subsequent tude of the temperature sensitivity vector changes construction of response surface and global sensi- simultaneously with the rate-of-change of temper- tivity analysis is reduced by several orders. With the ature, which remains relatively small and increases above efficient approach for computing the gradi- dramatically near the ignition point. ent of IDT, the total cost will be further reduced Figure 2 shows evolution of the sensitivity di- and the speed up factor is proportional to the num- rections of temperature and several major inter- ber of reactions. mediate species for methane and DME. The pro- In the following, we shall first investigate the files for n-heptane and iso-octane with two-stage similarity in the evolution of the sensitivity direc- ignition are similar to DME and are provided in tions in both single-stage ignition and two-stage ig- the supplemental material. To facilitate the com- nition. We then show that the sensitivity vector of parison, the inner product ui (t), uT, ign , first sug- IDT is parallel to the composition sensitivity vector gested in [4], is employed to represent the angle at the instant of ignition. Finally, an efficient ap- between the directions of the composition sensi- proach for computing the sensitivity of IDT based tivities and that of temperature sensitivity at the on the correlation is proposed and validated. ignition point, with ui (t), uT,ign  = ±1 represent- ing perfect alignment. Discussions on the switch- ing between positive and negative correlations can 2. Evolution of composition sensitivity be found in [4]. For methane with single-stage ig- nition, it is shown that the temperature sensitivity Equations (1) and (3) are integrated together direction quickly aligns with uT, ign , with an inner using Cantera [17] to obtain the evolution of product of 0.90 at NT = 0.5, and 0.99 at NT = 0.9. both composition and sensitivities. The normalized For DME with two-stage ignition, two-stage align-  kj ∂ ϕi ments of the temperature sensitivity vector are ob- composition sensitivity is given by S i j = ϕ ∂ k . For i j served, corresponding to the first and second stage the sensitivity vector of the ith composition vari- ignition. It quickly aligns with uT, ign with an inner  i ≡ [S able, e.g., S T i1 , . . . . . . , Sinr ] , its direction is the product of 0.95 at NT = 0.7, and 0.99 at NT = 0.9. same as that of the unit vector ui ≡ S i /||Si ||2 and its Therefore, the global similarity defined in [3] is ob- magnitude is given by λ ≡ ||Si ||2 .  served where approaching ignition for both the first In this study, the ignition state is defined as the and second stage ignition. time instant of the maximum heat release rate. The It is also shown that the sensitivity directions mechanisms employed are summarized in Table 1. of the fuel, the final products CO2 , H2 O and the While the methane mechanism involves only high- major intermediate species CO, OH change simul- temperature chemistry and single-stage ignition, taneously along the temperature sensitivity direc- the other three fuels involve low-temperature chem- tion, which corresponds to the local similarity de- istry and may result in two-stage ignition. Detailed fined in [3]. Furthermore, the directions of species mechanisms are used for methane and DME, while sensitivities and temperature sensitivity are largely skeletal mechanisms are used for n-heptane and correlated with each other, although deviations are iso-octane to reduce the computation cost of finite observed for the species CH2 O, H2 O2 and OH in difference approach. The autoignition simulation is the DME case. Note that the absolute value of the performed under constant volume; constant pres- inner product is presented in Fig. 2, and the sign sure simulation leads to the same conclusion and for the inner product between OH sensitivity and hence is not shown here. the reference direction changes due to its negative Representative evolution profiles of the magni- net production rate during the transition from the tude of the temperature sensitivity vector as well first stage to the second stage ignition. Typical evo- as the rate-of-change of temperature are shown in lution of the OH profile during the transition can 810 W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 a b Fig. 1. Evolution of temperature, its rate of change, and the magnitude of its sensitivity vector to kinetic parameters against the normalized time. The unit of dT/dt is K/s. The IDT is 38.4 ms in (a) and 0.7 ms in (b). a b Fig. 2. Evolutions of the inner product (absolute value) between the sensitivity directions of composition and uT, ign . be found in [22]. During the later phases of the sec- trajectory to rate constants only has a component ond stage ignition, the sign is reversed to be posi- parallel to the attracting manifold when approach- tive as the OH concentration increases again. The ing equilibrium. At the ignition state, if the at- same reason accounts for the behavior of H2 O2 and tracting manifold is one-dimensional and its sen- CH2 O near the second stage ignition. Neverthe- sitivity vector to rate constants is predominantly less, all the composition sensitivity vectors evolve in the tangential direction of the manifold, then toward aligning with uT, ign when approaching the the responses of different composition components ignition state. to the rate constants are also parallel with each Zsely et al. [3] have shown that the similar- other and this explains the alignment of the species ity of sensitivity functions is related to the exis- sensitivity directions. As shown in Fig. 1(a), the tence of low-dimensional manifolds in chemical ki- temperature sensitivity remains relatively small netic systems; they are briefed here for the later throughout the induction period and then increases discussion on the correlation between the ignition dramatically near the ignition point. It is antic- delay time and local temperature sensitivity. In ipated that when approaching the ignition state, the composition space, the one-dimensional trajec- temperature is insensitive to those sensitive reac- tory from autoignition quickly approaches the at- tions for the induction period and its evolution tracting manifold. König and Maas [23] showed and sensitivity (including direction and magnitude) that the sensitivity of the attracting manifold to is dominated by the local kinetics around the ig- rate constants vanishes and the sensitivity of a nition state. This explains why the direction of W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 811 a b Fig. 3. The most sensitive reactions for temperature at the ignition point. temperature sensitivity reaches a plateau when ap- CH2 OCH2 O2 H + O2 = O2 CH2 OCH2 O2 H (R273) proaching ignition. and CH3 OCH2 = CH2 O + CH3 (R248). The iso- Next, we shall investigate evolution of merization reaction R271 and the second oxygen the sensitive reactions as revealed by the addition reaction R273 are the key steps leading to components of uT, ign . Figure 3(a) shows low-temperature chain branching reactions. Reac- the most sensitive reactions for temperature tions R272 and R248 are the β-scission reactions at the ignition point for the stoichiometric competing with the low-temperature chain branch- methane/air mixture at 20 atm and 1000 K. ing reactions. Their evolutions in Fig. 4 show that The reactions of CH3 + HO2 = CH3 O + OH R271 is dominant during the initial stage of the (R119), CH2 O + O2 = HCO + HO2 (R32) induction period. With increasing concentration and CH3 + O2 = HCO + HO2 (R156) are of CH3 OCH2 O2 and CH2 OCH2 O2 H, the relative the most sensitive chain branching reactions sensitivities of R272 and R273 gradually increase while 2CH3 (+M) = C2 H6 (+M) (R158) and and approach a plateau when approaching the 2HO2 = H2 O2 + O2 (R116) are the most sen- first stage ignition point. After the temperature in- sitive chain termination reactions. Evolution of creases during the first stage ignition, β-scission of the corresponding components is then shown in the fuel radical, R248, and the subsequent reaction Fig. 4(a). It is seen that during the initial stage, R51, which are well known to be important in the the major competition occurs between R156 and intermediate-to-high temperature regime [25, 26], R158 in consuming the CH3 radical, which is start to be important. Interestingly, the relative consistent to the rate of production analysis (ROP) sensitivity of R271 is significantly reduced, while in [24]. Then the sensitivity of R156 gradually the sensitivity of the other key low-temperature decreases relative to R119, which is attributed reactions, R272 and R273, are promoted. This is to the shift of equilibrium due to the increasing possibly a consequence of R271 reaching partial CH2 O concentration [24]. equilibrium and the net reaction rate is then deter- Regarding the case of DME with two-stage mined by the subsequent rate-limiting steps, such ignition as shown in Fig. 3(b), the most sen- as R272 and R273. sitive reactions for temperature at the igni- In short, we have demonstrated that evolution tion point are CH3 OCH2 O2 = CH2 OCH2 O2 H of the composition sensitivity can help analyze the (R271), CH2 OCH2 O2 H = 2CH2 O + OH (R272), corresponding evolution of the rate-limiting steps 812 W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 ence approach with a perturbation factor, e.g., δ = 5%, i.e., ln τ ( (1 + δ)ki ) − ln τ (ki ) S τ,i ≈ (5) ln (1 + δ) The perturbation factor is chosen to be large enough to avoid numerical noise (discretization er- ror) in computing IDT, and small enough to avoid nonlinear effects. It is found that the direction of sensitivity for IDT and temperature sensitivity at ignition are al- most the same, i.e., uτ aligns with uT, ign . uT, ign is the sensitivity vector of temperature at the ignition point with respect to the kinetic parameters. The level of alignment is also assessed by the inner prod- uct between the two sensitivity directions, which is larger than 0.999. The largest 25 components (in magnitude) of the unit vector uτ for the stoichio- metric mixture of methane/air under the pressure of 20 atm and initial temperature of 1000 K are shown in Fig. 5. Also shown are the component- wise relative difference between uτ and uT, ign , de- fined as i ≡ (uτ,i − uT,ign,i )/uτ,i for the ith compo- nent. As shown, the relative errors are within 5% for the top 16 reactions, and become larger for the small components as the numerical error for the insensitive reactions using the finite difference ap- proach is larger. Such numerical error can be re- duced by using an adaptive perturbation factor for those insensitive reactions in the finite difference approach. However, since most of the sensitivity Fig. 4. Evolution of the most sensitive reactions for tem- analysis is focusing on the sensitive reactions, we perature. did not further refine the perturbation factor. Such excellent alignment can be explained by Fig. 6, in which the rate constant for R119, the most during the ignition delay period. The similarity sensitive reaction for IDT, is increased by 0.005% among the sensitivity vectors observed in previ- and leads to a shorter IDT. According to the trian- ous work [3] have been extended to both the first gle in Fig. 6, we have and second stage ignition, i.e., (I) the sensitivity for temperatures and most of the species concentra- ∂τ ∂τ ∂T τ T = = (6) tions are largely correlated with each other. (II) The ∂ ki ∂T ∂ ki T ki sensitivity directions for the major intermediate Equation (6) implicitly assumes that the changes species concentration and temperature align with ∂τ of ∂T at the ignition state is insignificant compared each other when approaching the ignition state. Next, we shall study the correlations between the to the change of temperature, ∂∂Tki . The term ∂T ∂τ is in- sensitive directions of composition and the ignition versely proportional to the rate-of-change of tem- delay time. perature and thereafter the heat release rate at the ignition state. The term ∂∂Tki is attributed to the ac- cumulated sensitivity during the entire ignition de- 3. Efficient approach for computing the sensitivity lay period. The assumption for Eq. (6) is justified of IDT by the analysis in Section 2, in that the evolving of the autoignition trajectory near the ignition state is The normalized sensitivity of the ignition delay not sensitive to those rate-limiting steps for the in- time τ is defined as duction period. Note that the correlation between ki ∂τ ∂ ln τ the normalized sensitivity vectors can be readily de- Sτ,i = = (4) rived from Eq. (6). τ ∂ ki ∂ ln ki The correlation of Eq. (6) is further validated Similarly, the sensitivity direction for IDT is the against the results from the finite difference ap- same as the unit vector uτ ≡ Sτ /||Sτ ||2 . The sensi- proach for the autoignition process of various fuels, tivity of IDT can be evaluated via the finite differ- covering both fuel-lean and fuel-rich conditions, W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 813 Fig. 5. Components of the sensitivity direction for IDT and the component-wise relative differences between uτ and uT, ign . Specifically, the engine speed is 1000 rpm; compres- sion ratio (CR) is 16.5; piston diameter and stroke length are 12.065 cm and 14.005 cm, respectively; ratio of the length of the engine connecting rod to the crank radius (LOLR) is 3.71; and the in- let valve close crank angle (IVC) is 142° BTDC. The initial mixture is natural gas in air; the com- position by volume are CH4 :0.0350, C2 H6 :0.0018, C3 H8 :0.0012, O2 :0.1824, CO2 :0.0326, H2 O:0.0609, N2 :0.6861; and the initial temperature and pres- sure at IVC are 447 K and 1.065 atm, respectively. The mechanism employed is GRI3.0 [18]. The in- ner product between the sensitivity direction for the ignition delay time (crank angle) and the tempera- ture sensitivity at ignition is larger than 0.999. The components of the sensitivity directions and the component-wise difference between uτ and uT, ign Fig. 6. Relations between changes of the ignition delay are further shown in Fig. 7. Similar to the case of τ and temperature at ignition due to perturbation of the methane in Fig. 5, the relative differences are within rate constant. 5% for the top ten components. Based on the correlation of Eq. (6), we can across a wide range of pressures and temperatures determine the sensitivity of IDT from the tem- as listed in Table 1. Typical results for DME, n- perature sensitivity at ignition. However, it is heptane and iso-octane are provided in the supple- challenging to accurately determine the propor- mental material. The inner product between the di- tionality constant via the temporal derivative of rections of the sensitivity for IDT and the tempera- temperature as it changes sharply at ignition. ture sensitivity at ignition is always larger than 0.99 Instead, we propose the following procedure: first for all of the cases listed in Table 1. It is worth men- calculate the local temperature sensitivity at igni- tioning that the correlation is also valid for the first tion with one run of autoignition integration, and stage ignition delay time, as long as the two-stage then calculate the sensitivity of IDT to the highest ignition behavior can be clearly identified. Other- sensitive reaction in uT, ign using the finite difference wise, it might show large difference due to the large approach, denoted as ∂∂lnlnkτj . Then the sensitivity of uncertainty in determining the ignition state for the IDT to the rest of reactions is given by first stage ignition. ∂ ln τ ∂ ln τ ∂ ln T ∂ ln T In addition to the constant volume simulation, / = / (7) the correlation is also validated in a simplified ∂ ln ki ∂ ln k j ∂ ln ki ∂ ln k j zero-D HCCI simulation with the configurations The finite difference approach on the most sen- based on the experiments and simulations in [27]. sitive reaction kj is used to estimate the ratio 814 W. Ji et al. / Proceedings of the Combustion Institute 37 (2019) 807–815 Fig. 7. Components of the sensitivity direction for IDT in the HCCI case and the component-wise relative differences between uτ and uT, ign . between the sensitivity of IDT and the temperature between the sensitivity direction of the ignition de- sensitivity. The most sensitive reaction is chosen to lay time and the temperature sensitivity direction avoid the discretization error when a small pertur- at the ignition state is always larger than 0.99 un- bation factor is used. If only the relative sensitivity der all the test cases. We then proposed an effi- is desired, the sensitivity direction is sufficient and cient approach to compute the sensitivity of the ig- the estimation of the ratio can be omitted. nition delay time from the temperature sensitivity. The approach only requires two auto-ignition inte- grations while the number of runs required by the 4. Conclusions conventional finite difference approach is propor- tional to the number of parameters. This approach We have studied the time evolution of the com- suggests extensive opportunities for efficiently op- position sensitivity during autoignition. The simi- timizing large reaction mechanism using gradient- larity among the sensitivity vectors studied in [3] is based algorithms. extended to the two-stage ignition, i.e., the sensi- Recognizing that the similar behavior has been tivity directions of temperature and most of the studied in the laminar freely propagated premixed intermediate species concentration are similar to flames in the suggested works [3], i.e., the similar- each other during the ignition delay period, and ity between the sensitivity vectors of the composi- consequently converge to the same direction when tions and the laminar flame speed, the current work approaching ignition during both the first-stage found analogous behavior in the auto-ignition sys- and second-stage ignition. An explanation based tem, and will help build a more comprehensive pic- on the attracting manifold and its sensitivity is pre- ture of the similarity in homogenous auto-ignition sented. Specifically, the sensitivity of the attracting and one-d flame system. manifold vanishes when approaching equilibrium, and the sensitivity of the autoignition trajectory is parallel to the intrinsic low-dimensional manifold Acknowledgments (ILDM) when approaching ignition. Furthermore, it is found that the sensitivity of This work was supported by the National the ignition delay time is proportional to the tem- Natural Science Foundation of China No. perature (and species concentrations) sensitivity. 91441202 and 51476087. The correlation is validated using various fuels, in- cluding methane, DME, n-heptane, and iso-octane, under both fuel-lean and fuel-rich conditions, cov- Supplementary materials ering both single-stage and two-stage ignition. In addition to the constant volume and constant pres- Supplementary material associated with this sure simulations, the correlation is also validated article can be found, in the online version, at in zero-D HCCI simulation. 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