Article pubs.acs.org/EF
Flux Projection Tree Method for Mechanism Reduction Ai-Ke Liu,† Yi Jiao,‡ Shuhao Li,§ Fan Wang,*,‡ and Xiang-Yuan Li*,† †
College of Chemical Engineering, Sichuan University, Chengdu, Sichuan 610065, People’s Republic of China College of Chemistry, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China § School of Aeronautics and Astronautics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China ‡
S Supporting Information *
ABSTRACT: Merits and demerits of the directed relation graph (DRG) method are analyzed. On the basis of these analyses, a flux projection tree (FPT) method for mechanism reduction is proposed. A tree-type structure is constructed in FPT based on the contribution of each species to the global flux; that is, the importance of each species is quantified by normalized projection of its participation flux vector upon the total species flux vector. Because a tree-type structure is simpler than a graph-type structure, FPT tends to be more efficient than DRG and path flux analysis (PFA) in computation. Additionally, the significance of each species in a mechanism is estimated on the basis of its contribution to the global species flux, instead of its contribution to the flux of a single species in a pre-chosen important species set, as in DRG and PFA. Thus, a reduced model obtained by FPT is more accurate in most cases. Detailed mechanisms for oxidation of ethylene, n-heptane, and PRF50 were reduced with FPT, and the reliability of the resulting skeletal mechanisms is comparable or even better than that of the skeletal mechanisms obtained by DRG or PFA with similar size. Because of its high efficiency, FPT can be used as the first-step reduction method or on-the-fly mechanism reduction approach in numerical simulations of reaction flow.
1. INTRODUCTION Numerical simulation has become a good observation and a powerful tool to investigate engine combustion and to develop control strategies because of its greater flexibility and lower cost compared to experiments.1 In the process of combustion simulations, molecular diffusion, turbulent transport, and chemical reactions occurring across the flame are three major factors. Lots of studies show that chemical kinetics is critical to achieve a reliable prediction on combustion features.2−4 Many reaction mechanisms have been developed to understand combustion processes and to facilitate numerical simulations. Recent advances in experimental and theoretical studies on real fuel chemistry have led to more accurate chemical mechanisms.5 To provide a reasonable description on combustion chemistry of real fuels, a large number of species and reactions are usually included in these mechanisms. For example, many reaction pathways are needed to capture the two-stage ignition behavior under low temperature or elevated pressure, and different chain-branching reactions are responsible for ignition in different conditions.6 However, it is still impractical to use a detailed mechanism with a large number of species in engine simulation because of its high computational cost. Furthermore, stiffness generated by some elementary reactions with a very short time scale is also a challenging problem in numerical simulations. Extensive size reduction as well as stiffness removal is thus essential to incorporate these mechanisms in reactionflow simulations. In the past few decades, extensive studies on mechanism reduction have been carried out and various methods have been proposed. These methods can be roughly classified into three categories as follows: skeletal reduction, lumping, and timescale analysis. Skeletal reduction is designed to produce a skeletal mechanism from a detailed mechanism by systemati© 2014 American Chemical Society
cally removing unimportant species and reactions. Detailed reduction,7 element flux analysis,8 direct relation graph (DRG),9 and DRG-based methods,10,11 path flux analysis (PFA),12 and simulation error minimization (SEM) methods13 are typical representatives of skeletal reduction. In lumping,14,15 species with similar chemical structures and reaction pathways are combined, such that the number of variables in differential equations is reduced. In time-scale analysis,16−18 species with short time scales are identified and differential equations for these species are replaced by algebraic equations. Stiffness of the system can thus be reduced. Performance of these methods has been reviewed by Lu and Law,19 Pitz and Mueller,20 and Hiremath and Pope.21 Skeletal reduction is usually used as the first-stage reduction to achieve chemical mechanisms with much smaller size while retaining its reliability on describing combustion characters of the fuel involved. Among the skeletal reduction methods, DRGbased methods are the most popular because of their conceptual simplicity and low computational cost. The mechanism analysis method based on the graph structure was first introduced by Bendtsen et al.22 in 2001 and later extended by Lu and Law9 in 2005. The idea of graph structure analysis has been explored extensively by many authors since then. However, there is still some weakness in the DRG-based methods. First of all, a set of important species, i.e., target species, needs to be chosen initially for DRG-based methods, and the resulting skeletal mechanism may depend upon these target species. Although DRG is found to be insensitive to this set of species, some DRG-based methods, e.g., DRG with error Received: January 26, 2014 Revised: July 11, 2014 Published: July 11, 2014 5426
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propagation (DRGEP),10 are more sensitive to it. In addition, a graph is a multi-layer structure, and relations between species become rather complicated in some DRG-based approaches, such as DRGEP. In these cases, some specially designed graph search techniques are required to generate a unique skeletal mechanism.23 Mechanism reduction can be applied in advance or on-the-fly during numerical simulations.24 Although many schemes have been developed to obtain reduced mechanisms from detailed mechanisms, the achieved reduced mechanisms are still too large to be adopted in engine simulations in most cases. Alternatively, an on-the-fly mechanism reduction scheme25−27 or adaptive chemistry24,27−30 may solve this problem. In these approaches, a series of reduced models are used in a reactionflow simulation, and each reduced model reproduces results of the detailed mechanism only under certain reaction conditions. In this way, the size of these models can be reduced even further. However, only the reduction approaches with high efficiency are applicable for on-the-fly reductions, and DRGbased methods have been used previously for this purpose.31 In this work, we proposed a flux projection tree (FPT) method for mechanism reduction. FPT is more efficient than DRG-based methods, which renders its application to on-the-fly reduction.
smaller than a given threshold, as the dash lines shown in Figure 1, are removed. Species in the connected part of the graph containing the target species will be retained, and a skeletal mechanism is obtained by including these species only and reactions involving them. As illustrated in Figure 1, species A1, A2, An, B2, Bp, and Bq will be retained. In practical implementations, both depth-first search (DFS) and breadth-first search (BFS) can be employed to search the graph and to generate skeletal mechanisms. From the point of view of reaction flux, the denominator in eq 1 is the flux of species A through all reactions based on the absolute reaction rate. This equation can thus be rewritten based on flux of the species rAB =
fA =
element reaction A1 A1 A2 A2
∑
|vA, iωi| ,
fA,B =
i = 1, I
∑
|vA, iωiδ Bi |
i = 1, I
→C+E +F→B →F+E +C→D
net reaction rate (mol cm−3 s−1) 0.4 0.6 0.6 0.4
mechanism and corresponding net reaction rates at a certain state point. A1 and A2 are chosen as the target species. According to DRG, the dependence of A1 upon B (0.6) is larger than that of A1 or A2 upon C (0.4), which indicates that B is more important than C based on analysis of DRG, and C may even be deleted if a threshold of 0.5 is used. The detailed analysis process can be found in the Supporting Information. However, C serves as an intermediate species in a chain reaction, and the total contribution of C to A1 and A2 is actually larger than that of B to A1 and A2. This shows that considering only the maximum contribution as in DRG may be problematic in some cases and a small threshold has to be used to keep all important species. However, some redundant species may also been retained at the same time. Some approaches, such as DRGEP and PFA, have been proposed to improve upon DRG. The difference among these approaches lies in the definition of the dependence of a species upon another species. In DRGEP, this dependence is defined as the maximum product of intermediate edge weights, and DRGEP has been shown to perform better than DRG in some cases.10,11 However, relations between species in DRGEP are rather complicated, and the resulting skeletal mechanism becomes more sensitive to the target species than that with DRG.23 In addition, more complicated algorithms, such as Dijkstra’s algorithm,32 are required to generate a unique skeletal mechanism with DRGEP. The first-generation PFA closely resembles DRG, while the second-generation PFA is also more involved in calculating the relation between two species.12 In all of these approaches, the importance of a species is evaluated on the basis of its maximum contribution to a single important species, instead of the whole set of important species. To circumvent this problem, a FPT method is proposed as described in the following. 2.2. FPT Method. In FPT, the importance of a species is evaluated according to its contribution to the flux of all of the species, which can be represented by a N-dimensional vector F, where N is the number of species in the detailed mechanism, and each element in the vector F corresponds to the flux of a certain species. The contribution of species B to the flux of species A is determined by fA,B in eq 3. Similar to F, a N-dimensional vector FB is introduced to describe the contribution of species B to flux of all species in the mechanism, i.e., the participation
∑i = 1, I |vA, iωiδ Bi |
⎧1 if the i th elementary reaction involves species B δ Bi = ⎨ ⎩ 0 otherwise
,
Table 1. Artificial Mechanism and Corresponding Net Reaction Rate at a State Point
2.1. DRG Method. In DRG, relations between each pair of species are represented in a relation graph. Each node in this graph is uniquely mapped to a species in the detailed model, and an edge from node A to node B represents the dependence of species A upon species B. The essence of DRG is to identify species that are important to the target species by searching this relation graph. In original DRG,9 the following normalized value rAB is used to quantify the dependence of species A upon species B
∑i = 1, I |vA, iωi|
fA
(3) where fA,B is the flux of A through all reactions, including species B, and fA represents the overall flux of A. Contribution of B to the flux of a single species, instead of that to the flux of all important species, is used to evaluate the importance of B in DRG. Table 1 gives an artificial
2. FPT METHOD
rAB =
fA,B
(1)
(2)
where vA,i is the net stoichiometric coefficient of species A in the ith reaction and ωi is the net reaction rate of the ith reaction. There are many species in a detailed mechanism, and some of them are strongly coupled. Species (nodes) are connected with each other by normalized values (edges), and a multi-layer graph can thus be constructed as shown in Figure 1. In this figure, An represents the nth target species and Bp is the pth species coupled directly or indirectly to these target species. Dependence of A upon B is different from that of B upon A; i.e., it is a directed graph. The total number of edges to be evaluated is the square of the species number in the detailed mechanism. After a directed relation graph is constructed, those edges
Figure 1. Directed graph showing typical relations between the species. 5427
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flux vector of B. The importance of species B in the detailed mechanism is determined on the basis of the normalized projection of the participation flux vector on the global flux vector
RB =
With these vectors, importance of each species can readily be obtained using eq 4 as follows:
RF = FTFF/FTF = 0.5
FTFB FT F
RR = FTFR /FTF = 1
(4)
F = [f1 , f2 , ..., fA , ..., fN ]T
(5)
FB = [f1,B , f2,B , ..., fA,B , ..., fN ,B ]T
(6)
RP = FTFP/FTF = 0.5 RP ′ = FTFP ′/FTF = 0.0003
(9)
It can be seen that species P′ has negligible influence on the global species flux, and it can be safely removed, while F, R, and P will be kept in the reduced mechanism. This example shows that FPT can deal with QSS species reliably. It can also be shown that FPT is able to afford reasonable analysis on partial equilibrium (PE) problems based on the sample PE mechanism proposed in ref 33, which is due to the appropriate definition of eq 4, where the net reaction rate is employed in FPT. Heat release is usually considered as important information in combustion of fuels, and it has been used in detailed reduction7 and directed relation graph with expert knowledge (DRGX).34 In this work, heat release is also incorporated to enhance the performance of FPT. Heat release of the ith reaction Qi is treated as a special “species”, and FB and F in eqs 4−6 will become N + 1-dimensional vectors with the additional element defined as
where fA in eq 5 and fA,B in eq 6 are calculated with eq 3. It can be seen from eq 4 that RB is closely related to the relative error induced to the global species flux by removal of species B. With the definition of RB, a FPT, as shown in Figure 2, can be constructed on each sampling state point. The root in Figure 2
max (|νn , iωi|)
n = 1, N
Figure 2. FPT showing typical relations between participation flux vectors of certain species and the global flux vector.
β=
i = 1, I
max(|Q i|)
(10)
i = 1, I
represents the global flux vector of all species F, and each leaf (a species) is the contribution of one species to this global flux vector, i.e., FB. Every leaf is connected with the root (the global flux vector) through an edge (RB). After a FPT has been constructed, the species with RB larger than a given threshold will be chosen as the important species, and a set of important species on this state point is thus obtained. The final skeletal mechanism is achieved from union of the important species sets on each sampling state point. Because of its simplicity in structure, the total number of edges to be evaluated is identical to the number of species in the detailed mechanism. Species searching is thus more efficient in FPT than that in DRG-based methods. For the artificial mechanism demonstrated in Table 1, RB is calculated to be 0.3 and RC is 0.36 using eq 4, which indicates that C is more important than B. FPT can thus provide a more reasonable description on the relative importance of a certain species based on its contribution to the flux of all of the species in the mechanism. To further demonstrate the reliability of FPT on mechanism reduction, the following artificial mechanism including a quasi-steadystate species (QSS) proposed in ref 33 will be analyzed with FPT:
(R1)
F → R,
ω1 = 1F
(R2)
R → P,
ω2 = 103R
(R3)
R → P′ ,
ω3 = 1R
fN + 1 =
fN + 1,B =
|βQ i|
∑ i = 1, I
(11)
|βQ iδBi |
(12)
where the scale factor for heat release β is the ratio between the maximum species flux and the maximum heat release among all of the reactions. The contribution of heat release to the flux is thus of similar magnitude to that of a species. Although the definition of the species important coefficient in FPT is different from that in DRG, the overall framework of FPT is similar to DRG. The key steps of FPT can be summarized as follows: (1) State points are first sampled from combustion simulations with the detailed mechanisms. (2) The global flux F and RB for each species are calculated according to eqs 4−6 and eqs 10−12 on each state point. (3) An important species set is obtained on each state point by retaining the species with RB larger than a given threshold. (4) The final skeletal mechanism is achieved from the union of all of these important species sets with corresponding reactions.
3. MECHANISM REDUCTION The detailed USC-II mechanism for ethylene oxidation35 was used as the first example to demonstrate the reduction procedure and performance of FPT. This mechanism consists of 111 species and 784 elementary reactions and was extensively validated against experimental data. Although this mechanism is not large, it is still prohibitive for threedimensional (3D) simulation of ethylene flames, and a substantial reduction is required. The perfectly stirred reactor (PSR) and autoignition simulations were used as the data sources for reduction, with pressure ranging from 0.5 to 30 atm, equivalence ratio ranging from 0.5 to 2.0, initial temperature ranging from 1000 to 1800 K for autoignition and 700 K for PSR, and residence time covering the entire ignition and extinction ranges. SENKIN36 is used to calculate the homogeneous autoignition delay time, which is defined as the
(7)
In this mechanism, F, R, P, and P′ represent fuel, radical, and two products, respectively. Using the fact that R is a QSS species, the rates of the reactions were evaluated as ω2 ≈ ω1 and ω3 ≈ (ω1/103). The global flux vector F for this mechanism as well as the participation flux vector of each species can thus be obtained on the basis of eqs 3, 5, and 6.
F = [ω1, 2.001ω1, ω1, 0.001ω1]T FF = [ω1, ω1, 0, 0]T FR = [ω1, 2.001ω1, ω1, 0.001ω1]T FP = [0, ω1, ω1, 0]T FP ′ = [0, 0.001ω1, 0, 0.001ω1]T
∑ i = 1, I
(8) 5428
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To further demonstrate reliability of FPT, skeletal reduction was also carried out on a detailed n-heptane oxidation mechanism developed by the Lawrence Livermore National Laboratory (LLNL).40 This detailed mechanism consists of 561 species and 2539 elementary reactions. A large number of species and reactions are included in this mechanism to provide a reasonable description on the negative temperature coefficient (NTC) and the two-stage ignition behaviors in the lowtemperature regime. The PSR and autoignition were used as the data source for reduction, with pressure ranging from 1 to 40 atm, equivalence ratio ranging from 0.5 to 1.5, initial temperature ranging from 600 to 1800 K for autoignition and 300 K for PSR, and residence time covering the entire ignition and extinction ranges. The same sampling range was used again for FPT, DRG, and PFA. For DRG and PFA, N2, O2, and n-heptane were selected as the starting important species. A sequence of threshold values from 0.0925 to 0.175 with a step of 0.025 was chosen for DRG, and a sequence of threshold values from 0.16 to 0.33 with a step of 0.01 was used for PFA. The thresholds adopted in FPT are from 0.0002 to 0.0045 with a step of 0.0001. Dependence of the autoignition error upon the number of species in the resulting skeletal mechanism is demonstrated in Figure 4.
time when the temperature is increased by 400 K from the initial condition. PSR37 is used for perfectly stirred reactor simulation. To facilitate performance comparison between FPT, DRG, and PFA, the same sampling range was used. For DRG and PFA, N2, O2, and C2H4 were selected as the initial important species. A sequence of threshold values from 0.14 to 0.5 with a step of 0.005 was adopted for DRG, and another sequence of threshold values from 0.16 to 0.7 with a step of 0.005 was chosen for PFA. The Princeton Chem-RC program12,38,39 was employed to generate skeletal mechanisms with DRG and PFA. On the other hand, the thresholds from 0.001 to 0.035 with a step of 0.001 were used in FPT, and target species are not required. A series of skeletal mechanisms were thus achieved, and errors of autoignition time of these skeletal mechanisms versus the number of species in these mechanisms are illustrated in Figure 3.
Figure 3. Autoignition errors with respect to the number of species in the skeletal mechanisms generated by DRG, PFA, and FPT for combustion of ethylene.
It can be seen from Figure 3 that the autoignition errors of skeletal mechanisms obtained by FPT are smaller than those of skeletal mechanisms from DRG and FPT with the same size when the number of retained species is larger than 36. On the other hand, PFA performs the best for skeletal mechanisms with species number less than 36. With redundancy species being removed gradually, the second or even higher generation flux12 between the remaining species becomes more significant. A multi-generation flux analysis has been adopted in PFA,12 while only the first-generation flux has been adopted in DRG and FPT. This may explain that PFA performs the best when the number of species is small. To generate a skeletal mechanism containing 38 species from the detailed mechanism, it takes about 5, 10, and 20 s with FPT, DRG, and PFA, respectively. It should be noted that computational time of these approaches depends upon details of implementations. However, FPT is more straightforward to obtain the set of important species than DRG and PFA, and it is thus computationally economical. On the other hand, computational effort of the first-generation PFA is similar to DRG, while the second-generation PFA, which is adopted in this work, is more time-consuming.
Figure 4. Autoignition errors with respect to number of species in the skeletal mechanisms generated by DRG, PFA, and FPT for combustion of n-heptane.
According to this figure, skeletal mechanisms with species number larger than 200 from FPT agree better with the detailed mechanism on ignition time than those mechanisms generated from PFA and DRG with the same size. Similar to the case of the USC-II mechanism, mechanisms provided by PFA are the best when species number is less than 200. It should be noted that all of these approaches fail to provide a reasonable skeletal mechanism with a minimum number of species and the more time-consuming directed relation graph aided sensitivity analysis (DRGASA) has to be employed.41 To obtain a skeletal mechanism of about 200 species from the detailed mechanism, FPT, DRG, and PFA take about 2, 11, and 240 min, respectively. This shows that FPT is more efficient that DRG and PFA and it can be used in on-the-fly reduction. FPT was applied to the reduction of the oxidation mechanism of primary reference fuels (PRFs),42 which contains 5429
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It shows that the advantage of FPT on computation cost will be more pronounced for mechanisms with a larger size.
1034 species and 4236 elementary reactions. PRF refers to a fuel mixture of n-heptane and isooctane. PRF50 is composed of 50% isooctane and 50% n-heptane by liquid volume. In the present work, a series of skeletal mechanisms was generated for PRF50. To carry out mechanism reduction, the PSR and autoignition were used as the data source with pressure ranging from 1 to 50 atm, equivalence ratio ranging from 0.3 to 0.7, initial temperature ranging from 750 to 1800 K for autoignition and 300 K for PSR, and the corresponding residence time covering the entire ignition and extinction ranges. The same sampling range was used again for FPT and PFA. It should be noted that there are bugs in DRG.exe, which is enclosed in Princeton Chem-RC.12,38,39 This executable program fails to carry out the reduction of such huge mechanisms. It is very time-consuming to obtain a skeletal mechanism by PFA from such a huge mechanism. To facilitate the mechanism reduction process, a two-stage PFA reduction strategy was applied. In the first step, a skeletal mechanism with 414 species is generated with PFA using N2, O2, IC8H18, and N−C7H16 as target species and a threshold of 0.2. The worst case autoignition error of this mechanism is 6.4%. This skeletal mechanism was further reduced in the second-stage PFA with threshold values from 0.22 to 0.4 with a step of 0.005 to produce a series of skeletal mechanisms. On the other hand, skeletal mechanisms were also achieved with FPT directly from the detailed mechanism. Figure 5 shows the autoignition time errors of the skeletal mechanisms whose retain species numbers are between 250
4. VALIDATION AND ANALYSIS The difference between reduced mechanisms and detailed mechanisms on ignition delay times, extinction times, and laminar flame speeds is usually used to examine the reliability of the reduced mechanism. In fact, a reduced mechanism can usually provide reasonable extinction time and laminar flame speed if it affords ignition delay times that closely resemble those with the detailed mechanism. We will only validate the skeletal mechanism for oxidation of ethylene in the following and skip that for n-heptane and PRF50. The performance of the skeletal mechanism with 36 species and 234 elementary reactions generated by FPT with a threshold of 0.03 for combustion of ethylene will be compared to that of the detailed mechanism. Dependence of autoignition time upon the initial temperature under constant pressure and enthalpy at various pressures and equivalence ratios with the skeletal and detailed mechanisms is demonstrated in Figure 6. The figure shows
Figure 6. Autoignition time with respect to the initial temperature for the ethylene/air mixture at various pressures and equivalence ratios with the detailed and 36 species skeletal mechanisms.
that the skeletal mechanism reproduces results of the detailed mechanism closely. The worst-case autoignition error is 8%, which occurs at a pressure of 30 atm, an equivalence ratio of 2.0, and an initial temperature of 1250 K. The extinction time provides valuable information on combustion characteristics and is important for development and validation of combustion models. Figure 7 shows the comparison between the extinction time calculated with PSR37 using the detailed and 36 species skeletal mechanisms at the pressures of 0.5, 5, and 30 atm, various equivalence ratios, and initial temperature of 700 K. It can be seen that the extinction time profile with the skeletal mechanism agrees rather well with that using the detailed mechanism. The largest relative error is about 18% at a pressure of 5 atm and an equivalence ratio of 2.0. Temperature profiles as a function of the residence time
Figure 5. Autoignition errors with respect to the number of species in the skeletal mechanism generated by PFA and FPT for combustion of PRF50.
and 400. Both PFA and FPT fail to obtain a credible skeletal mechanism with retain species number smaller than 250. Other research shows that DRGASA43 is more suitable for further reduction. It can be seen from this figure that the skeletal mechanism based on FPT has lower autoignition errors than the same size skeletal mechanism from PFA when the number of species exceeds 310. In other cases, FPT is comparable to PFA. As for computational efficiency, FPT is much less demanding than PFA. To generate a skeletal mechanism with about 300 species from the detailed mechanism, it takes about 4 min with FPT, while approximately 36 h is required with PFA. 5430
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Figure 7. Extinction time in PSR at various pressures and equivalence ratios with the 36 species skeletal and detailed mechanisms.
Figure 9. C species flux analysis of ethylene (C2H4) during PSR simulation under 1 atm, equivalence ratio of 1.0, initial temperature of 700 K, and residence time of 3 × 10−5 s. The first and second data are percentages of conversation corresponding to the detailed and 36 species skeletal mechanisms, respectively.
with PSR simulation over wide pressure and equivalence ratio ranges are illustrated in Figure 8. One can see from this figure that temperature profiles with the skeletal mechanism are consistent with those of the detailed mechanism.
Figure 10. Laminar flame speed on the equivalence ratio with the detailed and 36 species skeletal mechanisms.
to provide results that are in good agreement with those of the detailed mechanism, with the largest error of about 2 cm/s at low-pressure and fuel-rich conditions. To further check reliability of the achieved skeletal mechanism with FPT, its performance on species and temperature profiles are also investigated. Figure 11 shows the temperature and species profiles in a laminar premixed flame of a stoichiometric ethylene−air mixture with the skeletal and detailed mechanisms. One can see from this figure that profiles of temperature, some major species, and important radicals of the detailed mechanism are reproduced with high accuracy by the skeletal mechanism. It should be noted that some species, such as CH2O and CH3O, have been affected to some extent, but these species do not seem to have a large effect on the global performance of the mechanism.
Figure 8. PSR extinction temperature profiles with the detailed and 36 species skeletal mechanisms, under various pressures and equivalence ratios.
Element flux analysis44 has been used as an analysis45 tool. This method was employed to identify reaction pathways of detailed and 36 species skeletal mechanisms during PSR simulation under 1 atm, equivalence ratio of 1.0, initial temperature of 700 K, and residence time of 3 × 10−5 s. Figure 9 gives the results of flux for carbon with the detailed and skeletal mechanisms. According to this figure, the flux of carbon with the skeletal mechanism closely resembles that with the detailed mechanism. This indicates that the 36 species skeletal mechanism preserves all of the important species and reaction pathways of the detailed mechanism. This result demonstrates reliability of the skeletal mechanism generated by FPT. Laminar flame speed calculated by PREMIX46 with respect to the equivalence ratio for ethylene/air flames at 0.5, 5, and 30 atm with the detailed and 36 species skeletal mechanisms are shown in Figure 10. Once again, the skeletal mechanism is able
5. CONCLUSION A FPT method to generate a skeletal mechanism is presented and validated in this work. Target species are unnecessary in FPT to achieve a skeletal mechanism. A tree-type relation is established in FPT, and a skeletal mechanism can be generated from it. It is thus more straightforward to reach a skeletal mechanism than those based on a multiple-layer graph structure, such as DRG-based methods and PFA. In addition, significance of a species is evaluated on the basis of its 5431
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Figure 11. Spatial profiles in a 1D planar freely propagating laminar premixed flame with the detailed and 36 species skeletal mechanisms for the temperature, major species, and some important intermediate species.
contribution to the global flux, instead of flux of a single important species. In the implementation, importance of a species is quantified with the normalized projection of its participation flux vector upon the global species flux vector. Performance of FPT is further enhanced by incorporating heat release information into the flux vector. Autoignition delay times of skeletal mechanisms with different sizes generated by FPT, DRG, and PFA are compared for ethylene, n-heptane, and PRF50. Results show that skeletal mechanisms generated by FPT are comparable or even better than those generated by DRG and PFA over a wide parameter range. A 36 species skeletal mechanism for oxidation of ethylene generated by FPT was further validated, and results indicate that it can provide reasonable description on the extinction time, laminar flame speed, and species profiles. FPT can thus serve as a fast preliminary reduction method for mechanisms with large size or be adopted as an on-the-fly reduction approach in reaction flow simulations.
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ASSOCIATED CONTENT
S Supporting Information *
Details of analysis based on DRG and FPT based on an artificial mechanism and the 36 species skeletal mechanism for ethylene oxidation in CHEMKIN format. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (91016002 and 20973118). REFERENCES
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