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. 2024 Feb 15;9(8):9123–9136. doi: 10.1021/acsomega.3c07936

Hybrid Algorithm for Development of Reduced Skeletal Kinetic Mechanisms with Specific Species Targeting

Abbas Babaei Zarch †, Karim Mazaheri †,*, Meysam Khademorezaeian †
PMCID: PMC10905699  PMID: 38434819

Abstract

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A novel algorithm is introduced to reduce the number of detailed kinetic mechanisms. The algorithm uniquely employs a classification of species and a defined parameter that quantitatively identifies the contribution of each species under a combustion condition of interest with specific species targeting. It also incorporates sensitivity analysis, the directed relation graph (DRG) method, and dynamic refinement. The proposed procedure is applied to the GRI-3.0 mechanism with atmospheric pollutants (CO, CO2, NO, and NO2) targeted in a lean methane-air flame at high pressures. The performance of the reduced mechanism is assessed, and good accuracy with considerable computational cost reduction is achieved. Investigated properties by the perfectly stirred reactor (PSR) model incorporated adiabatic flame temperature and the concentrations of targeted pollutants versus equivalence ratio (0.5–1.2), pressure (1–14 bar), and residence time (0.001–1 s) variations. The maximum prediction errors of temperature and greenhouse gas (GHG) mole fractions’ profiles were less than 1%, while NOx versus residence time showed errors of 11.7 and 4%, respectively. Additionally, the flame speed yielded a maximum deviation of less than 2%. The computational cost in a 2D (axisymmetric) simulation revealed a 61% reduction. It is shown that the introduced algorithm is effective and can be applied to large mechanisms aiming for particular species predictions under specified operating conditions with lower computational costs.

1. Introduction

Despite the growth of renewable energies, combustion still produces 90 percent of the total energy generated on the earth. Thus, it can be predicted that combustion will remain the main source of energy for a long time.1 The performance of combustion systems is tied to numerical combustion simulations. The science of combustion modeling has witnessed significant advancements over the years, contributing to our understanding and control over thermal systems, thereby enhancing energy efficiency and curbing emissions. Regarding the application and purpose of these simulations, the required output data and accuracy may vary. Thus, the availability of models with a variety of required computational costs and hardware becomes a matter of significance.

Pollutant-related legislation to mitigate atmosphere contamination urged different industries to accurately predict NOx and GHG production utilizing numerical simulations.2 Chemical mechanisms play a significant role in predicting combustion pollution in the numerical simulation of reacting flows. The more detailed the chemical mechanism, the more accurate the results will be. Utilizing detailed mechanisms that incorporate hundreds of species and thousands of elementary chemical reactions leads to the iterative treatment of a huge and stiff system of ordinary differential equations with different chemical and hydrodynamic time scales. This procedure entails extensive CPU time and expensive hardware for two- or three-dimensional combustion simulations. Developing a reduced mechanism to estimate pollutants for a range of parameters suitable for a specific application would be very useful for the industry.

However, the implementation of these schemes in CFD software is not straightforward as discussed in.3 Computational singular perturbation (CSP),4−8 intrinsic low-dimensional manifold methods (IDLM),9,10 repro modeling,11 rate-controlled constrained equilibrium (RCCE),12−14 flame-generated manifolds method (FGM),15 Roussel and Fraser algorithm (RF),16 and S-step algorithm17 are the main algorithms developed during the last decades to achieve reduced versions of the detailed mechanisms. Generally, in these algorithms, one may select a number of species and steps and reduce them to an acceptable level of lower accuracy. In 2010, Massias et al. used CSP to develop a ten-step mechanism based on GRI-2.1118 for ethane oxidation.6 CSP can be used to identify and eliminate only the elementary reactions that are not important for any species through the use of the importance index.19 Their mechanism could reproduce accurate data for flame speed, temperature, and major species concentrations for the full flammability range of methane-air mixtures and could be used for diffusion flames.

Reduced mechanisms based on the techniques employed in the reduction process can be divided into global and skeletal mechanisms. On the one hand, global reduced mechanisms use nonphysical kinetics, that is, the reaction rates of reduced mechanisms are algebraic functions of the primary mechanism. In 1985, Peters used steady state and equilibrium assumptions to develop a reduced global mechanism for the oxidation of lean methane-air mixtures.20 First, he introduced a four-step global mechanism in which each reaction rate was an algebraic function of 18 elementary reactions. In the validation step using the flame speed parameter, he found that this mechanism yields an excessive H atom production that causes unreasonably high flame speeds. To correct the proposed mechanism, he equated the reaction rate of one of the elementary reactions to zero and acquired a flame speed of 40 cm/s for the stoichiometric condition.

On the other hand, to have a reduced skeletal mechanism, one must select a subset of the main mechanism or another skeletal mechanism. This method includes the quantification of the species and reaction importance based on a quantitative consideration (and probably for a particular application) and then eliminates unimportant species and reactions from the detailed mechanism.

Many schemes are introduced to find a suitable skeletal mechanism, for example, directed relation graph (DRG),19 directed relation graph with error propagation (DRGEP),21 and sensitivity analysis.22 Although these methods were introduced decades ago, they directly or as a portion of a larger hierarchical procedure are still being utilized to develop new reduced skeletal mechanisms for various fuel surrogates23−26 On the other hand, new methods are still being developed for skeletal reduction of large detail kinetic mechanisms. In 2023, Liu et al.27 developed a method based on entropy production analysis in homogeneous autoignition processes and laminar premixed flames. Their method included the consideration of entropy production from chemical reactions, mass diffusion, and heat conduction. Tuo et al.,28 introduced a skeletal reduction method based on the hierarchical sensitivity analysis and used it to reduce the size (450 species and 4569 reactions to 46 species and 197 reactions) of the 135TMB mechanism.29 While their method is implemented in terms of carbon levels, it is able to select targets such as autoignition delay time, laminar flame speed, flame extinction, speciation, and so forth. They also incorporated the quasi-steady-state approximation and reaction lumping for further reduction. Li et al.,30 introduced a method in which species are ranked according to their presence in flame; then, reactions are ranked in accordance with the generation of those species. Finally, a set of influential species and an extracted database from laminar premixed flame simulation determine the elimination of reactions and the accuracy of the concerned predictions. They called their method the object-oriented directed path screening (OoDPS) method. They claim that the assessment resulted in a more accurate prediction with respect to DRGEP that reflects relative superiority. They also mentioned that in high reduction levels, a discrepancy in predicted OH concentration is observed with changing the concentrations of ammonia in the ammonia-methane fuel mixture. In 2024, Liu et al.,31 developed a so-called “forced-optimally time-dependent (f-OTD)” methodology in which local sensitivities of the thermo-chemical variables with respect to the reaction rates are compared. Similar to the present case, with the aim of performing validation of the resulting reduced skeletal mechanism, they applied their algorithm to methane combustion.

Recalling the skeletal reduction of large mechanisms incorporating methane combustion, in 1994, Frenklach and Kazakov32−35 used the GRI-2.11 mechanism as a reference to introduce a reduced skeletal mechanism with 84 reactions and 19 species (DRM-19). In the reduction procedure, a better and safer strategy, they simultaneously used two criteria to assess the contribution a given reaction makes to the main chain branching (criterion one) and heat release (criterion two). These criteria were examined, while a zero-dimensional constant-pressure model (with the parent mechanism) was used. They asserted that criterion 1 is required for the reduction analysis of flame-initiating reactions of the preheated zone, while criterion 2 provides a better selection for reactions describing the main reaction zone of a premixed flame. Finally, the assessment of their iterative procedure was based on the flame temperature profiles in a laminar premixed atmospheric stoichiometric adiabatic methane-air flame when the resulting skeletally reduced mechanism was employed. The mentioned method was applied with uncertainty quantification in 2021 by Li et al.36 They addressed uncertainty quantification and model discrepancy by utilizing a novel version of the bound-to-bound data collaboration (B2BDC) method to offer a more comprehensive understanding of uncertainty in predictions. The application of this extended framework to a hydrogen combustion model demonstrates its effectiveness in recovering data set consistency and improving prediction accuracy through scenario-dependent model corrections.

Senkaran et al.37 also extracted a skeletal mechanism using DRG and sensitivity analysis with 73 steps (reactions) and 17 species. However, the incorporation of nitrogen oxide chemistry to add the capability of NO production prediction into a reduced skeletal mechanism demands additional considerations. In 2013, Karalus et al.38 proposed a 177-step reduced skeletal mechanism to predict NOx generation in methane oxidation. They employed the PSR model in CHEMKIN-Pro software and integrated sensitivity analysis and the DRG method to reduce the GRI-3.039 mechanism.

The problem of pollutant emission prediction in the lean combustion of methane-air mixtures has been found to be well-addressed in the literature. Thus, the rich background regarding the detail, physics, and results creates a valuable basis for the validation of the newly developed algorithm here.

In 1992, to predict the production and consumption of hydrocarbon radicals in a PSR model, Glarborg et al.40 used steady state and partial equilibrium assumptions to reduce a 77-reaction skeletal mechanism to a four-step global mechanism. They showed that this model predicted NOx generation in methane oxidation well for different values of equivalence ratios, inlet temperature, and residence times; however, results are not acceptable for rich or low-temperature inlet conditions. Sung et al.3 used a computer program called CARM (computer-assisted reduction mechanism) to design a reduced global mechanism comprising nitrogen chemistry based on the GRI-3.0 mechanism. By adding different NOx-generating mechanisms such as prompt, nitrous oxides, and ammonia, they introduced new 14-, 15-, and 17-step reduced mechanisms. CARM was developed by Montgomery et al.41 and is linked to CHEMKIN-Pro. This program inverts a full mechanism to a reduced global mechanism based on the given inlet conditions, and the reaction rates are the algebraic functions of the elementary reaction rates. Lu and Law42 also used QSS analysis and the DRG method to reduce GRI-3.0 to 19 species and 15 lumped steps, providing a fairly accurate prediction mechanism for various phenomena, including perfectly stirred reactors, autoignition, premixed, and nonpremixed reactors. They also added a few reactions to the model to enhance its capability for NOx prediction. Systematic development of 14-step and 16-step mechanisms to predict NOx emission from methane combustion was carried out by Chen and co-workers.43 The 14-step mechanism did not lead to satisfying results to predict NOx emissions, while the 16-step mechanism was relatively accurate for mixture fractions of less than 0.1. They also contended that the presence of C2H6 and C2H4 impacts the accuracy of the mechanism, in terms of species prediction. Since this method uses global steps, its computational time is still much longer than that of skeletal mechanisms.

Many people have used the DRG method, sensitivity analysis, and CSP to find reduced skeletal mechanisms. In 2014, a 46-step mechanism including 34 species was introduced by Azevedo et al.44 Their mechanism was reasonably accurate in predicting the adiabatic temperature, flame front, and adiabatic flame velocity, while the accuracy of nitrogenous species prediction was not entirely satisfying. In 2020, based on the GRI-3.0 mechanism, utilizing the DRG method and sensitivity analysis, a 65-step mechanism with 22 species was introduced to predict NOx emissions in turbulent methane flames.45 In 2006, Valorani et al.46 used the CSP scheme to introduce a repeatable algorithm to reduce a detailed mechanism to a skeletal mechanism. This algorithm removes less significant fast or slow reactions. They proved the efficacy of this method by comparing the results obtained from autoignition analysis in a constant-pressure reactor and the simulation of a premixed flame and a counter-flow diffusion flame.

In the quest for an optimal balance between model fidelity and computational efficiency, we introduce a hybrid reduction algorithm that combines the strengths of the DRG method and sensitivity analysis. This also introduces a delta δrel parameter to quantitatively identify the species’ contribution to fuel decomposition and targeted species production/consumption rates under specified combustion conditions. Thus, this work presents a new strategy for mechanism reduction, offering a robust, fairly accurate, and computationally efficient approach to combustion modeling.

The novelty of this work lies in the incorporation of a kinetic-based classification of species and allocation of independent reduction criterion to each group’s chemistry followed by a production/consumption analysis employing a novel parameter (δrel). The algorithm is able to systematically aim the prediction of the concentration of particular minor species that do not strongly determine the major characteristics of the combustion process (like temperature distribution, ignition, flame speed, etc.), but their concentration/emission is of concern (e.g., NOx). This is feasible when the corresponding responsible submechanism is shielded from the main reduction procedure by selecting additional but proper criteria for the elimination of its components (species and reactions). In other words, different assessment/elimination criteria are allocated to submechanisms of interest and the whole parent mechanism itself. Otherwise, the important species and reactions that determine the formation of the concerned species may be eliminated after an unsuitable filtration (or identified to be unimportant by general criteria that consider only the general characteristics of the combustion process). This is not exclusive to the present algorithm to systematically aim for particular species concentrations during skeletal reduction. DRM is also capable of aiming for the prediction of a minor species concentration by defining a new criterion (as a mathematical relation) based on one of the effective reactions. However, the introduced method here uses a new technique and prospect to fulfill this goal.

Here, to demonstrate the effectiveness of the proposed algorithm, the lean combustion of preheated methane-air mixtures in elevated pressures is studied. The GRI-3.0 mechanism39 which includes NOx and greenhouse gas production mechanisms, including 53 species and 325 reactions is used as the parent mechanism. Additionally, retaining reduced mechanism accuracy in the prediction of NOx and GHG concentrations was the goal during the reduction procedure. The algorithm’s potential to reduce the skeletal mechanism while consistently replicating the results of the original mechanism underscores its effectiveness and potential application in the field of combustion modeling. The comparison between the results obtained from two skeletally reduced mechanisms (with different techniques, but with the same parent mechanism and goal) was also conducted to assess the superiority of the introduced algorithm.

2. Criterion to Determine Species Contribution Significance

Regarding a particular operating/combustion condition (i.e., equivalence ratio, pressure, unburnt mixture temperature, etc.), production rate analysis is the main part of many mechanism reduction algorithms.19,21 The analysis determines each species’ production and consumption rate and identifies the species with similar production and consumption rates. Sometimes, the results of this analysis are used to find simple algebraic equations in order to reduce the stiffness of a system of equations. This analysis aims to identify steady-state species that can then be removed during a reduction process. Here, δabs is a measure of each species’ net production/consumption rate:

2. 1

where Inline graphic and Inline graphic are the production/consumption rates of the i-th species, including all reactions, respectively.

This gives the normalized absolute value of the production/consumption rate. The higher values of δabs can be translated to the higher level of a species’ contribution significance to the overall chemistry of interest, regardless of its abundance within the physical domain. Meanwhile, its small values point at (quasi-steady-state assumption) QSSA for the species.

Since different species have different concentrations (mole fractions here) of χi, to consider the relative significance of each species, a modified form of δabs,i is used, where

2. 2

Here, χi is the mole fraction. Evaluation of this parameter in specified combustion conditions (utilizing a PSR in steady-state) is the first step that entangles the reduction procedure with the specified conditions.

Løvs et al.,47 introduced a similar parameter “LOI” (Level of Importance) for every species in a detailed mechanism for reduction purposes. However, LOI is calculated via complex mathematics and notions, while δrel is simply a normalized and scaled consumption/production rate of species in every reaction in which it participates. Additionally, result of the LOI analysis is the elimination of the selected species from the mechanism; while in the algorithm introduced here, the species that do not satisfy the criterion are only omitted from particular steps of the algorithm. This means that the final mechanism can still contain the species that did not satisfy the criterion defined by δrel.

3. Introducing the Reduction Algorithm

To have a clearer vision of how the reduction procedure works, a chart is presented in Figure 1 for more clarification. The five steps of the algorithm are outlined as follows:

  • 1.

    Specify fuel, targeted species, and combustion conditions (e.g., pressure, unburnt mixture temperature, equivalence ratio, etc.). Classify the species of the original mechanism into different groups which contribute to the fuel decomposition, and the submechanisms responsible for the production/consumption rate of the targeted species.

  • 2.

    Considering the criterion (δrel) for all species and selecting the critical species from each group that plays the major role in combustion conditions specified in step one. In this step, a pool of species is obtained (pool number one).

  • 3.

    Perform a sensitivity analysis and obtain important reactions entangled to the selected species in step two. (These reactions may include species other than the ones that satisfied the δrel criterion). In this way, pool number two is comprised.

  • 4.

    Using the DRG method eliminates the species, which simultaneously has a weaker impact on the decomposition of the fuel and consumption/production rates of the targeted species.

  • 5.

    For further reduction, examine the performance of the mechanism after applying different cut-off criteria for the filtering steps (Steps II, III, and IV) as a closed-loop algorithm.

Figure 1.

Figure 1

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Reduction algorithm flowchart (boxes with solid boundaries) and the corresponding outputs after being applied to the GRI-3.0 mechanism with SGT600 gas turbine operating conditions (boxes with dashed boundaries).

The presented algorithm employs a systematic and targeted approach, leading to reduced skeletal mechanisms. In the first step, by imposing specified combustion conditions to δrel calculations, the reduction procedure will be guided toward the more relevant output. Classifying the species also helps to protect the critical submechanisms from automatic removal based on general assumptions. This step is pivotal because it creates a platform for our targeted analysis. The next step involves the application of the δrel criterion to identify the critical species that play a significant role under the specified combustion conditions. Next, a sensitivity analysis is performed to identify important reactions tied to the selected species from the previous step. This process helps identify the species that, while they may not satisfy the δrel criterion, are part of significant reactions. Therefore, it acts as a safeguard, preventing the inadvertent elimination of such species and ensuring the integrity of the mechanism. Following this, the DRG method is used to eliminate species that have a minimal impact on both the decomposition of the fuel and the consumption/production rates of the targeted species.

As a case study, we have applied the above algorithm to GRI-3.0 mechanism for lean and preheated methane-air combustion in elevated pressures and assuming NO, NO2, CO, and CO2 species as the targeted species. Figure 1 illustrates the algorithm flowchart to provide a more conceivable procedure for applying this reduction algorithm to the GRI-3.0 mechanism. In this chart, boxes with solid lines describe the reduction procedure steps, and boxes with dashed borders describe the output.

4. Applying the Reduction Procedure

Assessment of the reduced mechanism performance generated by the introduced algorithm entails solving a problem with a known answer. The rich literature and well-known physics of methane oxidation in lean regimes urge us to apply the reduction algorithm to a validated detailed mechanism developed for such a problem. Thus, GRI-3.0 is selected as the detailed mechanism, and the operating condition of the SGT600 industrial ground gas turbine (as a representative of real practical procedure) is the desired combustion condition while NO, NO2, CO, and CO2 are the targeted species.

4.1. Species and Combustion Condition Targeting

The first step of the algorithm incorporates aiming for targeted species and specifying combustion or operating conditions. This is an essential step in the reduction procedure due to its impact on multiple subsequent steps. The introduced algorithm will be used to target pollutant emission prediction in 14 bar, an unburned lean (equivalence ratio equal to 0.65) mixture of methane-air with 640 K temperature, a mass flow rate of 4 kg/s, and a residence time of 10 ms.

4.2. Classification of Species and Calculating Species Contribution Significance

The next step is to classify the species of the detailed mechanism into groups that comprise the submechanisms responsible for the chemistry of fuel decomposition and consumption/production rates of targeted species.

Here, NO, NO2, CO, and CO2 are targeted species. On the one hand, the chemistry of fuel decomposition, as well as CO, and CO2 production/consumption rates are highly coupled. On the other hand, NO and NO2 submechanisms are not separable. Thus, all the species are divided into the C group (shown partially in Table 2) and N group (shown in Table 3). The C group species are responsible for CH4 decomposition and CO and CO2 production/consumption rates, and the N group holds the same stance for nitrogen oxide chemistry.

Table 2. First 21 Species of Group C Sorted Based on Relative Combustion Rates.

no. species δrel,i
1 O2 0.001506
2 H2O 0.000694
3 CO2 0.000573
4 CH4 3.02 × 10–05
5 OH 3.77 × 10–08
6 CO 7.98 × 10–09
7 H2 1.49 × 10–09
8 O 1.69 × 10–11
9 H 3.71 × 10–12
10 HCCOH 2.45 × 10–12
11 HO2 1.01 × 10–12
12 H2O2 9.74 × 10–13
13 CH2O 8.8 × 10–13
14 CH3OH 6.78 × 10–13
15 CH3 1.04 × 10–13
16 C2H2 6.24 × 10–14
17 C2H4 1.74 × 10–14
18 CH2CO 9.4 × 10–15
19 C2H6 7.06 × 10–15
20 HCO 2.32 × 10–15
21 CH2 1.67 × 10–15
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Table 3. Species of Group N Sorted Based on Relative Combustion Rates.

no. species δrel,i
1 N2 0.008417
2 NO 9.98 × 10–08
3 N2O 1.03 × 10–09
4 HCNO 8.36 × 10–11
5 NH3 9.64 × 10–12
6 HNCO 5.45 × 10–12
7 NO2 3.09 × 10–12
8 HOCN 1.50 × 10–12
9 HCN 5.21 × 10–13
10 NCO 6.63 × 10–15
11 HNO 1.76 × 10–15
12 NNH 4.99 × 10–16
13 NH2 3.64 × 10–16
14 NH 4.51 × 10–17
15 CN 9.74 × 10–18
16 N 6.18 × 10–18
17 HCNN 1.79 × 10–19
18 H2CN 5.76 × 10–20
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It is critical to eliminate as many species as possible when it comes to computational cost reduction because it is more dependent on the number of species that a kinetic mechanism involves rather than the number of reactions.20 The PSR module in the CHEMKIN-Pro software is used with the GRI-3.0 mechanism to sort all species based on their δrel value (see Table 1).

Table 1. Sorted Species Are Based on Relative Combustion Rates.

no. species δrel,i
1 N2 0.008417
2 O2 0.001506
3 H2O 0.000694
4 CO2 0.000573
5 CH4 3.02 × 10–05
6 NO 9.98 × 10–08
7 OH 3.77 × 10–08
8 CO 7.98 × 10–09
9 H2 1.49 × 10–09
10 N2O 1.03 × 10–09
11 HCNO 8.36 × 10–11
12 O 1.69 × 10–11
13 NH3 9.64 × 10–12
14 HNCO 5.45 × 10–12
15 H 3.71 × 10–12
16 NO2 3.09 × 10–12
17 HCCOH 2.45 × 10–12
18 HOCN 1.5 × 10–12
19 HO2 1.01 × 10–12
20 H2O2 9.74 × 10–13
21 CH2O 8.8 × 10–13
22 CH3OH 6.78 × 10–13
23 HCN 5.21 × 10–13
24 CH3 1.04 × 10–13
25 C2H2 6.24 × 10–14
26 C2H4 1.74 × 10–14
27 CH2CO 9.4 × 10–15
28 C2H6 7.06 × 10–15
29 NCO 6.63 × 10–15
30 HCO 2.32 × 10–15
31 HNO 1.76 × 10–15
32 CH2 1.67 × 10–15
33 CH2OH 1.56 × 10–15
34 CH3O 1.47 × 10–15
35 NNH 4.99 × 10–16
36 NH2 3.64 × 10–16
37 C2H3 1.2 × 10–16
38 CH3CHO 1.16 × 10–16
39 C2H5 1.07 × 10–16
40 CH2(S) 7.55 × 10–17
41 HCCO 5.68 × 10–17
42 NH 4.51 × 10–17
43 C2H 3.6 × 10–17
44 CN 9.74 × 10–18
45 N 6.18 × 10–18
46 CH2CHO 4.46 × 10–18
47 CH 1.65 × 10–18
48 C3H8 1.37 × 10–18
49 HCNN 1.79 × 10–19
50 H2CN 5.76 × 10–20
51 C3H7 3.05 × 10–20
52 C 7.43 × 10–21
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Using a threshold value of 10–9 for the δrel parameter, six species were selected (O2, H2O, CO2, CH4, OH, and CO) from group C and three species (N2, NO, and N2O) from group N (see Tables 2 and 3). Here, the allocated threshold of the δrel parameter can vary for each group if the calculated values for major species in each group are not in the same order.

In the last loop (step 5) of the algorithm, H2 from group C is removed to not get involved in the sensitivity analysis. Subsequently, these nine species are used for sensitivity analysis in the next section in order to decide which reactions must be retained.

Since in this section, one only decides to retain particular species for sensitivity analysis and no species are eliminated yet. In the next step, the sensitivity analysis can identify important reactions that involve species other than these nine species thereof.

4.3. Sensitivity Analysis

A sensitivity analysis is conducted based on the first-order sensitivity coefficient defined in the reference.17 In this definition, the dependent variable is either the concentration of selected species χi or adiabatic flame temperature T, and the independent variable is the reaction rate coefficient in the Arrhenius equation, Inline graphic. The sensitivity coefficients are defined as follows:

4.3. 3

A value of 0.01 for beta means that increasing the rate of reaction j by 100% results in 1% increase of species i. The sensitivity analysis was carried out by CHEMKIN-Pro48 using a PSR model and GRI-3.0 mechanism for equivalence ratios between 0.5 and 0.8. The inlet pressure and temperature are again 14 bar and 640 K, respectively, with the same residence time of 10 ms.

The analysis is performed for all nine species selected above and for all 325 reactions of GRI-3.0 (i.e., 2925 cases are studied). Table 4 demonstrates the cut-off values allocated to each species for this analysis. These cut-off values were selected in order to have the smallest possible number of reactions, while the accuracy of the resulting mechanism in terms of concentration prediction in a PSR with the specified conditions is still acceptable. Finally, this sensitivity analysis shows the relative significance of each reaction. For species of group C, 39 reactions are selected, including 19 species (see Table 5). Similarly, this procedure resulted in the identification of seven reactions for species of group N with six species. These 25 species and their relative consumption rates are listed in Tables 5 and 6 based on the value of relative consumption rates. Up to this point, this reduced mechanism includes 25 species and 46 reactions. In the next section, the mechanism is further reduced using the DRG method.

Table 4. Cut-Off Values in Sensitivity Analysis for Different Species in Groups C and N.

species CH4 O2 H2O, CO2, OH, CO N2 NO, N2O
cut-off 0.005 0.001 0.0001 0.00001 0.1
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Table 5. List of 19 Species Resulting from the Sensitivity Analysis for Group C.

no. species group C δrel
1 O2 0.001506
2 H2O 0.000694
3 CO2 0.000573
4 CH4 3.02 × 10–05
5 OH 3.77 × 10–08
6 CO 7.98 × 10–09
7 H2 1.49 × 10–09
8 O 1.69 × 10–11
9 H 3.71 × 10–12
10 HO2 1.01 × 10–12
11 H2O2 9.74 × 10–13
12 CH2O 8.8 × 10–13
13 CH3OH 6.78 × 10–13
14 CH3 1.04 × 10–13
15 HCO 2.32 × 10–15
16 CH2 1.67 × 10–15
17 CH2OH 1.56 × 10–15
18 CH3O 1.47 × 10–15
19 CH2(S) 7.55 × 10–17
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Table 6. List of 6 Species Resulting from the Sensitivity Analysis for Group N.

no. species group N δrel,i
1 N2 0.008417
2 NO 9.98 × 10–08
3 N2O 1.03 × 10–09
4 NO2 3.09 × 10–12
5 NH 4.51 × 10–17
6 N 6.18 × 10–18
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4.4. DRG Method Reduction

DRG is a methodology used to develop reduced mechanisms. The graph shows how each species contributes to the production of other species. It consists of a few nodes (species), and directed arrows show which species (arrow tail) contributes to the production of another species (arrowhead), and the thickness of each arrow shows the consumption/production rate. The thickness can be replaced by the number on the corresponding arrow that identifies the production/consumption of each species per unit time.49

Figure 2 shows the main path of methane decomposition and the arrows toward the product species in important reactions by which CH4 was decomposed (not the species that participated in the decomposition reaction as reactants). This figure shows that species CH2, CH2(s), and CH3 play a significant role in the decomposition of methane, while they are located in the lowest rows of Table 5. The relative significance of methane decomposition to CH3 in comparison to H and OH through reactions CH4 + M → CH3 + H + M and CH4 + O → CH3 + OH, respectively, was also reported by Peters.20 Therefore, based on DRG, they are considered important. Figure 3A,B shows the CO- and CO2-dominated production/consumption rate paths, respectively. Regarding the content in Table 5, one observes that HCO and CH2O are also located in the lowest rows, but Figure 3 illustrates that they play a major role in the production of CO and CO2. Thus, they are not removed from the reduced mechanism.

Figure 2.

Figure 2

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Main path of methane decomposition in GRI-3.0 with the consumption/production rate for each path (results of the DRG method).

Figure 3.

Figure 3

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Main path of production of CO (A) and CO2 (B) with the consumption/production rate for each path (results of the DRG method).

Based on the DRG analysis here, the smallest contribution to methane consumption or the production of the main products can be attributed to CH3O, CH2OH, and CH3OH. Therefore, these three species and the reactions in which they participate (8 reactions in Table 7) can be eliminated from the reduced mechanism.

Table 7. List of Reactions Corresponding to the Less Significant Species of Group C in Table 5.

no. reaction A b E
21 OH + CH3OH ⇔ CH3O + H2O 6.30 × 1006 2 1500
22 HO2 + CH3 ⇔ OH + CH3O 3.78 × 1013 0 0
30 CH3 + O2 ⇔ O + CH3O 3.56 × 1013 0 30,480
35 CH3O + O2 ⇔ HO2 + CH2O 4.28 × 10–13 7.6 –3530
8 H + CH2OH ⇔ OH + CH3 1.65 × 1011 0.7 –284
20 OH + CH3OH ⇔ CH2OH + H2O 1.44 × 1006 2 –840
14 OH + CH3(+M) ⇔ CH3OH(+M) 2.79 × 1018 1.4 1330
27 CH2(S) + H2O(+M) ⇔ CH3OH(+M) 4.82 × 1017 0.2 1145
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The DRG method was utilized to investigate the species in group N for further reduction. This corresponds to the production and consumption rates of NO and N2O. A complete list of these reactions is given in Table 8. The paths are shown in Figure 4A,B. Based on the results of DRG analysis for these pollutants, the rates of reactions resulting in N2O are very low and N2O is not considered important in the final products. The main species produced from N2O is NH, which is not a significant species based on production rate analysis. Therefore, reactions related to N2O may be removed. As a result, three reactions in addition to species NH and N2O from the last lists can be eliminated with no major impact on the main product prediction. In the final stage, the Zeldovich main reactions50 and a reaction for NO2 are left. In the operational conditions considered in this article, NOx is mostly produced by this mechanism. In the end, 35 reactions and 21 species remain in the reduced mechanism, as listed in Table 9.

Table 8. List of Reactions Corresponding to the Species in Group N.

no. reaction A b E
40 H + O2 ⇔ O + OH 2.65 × 1016 –0.7 17,041
41 N + NO ⇔ N2 + O 2.70 × 1013 0 355
42 N + O2 ⇔ NO + O 9.00 × 1009 1 6500
43 N2O + O ⇔ 2NO 2.90 × 1013 0 23,150
44 N2O + H ⇔ N2 + OH 3.87 × 1014 0 18,880
45 HO2 + NO ⇔ NO2 + OH 2.11 × 1012 0 –480
46 NH + NO ⇔ N2O + H 3.65 × 1014 –0.5 0
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Figure 4.

Figure 4

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Main path of production/consumption rates of NO (A) and N2O (B) in GRI-3.0 with the consumption/production rate for each path (results of the DRG method).

Table 9. Final List of Reactions Retained in the Reduced Mechanism.

no. reaction A b E
1 O + HO2 ⇔ OH + O2 2.00 × 1013 0 0
2 O + CH3 ⇔ H + CH2O 5.06 × 1013 0 0
3 O + CH4 ⇔ OH + CH3 1.02 × 1009 1.5 8600
4 H + O2 + M ⇔ HO2 + M 2.80 × 1018 –0.9 0
5 H + 2O2 ⇔ HO2 + O2 2.08 × 1019 –1.2 0
6 H + O2 + H2O ⇔ HO2 + H2O 1.13 × 1019 –0.8 0
7 H + O2 + N2 ⇔ HO2 + N2 2.60 × 1019 –1.2 0
8 OH + H2 ⇔ H + H2O 2.16 × 1008 1.5 3430
9 2OH(+M) ⇔ H2O2(+M) 7.40 × 1013 –0.4 0
10 2OH ⇔ O + H2O 3.57 × 1004 2.4 –2110
11 OH + HO2 ⇔ O2 + H2O 1.45 × 1013 0 –500
12 OH + H2O2 ⇔ HO2 + H2O 1.70 × 1018 0 29,410
13 OH + CH3 ⇔ CH2 + H2O 5.60 × 1007 1.6 5420
14 OH + CH3 ⇔ CH2(S) + H2O 6.44 × 1017 –1.3 1417
15 OH + CH4 ⇔ CH3 + H2O 1.00 × 1008 1.6 3120
16 OH + CO ⇔ H + CO2 4.76 × 1007 1.2 70
17 OH + CH2O ⇔ HCO + H2O 3.43 × 1009 1.2 –447
18 CH2+O2 ⇒ OH + H + CO 5.00 × 1012 0 1500
19 CH2(S) + N2 ⇔ CH2 + N2 1.50 × 1013 0 600
20 CH2(S) + O2 ⇔ H + OH + CO 2.80 × 1013 0 0
21 CH2(S) + O2 ⇔ CO + H2O 1.20 × 1013 0 0
22 CH2(S) + H2O ⇔ CH2 + H2O 3.00 × 1013 0 0
23 CH2(S) + CO2 ⇔ CO + CH2O 1.40 × 1013 0 0
24 CH3 + O2 ⇔ OH + CH2O 2.31 × 1012 0 20,315
25 HCO + H2O ⇔ H + CO + H2O 1.50 × 1018 –1 17,000
26 HCO + M ⇔ H + CO + M 1.87 × 1017 –1 17,000
27 HCO + O2 ⇔ HO2 + CO 1.34 × 1013 0 400
28 O + CH3 ⇒ H + H2 + CO 3.37 × 1013 0 0
29 OH + HO2 ⇔ O2 + H2O 5.00 × 1015 0 17,330
30 CH2 + O2 ⇔ 2H + CO2 5.80 × 1012 0 1500
31 CH2 + O2 ⇔ O + CH2O 2.40 × 1012 0 1500
32 H + O2 ⇔ O + OH 2.65 × 1016 –0.7 17,041
33 N + NO ⇔ N2 + O 2.70 × 1013 0 355
34 N + O2 ⇔ NO + O 9.00 × 1009 1 6500
35 HO2 + NO ⇔ NO2 + OH 2.11 × 1012 0 –480
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Here, the N2O path is eliminated by the algorithm. In the previous works, N2O was considered a major species for lean methane-air premixed flames at high pressures51,6,52 not because its pathway contribution is more significant than prompt or thermal NO production but for its major role as an inert species and its slow contribution in the system. However, manual analysis (not manual elimination) by the author done here, in terms of retaining or eliminating the N2O path in the reduced mechanism, showed that by retaining the N2O path, the resulting NO mole fraction curve with respect to pressure variations will not have the same trends as those when GRI-3.0 was employed. It means that, in a particular operating pressure, the presence of the N2O path can result in a more accurate NO prediction, while in other pressures, it can lead to worse prediction. Here, it seems that elimination of the N2O path will result in the predictable underprediction of NO concentrations with respect to pressure variations.

5. Result and Discussion

Kuligowsky and co-workers53,54 employed the Sandia steady, laminar, one-dimensional, premixed flame code55 and GRI-3.0 mechanism to obtain NO concentration of lean premixed methane-air flames. The results of their comparison between numerical and experimental data are shown in Figure 5. This figure compares the experimental and numerical amount of NO production obtained from the oxidation of the lean premixed methane-air mixture at an equivalence ratio of 0.8 for different pressures. The consistency of the two sets of data confirms the validity of the GRI-3.0 mechanism. Thus, the obtained data from numerical simulations using the GRI-3.0 mechanism at lean conditions and pressures up to 14 atm is an appropriate reference to validate the functionality of the introduced reduced mechanism.

Figure 5.

Figure 5

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Comparison of NO concentration based on GRI-3.0 with experimental results.3,40

5.1. PSR Analysis in SGT600 Combustion Condition

In order to examine the performance of the introduced 35-step mechanism against other reduced mechanisms that aimed at NOx prediction, two reduced mechanisms with 17738 and 6545 steps were selected. This comparison seems to be logical because all of the mechanisms have the same parent mechanism and are aimed to keep the capability of NO concentration prediction after size reduction. This can also be referred to as a comparison among kinetic mechanism reduction methods in which two of them used DRG and sensitivity (177-step and 65-step) and the one that is extended by novel steps (the introduced 35-step). Utilizing the PSR model in CHEMKIN-Pro software, the distribution of the temperature and mole fractions of CO, CO2, NO, and NO2 (if applicable to the mechanism) were obtained with respect to variation of equivalence ratio (0.5–1.2), pressure (1–14 bar), and residence time (0.01–1 Sec). During this procedure, if a parameter was desired to be kept constant, values of 14 bar, 640 K, 0.65, 4 kg/s, and 0.01 s were selected as reference for pressure, inlet temperature, equivalence ratio, mass flow rate, and residence time, respectively.

In Figure 6, similar distributions of temperature with respect to the equivalence ratio, residence time, and pressure variations in the steady state demonstrate an agreement between the results obtained by all reduced mechanisms and GRI-3.0. The 65-step mechanism slightly (less than 1%) underpredicts the temperature in high residence times. This illustrates that in this mechanism, a portion of reactions responsible for the regulation of heat release in conditions wherein the mixture is subjected to a favorable combustion condition might be removed.

Figure 6.

Figure 6

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Steady-state temperature, CO, CO2, NO, and NO2 distributions of methane-air oxidation in a PSR utilizing the developed mechanism here (35-step), the 65-step mechanism,45 the 177-step mechanism,38 and the GRI-3.0 mechanism39 with SGT600 combustion conditions.

More discrepancies can be witnessed in the distributions of CO and CO2 in the results obtained by the 65-step mechanism in comparison to the others. The CO2 deviation with respect to residence time has a trend similar to that of the temperature distribution. Since the reactions in which CO and CO2 take parts as reactants and products, respectively, are a portion of reactions responsible for heat release, this deviation could be expected regarding the temperature curves. However, there are negligible differences in temperature, CO, and CO2 curves resulting from the 35-step skeletal mechanism introduced here in comparison with the 177-step and the GRI-3.0 in rich combustion conditions (higher equivalence ratios), high residence times, and elevated pressures.

NOx distributions show relatively larger discrepancies in lean combustion with equivalence ratios close to 0.94, high residence times, and elevated pressures. Since the other selected reduced mechanisms (i.e., 65-step, and 177-step mechanisms) do not incorporate NO2 species, the final graphs are only plotted for the GRI-3.0 and the 35-step mechanism. NO distributions obtained from all mechanisms show the same trends with respect to the equivalence ratio and residence time variations. The maximum error between the NO distributions obtained from the 35-step mechanism with respect to the GRI-3.0 results is 11.7% (at equivalence ratio ∼ 0.94) and 4% (at residence time = 1), while the same errors were 35 and 12.5% for the 65-step mechanism. The maximum errors for NO2 predictions are also 11 and 2.4% in equivalence ratio and residence time data.

The curves obtained for NOx mole fractions versus pressure variations show different trends in some cases. The NO curve obtained from the 65-step mechanism and the NO2 curve from the 35-step is of this category, while the 35-step mechanism always underpredicts the NO mole fraction (with 34% error) with a similar trend for the GRI-3.0 mechanism. The 65-step mechanism underpredicts NO in pressures below ∼3.5 bar and overpredicts this quantity (errors of −16% in 1 bar and +47.7% in 14 bar). The resemblance between the curve’s trends with a specified offset makes the final data obtained from the 35-step mechanism still valuable as a rare feature among reduced mechanisms with the same level of reduction.

5.2. Flame Speed Analysis

Using the flame speed module in CHEMKIN-Pro software, the methane-air flame speed with respect to equivalence ratio variations (0.6–0.8 in accordance with the targeted equivalence ratio) for all of the four kinetic mechanisms used here was calculated (see Figure 7).

Figure 7.

Figure 7

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Flame speed calculation with respect to equivalence ratio variations for P = 1 bar and TUnburnt = 300 K (left graph) and SGT600 operating conditions (i.e., P = 14 bar and TUnburnt = 640 K) (right Graph) using the GRI-3.0,39 177 steps,38 65 steps,45 and introduced kinetic mechanisms.

This confirms that the introduced mechanism can accurately predict the flame speed at arbitrary pressures if the mixture equivalence ratio stays low.

5.3. Computational Cost Evaluation

A two-dimensional (axisymmetric) steady-state simulation of Sandia flame D56,57 was conducted to determine the computational cost efficiency of the introduced mechanism in comparison with two other reduced mechanisms. In this simulation, the ANSYS-Fluent 1958 software was used, and all configurations were set similar to the literature.59,60 Since the primary purpose of this simulation was only the determination of computational costs, a relatively coarse grid was selected with 0.3 × 0.3 mm cells in the flame zone. This entails a very small deviation in the temperature and axial velocity profile from the finer mesh in the reference.59 The EDC model61,62 and Reynolds stress model (RSM)63−65 were used to model turbulence-chemistry interactions and turbulence, respectively. The coupled pressure–velocity coupling scheme and the PRESTO pressure interpolation schemes were also employed with second-order discretization for species. Contours of temperature obtained after using the GRI-3.0 and 35-step mechanism and the obtained temperature distributions along the axis utilizing all of the mechanisms are given in Figure 8A,B.

Figure 8.

Figure 8

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Sandia flame D temperature contours, employing the GRI-3.039 (the bottom half of (A)) and 35-step (the top half of image (A)) and the experimental temperature distribution on the axis represented in reference (59) alongside the data obtained from present simulation (B) utilizing 177 steps,38 65 steps,45 and introduced kinetic mechanisms.

The criterion for convergence was continuity residuals of less than 1e-4 for continuity and 1e-7 for velocity components, in addition to maximum temperature variations of 1% on the axis. Results showed that the normalized time spans needed for 100 iterations after convergence were 1, 0.45, 0.33, and 0.39 when GRI-3.0, 177-step, 65-step, and 35-step mechanism were utilized. Considering the method by which the two selected mechanisms from the literature were developed, using both the DRG method and sensitivity analysis, the comparison can indicate that the present algorithm is superior in terms of reducing computational costs while maintaining accuracy.

6. Conclusions

The primary purpose here was to introduce a hybrid algorithm comprising conventional and novel methods to develop reduced skeletal mechanisms. The novel parts of the algorithm are the ability of species-specific targeting by, first, chemistry-based classification of the species of the detailed mechanism and, second, the introduction of the δrel parameter which is able to quantitatively identify the species with a dominant contribution to each group’s chemistry. Furthermore, the utilization of sensitivity analysis and the DRG method in the context of a hybrid algorithm with a refining feedback loop are a portion of newly introduced notions here.

To assess the performance of the algorithm, it was applied to the GRI-3.0 mechanism, while NOx (NO and NO2) and GHG (CO and CO2) were selected as the targeted species in lean combustion (Φ = 0.65) of preheated (640 K) methane-air mixtures at elevated pressures (up to 14 bar). In the starting step, all of the species in the GRI-3.0 mechanism were sorted into two groups. Group C included species responsible for methane decomposition and consumption/production rates of greenhouse gas, and group N incorporated the species that form the submechanism of nitrogen oxide chemistry. Calculating δrel for each species in two groups led to the selection of nine species from the two groups as the inputs of the sensitivity analysis (no species elimination so far). The sensitivity analysis for these nine species resulted in 25 species and 46 reactions. Finally, the DRG method based on targeted species production/consumption rates and the main fuel decomposition, in addition to the subsequent iterative feedback loops, led to the introduction of the final reduced mechanism comprising 21 species and 35 reactions, as shown in Table 9.

The mechanism performance was assessed to confirm the effectiveness of the introduced algorithm. PSR and flame speed calculator models in CHEMKIN-Pro software were utilized, and the results, including substantial species concentrations, adiabatic flame temperature, and flame speed, were compared against the GRI-3.0 and two other reduced mechanisms from the literature. All the trends and the accuracy of the adiabatic flame temperature and the GHG concentrations for the whole range of this study (equivalence ratio 0.5–1.2, pressure 1–14 bar, residence time 0.001–1 s) were excellent (less than 1% error).

The resulting NOx mole fraction obtained from the PSR model utilizing the introduced (35-step) mechanism showed a relatively larger error (with GRI-3.0 results as the reference) in equivalence ratios near 0.94 (error of 11.7%), the residence time of 1 s (error of 4%), and 14 bar pressure (error of 34%). The trend of the NO curves resembled the GRI-3.0 results in all the calculations, while a discrepancy was observed in the NO2 distribution with respect to pressure variations. Regarding the level of reduction, the constant underprediction offset of the NO curves can result in valuable data. Finally, the flame speed calculations revealed excellent accuracy for lean methane-air flames of arbitrary pressures and unburned mixture temperatures.

To evaluate the level of computational cost reduction, a two-dimensional (axisymmetric) simulation of Sandia flame D was conducted. The computational investigations after convergence (in steady-state) showed that normalized time of 1(for the GRI-3.0), 0.45 (for the 177-step), 0.33 (for the 65-step), and 0.39 (for the 35-step) is required for 100 iterations. This can be deduced from the fact that employing the 35-step mechanism can reduce the computational costs to 39% (in comparison to GRI-3.0), whereas the results are more accurate and reliable (in addition to containing NO2 species) than the 65-step mechanism. The comparison of the results obtained from four mechanisms can demonstrate that the present algorithm is superior to algorithms that employ DRG, sensitivity analysis, or their simultaneous combination. In light of these results, the developed reduction algorithm demonstrates its potential as an effective tool in kinetic mechanism reduction.

Although the present study resulted in a final prospect demonstrating the effectiveness of the proposed algorithm, a comprehensive conclusion regarding the performance of the introduced algorithm demands further investigation and assessments against other powerful techniques. DRM was used to develop skeletal reduced mechanisms that were found to be strongly satisfying in terms of the goals of the reduction procedure. As mentioned in the introduction, there are similarities between DRM and the present method. Both of the skeletal reduction techniques are capable of considering the contribution significance of different influential factors to a desired characteristic of the reduced kinetic mechanism (e.g., the contribution significance of a reaction in NOx chemistry). Both are capable of focusing on particular chemical attributes that must be retained during the reduction procedure. Additionally, both of them use zero-dimensional numerical models to extract the parameters that can be used to assess whether the reduction procedure is being conducted with regard to specified criteria.

However, the incorporation of other techniques like DRG and sensitivity analysis after newly proposed preparations for creating proper inputs in the present method is the facet that distinguishes the two algorithms. The present method can be referred to as an algorithm with partially innovative and partially classical parts with interconnections forming a new algorithm.

To understand and compare the effectiveness of the two techniques, just like the procedure used in this study (assessing the performance of 35-step, 65-step, and 177-step mechanisms), one needs to compare the performance of two skeletally reduced mechanisms with the same parent mechanisms and the same reduction goal, while both methods are utilized separately. This may require further effort beyond the present study and will put forth an open question for future investigations.

The authors declare no competing financial interest.

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