Deliverable 38. Reduced Kinetic Models for Different Classes of Problems

Transcription

1 SAFEKINEX SAFe and Efficient hydrocarbon oxidation processes by KINetics and Explosion expertise and development of computational process engineering tools roject No. EVG1-CT Work ackage 5 Kinetic Reduction Software Deliverable 38 Reduced Kinetic Models for Different Classes of roblems University of Leeds Leeds LS2 9JT UK M. Fairweather J. F. Griffiths K. J. Hughes M.J. illing R. orter A. S. Tomlin

2 Contents 1 Introduction 3 2 Summary of Model Reduction Methods a) Background 3 b) Sensitivity-based reduction procedures for kinetic models 4 c) Timescale-based reduction procedures for kinetic models 5 3 C 1 C 3 Alkanes a) Reduced models for propane derived in zero-dimensional simulations 7 b) Application to and further development from one-dimensional simulations 11 4 C 4 C 1 Alkanes 14 5 Alkenes 17 6 Cyclic Alkanes (Naphthenes) 19 7 Aromatic Compounds 24 8 rediction of the Minimum Autoignition Temperature (MIT) a) Background 26 b) Autoignition temperature (AIT) predicted from comprehensive schemes 27 for a range of alkanes c) rediction of AIT using reduced kinetic models 32 9 Discussion and Conclusions a) Relationships between redundant species identified in the comprehensive 33 schemes b) Redundant reactions patterns 36 c) Quasi steady state species 36 d) Summary of the scale of reduction of kinetic mechanisms and the gain in 37 speed-up of computation e) The status of chemical computation with CFD analyses, with reference to 38 AIT prediction. 1 References ublications and Conference resentations 4 12 Appendices a) ropane oxidation reduced to 42 species in 166 reactions 42 b) N-butane oxidation reduced to 74 species in 3 reactions 45 c) Cyclohexane oxidation reduced to 47-1 species in reactions 52 d) Classes of reactions removed from cyclohexane and n-butane schemes 6 2

3 1. Introduction This report constitutes the background and commentary on Deliverable 38 as a contribution to W 5 Reduction Methods and Validation. The purpose of Deliverable 38 is to provide reduced kinetic models for different classes of hydrocarbons that may be used for the prediction of autoignition hazards. The models have been derived using the methodology discussed in Deliverable 37, Kinetic reduction software: report on reduction techniques. The numerical programs can be accessed from the SAFEKINEX website ( In this report we describe the models appropriate to each of the classes considered and what is achieved in terms of the extent of reduction that is possible from the comprehensive schemes derived using the automatic generation methods of EXGAS (Deliverables 35 and 36) and the speed-up in computation that can be achieved. Results illustrating the application of the reduced schemes are also presented here and representative reduced schemes are given in Appendices. The classes of compounds that have been considered are C 1 C 3 hydrocarbons, alkanes from C 4 C 1, cyclic alkanes (naphthenes), alkenes and aromatic compounds. The development and performance of comprehensive kinetic models for these classes of compounds were part of W3 and are reported in the preliminary Deliverables and the final Deliverables In Deliverable 38 we refer to a comprehensive kinetic scheme as one that contains all of the kinetic information that is available to represent the behaviour of a particular substance over specified ranges of conditions. For the present purpose, each of these schemes will have been generated by use of EXGAS. A mechanism that has then undergone some manipulation to reduce its complexity is generally referred to as a reduced scheme. A limiting condition for this reduction procedure, creating a kinetic model that has been reduced only by removal of redundant species and reactions and cannot be reduced further without losing the kinetic form of the mechanism (that is, it is the shortest form of the reaction scheme that can be achieved whilst still retaining the format of the original elementary reactions) is called a skeleton scheme. As noted below (Section 2c) and illustrated in Sections 4 and 6, quite considerable further reductions of the skeleton scheme can be achieved, but it is at the expense of the explicit mechanistic structure represented by elementary reactions. We might recognise the anatomical connection of reducing the comprehensive scheme to its bare bones to generate the skeleton scheme, but with all of its main connections intact. Unfortunately, there is no analogous anatomical state to correspond to that which constitutes the lumping of species and reaction parameters in subsequent stages. a) Background 2. Summary of Reduction Methods Except in the simplest, idealised scenarios, the need for reduction of comprehensive kinetic models that are applied to combustion problems and hazards arises because the detail and complexity of the models (incorporating hundreds of chemical species and thousands of reactions) makes their computational application too inefficient, or even not viable when the complementary physical processes such as heat and mass transfer or gas motion are also embodied in the simulations. In order to bring about reduction in the numbers of reaction species and reactions, without losing the quantitative capability to predict the required information about the combustion system or chemical process, it is necessary to exploit formal mathematical methods and, 3

4 where practicable, to incorporate them in automated numerical programs in a format that is widely applicable. The format in which we have set up the programs includes thermodynamic data in the form of 14 NASA polynomial coefficients for each chemical species and a list of reactions and associated Arrhenius parameters. The mechanisms are in a standard CHEMKIN [1] format (see Deliverable 37). The focus of Work ackage 5 is on the behaviour of the chemical models. Consequently, because the starting point for mechanism reduction is kinetically complex, the developments themselves have been made within the idealised context of a spatially uniform chemical reactor. This enables gas motion, either intentionally induced or arising from natural convection in practical circumstances, to be disregarded and for heat transport from the system to be characterised by a Newtonian heat transfer coefficient. The implications of such simplifications (which do not themselves impinge on Work package 5) are discussed in Deliverable 18. Nevertheless, an example to illustrate the application and use of the reduced model in a more complex physical environment involving heat and mass transport by diffusion, which requires a spatio-temporal computation in at least one dimension, is also included here. This example, pertaining to the spontaneous combustion and ignition of propane [2], highlighted some of the potential limitations in the fundamental data that are incorporated in comprehensive mechanisms themselves (regardless of their scientific origin). Moreover, through this particular example, by application of new methods for uncertainty analysis we have been able to inform the combustion community the extent to which discrepancies may exist in the assigned data. One publication [3], in the journal hysical Chemistry Chemical hysics, was announced as a hot topic in June 26 by the Royal Society of Chemistry for its timeliness and importance ( Subsequently the same methods were applied by us to another topical area for potential combustion hazards involving carbon monoxide and hydrogen mixtures [4]. b) Sensitivity-based reduction procedures for kinetic models The foundation for kinetic model reduction used here is local sensitivity analysis, which embodies a series of stages. The purpose is to reduce the numbers of species and reactions of the mechanism while retaining the desired quantitative output from the model. In sensitivity analysis, the effect of making small changes in parameters or variables on the magnitude of other variables of the system is investigated, for example as the effect of small changes of concentration of each species on rates of product generation. The effect of perturbations applied to each variable can be quantified and ranked in importance, such that thresholds can then be applied to decide which species (or reactions) can be retained and which can be discarded [5]. The conventional procedure is first to identify redundant species via investigation of the Jacobian matrix. Subsequently, reactions can be removed using either the overall sensitivity of production rates of necessary species to changes in rate parameters using a least squares objective function, or through principal component analysis of the rate sensitivity matrix [6]. There is a range of chemical kinetic based software available to aid the chemist or engineer. The most common of these are based on the CHEMKIN family of numerical codes [1] which are used worldwide in combustion modelling and have had a large impact on the research community by providing a common framework for communicating work via a CHEMKIN format [7]. The KINAL [8, 9] or KINALC [6, 1] packages, the latter of which 4

5 uses the CHEMKIN format, are examples for the application of sensitivity methods and have been used in W5. Just as the manual construction of comprehensive mechanisms is laborious and prone to error, and has been circumvented by the development and application of EXGAS (W3, Deliverable 35), so the reduction of schemes in an intensive interactive way is both laborious, prone to potential error and requires a detailed understanding of the principles involved. Thus the reliability of the results, and the scope of application are enhanced, and many man hours saved by the use of automatic reduction software. The codes are based on the use of UNIX shell scripts to completely automate the utilisation of numerical integration and local sensitivity analysis software in a CHEMKIN format as described in Deliverable 37. The main requirement of the user is the definition of sensitivity thresholds which define the eventual size and resulting accuracy of the reduced schemes. The software has been designed to allow the user to specify several thresholds so that schemes of varying size can be easily developed depending on the required accuracy of the final application. The reduced mechanisms must be validated at the various stages of reduction and this is done through comparison to output predictions obtained from the comprehensive mechanism. With regard to the prediction of autoignition temperature (AIT), as a primary goal of SAFEKINEX, comparisons were made between the predicted temperature - time profiles at specified initial and boundary conditions and also the automatically generated ignition diagram, which shows the conditions within which a range of complex modes of combustion phenomena occur (see Deliverable 37). The precision with which quantitative agreement between the reduced and full models is established becomes a determinant of the extent of the reduction that can be achieved. This is controlled by tests at different thresholds. The overall target becomes a balance between computational efficiency and accuracy of reproduction of the output of the model. Within this environment, models retain mass balance and compatibility with any CHEMKIN based software. The computational time taken to obtain a numerical solution from a mechanism of given size is typically N 2, where N is the number of species, or n, where n is the number of reactions. Therefore large computational savings can be made, especially when the number of species is reduced, as noted later. c) Timescale-based reduction procedures for kinetic models Further reduction may be achieved by the exploitation of the range of time scales present in the system via the application of the Quasi Steady State Approximation (QSSA) combined with reaction lumping [11, 12]. The QSSA method is employed in mechanism reduction to identify species which react on a very short time scale and locally equilibrate with respect to species whose concentrations vary on a slower timescale. These fast reacting species are known as quasi-steady state (QSS) species and their removal can reduce the stiffness of the resulting reduced models. The main assumption of the QSSA is that the equilibration of the QSS species is instantaneous. The concentration of the QSSA species can then be determined (to good approximation) from a local algebraic expression rather than a differential equation. The algebraic expression is derived by setting the QSS species rate of production to zero. revious applications of the QSSA to complex kinetic schemes have tended to employ iterative methods to solve the algebraic expressions for the concentrations of coupled QSS species. Although this results in a reduction in the number of differential equations that need to be solved, additional computational effort is required to solve for the QSS species, which limits the speed-ups that can be achieved. Substantial computational savings can be made 5

6 when the QSS species is removed via reaction lumping [11, 12]. In its simplest form a reaction scheme or subset consists of a set of reactions from reactants going to intermediates, or a coupled set of intermediates that are QSS species, which then form products. Via reaction lumping, this set of reactions is changed to a single reaction involving only reactants going to products. QSS species intermediates are therefore eliminated. The rate constants of the lumped reactions will be algebraic combinations of the rate parameters of the original reactions and, in many cases, also intermediate species concentrations and are derived subsequent to the application of QSSA. This is illustrated in the following, where B is a QSS species linking the reactant A to the product C. k 1 A B C k -1 k 2 When the QSSA is applied, (1) d[b ] = dt (2) so that [ B] = Hence k 1 k1 + k 2 [ A] (3) d[ dt k k [ A] k C ] = 1 2 k2[ B] = k (4) where k ' k1k2 = (5) k + 1 k 2 Therefore the above set of reactions can be replaced by a single reaction of the form A C (6) with the effective rate coefficient, k, as defined. The quantitative kinetic involvement of the intermediate species B in the overall reaction is encapsulated in k. Although it may then be possible to parameterise the k terms with a conventional Arrhenius type expression, this is not generally the case, and therefore the resulting scheme no longer complies with traditional CHEMKIN formulations. However, mass balance is retained in the scheme and simple subroutines describing the chemical rate equations can be automatically developed. The savings, in terms of the number of species eliminated, make this a worthwhile procedure, as shown in the examples below. If the concentrations of any of the QSS species need to be established, they can be regenerated using the appropriate 6

7 algebraic expressions, although usually only the major products and temperature are required from the simulation. The versatility of the automated reduction codes and their application to different classifications of fuels is illustrated next, and considerably reduced chemical models which can be used in higher dimensional simulations are obtained as output (see Appendices). Application of the QSSA to the reduced mechanisms is also shown, making substantial computational savings whilst incurring very little error in the kinetic model performance. The removal of redundant species and reactions provides the potential for further exploitation of the reduced schemes to investigate the consequences of uncertainty in parameter values. Such tests are important because quantitative discrepancies between model predictions and experimental results may be attributed to incorrectly assigned parameter values. Not all parameters control the response of the system equally, so it is important to identify those which play the most important part and to rank the order of their significance. This procedure is classified as uncertainty analysis, as noted in Section 2a, and is illustrated in Section 3b with reference to the performance of a reduced model for propane combustion. 3. C C Hydrocarbons 1 3 a) Reduced models for propane derived in zero-dimensional simulations The predicted behaviour of the smallest hydrocarbon molecules is illustrated here with respect to propane. The starting point was the comprehensive model for propane combustion, developed at CNRS-DCR, Nancy, comprising 122 species in 1137 irreversible reactions and validated elsewhere over the temperature range 6 2 K (Deliverable 35). Kinetic parameters were obtained from literature values or derived using additivity rules [13, 14]. The thermochemistry, equilibrium constants and hence the reverse reaction rate coefficients were derived from NASA polynomial functions. 12 ssure/ka re stage 2 stage ignition 4 stage 5+ Multiple cool flames 2 Slow reaction 1 cool flame T a /K Figure 1. Full Nancy scheme simulated p-t a ignition diagram for equimolar C 3 H 8 + O 2 in a closed vessel under spatially uniform conditions. This, and other, C 3 H 8 ignition diagrams were computed at 2K intervals using an automatic generation procedure, giving precision of the boundary to + 7 a (+.5 torr). Black squares denote the user selected operating conditions for model reduction. 7

8 The model, in a CHEMKIN format, was tested for its ability to predict cool flame and ignition phenomena under spatially uniform, closed vessel conditions over the vessel temperature range 5 65 K at sub atmospheric pressures. All of the qualitative structure of the ignition diagram, including the complex multiple-stage ignition phenomena, was recovered [15], as shown in Fig. 1. As outlined above, the reduction of this comprehensive kinetic model was performed by a combination of local Jacobian analysis for the reduction of species, followed by the use of rate sensitivities for the reduction of reactions as available in KINALC, subject to the improvements made within this project and described in Deliverable 37. The variation in performance of the resulting reduced schemes comprising 63 down to 41 species is shown in Figure 2 as a function of the predicted temperature time record for two-stage ignition of equimolar propane + oxygen in a closed vessel at 62 K and a total pressure of 85.3 ka. 14 (b) (a) (c) 12 T/K t/s Figure 2. Illustration of the predicted 2-stage ignition of an equimolar propane +oxygen mixture at 62 K and 85.3 ka. When the full scheme (solid line) is reduced to a 63 necessary species scheme (circles) the agreement with the full scheme is nearly perfect. The other simulations relate to reduced models, as discussed below. As can be seen in Figure 2, even reducing the number of species in the kinetic model to 5% maintains an excellent quantitative agreement with the performance of the comprehensive scheme. Further reductions begin to introduce quantitative discrepancies in the time dependence, and in a complex way. Two species removal strategies were tested. In the first, a single set of thresholds were applied with increasing tolerances producing smaller and smaller schemes. In the second approach, a two stage reduction was attempted. The first stage reduction to 63 species was achieved with excellent agreement, as shown in Figure 2. Further species removal at this stage by relaxing the tolerances results in a 62 species scheme (Fig. 2 curve a) whose output has incurred significant error when compared to the full scheme. Alternatively a second stage reduction is performed with the 63 necessary species scheme as the starting point, to produce schemes of 42 species (Fig. 2 curve b) and 41 species (Fig. 2 curve c). The agreement of the 42 species scheme is deemed acceptable but in the case of the 41 species scheme it is deemed unacceptable. That the performance of a 62 species model (Fig.2, curve a) appears to be inferior to that of the 42 (Fig. 2, curve b) species model 8

9 arises because the application of a two stage reduction strategy leads to slightly different couplings of the species at different reduction stages. In the single stage reduction, further species removal at each stage using relaxed tolerances resulted in poorer quality reduced mechanisms, this being reflected in the temperature profiles and also in the ignition diagrams. The need to perform a second stage reduction arises because, during the iterative Jacobian analysis, redundant species can be incorporated into the reduced mechanism before all necessary species have been incorporated. These incorporated redundant species can be more easily removed by performing a second stage reduction due to the altered couplings. The need for a two-stage reduction strategy is not unique to Jacobian analyses. It has also been used by other researchers in the reduction methods of direct relation graphs [16]. Nevertheless some deviation from the prediction of the comprehensive scheme may be acceptable in the interests of more efficient computation without significant loss of precision in the prediction. In the present example (Fig. 2, curve c), the reduction to 41 species in the model has affected the maximum temperature reached in the second stage of two stage ignition. This would not be of particular concern when the purpose is solely to predict the occurrence of ignition or its time dependence. The abscissa of Figure 2 is scaled to highlight the discrepancy, but the (important) elimination of over 6% of the original species from the comprehensive kinetic scheme has lengthened the predicted overall ignition delay by only 1%. The time to the occurrence of the first stage of ignition, and the temperature reached in it has not been affected by the model reduction. Consequently, if the interest was solely in a prediction to avoid the possible onset of twostage ignition (that is, via the cool flame stage) and the time at which this is attained, it might be possible to gain further, significant, kinetic model reductions. As discussed in later Sections, further reductions in the propane model applied to autoignition, could certainly be attained by application of QSSA. The success of the reduced scheme for propane combustion comprising 42 species involved in 545 irreversible reactions, as measured with respect to the predicted p- T a ignition diagram from the full scheme, is shown in Figure a 8 ressure/k stage ignition Multiple stage ignitions and cool flames 2 Slow reaction T a /K Figure 3. Comparison of the p T a ignition diagrams produced by species reduced mechanisms for equimolar propane +oxygen. Solid line: full mechanism, 122 species and irreversible 1137 reactions. Dashed line: reduced mechanism, 42 species and 545 irreversible reactions. Dotted line: further species reduced mechanism, 41 species and 57 reactions. 9

10 Having identified the subset of necessary species, principle component analysis (CA) of the rate sensitivity matrix was then applied to the partially reduced mechanism in order reduce the numbers of reactions in the scheme whilst retaining a good agreement with the output of the full scheme. Smaller subsets of necessary species, specific to each time point, found from local Jacobian analysis were used in the objective function of the sensitivity measure. This method has been found to achieve a better reduction than the conventional alternative of using the combined list of necessary species in the objective function. The combined list is collated from all time points considered for sensitivity analysis and is used to directly construct the reduced mechanism (see deliverable 37 for further details). The investigation of a variety of thresholds yielded a mechanism consisting of 42 species and 166 irreversible reactions which gave the best trade off between minimum number of variables and good agreement with the output from the full mechanism. Altered thresholds for the CA tolerances resulted in further reaction removal down to 42 species and 145 irreversible reactions. However, the error induced by the removal of these extra 21 reactions (see Figure 4) over the range of operating conditions was deemed to be unacceptable when the limited computational speed-up achieved was taken into account ressure/ka stage ignition Multiple stage ignitions and cool flames 2 Slow reaction Figure 4. Comparison of the p-t a ignition diagrams produced by reaction reduced mechanisms equimolar propane +oxygen. Solid line: Species reduced mechanism, 42 species and 545 reactions. Dashed line: CA reaction reduced mechanism, 42 species and 166 reactions. Dotted line: further CA reaction reduced mechanism, 41 species and 145 reactions. The p-t a diagram of skeleton mechanism consisting of 42 necessary species and 166 irreversible reactions shown in Figure 4 displays all the types of dynamic behaviour exhibited by the full system. Quantitatively, the agreement with the full scheme is excellent over the range of operating conditions, with only very minor differences. The agreement between the low temperature ignition boundaries produced by the full and final reduced mechanisms is nearly perfect. Good agreement can also be found between the temperature profiles of the full and final reduced schemes. After species reduction to 42 species and 545 reactions, the computational time for a single zero-dimensional calculation is faster and is 37% that of the full scheme. The final reduced scheme has a 15% run time compared to that of the full kinetic model. It is known that the run time of the reduced scheme should be N 2, where N is the number of species. So our calculated run time compares well to the estimated run time of 12% of the full mechanism. T a /K 1

11 b) Application to and further development from one-dimensional simulations As part of the present study, relating to a system involving heat and mass diffusion in a one-dimensional calculation, some additional quantitative changes were then made within the comprehensive mechanism in order to incorporate both modified rate parameters and reaction paths for oxidation reactions involving n-c 3 H 7, i-c 3 H 7 and C 2 H 5, calculated using a master equation model by DeSain et al [17], leading to a final model of 122 species and 765 reversible reactions. These data relate to all of the reaction channels in which the respective alkylperoxy and hydroperoxyalkyl radicals are involved, leading to OH, HO 2, oxygenated molecular intermediates and alkenes. The kinetic data, as presented by DeSain et al [17], do have a constraint in that the pressure dependence was reported through tabulations restricted to the three pressures of 3 and 76 torr, and 1 atm, the 76 torr data set being implemented here. However, these data are appropriate for application to the simulation of minimum ignition temperature (MIT), as discussed in Section 8. Additionally, slight adjustments to the heats of formation of seven alkylperoxy and alkylhydroperoxy radicals (appropriate to the DeSain et al data [17]) were made to give a consistent set of parameters throughout the entire scheme. The performance of this model was tested when heat and mass transport occur solely by diffusion, involving a onedimensional simulation [2]. This physical environment is established in microgravity and avoids the uncertainty associated with the need to assume an idealised heat transport process. 1 8 p/ka t/s Figure 5. Simulated pressure time records (solid lines) respectively at 2, 25, 3 and 35 ka initial pressures for an equimolar propane + oxygen mixture under zero gravity, and experimental pressure time records (dashed lines) respectively at pressures of 51.6, 64.2 and 78.6 ka at 593 K under microgravity [2]. These conditions confer a sound foundation for a quantitative comparison with appropriate experimental measurements, as shown in Figure 5 in terms of pressure time profiles at different initial pressures. The significance of these results is that, in order to reproduce similar ignition delay and times to the cool flame as those measured experimentally, it was necessary to reduce the initial pressure for the simulations. There 11

12 is an inescapable conclusion that there is a greater reactivity of predicted behaviour with respect to reactant concentration (or pressure) than that found experimentally, which is reflected in the quantitative comparison of the predicted ignition boundary of Figure 1 with that found experimentally [15], and illustrated in Figure 6. From supplementary tests, the temperature dependence that is built into the parameters of the kinetic model seems to be satisfactory. For this reason it was deemed necessary to explore uncertainties in the magnitudes of assigned parameters, with particular reference to thermochemistry of intermediate species ressure/ka T/K Figure 6. Experimental [CC 4] and modeled 2-stage ignition and cool flame boundaries for a 1:1 C 3 H 8 :O 2 mixture in a reaction volume of 3 cm 3. The simulated heat loss rate was 2 mw cm -3 K -1, which corresponds to a heat transfer coefficient of 28 W m -2 K -1. Experimental 2-stage ignition boundary Experimental cool flame boundary Simulated 2-stage ignition boundary. Simulated cool flame boundary An example of uncertainty analysis using the Morris one-at-a-time method [18] is illustrated in Figure 7. These results reveal the most important species of the reduced model the thermochemistry of which controls the time to the first stage of two-stage ignition, the temperature reached in the first stage and the time of evolution from the first to the second stage. Those species with the highest absolute mean perturbations to each output can be interpreted as having the highest influence and the species can be ranked in terms of their order of importance. The standard deviation expresses the nonlinearity of the output response with respect to perturbations in the parameters. It is conspicuous that the thermochemistry of the same few species predominates in the determination of these three very important parameters. The extent of interactions between the variation of individual heats of formation for these species is embodied in the results of the Morris analysis as the standard deviation, but the extent to which changes in the parameters will affect the output has to be tested using Monte Carlo methods. The results of these analyses are discussed fully elsewhere [3], but we should note that the changes that are necessary for certain thermochemical parameters in order to improve the quantitative performance of the kinetic models do not fall outside the errors that would be assigned to those parameter values on the basis of current knowledge. In summary, there has to be an on-going investigation and refinement of (potentially quite limited numbers of) parameter values used in comprehensive and reduced models in order to improve the quantitative prediction of combustion behaviour. It is also important to note that 12

13 the application of the quantitative approaches to uncertainty analysis requires the development and use of reduced schemes in order for the techniques to be computationally viable O 2 n-c 3 H 7 O 2 OOH Standard Deviation i-c 3 H 7 O 2. OOH τ absolute mean perturbation/s 1 Figure 7a. Morris analysis for the species heats of formation with respect to time to the first stage of two stage ignition, τ 1. Specified variations are made in the magnitude of H o f for each species O 2 OOH n 7 Stan dard Dev iatio C 2 H 5 O 2 i-c 3 H 7 O 2 n-c 3 H 7 O τ 2 absolute mean perturbation/s Figure 7b. Morris analysis for the species heats of formation with respect to time to the second stage of two-stage ignition, τ 2. Specified variations are made in the magnitude of H o f for each species.. 13

14 OOH O 2 OOH S tan dard Deviation C 2 H 5 O 2 i-c 3 H 7 O 2 C 2 H 5 OOH n-c 3 H 7 O T 1 absolute mean perturbation/k Figure 7c. Morris analysis for the species heats of formation with respect to temperature reached in the first stage of two-stage ignition, T 1. Specified variations are made in the magnitude of H o f for each species. 4. C 4 C 1 Alkanes Here we show that such methods can be successfully applied to highly complex reaction sequences such as those occurring in alkane combustion. An EXGAS generated n-heptane scheme was simplified by the reduction procedures outlined in Section 2b and illustrated in Section 3. The comprehensive scheme included 358 species in 2411 irreversible reactions. The reduction of this comprehensive model to its skeleton form comprised 218 necessary species in 81 reactions. This scheme gave an excellent quantitative match to that from the comprehensive mechanism, when the overall ignition delay was computed as a function of reaction vessel temperature, for an n-heptane + air mixture at φ =.65, as shown in Figure T/K t/s Figure 8. Typical temperature profiles calculated from the comprehensive (358 species in 2411 reactions - solid line) and skeleton (218 necessary species in 81 reactions) - dashed line) n-heptane reaction mechanisms (T a = 55 K, p = ka (1.711 atm). 14

15 Over 1 QSS species can be identified amongst the 218 necessary species of the skeleton scheme but many of them are involved in highly coupled reaction sequences. One example is the reaction mechanism of the symmetrical heptylperoxy isomer as illustrated in Figure 9, which proceeds via a set of connected QSS species to 13 different product channels. C 7 H 15 OO O 2 2 product 2 product channels channels Q 1 OOH Q 4 OOH Q 2 OOH Q 3 OOH 4 product channels 4 product channels O 2 Q 1 OOH O 2 Q 4 OOH O 2 Q 2 OOH O 2 Q 3 OOH OH + C 7 H 14 O 3 decomposition products Figure 9. Reactions of the symmetrical heptylperoxy isomer via a set of connected QSS species to products. Here, all of the Q OOH and O 2 Q OOH species are QSSA candidates, but it is not a straightforward matter to remove them via reaction lumping, as the set of simultaneous algebraic equations defining the concentrations of the connected QSS species obtained from such a complex set of interactions are too intractable to easily be solved manually. However, by the use of an algebraic equation manipulation package such as MALE to solve the set of simultaneous equations resulting from the application of the QSSA, this problem can be overcome. The solutions produced by MALE are of the form: [Q 1 OOH] = f 1 (k) [C 7 H 15 OO] (7) Similar relationships exist for Q 2 OOH etc., where f 1 (k), f 2 (k) etc. are complicated functions of all of the individual rate coefficients involving the QSS species in Figure 9. Thus, if there is a reaction leading to a product species, such as: Q 1 OOH A (8) with a rate coefficient k a, then it can be replaced with a direct reaction of C 7 H 15 OO to product A with an effective rate coefficient given by the product k a f 1 (k). A similar procedure can be performed for all 13 individual product channels in Figure 9 to give a final mechanism in which all of the QSS species in Figure 9 along with the individual reactions involving these species have been removed and replaced by direct reaction of C 7 H 15 OO to each of the 13 product channels. 15

16 By systematically applying this procedure to most of the identified QSS species, more than half of the species in the skeleton n-heptane scheme can be removed, with a minimal effect on the temperature profile, as illustrated in Figure 1. Not all QSS species have been removed, the simplest QSS species, principally small radicals such as H, O, OH etc. are involved in so many reactions that even with the use of MALE, their removal is not practical. As noted previously, such a scheme is no longer compatible with standard CHEMKIN codes due to the need to compute the complex f(k) expressions that cannot necessarily be parameterised using the standard functional forms available within CHEMKIN T/K t/s Figure 1. Comparison of the typical temperature profile to that from the full (358 species in 2411 reactions solid line) and skeleton (218 necessary species in 81 reactions dashed line) mechanisms calculated from the reduced scheme generated after applying the QSSA (117 species in 571 reactions dotted line). Yet further reductions in the mechanism obtained after application of the QSSA are possible. As the structure of the scheme is now radically different from the starting comprehensive scheme and also the skeleton scheme, then the methods of Section 2b can be re-applied to identify yet more redundant species and reactions, reducing the scheme to 11 species and 452 reactions while still retaining excellent agreement. In addition, the issue of species that are now only present as products can be addressed. This arises because in the residual scheme there are a number of species the consuming reactions of which have been removed, and hence exist only as reaction products. Nevertheless these species cannot be removed because the reactions that produce them are important with respect to prediction of the temperature and/or other important species. However, a further approximation can be made by amalgamating these product species into a single dummy product. We have assigned the thermodynamic properties of carbon dioxide to the dummy product since, amongst these product-only species, CO 2 is present in the highest concentration. This produces a scheme of 81 necessary species and 452 reactions which, in addition to the incompatibility with CHEMKIN introduced by the QSSA rate coefficient expressions, strict mass balance and an accurate heat release calculation are no longer preserved. However in practice, this does not prevent the mechanism producing an excellent reproduction of the temperature profile, as illustrated in Figure

17 T/K t/s Figure 11. Comparison of typical temperature profiles produced by all the n-heptane schemes, full 358 species in 2411 reactions (solid line), skeleton 218 necessary species in 81 reactions (dashed line), QSSA 117 species in 571 reactions (dotted line), and reduced QSSA scheme 81 necessary species in 452 reactions (dash-dot line). The computed ignition diagrams from the comprehensive, skeleton, and minimal scheme of 81 necessary species in 452 reactions, are shown in Figure Ignition /ka Cool flames Slow reaction T/K Figure 12. Complete ignition diagram produced by the comprehensive (358 species in 2411 reactions solid line), skeleton (218 necessary species in 81 reactions dashed line), and reduced QSSA scheme (81 necessary species in 452 reactions dash-dot line). 5. Alkenes A comprehensive scheme for 1-hexene was developed by the research group at Nancy, consisting of 1128 species and 55 mixed reversible and irreversible reactions (6437 irreversible reactions). The ignition diagram generated for a stoichiometric 1-hexene/air mixture is shown in Figure 13 and, with it for comparison, is a predicted ignition diagram 17

18 from a skeleton reduced scheme of 816 species in 252 irreversible reactions. This shows an excellent agreement between the full and skeleton scheme at this extent of reduction Ignition a /k Cool flames 2 Slow reaction T/K Figure 13. Comparison of predicted ignition diagrams from comprehensive (solid line), and Skeleton (dashed line) 1-hexene schemes. The square symbols denote the sets of initial conditions of temperature and pressure for which the mechanism reduction procedure was performed. However, there are still a large number of species and reactions in this skeleton scheme, and very little further progress can be made by the conventional methods of species and reaction removal before the agreement between comprehensive and reduced schemes breaks down Ignition 14 /ka Cool flames Slow reaction T/K Figure 14. Comparison of ignition diagrams from the comprehensive (solid line) and species-reduced skeleton scheme (dashed line) when only conditions close to the low temperature boundary were chosen to perform the reduction. The square symbols denote the initial conditions chosen for the reduction in this instance. By concentrating solely on reduction conditions near the low temperature ignition boundary, in a similar manner to that discussed in Section 6 with respect to cyclohexane, an improved reduction could be obtained that would still reproduce this low temperature boundary, but deviate in other regions of the ignition diagram. This is illustrated in Figure 14, 18

19 in which the species and reaction reduction was performed only at the two points indicated close to the low temperature ignition boundary. The resultant skeleton scheme comprised 617 species in 264 reactions, the higher number of reactions than that involved in Figure 13 arising from the adoption of different reduction thresholds. QSS candidate species were also identified in the skeleton 1-hexene mechanism, as discussed in previous and subsequent Sections. Although these QSS species have not been removed by analytical methods from the mechanism, it is possible to adjust the numerical integration code to force their time derivative to be zero in order to simulate the effect of actually applying the QSSA to these species. The predicted temperature profiles are shown in Figure 15, obtained from simulations at conditions corresponding to those marked in Figure 14. The temperature profile from the skeleton scheme is compared against profiles obtained with 2 levels of approximation (a 1 or 15% threshold of instantaneous QSSA error) in identifying the QSSA species. These show that, in principle, over 35 of the 617 original species in the skeleton scheme could be removed with negligible consequences on the predicted behaviour. The presence of large numbers of intermediate QSS species is, perhaps, an indication that some lumping of the reaction mechanism at the generation stage might facilitate subsequent mechanism reduction T/K t/s Figure 15. Comparison of a temperature profile prediction from the skeleton scheme of 617 species (solid line) against the predictions obtained by applying the QSSA to 345 (1% threshold, dashed line) or 376 (15% threshold, dotted line) of these species. 6. Cyclic Alkanes (Naphthenes) The full cyclohexane mechanism generated by EXGAS comprises 499 species in 125 reversible reactions and 1298 irreversible reactions (giving 2323 reactions equivalent to 3348 irreversible reactions in total). As in previous examples, the initial mechanism reduction was performed on the simulation of autoignition phenomena in the temperature range 5 8 K, with particular reference to the location of the p- T a ignition boundary. Based on three conditions in this ignition diagram (Figure 16), three stages of species reduction gave a skeleton scheme comprising 16 necessary species in 541 reversible and irreversible reactions which enabled a very accurate prediction of the ignition diagram (Figure 17) with commensurate accuracy in the time dependent response (Figure 18). 19

20 The reduction of the cyclohexane mechanism thus far retains excellent agreement with the low temperature ignition boundary generated with the full mechanism over the temperature range of 5 55 K. The comparisons of temperature profiles as output from all of the presented full and reduced mechanisms in this region of operating conditions also show excellent agreement. Therefore, further reductions were carried out in order to assess what is the minimum size of mechanism which will retain these characteristics. The resulting reduced mechanisms enable a quick calculation of AIT and are useful in evaluating the underlying kinetics driving the transition from slow reaction to 2-stage ignition behaviour when initial ambient conditions vary ressure/ka Slow Reaction 2 - stage ignition Cool flame 2 cf T a /K Slow Reaction Figure 16. Simulated p-ta ignition diagram for stoichiometric c-c6h 12 + air in a closed vessel under spatially uniform conditions derived from the comprehensive EXGAS scheme. The black squares are the user selected operating conditions for reduction of the scheme a ressure/k Slow Reaction 2 - stage ignition Cool flames T a /K Slow Reaction Figure 17. Comparison of the p-t a ignition diagrams produced by the full and species reduced mechanisms for stoichiometric cyclohexane in air showing the full mechanism (solid line) and reduced mechanism comprising 16 species and 541 reactions (open circles). 2

21 14 12 T/K t/s Figure 18. Comparison of the temperature profiles output from the full and species reduced cyclohexane mechanisms in a stoichiometric c-c 6 H 12 + air mixture at 65 K and ka, symbols as in Figure 17. The reduced mechanism comprising 16 necessary species and 541 reactions was used as the starting point for the further reductions. Two alternative operating conditions for sensitivity analysis were selected close to the low temperature ignition boundary. The objective of the reduction was to produce a further reduced scheme which would retain the characteristics of the low temperature ignition boundary and low temperature time dependent behaviour produced using the full mechanism. The reduction proceeded by performing further multi-stage Jacobian analyses. This resulted in the creation of reduced mechanism comprising 59 necessary species in 29 reactions. Next, CA methods were employed to remove reactions. Once again small subsets of necessary species were used in the objective function at each time point to which they are specific. The reactions of the 59 necessary species mechanism were converted to irreversible form and this resulted in 493 irreversible reactions. The number of reactions was successfully reduced and this resulted in a mechanism comprising 59 necessary species and 238 irreversible reactions. This mechanism is shown schematically in Figure 21a, illustrating the major reaction pathways. The reduced mechanism was then validated by the construction of the p-t a ignition diagram and by making a comparison with the same diagram generated with the full mechanism. The comparison of the p-t a ignition diagrams produced by the full mechanism and reduced mechanism (Figure 19) shows excellent agreement between their low temperature ignition and cool flame/slow reaction boundaries, however significant error has been incurred above the temperature of 54 K to the ignition/cool flame boundary in the reduction. Further species and reaction removal using the sensitivity techniques alone results in a scheme comprising 48 necessary species whose p-t a ignition diagram gives reasonable agreement with that of the full mechanism, but whose calculated ignition delay times have incurred significant error. Both of these mechanisms are summarised in the Appendix. 21

22 35 3 ressure/ka stage ignition 5 Slow Reaction Cool flames T a /K Figure 19. Comparison of the p-t a ignition diagrams produced by the full and mechanism C for stoichiometric cyclohexane in air. Solid line: full mechanism, 499 species and 2323 reactions. Circles: reduced mechanism comprising 59 species and 238 irreversible reactions, post CAF. The QSSA combined with reaction lumping was applied to the 59 necessary species mechanism to further eliminate a number of intermediate species whilst incurring little error to output ignition delay predictions. Thresholds were applied to the calculated instantaneous QSSA error for each species over all considered time points, thus providing an automatic way of identifying QSS species. These procedures gave a successful prediction of the ignition diagram based on a mechanism with only 33 necessary species in 323 reactions (Figure 2) ka ressure/ stage ignition 5 Slow Reaction T a /K Cool flames Figure 2. Comparison of the p-ta ignition diagrams produced by the full mechanism and reaction lumped 35 species mechanism with product species lumped into 1 dummy species. Conditions are for a stoichiometric cyclohexane + air mixture. Solid line; full mechanism. Open circles; QSSA reduced mechanism comprising 33 species and 323 reactions. This mechanism requires a dedicated integration code in order to solve the resulting new algebraic formulations for the lumped reaction rate parameters. The mechanism and code are 22

23 available in electronic form from the authors. The reduced QSSA mechanism is shown diagrammatically in two forms in Figure 21. +O 2 HO OO HOO HOO HOO HOO O +O 2 +O 2 O O HOO OO HOO OO H +O 2 O H OO O H OOH O H OOH O H OOH C 2 H 4 Ketohydroperoxides O H OOH +O 2 O H O O OOH +O 2 O H O O OOH O H OOH O CH 2 CO O O CO CH 2 O Figure 21a. Major fluxes of carbon atoms during simulated isothermal oxidation of cyclohexane in air using the 59 species 238 reactions mechanism. Conditions relate to the molar proportions 1:2, 47 K and 1 atm. Arrow thickness is scaled to magnitude of element flux. 23

24 +O 2 HO OO HOO +O 2 +O 2 +O 2 +O 2 HOO HOO OO OO O OOH C 2 H 4 Ketohydroperoxides H O CH 2 CO O O CO CH 2 O Figure 21b. Schematic of major carbon fluxes during isothermal oxidation of the QSSA reduced mechanism comprising 33 species and 323 reactions. Simulated at 47 K, 1 atm and cyclohexane + air in the molar proportions 1:2. 7. Aromatics Comprehensive schemes for o-xylene (188 species, 1238 reversible and irreversible reactions), m-xylene (186 species, 1238 reversible and irreversible reactions), and p-xylene (19 species, 1236 reversible and irreversible reactions) were provided by Nancy. An ignition diagram generated for a stoichiometric o-xylene + air mixture is shown in Figure 22. The points indicated on the figure represent the initial conditions at which redundant species and reactions were identified in the construction of a reduced mechanism. An interesting, ressure/ka T/K Figure 22. redicted ignition diagram for o-xylene from the full mechanism: solid line, ignition boundary; dashed line, cool flame boundary; dotted line, multiple cool flame boundary. Numbers represent the initial conditions at which redundant species and reaction identification was performed. 24