1

SAFEKINEX

SAFe and Efficient hydrocarbon oxidation processes by KINetics and

Explosion eXpertise and development of computational process engineering

tools

Project No. EVG1-CT-2002-00072

Work Package 5

Kinetic Reduction Software

Deliverable 37

Repor t on reduction techniques

University of Leeds

M. Fairweather J. F. Griffiths K. J. Hughes

R. Porter A. S. Tomlin

2

Contents

1 Introduction 2

2 A brief overview of mechanism reduction 3

3 Numerical Integration Code 5

4 Ignition diagrams 6

5 Reduction of comprehensive schemes 8

a Automatic selection of time points for sensitivity analysis 9

b Removal of redundant species 10

c Removal of redundant reactions 10

d Principal component analysis 11

e Rate of production analysis 13

6 Model validation and application of sensitivity and principal component analysis for the lean n-butane + air system

13

7 Further reduction techniques 15

8 Application of the Quasi-Steady State Approximation 17

a Illustration with n-heptane 21

9 Constraints involved in model application and further developments 22

10 Conclusions 25

11 References 25 Appendix 1 Species and reaction elimination patterns between fuels 26 1. Introduction

Deliverable 37 is part of WP 5 “Reduction methods and validation” . It presents a review of reduction methodologies currently at our disposal, and is illustrated here mainly by the prediction of ignition diagrams of n-butane + air. By the development and use of UNIX shell scripts to manipulate the individual codes and their output; many of the procedures are automated, leading to a large saving in the user time required, and reducing the scope for user error.

Detailed combustion kinetic mechanisms contain hundreds of chemical species and thousands of reactions, making them too computationally expensive to be solved in computational fluid dynamics (CFD) codes. In addition, applications in even simple physical geometries that require many simulations, such as the generation of ignition diagrams over wide ranges of conditions, may be computationally too expensive using full schemes. As a consequence there is currently a great interest in reduced kinetic models to represent hydrocarbon combustion. To be valid, such models must be capable of reproducing the various autoignition phenomena seen in experiments over a wide range of operating

3

conditions in both full and reduced form. By adopting formal mathematical procedures, more compact and computationally efficient kinetic models can be generated by reducing the numbers of species and reactions from the detailed mechanisms. Currently, this involves running full kinetic models with multiple initial conditions in a non CFD-based environment, interpreting the results using local sensitivity and time-scale based methods, identifying and removing redundant species and reactions, and then testing the reduced mechanisms. To perform these tasks manually requires that, ideally, the user has a detailed understanding of the principals involved. It is also very time consuming, and is prone to user error. Thus the reliability of the results could be improved and many man hours saved by automating these tasks using programming techniques.

In this report we describe software which has been set up to minimise the numbers of chemical species and reactions without loss of important kinetic detail. The codes are based on the use of UNIX shell scripts to completely automate the utilisation of numerical integration and local sensitivity analysis software. Reduced chemical models which can be used in higher dimensional simulations are obtained as output. The bench-mark is set by the performance of the full scheme and the criteria for performance of the reduced models are matched to this. The prediction of ignition diagrams is used to illustrate the techniques, although they are equally applicable to the prediction of autoignition delay times and flame propagation.

As well as being fundamental to the potential autoignition hazards encountered in chemical industry, an important basis for validation of the models is the ignition diagram, which is a representation of the ignition phenomena as a function of mixture composition or total pressure versus ambient temperature, typically between 500 K and 900 K. Three major classes of ignition phenomena occur. Slow reaction, in which there is no sudden temperature jump, and typically a limited change in temperature. Cool flames, either single or multiple, in which partial fuel and oxidant consumption occurs with temperature spikes in the range 20 K < T/K < 300K. Ignition, during which either fuel or oxidant consumption is complete, and (except in very fuel rich or lean conditions) is accompanied by a sharp temperature spike of many hundreds of degrees. Further subdivisions of ignition may be made, depending on the number of associated cool flames, for instance one cool flame followed by ignition would be classified as 2-stage ignition, 2 cool flames and then ignition would be 3-stage ignition etc. The construction of the numerically predicted ignition diagram involves identifying the location of the boundaries between these different types of behaviour, and is itself a laborious process which is amenable to automation.

2. A br ief overview of mechanism reduction In any reduction analysis it is important to develop a reduced chemical model that is capable of representing the dynamics and kinetic features of comprehensive and detailed kinetic schemes describing the chemical process, whilst minimising the number of variables required. The most advanced, automated procedures available at present for the construction of comprehensive, kinetic mechanisms are those developed and applied in WP4. Such work enables the construction of mechanisms and estimation of appropriate rate coefficients based on structural relationships. This point is very important for systems where kinetic studies of elementary reaction rates have not been performed. In the development of reduced models it is important to begin with such detailed chemical schemes so that prior assumptions about which processes are important are not part of the reduction process. Rather, a series of formal or mathematical procedures should be employed to determine the important species and

4

kinetic steps from the comprehensive scheme over a wide range of process conditions. A brief overview of such reduction methods will be given here with detailed descriptions of the individual techniques and examples described in later sections. Following the development of sensitivity analysis as applied to chemical kinetic mechanisms [1], there has been considerable progress in the development of mathematical tools for the formal reduction of kinetic schemes to encapsulate the essential behaviour of systems [2]. Sensitivity based methods fall into the class of procedures aimed at determining which species and reactions within the initial scheme can be classed as redundant over the selected range of operating conditions. The methods therefore result in the removal of species and reactions from the comprehensive scheme resulting in a reduced mechanism. Whilst such methods can lead to substantial computational speedups in some cases, the reduced schemes developed often still contain fairly large numbers of species. In addition, due to the range of time-scales present in kinetic systems, the resulting chemical rate equations are classed as being “stiff” . The stiffness of the system is classified by the ratio of the slowest and fastest time-scales within the model and the presence of reactive radicals often leads to a high level of stiffness. The stiffness of the system has an impact on the computational methods that are used to solve the chemical rate equations, and therefore it is desirable to reduce not only the number of variables within the reduced model, but also the stiffness of the system. As a result, much of the subsequent development of mechanism reduction is based on an exploitation of the range of characteristic chemical time scales that prevail within a complex kinetic network, especially those relevant to combustion or to chemical and process engineering. Extensive analysis of combustion systems has shown that the longer time dynamics can be described by mainly slower processes since the fast processes locally equilibrate very quickly. Therefore a range of techniques have been developed to identify fast processes and to attempt to describe the overall system behaviour in terms of only the slow variables [2]. Such methods include computational singular perturbation (CSP) analysis [3], intrinsic low-dimensional manifold (ILDM) analysis [4], repro-modelling [5], and lumping with time scale separation [6]. The more traditional quasi-stationary state analysis (QSSA) is also employed extensively [7] and falls into this class of time-scale based techniques. All of these methods exploit the fact that perturbations of any fast-changing variable will lead to a rapid relaxation back to the trajectory of the slow chemical or physical variables in phase-space. The slow variables tend to dominate the overall evolution of the system and, to a very good approximation, the fast-changing variables can be tied directly to these (as in chemical applications of QSSA), or lumped to a single variable to be considered simultaneously with the slow variables. A major quantitative need is to establish the accuracy of such approximations. Whilst based on similar principles of time-scale separation, the practical application of the above techniques differs quite widely. Most applications of QSSA maintain some kind of kinetic structure within the reduced model i.e. the model has the form of a reduced set of chemical reactions, although the stoichiometries may be non-integer in some cases. Many previous applications of the QSSA also involve additional iterative numerical methods to solve for the concentrations of the QSSA species. The QSSA method has obvious kinetic foundations because time-scales are associated with individual species, such as radicals, although in some cases this may limit the extent of reduction that can be achieved. In contrast, ILDM and repro-modelling based methods replace the kinetic scheme with a model that cannot directly be related back to its original kinetic form. Such models may be based on fitting methods such as polynomial or neural network functions [8-11] or the tabulation of reduced numbers of variables [12]. The level of reduction achieved using such methods can be

5

extensive, although it then becomes difficult to explore the driving mechanisms behind kinetic features of the systems being modelled. In addition, where changes to the original model are made, such as amendments to important rate constants, the reduced model has to be redeveloped, which is a time consuming process. The choice of reduction method therefore depends on the intended application of the reduced model. Within the current project the QSSA will be employed as the method of exploiting time-scale analysis in order to maintain the kinetic foundations of the reduced model. The examples will show that substantial reductions in species numbers can be achieved using the QSSA and that reaction lumping can be employed to develop reduced kinetic schemes with simple structures. The reduced models can therefore be employed to explore the kinetic processes driving key features of the intended applications, such as in establishing the autoignition limit. The procedures outlined above constitute the base from which the present project has been developed, working from the comprehensive schemes derived in WP4 through EXGAS [13]. In order to establish general application, the representation of chemical mechanisms and the thermodynamic properties of species are compatible with CHEMKIN [14], which is in common usage. Within this environment other web-based facilities can be exploited, such as MECHMOD [15], a program to manipulate CHEMKIN format mechanisms, for example by removing individual species and their associated reactions, and KINALC [16], a comprehensive package to process sensitivity analysis data, typically used to identify redundant species and reactions. The methods reported here are established whereby formal and rigorous reduction of comprehensive kinetic schemes can be made to enable their incorporation in numerical programs to predict auto-ignition hazards applicable to industrial environments and for the scaling of results from laboratory studies to commercial plant operation. 3. Numer ical Integration Code

The time evolution of a kinetic system in a zero-dimensional model at specified initial ambient temperature, pressure and composition are performed by us using the SPRINT integration package [17] adapted to interpret CHEMKIN format mechanisms in order to simulate the evolving chemistry and combustion phenomena. Adiabatic or non-adiabatic conditions in closed or open vessels can be explored. Employing standard numerical methods, such as Gear and backward differentiation, SPRINT solves the coupled differential equations describing the rate of change of concentration of each chemical species and energy conservation. The chemical rate of change of concentration in a closed vessel is given by:

[ ]nj

jij

i fRvdt

cd == � , (1)

where ci is the concentration of species i, vij is the stoichiometric coefficient of the species i in the reaction j and Rj is the jth reaction rate. Energy conservation is described by:

( ) )( ajj

oj TT

V

UARH

dt

dTCv −−∆=� , (2)

where Cv is heat capacity,

�Hj is the enthalpy of reaction j, V the volume, A the reactor

surface area, U the heat transfer coefficient Ta is the ambient temperature. The heat loss rate is calculated on the basis of Newtonian cooling through the walls. Inflow and outflow mass

6

and energy terms are added for open vessel applications. The outputs of the code include temperature, pressure, species profiles, heat release and reaction rates. 4. Ignition diagrams

The quantitative tests for the validity of combustion models include isothermal comparisons of the predicted yields of intermediates and products with experiment, and of ignition delays during non-isothermal reaction, mainly at high pressure. Only very rarely have there been tests based on the cool flame and ignition phenomena mapped in the pressure - temperature (p-Ta) plane for a given composition, as is exemplified in Fig. 1 for a stoichiometric n-butane + air mixture, or in the composition - temperature (� -Ta) plane at a given total pressure. Yet in many respects these are the primary data which may be regarded to be the first indication for the existence of a potential industrial combustion hazard. In terms of establishing appropriate reduced models for the purpose of combustion hazard prediction, the preservation of the quantitative features of the ignition diagram is very important. Thus the comparison of ignition diagrams produced by full and reduced schemes constitutes the main test of the validity of the reduced schemes. The manual construction of simulated ignition diagrams is very laborious, and is especially slow when very large comprehensive kinetic models are being used. So the first requirement is for an automated procedure for the generation of the diagram.

In this project the (� -Ta) or (p-Ta) ignition diagrams are constructed automatically using

UNIX shell scripts to control the execution of a modified version of SPRINT in which the various non-isothermal behaviour i.e., ignition, cool flames, and slow reaction are characterised. This characterisation of the reaction modes is based on monitoring the temperature increase and temperature gradient within the simulated temperature - time

550 600 650 700 750

1

2

3

4

5

6

p /

atm

Ta / K

Cool flamesSlowReaction

SlowReaction

2-stage ignition

Figure 1. Experimental p-Ta ignition diagram for stoichiometric n-C4H10 + air, showing regions of 2-stage ignition, cool flames and slow reaction [18].

7

profiles from given initial conditions, just as one would examine a measured experimental temperature-time record and make a judgement as to the qualitative nature of the phenomenon that is displayed, based on the observed temperature record. The fundamental difference between this direct assessment by inspection and the automatic generation of a complete ignition diagram is that criteria to represent each class of behaviour have to be established on a quantitative basis within the program in order for the qualitative distinction to be made. The conditions for the location of the boundary between each kind of behaviour can then be identified. This part of the process is interactive and requires the user to have some physical perception of the nature of the phenomena that are being simulated so that appropriate criteria are defined. It is prudent also to make “spot checks” of the categorisation that has been made following execution of the program in order that errors or misconceptions are not perpetrated. Typical criteria adopted are that during slow reaction ∆T does not exceed 20 K, . Ignition is deemed to have occurred if T exceeds 1100 K. The identification of a cool flame or multiple cool flames requires not only that ∆T > 20 K, but also that a maximum is identified before T exceeds 1100 K.

Implementation of the program requires that a temperature range and temperature

increment must be defined, e.g. 550 – 750 K at 5 K increments. In the case of a p-Ta diagram, a mixture composition and initial pressure are defined, and simulations performed for each temperature increment within the prescribed range. That is, the initial pressure must be high enough that throughout the entire temperature range, slow reaction or ignition are the only processes that occur. Then the temperature range is repeatedly traversed, and a bisection method is employed in which 50% of the total initial pressure is taken, and progressively halved again until a distinction at a given temperature between an ignition and cool flame or slow reaction behaviour is detected in successive simulations. The region between the two is then further bisected so that the conditions that represent the two modes of behaviour are refined to a required degree of precision to locate the boundary, typically 0.5 torr. A similar procedure is used to distinguish cool flame phenomena from slow reaction in order to locate the cool flame/slow reaction boundary. At the end of the simulations, a set of files are produced giving the temperature – pressure coordinates that define the various boundaries found. A similar procedure can be set up to determine � -Ta ignition diagrams. In this instance the bisection method is applied on the fuel partial pressure to a desired level of accuracy while keeping total pressure constant by compensating adjustments being made to the other mixture components. For the purpose of developing and testing reduced models, the first requirement is to set up the � -Ta ignition diagram that is predicted from the full scheme as the benchmark against which the reduced models are tested. Figure 2 shows an example from the full n-butane oxidation scheme for its percentage composition in air at 0.2 MPa as a function of temperature

The number of temperature profiles generated by the numerical simulation that are required for the accurate derivation of an ignition diagram will depend on the options selected by the user, such as the temperature range, the desired level of accuracy and the extent of detail required. (That is, one can choose the detail with which the multiple cool flame or multiple stage ignition regions are represented.) For the simpler (� – Ta) diagrams presented for n-butane + air and presented in this report 1– 1.5k simulations were made. For more detailed ignition diagrams of (p-Ta) format with up to 10 cool flames and 7 stage ignitions detected the number of simulations would be 5 to 10k.

As may be inferred from Fig. 2, a horizontal traverse of the (� -Ta) plane is most

appropriate for locating where the lean limit for autoignition lies at a given temperature

8

because it is possible to unambiguously identify the existence of an ignition “peninsula” , such as that existing between T ~580 and 725 K. An alternative approach is to make a vertical traverse of the ignition diagram. That is, a composition range and increment are selected and, starting at the highest desired temperature, the bisection method is applied to the ambient temperature in order to locate the various boundaries. This method is useful to obtain a predicted minimum autoignition temperature (AIT) of the fuel at a given pressure and composition. However, the peninsula shown in Fig. 2 would not readily be identified following this procedure.

0.0 0.5 1.0 1.5 2.0 2.5550

600

650

700

750

Ta/

K

% n-C4H

10 by volume in air

Cool flame

SlowReaction

Ignition

Figure 2. A simulated ( � -Ta) ignition diagram at 0.2 MPa from the full, EXGAS n-butane mechanism, showing regions of ignition, cool flames, and slow reaction. 5. Reduction of comprehensive schemes

The purpose of this Section is to describe the mathematical techniques employed, and the information required in order to be able to reduce the numbers of species and reactions in a mechanism while still retaining satisfactory agreement with the predictive behaviour of the full scheme. The methods used in this work for initial reduction to a skeleton scheme pertain to the local sensitivity analysis methods available in the KINALC package [16]. In its simplest form a local sensitivity analysis method uses the information from a single point in time or composition space during a simulation, and assesses the effect of perturbations in variables such as species concentrations and rate parameters on specified outputs, either individually or collectively. Because large changes in concentrations and conditions are encountered as time elapses from the start to the end of a combustion process, or as initial ambient conditions are changed, the local sensitivity information acquired from a single point would be insufficient to determine the importance of chemical reactions or species over the entire range of conditions encountered. Therefore a range of time points with varying conditions are needed encompassing the start through to the completion of an oxidation process at a number of initial conditions but without the unnecessary duplication

9

of very similar conditions. Thus the collation of information from representative local conditions allows us to gain insight into the full chemical kinetic mechanism and enables mechanism reduction to a scheme that incorporates only the relevant components for the purpose intended. The range of conditions selected will determine the domain to which the resulting reduced mechanism will be applicable and it follows that a reduced scheme that has been derived for a particular use will not necessarily serve another purpose.

5a. Automatic selection of time points for sensitivity analysis Using the simulated ignition diagram in Fig. 2 as a reference, a number of different operating conditions are selected covering a representative range of the temperature/composition space at which sensitivity analysis and mechanism reduction are to be performed. The procedure has then been automated by the use of a UNIX shell script to run the integration code at each chosen condition, and manipulate the output data files. Time points from the calculated temperature profiles at the chosen operating conditions are automatically selected, as shown in Fig. 3 based on the following criteria. To capture the important species and reactions in the initial stages of reaction, the first time point is chosen to be the third integration output point. Subsequent time points are selected at ∆T values of 10, 50, 100, 200, 300 and 400 K and additionally at any temperature gradient maxima and minima. The combination of these criteria ensures that information covering the entire range of conditions applicable throughout the integration is used in the identification of important species and reactions.

0.0 0.1 0.2 0.3 0.4 0.5

600

800

1000

1200

T/K

t/s

Figure 3. Automatically selected time points (black squares) for sensitivity analysis during simulated 2-stage ignition at 630 K, 1.78 % n-C4H10 by volume in air and 0.2 MPa total pressure.

10

5b. Removal of redundant species

The initial stage in finding an appropriate sub-mechanism is the identification of redundant species. Information related to the automatically selected time point conditions and rate data from the mechanism are used to identify necessary species via the investigation of the Jacobian matrix [3]. The necessary species include selected important species as defined by the user, usually the reactants, and other species which have to be included in order to reproduce the concentrations of important species or important reaction features. The remaining species are the redundant species. Perturbations to the concentration of a species will change the rates of those reactions in which this species is a reactant. This may have a direct effect on the concentrations of other species which are reactants or products of these reactions. The primary concentration changes then may cause perturbations to the concentration of other species through indirect couplings. The aim of the procedure is to identify those species which are strongly coupled to the selected important species either directly or indirectly. The sensitivity of the rate of production of an N-membered group of important species to a change in concentration of species i, is given by:

( )2

1

ln/ln�=

∂∂=N

nini cfB , (3)

where fn is the rate of production of species n as shown in equation (1). The higher the Bi value the greater the direct effect of species i on the rate of production of important species. Necessary species with indirect effects on the important species are taken into account by an iterative procedure, whereby the Bi values are calculated for all species and the species outside of the N-membered group with the highest Bi value is incorporated into the N-membered group. This procedure is repeated until the vector B converges and a large gap appears between the ranked Bi values of necessary and redundant species as they form definite groups. Equation 3 is applied using algorithms incorporated into the SPRINT code originally implemented in the KINALC package [16]. A combination of the identified necessary species is taken at the selected time points and the irreversible consuming and reversible reactions of all redundant species are removed. After species removal there will be a certain amount of error when comparing the output from the reduced scheme to that of the full scheme and the larger the number of species removed the greater will be that error. The amount of allowable induced error of a reduced mechanism will depend on its application and so a judgment must be made on the basis of the performance of the reduced models with varying numbers of species obtained by applying varying tolerances to the Bi value in equation 3. However, the entire recalculation of equation 3 for each new tolerance tested is avoided by saving and manipulating the output sensitivity analysis files. After comparison of full and reduced models via ignition diagrams and temperature trajectories the reduced mechanism with the desired degree of accuracy and reduction is selected. The resulting mechanism is then converted to irreversible form for further analysis with respect to reaction removal and further species reduction. 5c. Removal of redundant reactions

Having obtained a partially reduced scheme in which the redundant species have been removed, methods are then employed to significantly reduce the number of reactions without further detriment to model performance. Local sensitivity analysis is used to identify redundant reactions by consideration of either the sensitivity of the concentration of

11

necessary species to perturbations in the rate parameters of each reaction, or the sensitivity of production rates of necessary species to perturbations in the rate parameters of each reaction [16, 19]. The latter is often preferred due to its easier computation. The rate sensitivity matrix F

~ is given by:

(4) ,~

j

i

i

j

k

f

f

kF

∂∂

=

where kj is the rate parameter of the jth reaction and fi is the rate of production of species i. The effect of a change of each rate parameter on the rates of production of necessary species can be quantified by a least-squares objective function Bj, where Bj is defined as:

(5)

2

� ��

�

�

��

�

�

∂∂=

i j

i

i

jj k

f

f

kB

A reaction is considered important if it has a Bj value above a user-specified threshold. The calculation is applied consecutively to the automatically selected time points at a number of user-selected ambient conditions, as described in Section 5a. The important reactions are collated from all of the considered time points and the redundant reactions (i.e. those reactions having no impact at any of the time points, as defined by the selection criteria) are then removed.

In order to be generally applicable, the reduced mechanism should contain species and reactions that have been collated over all local time-points considered. However, considerable improvement in the performance of the reduced models for a given size can be obtained by using time-point specific information about necessary species in the reaction reduction procedure. In order to achieve this, subsets of necessary species relevant for each specific time point are included within the objective function of equation 5, rather than the entire group of necessary species. These subsets of necessary species have already been identified via the investigation of the Jacobian matrix (equation 3) at each time point considered. This point is illustrated in Fig. 4 by comparing temperature profiles of reduced mechanisms obtained by applying Equation 5, with either the full set of species included in the summation i, or time point specific sets as identified by the local Jacobian matrix. Figure 4 shows that by using time-point specific information, for a given size of reduced mechanism (75 species), the performance is significantly improved compared to the equally sized reduced mechanism obtained using the combined species set. A similar result would follow with respect to principal component analysis discussed below.

5d. Principal component analysis

The main disadvantage of the method employed in equation 5 is that large amounts of data in the rate sensitivity matrix are not utilised. A more sophisticated method which examines the interactions between parameters stored in the rate sensitivity matrix is called principal component analysis (PCA) [16, 19]. PCA is a widely used statistical analysis tool for reducing the dimensionality of systems and finding subsets of variables which are highly correlated with each other. The analysis is based on the eigenvalue-eigenvector decomposition of the cross-product matrix FF T ~~

. The eigenvalues measure the significance of their respective eigenvector elements in the overall mechanism at the chosen time point.

12

0.35 0.40 0.45 0.50600

800

1000

1200

T/K

t/s

Figure 4. A comparison of the effects of reaction removal using all necessary species or a subset at each time point in the objective function at Ta = 630 K for a mixture of 1.78 % n-C4H10 by volume in air at 0.2 MPa total pressure. Solid line: species reduced, 75 species and 715 reaction mechanism. Dashed line: subset reduced, 75 species and 449 reaction mechanism. Dotted line: all necessary species reduced, 75 species and 449 reaction mechanism. Each eigenvector represents a set of coupled reactions whose relative contributions are shown by the relative size of the eigenvector elements. Thresholds are defined for the significant magnitudes of the eigenvalues and eigenvectors thus providing an automatic way of deciding which reactions can be eliminated. Typically threshold values for the eigenvalues are chosen between 1.0 × 10-2 and 1.0 × 10-4 and for the eigenvectors between 0.01 and 0.2. The choice of threshold governs the accuracy of the resulting scheme, although often for the eigenvalues a natural threshold exists where there is a large gap between successive eigenvalues. As before, the calculation is applied consecutively to several automatically selected time points at a number of user selected ambient conditions. Here also the important reactions are collated from all of the considered time points and redundant reactions are then removed. After selection of the most appropriate reduced reaction mechanism, at this stage the reduction is evaluated by comparison of ignition diagrams and temperature trajectories.

Because PCA is computationally intensive, especially for large schemes, the rational procedure is to apply it to schemes which have previously had species and reactions removed by the sensitivity methods described in Sections 5b and 5c. When investigating a variety of thresholds to get the correct sized mechanism with a balance between minimum number of reactions and accurate description of the full model, the recalculation of PCA should be avoided in order to save computational time and resources. This is achieved by saving and manipulating the output PCA results from the initial calculation.

13

5e. Rate of production analysis

Redundant reactions can also be removed by the method of rate of production analysis (ROPA) as implemented in KINALC [15]. If a species is formed by fast reversible reactions and if its concentration is low it is not a significant mass reservoir. These fast producing and consuming reactions can be identified at each of the automatically selected time points along the temperature profiles at the user specified ambient conditions and eliminated from the mechanism. The KINALC output gives an ordered list of reaction contributions to the production and consumption rates of each species. The percentage contribution of each reaction to production or consumption are also calculated, which helps the user to identify the fast pairs of producing and consuming reactions. The mechanism is then validated after removing each pair.

In addition to removing fast reversible pairs with high rates, ROPA techniques can be used to identify reactions that make negligible contributions to the rates of production and consumption of each necessary species. A reaction is considered important if it has a higher contribution than a threshold percentage to the production rate of a species. Other reactions are considered to be redundant. The value of the threshold can be altered and is typically 5 – 10%. However there is disadvantage with this method, in that the application of a uniform threshold for each time point and species can result either in redundant reactions being left in the scheme, or in the oversimplification of the mechanism, to the detriment of its subsequent application. It should therefore be applied with some caution. Sensitivity based methods are generally favoured. 6. Model validation and application of sensitivity and pr incipal

component analysis for the lean n-butane + air system The comprehensive model to which the methods were applied was derived at CNRS-

DCPR, Nancy [13] for n-butane oxidation, comprising 125 species in 314 irreversible reactions and 417 reversible reactions. The reversible reactions can be expressed in pairs in an irreversible form, the equivalent full scheme then having a total of 1148 irreversible reactions. The resulting system of ordinary differential equations was solved using the SPRINT integration package [17] for reaction in a non-adiabatic closed vessel with spatial uniformity assumed. A composition - ambient temperature (� -Ta) ignition diagram was constructed using the developed software described in Section 4 to a resolution of ~0.03%. The resulting � -Ta ignition diagram shown in Fig. 2 was used as the benchmark against which the reduced models were tested.

The specific methods employed in the initial reduction were the identification and removal of redundant species as described in Section 5b, followed by the removal of redundant reactions using the PCA method described in section 5d. A typical temperature time profile showing the extent of agreement between the full and reduced models is given in Fig. 5 and the corresponding � -Ta ignition diagram is shown in Fig. 6.

14

0.00 0.05 0.10 0.15 0.20 0.25 0.30700

800

900

1000

1100

1200

T/K

t/s

Figure 5. Comparison of predictions from the full mechanism (solid line) and the reduced mechanism (dashed line), comprising 75 species and 300 reactions, following the Jacobian and sensitivity / principal component analysis.

0.0 0.5 1.0 1.5 2.0 2.5550

600

650

700

750

Ta/K

% n-C4H10 by volume in air

SlowReaction

Cool flames Ignition

Figure 6. Comparison of the predicted ignition diagram produced by the full mechanism (solid line) and the reduced mechanism (dashed line), comprising 75 species and 300 reactions, following the Jacobian and sensitivity / principal component analysis.

15

With this extent of species and reaction removal, there is still an excellent agreement

between the simulated temperature profiles calculated with the full and reduced schemes, which is supported by the agreement between the calculated ignition diagrams (Fig. 6). It is possible to apply different thresholds in the methods used so far in order to reduce the mechanisms still further, but at a cost of a reduced level of agreement with the full scheme. This is exemplified in Fig. 7 whereby the 75 species reduced mechanisms are compared at different levels of reduction post sensitivity and principal component analyses via ignition boundaries. When compared to the reduced mechanism without reaction removal (715 reactions), a reaction reduced mechanism with 300 reactions gives an overall error which is deemed acceptable over the range of operating conditions. At higher specified thresholds for the eigenvalues and eigenvectors of PCA, it is possible to reduce the number of reactions to 269. However, the increasing error induced by this further reduction is considered to be inappropriate since it leads to the removal of only a further 31 reactions giving little extra computational saving.

1.6 1.8 2.0 2.2550

600

650

700

750

Ignition

Cool flames

Ta/K

% n-C4H

10 by volume in air

Figure 7. Comparison of the ignition diagram produced by species reduced mechanism. Solid line: full mechanism, 75 species and 715 reactions. Dashed line: reduced mechanism, 75 species and 300 reactions, post Jacobian and sensitivity/principal component analysis. Dotted line: further reaction reduced mechanism, 75 species and 269 reactions. 7. Fur ther reduction techniques

As discussed in Section 2, further progress in terms of reduction can be made by the investigation of time scales in order to remove fast reacting species and ultimately reduce the stiffness. The simplest of these time-scale based techniques is based on QSS analysis and will be described in detail below. However, there are other related techniques that will be described briefly here in order to put subsequent analysis into context. One such technique put forward by Lam and Goussis is called computational singular perturbation analysis (CSP)

16

[3]. It can be used to categorise the various timescales within kinetic systems and therefore identify the fastest species which can be decoupled from the system. The method works by transforming the species in the original mechanism into variables with less direct kinetic meaning, but more directly associated with the system time-scales. The transformed variables can then be rearranged into an ordered series of vectors which represent ranked reaction groups with associated timescales of a certain magnitude. These reaction groups are known as modes. As the rates of the reactions in the modes approach zero and become exhausted, user selected thresholds are used to determine which of the modes can be categorised as ‘dead modes’ , i.e. not contributing to the overall system behaviour. Information related to the dead modes can then be retransformed to a system with physical meaning in order to identify species or variables which can be eliminated or approximated using QSSA type techniques. Using CSP, successful reductions have been undertaken on mechanisms for a methane/air flame system [20], and for that of methane combustion under mixed air-steam turbine conditions [21].

Intrinsic low dimensional manifold (ILDM) methods [4] rely on the same premise as CSP, in that reaction groups with a variety of time scales are separated into groups in order to decouple or eliminate the fast timescales, and thus reduce the dimensionality of the system. The fast timescales, which we wish to decouple, may occur on a scale as small as 10-10 s, and are therefore negligible in comparison to the duration of an ignition. As the reaction proceeds, each of the fast “modes” collapses and reaction trajectories eventually decay to an attractor trajectory in the concentration phase space, from a number of varying initial conditions. The attractor is itself, located in what is called a slow or inertial manifold of lower dimension than the full number of system variables. The manifold describes the long-time behaviour of the system, and therefore if the system dynamics can be described on this manifold, a very much reduced model can be developed. The reduced model is usually developed in either tabulated or fitted form [8, 10-12] and therefore has to be redeveloped when any improvements in the starting mechanism are made. However, the ILDM technique has been shown to substantially reduce computational cost when applied to methane autoignition in turbulent combustion calculations [22].

An alternative way to reduce the computational cost of CFD with incorporated chemical

kinetics is by using a repro-model [5] to describe the chemical changes. A repro-model is an alternative mathematical description of a larger, more computationally intensive mathematical model and utilises functions and algebraic equations to do this. In combustion modelling, the functions of a repro-model describing a chemical mechanism can, for example, take the form of a set of high order polynomials, which represent functional relationships between selected variables of concentation or temperature over chosen time steps. Before a repro-model is fitted, the influential variables that will be used to describe the low dimensional model must be selected. These can be found by using ideas related to the ILDM technique described above. Time-scale based methods can be used to identify the minimum number of variables required to describe the system on the slow manifold. Key variables are then chosen in order to parameterise the manifold and the fitted model developed for these variables using simulations of the full model. The term “repro model” now becomes clear since the reduced model is nothing other than a “ reproduction” of the behaviour of the full model over the chosen simulation conditions. The accuracy of the repro model therefore depends on the range of conditions chosen for the fitting, and the accuracy of the fitting procedure. A range of fitting techniques have been utilised including high order polynomials, second order polynomials, neural networks and tabulation/interpolation methods. At present the application of these techniques is relatively new. It has not therefore

17

become clear which of these methods is best suited to which types of applications. However, incorporating a repro-model into a CFD code is a viable option, because of the method of operator-splitting, whereby the physics and the chemistry are calculated separately and therefore the method of calculating the chemical changes due to reaction are unimportant to the transport phenomena calculations. A repro-model has been successfully fitted to a tropospheric chemical model and resulting in speedups of the order of a factor of 25 with errors less than 1% when compared to numerical solutions obtained from the original mechanism [8].

A further method of reduction is species lumping, where two or more species are grouped together thus reducing the number of ordinary differential equations required to describe the kinetics. One of the simplest types is chemical lumping, where chemical intuition of reaction rates, mechanism and species structure is utilised. It is often possible to lump isomers using this technique. Other lumping methods, using more formal mathematical procedures, require new lumped variables which are related to the original variables by a ‘ lumping function’ and this can be either linear or non-linear. Linear lumping is relatively straightforward and can be done in an automatic way. The new variables are linear combinations of the original ones. The simplest case here would be that the concentration of the new lumped variable is just the summation of the concentrations of its component species. The main drawback of linear lumping is that combustion systems are highly non-linear, so a linear representation for complete reaction over all conditions can be inaccurate. A potential solution to this problem is by using an ‘adaptive chemistry’ approach, where a number of small linear lumped schemes are produced for different narrow ranges of operating conditions. These are stored on file and switched as input when the conditions to which they pertain are encountered. However, the disadvantage of this is that the ‘switching’ process can hamper the calculation so much that the computational speed up gained from the reduction is cancelled out. The more mathematically rigorous method of non-linear lumping can potentially solve these problems, though the complexity of the required analysis can hinder its use. Non-linear lumping incorporating timescale separation techniques such as ILDM or CSP, can give insight into similar fast or slow variables which can be lumped and is a sophisticated reduction technique, applied in [23] to atmospheric chemistry [2,9,24]. 8. Application of the Quasi-Steady State Approximation

The application of the sensitivity methods described in Section 5 leads to a skeleton

mechanism with many redundant species and reactions removed. However, in most cases the extent of the reduction achieved by such methods is not sufficient for application of the chemical model within complex flow computations. As described above, subsequent reduction may be based on exploiting the time-scales present in the mechanism. A range of techniques were introduced in Section 7. However, many of these techniques lead to reduced models of a completely different form to the original kinetic description (such as polynomial equations or look up tables). For the exploration of kinetic features, it is desirable to develop a reduced model that exploits the time-scales within the system but maintains some kind of kinetic and mechanistic description of the reduced model. How QSSA based techniques combined with reaction lumping can achieve this are described in this Section. QSSA based methods are commonly used in kinetic model reduction by assuming that fast reacting species locally equilibrate with respect to the slower species within the system. The concentration of the QSSA species can then be approximated via an algebraic expression rather than a differential equation, obtained by setting the QSSA species rate of production to zero, i.e., 0=q

if where the superscript q denotes a QSSA species. This in itself does not

18

necessarily lead to any gain in computational time since iterative methods may be required to solve the resulting coupled equations. In order to achieve substantial savings the QSSA species should be removed from the mechanism. In many cases QSSA species can be removed via simple reaction lumping [2]. Alternatively, the concentration of species ci can be expressed in terms of the concentrations of other species in the system and the rate parameters. If such expressions can be solved analytically then iterative techniques are not required. The choice of species that are suitable for application of the QSSA was guided in this application by the instantaneous QSSA error for a single species [25], defined as

(6) ,1

ii

i

i

si J

f

cc =∆

where Jii is the diagonal element of the chemical Jacobian for species i. Other methods for the identification of QSSA species include the CSP approach outlined above. Calculation of the instantaneous QSSA error was incorporated into the UNIX shell scripts developed for the automatic mechanism reduction as discussed previously. Although the QSSA errors vary throughout the simulations, peaking during ignition, for many species the errors remain below a certain threshold throughout. Using a tolerance of 1% across all selected time-points for the QSSA error, 31 QSSA species can be automatically identified within the n-butane scheme. Simple methods for the removal of these species from the scheme should therefore be developed.

The simplest kinetic structures to yield QSSA species are those in which there is one producing and one consuming reaction. For example, in the n-butane oxidation mechanism there are several reactions of the generic type (class I): where O2QOOH is a QSSA species. In these cases it is a trivial matter to show that the QSSA species can be removed and the above sequence of reactions replaced by: where k1� = k1(1-k-1/(k-1+k2)). This reaction lumping is the direct equivalent of solving

0=qif for O2QOOH. Within the numerical integration code, k1� may be calculated as this

function of the 3 individual elementary reactions. Alternatively k1� may be calculated separately as a function of temperature and if possible fitted to a CHEMKIN compatible function such as an Arrhenius expression. This has the advantage of being computationally more efficient when the numerical integration is performed, and the mechanism is still completely compatible with CHEMKIN, but it has the disadvantage in that the parameters so generated for k1� are a mathematical representation and have lost a direct kinetic connection to the elementary chemical reactions. The application of the QSSA in the above case leads to the removal of one species and two reactions.

The next stage of complexity occurs with sequences of connected QSSA species, as illustrated by the generic reactions (class II):

QOOH O2QOOH OH + product1

-1

2+ O2

QOOH OH + product1’

+ O2

19

Class II is distinguished from Class I in the n-butane oxidation scheme insofar that RH represents some relatively minor reaction intermediates involving 4 or 5 membered oxygenated ring structures. Either the full level of complexity in terms of alternative R, RO2 and QOOH reactions for these peripheral structures was not created in the initial automatic mechanism generation code or they were deemed unimportant and thus removed by mechanism reduction methods. Solving the algebraic expressions resulting from the application of the QSSA for the highlighted species in Class II can be demonstrated to be equivalent to the lumping of several of the individual reaction steps, which results in the removal of RO2, QOOH and O2QOOH. The central part of the reaction sequence can then be replaced by:

product OH2O R '22 +→+

where

]O/[))))/(]O[]O[/(/(1( 254424243333222'2 kkkkkkkkkkkkk +−+−+−= −−−−−−

Then R can be removed to leave the final reaction sequence:

where

and

This illustrates the main issue that complicates the application of the QSSA method. The

rate coefficient expressions may be functions of individual component species concentrations, in this case oxygen, and therefore in general cannot be replaced by a simple CHEMKIN compatible parameterization. The consequence of this is that if the removal of these QSSA species is to be quantitatively correct, then the mechanism is no longer fully CHEMKIN compatible, and requires a version of the integration code applied outside CHEMKIN in order to cope with these details. The kinetic structure of the scheme is however maintained and the relative importance of the reaction rate parameters can be

XH, product OH X 2O RH

XH alkene R X RH8

2

'7

++→++

++→+

���

����

�

+=

6'2

617 kk

kkk

.6

'2

'2

18 ���

����

�

+=

kk

kkk

RH + X R RO2 QOOH O2QOOH

OH + productR' + alkene

1(-XH) 2 (+O2)

-2

6

3

-3

4 (+O2)

-4

5

20

automatically seen. In addition, if modifications to the parameters are made in later versions of the starting mechanism, these can be directly employed in the reduced scheme.

If CHEMKIN compatibility is deemed to be important, approximations may be made. In

the most rudimentary approach, k2’ is assumed to be a constant fraction of k2, and the ratio of k7 to k8 is fixed. The k2’ fraction is set at the value calculated in the region of maximum flux through R to OH + product. This region occupies a fairly narrow temperature window, and a rate of production analysis of the full scheme shows that this constant ratio assumption can be a good approximation. Applying it in the present example for n-butane oxidation gives simulated temperature profiles in excellent agreement with those obtained from the original scheme. The ratio of k7 to k8 is not constant, and changes significantly with temperature in a way that favours k8 at low temperatures and k7 at high temperatures. Even so, assuming a constant ratio based on that applicable at low temperatures (T < 800 K), very good agreement in the simulated temperature profiles is found, with only slight deviations at the later times and higher temperatures where this approximation is no longer valid.

Of the QSSA species identified in the reduced n-butane oxidation mechanism (75 species in 300 reactions), 14 fell into the two classes exemplified above as I and II. Their removal resulted in a mechanism comprising 61 species and 270 reactions. Focusing on the cool-flame/ignition boundary, where there is the greatest discrepancy between the various mechanisms, Fig. 8 shows the predicted ignition boundaries from the rudimentary and exact QSSA species removal implementations compared with that from the originating reduced scheme comprising 75 species in 300 reactions. It shows virtually perfect agreement for the exact QSSA species removal method, and still excellent agreement when the approximation of a constant value for k2’ is applied.

1.6 1.8 2.0 2.2550

600

650

700

750

T/K

% n-C4H

10 by volume in air

Cool flames

Ignition

Figure 8. Comparison of the predicted ignition diagram from the base 75 species and 300 reaction reduced scheme (solid line) to that predicted from the exact QSSA species removal implementation (dashed line) and the rudimentary QSSA species removal implementation (dotted line).

21

It is possible to remove many more QSSA species, both those already identified, and those which would be included by relaxing the QSSA error tolerance from 1%, but at the cost of increased complexity in the derived reaction rate coefficient expressions. Nevertheless the QSSA technique when coupled with reaction lumping provides a powerful method for exploiting time-scales within mechanism reduction. The mechanistic structure is maintained in the reduced models and the application of the reaction lumping reduces the need to incorporate complex iterative techniques in order to solve the QSSA expressions. The methods are expected to provide increasing levels of reduction relative to the full schemes for higher hydrocarbons where large numbers of QSSA species should exist. The oxidation of n-heptane is used to briefly illustrate this point in the next section.

8a. Illustration based on n- heptane oxidation

Another example to illustrate the extent of species and reaction reduction that may be obtained using QSSA is shown here with respect to an ignition diagram generated for an n-heptane + air mixture at φ = 0.65. The full mechanism derived using EXGAS consists of 358 species with 2411 irreversible reactions. Removal of redundant species and reactions by the methods described in sections 5b and 5d produced a ‘skeleton’ mechanism of 236 species with 810 irreversible reactions. The QSSA was then applied to remove all QSSA species and reactions of the class I and II type discussed in Section 9. In addition, further QSSA species and reactions were removed as follows: The class I example was extended to include cases where a QSSA species had 2 or more product channels, and the class II type was extended to include the removal of all R, QOOH and O2QOOH based on the parent C7 structure. Removal of heptylperoxy radicals was possible, but due to the relatively large instantaneous QSSA error of ~5% for these species, a significant deviation in predicted temperature profiles at low initial temperatures was observed, and therefore to keep the excellent agreement under all conditions, heptylperoxy radicals were not removed. This led to a complex set of reactions and species being removed, there being many possible isomerisations connecting individual R and QOOH isomers. In addition, these isomers themselves have large numbers of unique destruction paths. Due to the resulting level of complexity, the software package Maple was utilized to solve symbolically the algebraic equations obtained by setting 0=q

if for the QSSA species rates of production and therefore derive the replacement rate coefficient expressions. These solutions were then incorporated into a customized version of the integration code to work with a customized reaction mechanism with the QSSA species and their associated reactions removed. This led to a halving of the number of species from the ‘skeleton’ mechanism, to give a resulting mechanism of 117 species with 571 irreversible reactions. The ignition diagrams produced with these schemes are given in Fig. 9, showing excellent agreement. The time savings are illustrated in Table 1, expressed as the time taken to calculate one temperature – time profile with each mechanism. Compared to the full scheme, there is over a factor of 7 speed-up with the ‘skeleton’ scheme, and a further factor of 2 speedup with the QSSA reduced scheme. The possible time savings are potentially greater than this, since there is still scope for further QSSA species removal. One thing to note here is that the overall speedup achieved is greater than a factor of 15 even though the species are reduced by a factor of 3. The additional speedup over and above that which may be estimated by the relative reduction in species squared (i.e. 9) is likely to be attributed to the reduction in stiffness achieved. This allows potentially larger time-steps to be used during the numerical integration. The large reduction in the overall number of reactions also reduces the computational effort involved in generating the rate of production expressions. Additionally it may be possible to parameterize the effective rate coefficients introduced by removing the QSSA species, thus

22

removing the need to calculate them from the extremely complex algebraic expressions provided by Maple. The resulting scheme may also be amenable to further removal of the lumped reactions by reapplying the reaction removal technique of Section 5d.

500 550 600 650 700 750 800 8500.0

0.5

1.0

1.5

2.0

Slowreaction

Cool flames

P/a

tm

T/K

Ignition

Figure 9. Comparison of n-heptane + air (p – Ta) ignition diagrams at φ = 0.65 produced with the full mechanism (solid line), the reduced mechanism generated by removing redundant species and reactions (dashed line), and the QSSA reduced mechanism (dotted line).

mechanism species ir reversible

reactions time / s

Full 358 2411 140 Skeleton 236 810 19 QSSA reduced 117 571 9

Table 1. Comparison of the time taken to perform one simulation with the full, skeleton, and QSSA reduced heptane oxidation mechanisms. 9. Constraints involved in model application and fur ther developments

In this final Section we discuss the consequences and constraints that arise when the methods discussed here are used for the reduction of comprehensive kinetic models. That is, once reductions of a comprehensive scheme are established according to prescribed criteria, the application of that model becomes restricted to those criteria or, at best, to conditions that are very closely associated. We may illustrate this in the following way. The reduced scheme, derived for n-heptane oxidation under criteria that relate to the p- Ta ignition diagram at φ = 0.65, is shown to be well matched to the prediction from the full scheme (Figure 9). We may then make a comparison of the predictions from these two models under different conditions, for example for the ignition delay at a pressure of 13.5 bar in n-heptane + air at φ = 1, for which experimental data from a shock tube study are also available [26], and were used as

23

part of the validation of the comprehensive mechanism in WP4 (deliverable 35). This comparison is shown in Fig. 10.

700 800 900 10000

2

4

6

8

10t ig

n/m

s

Tinitial

/K

Fullmechanism

Reducedmechanism

Reduced-QSSAmechanism

Figure 10. A comparison of the predicted ignition delay for n-heptane + air at φ = 1 and a total pressure of 13.5 bar. The comprehensive and reduced models are those used to derive the predicted ignition diagrams shown in Fig.9.

There is very acceptable agreement between the predictions in the initial temperature range from 700 – 875 K, but there is a significant discrepancy emerging above 875 K, in which the reduced model leads to a prediction of lower reactivity (longer ignition delay). The origin of this outcome can be traced back to the selected time points for the initial model reduction, shown in Fig. 3. Although there are time points associated with temperatures exceeding 875 K, the chemical properties of the system at these conditions are derived from a kinetic evolution embracing low temperature branching and two-stage chemistry. This is also the case for the predicted ignition delays at Tinitial < 875 K (Fig. 10). However, the kinetic response of the system at Tinitial > 875 K is governed by an evolution that is wholly associated with the single-stage chemistry regime. Since this regime of kinetic development has not been captured in the criteria used for the initial reduction, the higher temperature prediction of ignition delay are a less satisfactory representation of those from the comprehensive scheme. Also included in Fig. 10 is the response that is predicted following the application of the QSSA to the reduced scheme (dotted line). The agreement between the predicted ignition delays from the reduced kinetics and reduced-QSSA scheme confirms that the QSSA procedure does not confer any further penalty in restriction of the scope of application of the model.

Progress to further model reduction imposes additional constraints as well as those discussed above. These are illustrated when the reduction methods described in Section 6 and 7 are further applied to the “ reduced-QSSA” scheme comprising 117 species in 571 irreversible reactions, as exploited in Fig. 10. There are three possibilities for further reduction: (a) removal of species, (b) removal of reactions, now in QSSA format, and (c)

24

amalgamation of “product only” species to a single component. Procedures (a) and (b) require the manipulations described in Sections 5b and 5c. “Product only” species are redundant in the sense that they do not contribute to the kinetic evolution, but they do affect the thermochemistry of the combustion system. By classifying redundant species into a single substance, the enthalpy change is modified. This may not necessarily be very important, especially if the identities of the two final products, H2O and CO2, are preserved outside this lumping. Moreover, mass balance is no longer preserved, with the consequence that all subsequent manipulations then have to be performed outside the KINALC package because a requirement of the CHEMKIN format has not survived.

The extent to which further reductions of the n-heptane oxidation mechanism were

achieved is shown in Fig. 11. Species and reaction removal down to 110 species and 410 reactions was viable with minimal loss of quantitative agreement with the original QSSA kinetic model. Amalgamation of “product only” species to a single (arbitrary) substance, with the exception of H2O and CO2 retained as specified products, results in a scheme comprising 83 species and 410 reactions. This amalgamation, which has the consequence of loss of energy and mass balance, gives no identifiable difference from the predictions of the 110 species / 410 reaction model. Further reaction removal leads to increasing quantitative discrepancies, as is illustrated by reduction to 367 reactions (Fig. 11).

700 800 900 10000

2

4

6

8

10

83 species367 reactions

110 / 83 species410 reactions

t ign/m

s

Tinitial

/K

QSSA 117 species571 reactions

Figure 11. A comparison of the predicted ignition delay for n-heptane + air at φ = 1 and a total pressure of 13.5 bar. Various extents of supplementary reduction of the QSSA model used in Fig. 9 are shown.

It may be inferred from this analysis that further reductions of the kinetic scheme below about 80 species must have implications for reduction in the scope of application of the model and significant loss off identity of chemical species, or even recognizable kinetic structures within the scheme.

25

10. Conclusions Techniques for mechanism reduction based on local sensitivity analysis methods have been incorporated into a general numerical integration code. UNIX shell scripts have been constructed to automate the application of these techniques to accomplish the efficient reduction of a full reaction model to encompass low temperature initiation of reaction, low and intermediate temperature cool flame phenomena, and high temperature ignition. By automation of the procedures the otherwise considerable and time consuming computational effort required is reduced to the user having only to make choices of appropriate tolerances and selection of the reduced scheme based on comparison of the ignition diagrams generated from the full and reduced schemes. This also is done in an automatic way. Further reduction is effectively achieved by coupling the application of QSSA with reaction lumping. Here also the QSSA species to be removed are identified through automated methods with user prescribed tolerances. For QSSA species involved in simple reaction structures, reduction can be achieved whilst maintaining the kinetic structure of the mechanism and CHEMKIN compatibility. For more highly coupled species, customised reaction mechanisms and integration codes are required in order for the highest accuracy to be maintained, although these can still be directly related back to the rate constants within the full mechanism. Complete CHEMKIN compatibility can be restored via the fitting of lumped rate constants. Application of these techniques has been demonstrated with respect to generation of n-butane + air and n-heptane + air ignition diagrams, using the highly automated numerical packages discussed in this report. In the case of n-heptane oxidation, the final number of species is reduced below one third of that of the full scheme, and there are less than one quarter of the original number of reactions. Consequently, a computational speedup of approximately a factor of 15 is achieved, while still retaining excellent predictive properties, when compared with the application of the full mechanism. 11. References 1. E.P. Dougherty, H. Rabitz, J.Chem.Phys. 72 (1980) 6571. 2. A.S. Tomlin, T. Turányi, M.J. Pilling, Comprehensive Chemical Kinetics, Vol. 35,

Elsevier, Amsterdam, 1997 p.293. 3. S.H. Lam, D.A. Goussis, Proc. Combustion Institute 22 (1988) 931. 4. U. Maas, S.B. Pope, Combust. Flame 88 (1992) 239. 5. T. Turányi, Proc. Combustion Institute 25 (1994) 948. 6. G. Li, A.S. Tomlin, H. Rabitz, J. Toth, J. Chem Phys. 101 (1994) 1172. 7. A.S. Tomlin, M.J. Pilling, J.H. Merkin, J. Brindley, N. Burgess, A. Gough,

Ind.Eng.Chem.Res. 34 (1995) 253. 8. R. Lowe, A.S. Tomlin, Env. Mod. Soft., 15 (2000) 611. 9. R.B. Brad. Reduced Kinetic Mechanisms for Chemical and Process Engineering

Applications. Phd Thesis, University of Leeds, The School of Process Environment and Materials Engineering, 2005.

10. J.A. Blasco, N. Fueyo, C. Dopazo, C. Ballester, Combust. Flame, 113 (1998) 38-52. 11.S.R. Tonse, N.W. Moriarty, N.J. Brown, M. Frenklach, Israel J. Chem. 39 (1999) 97-106. 12. S.B. Pope, Combust. Theory Modelling 1 (1997) 41–63. 13. www.ensic.u-nancy.fr/DCPR/Anglais/GCR/softwares.htm 14. R.J. Kee, F.M Rupley, J.A. Miller, Chemkin-II: A Fortran Chemical Kinetics Package

for the Analysis of Gas Phase Chemical Kinetics, Sandia National Laboratories Report No. SAND89-8009B, 1991

26

15. www.chem.leeds.ac.uk/Combustion/mechmod.htm 16. www.chem.leeds.ac.uk/Combustion/kinalc.htm 17. M. Berzins, R.M. Furzland, Shell Research Ltd, TNER 85058, (1985). 18. D.T.A. Townend, M.R. Mandlekar, Proc. Roy. Soc. (London) A141 (1933) 484-493. 19. T. Turányi, T. Berces, and S. Vajda, Int.J.Chem.Kinet. 21 (1989) 83. 20. A. Masaias, D. Diamantis, E. Mastorakos, D.A. Goussis. Combust. Flame, 117 (1999)

685. 21. G. Skevis, A. Chrissanthopoulos, D.A. Goussis, E. Mastorakos, M.A.F. Derksen, J.B.W.

Kok, Appl. Therm. Eng., 24 (2004) 1607. 22. Z.Ren, S.B. Pope, Proc. Combustion Institute 30 (2005) 1293. 23. B. Sportisse, R. Djouad, J. Comput. Phys., 164 (2000) 354. 24. J.F. Griffiths, Prog. Energy. Combust. Sci. 21 (1995) 25. 25. T. Turányi, A.S. Tomlin, M.J. Pilling, J. Phys. Chem. 97 (1993) 163. 26. H. Ciezki and G. Adomeit, Combust. Flame 93 (1993) 421

Appendix 1. Species and reaction elimination patterns between fuels

The purpose of this Appendix is to explore the potential for a generalised modelling

approach. In a preliminary overview we illustrate the common ground that has emerged in the two examples of mechanism reduction, involving n-butane and n-heptane oxidation. The nomenclature for species identification that evolves from EXGAS is retained. The structural details and their implications will be discussed in due course.

The first step in the reduction process is the elimination of species via the investigation

of the Jacobian matrix. Subsequently, reactions can be removed via sensitivity or principal component analyses. Kinetic insights could be gained into which species or reactions are consistently eliminated for each alkane fuel type which may be useful to the development of generalised mechanisms. Here, the species eliminations have been analysed for n-butane and n-heptane oxidation by the comparison of their respective full and species/reaction reduced mechanisms. As discussed above. the n-butane mechanism was reduced under lean conditions in air from 125 species in 314 irreversible reactions and 417 reversible reactions down to 75 necessary species and 300 irreversible reactions. As a basis of comparison, the n-heptane mechanism was reduced under lean conditions in air from 357 species in 594 reversible reactions and 1223 irreversible reactions down to 236 species and 810 irreversible reactions. A number of species were commonly found to be redundant in both schemes. They are; B3C, B4CH, C2H6O, C3H5CHOY, C3H5OHY, C3H8, C3H8CO, C4H7CHOY, C4H7OHY, C5H9OHY, C6H10Y2, C6H10Z#6, C6H12Y, C7H12Y2, C7H12Y2, C8H14Y2, R6CH2OH, R9C2HT, R16C2H4OOH, radical C3H7, radical C3H7OO, radicals of C3H6OOH, radicals of C4H8OOH, radicals of C3H6OOOOH, radicals of C4H8OOOOH and ZCOC2H3Z.

Next, reactions associated with the primary fuel were investigated to see if there were any parallels which could be drawn. In the EXGAS generated full schemes the primary fuel typically undergoes a number of reactions. They are; a number of unimolecular decomposition reactions (2 and 3 for n-butane and n-heptane respectively) to form alkyl radicals of various chain lengths e.g.

C7H16-1 � R19C3H7 + R20C4H9,

27

and H atom abstraction by O2, O, H, OH, HO2, CH3, CHO, CH2OH, CH3O, CH3O2, C2H5 and ROO• to form R•, e.g.

C7H16-1 + OH � R23C7H15 + H2O.

In both cases all of the unimolecular decomposition reactions were removed by the sensitivity type reductions. H atom abstractions by R5CHO, R11C2H5, and R6CH2OH were also consistently removed. Alkyl radicals are formed by H atom abstraction from the primary fuel, giving a number of isomers in each case. These were analysed for consistent elimination patterns. Alkyl radicals undergo a number of known reactions in the full schemes, namely; addition of O2, alkyl radical decomposition and H atom abstraction by O2. Of these, only the alkyl radical decompositions to form H + PRODUCT, e.g.

R20C4H9 � H + C4H8Z,

were ever removed. These had a strong likelihood of being removed as 83% of reactions of this type were removed in the cases tested. ROO• radicals were also examined. As well as the reactions already mentioned these radicals undergo isomerisations of the type

R34C7H15OO � R54C7H14OOH .

Reactions of this type were partially removed by the sensitivity reductions, with 31% of them being eliminated. No patterns regarding which of the types of isomerisations being eliminated were observed for n-butane, although some clear patterns emerged for n-heptane. The reverse reactions with the lowest and highest prefix to C7H14OOH were consistently removed from the n-heptane scheme, e.g. Full scheme

R36C7H15OO � R61C7H14OOH R36C7H15OO � R62C7H14OOH R36C7H15OO � R63C7H14OOH R36C7H15OO � R64C7H14OOH R36C7H15OO � R65C7H14OOH

Removed

R61C7H15OOH � R36C7H14OO R65C7H15OOH � R36C7H14OO

Thus patterns can be found for the types of species and reactions which are removed by

sensitivity analysis based reduction methods. As further mechanisms of different fuels are analysed, more patterns are likely to emerge as the kinetic schemes are examined through to the final products.