Chapter 5Simplified Reaction Models for Combustionin Gas Turbine Combustion Chambers

Dirk Lebiedz and Jochen Siehr

Abstract The aim of this work is an effective dimension reduction of chemicalcombustion mechanisms. Despite permanently growing computer power, the sim-ulation of a reaction-diffusion-convection system involving a large scale chemicalcombustion mechanism is still far from reach. On the other hand, prediction of e.g.soot, NOx, and other pollutants needs detailed mechanisms. Here, model reductionmethods can be used for generation of small models. At first, the focus of this workhas been on an efficient use of the intrinsic low dimensional manifold method. Later,a new method has been developed based on optimization methods. An efficient toolfor solving these optimization problems has been developed and reaction models upto the size of syngas combustion have been reduced.

Keywords Model reduction • Slow invariant manifold • Chemical kinetics •Nonlinear optimization

5.1 Introduction

The interplay between flow (convection), diffusion, and chemical reaction processesin a reactive flow simulation is fairly complex if the model is based on a detailedcombustion mechanism involving a large number of chemically reactive species.Here, complexity reduction and model reduction methods can be used effectively.

D. Lebiedz (�)Institute for Numerical Mathematics, University of Ulm, Helmholtzstraße 20,89081 Ulm, Germanye-mail: dirk.lebiedz@uni-ulm.de

J. SiehrInterdisciplinary Center for Scientific Computing (IWR), University of Heidelberg,Im Neuenheimer Feld 368, 69120 Heidelberg, Germanye-mail: jochen.siehr@iwr.uni-heidelberg.de

Fluid Mechanics and Its Applications 102, DOI 10.1007/978-94-007-5320-4 5,© Springer ScienceCBusiness Media Dordrecht 2013

161J. Janicka et al. (eds.), Flow and Combustion in Advanced Gas Turbine Combustors,

162 D. Lebiedz and J. Siehr

5.1.1 The Aim of Model Reduction

The aim of many model reduction approaches is an identification of so called slowinvariant manifolds (SIM). These manifolds are located in the chemical compositionspace. The general reaction transport equation in the variable —, which can be massfractions, partial density, temperature, or any variable describing the system statedepending on time t and space x, can be written as

@t � D S.�/ C T.�; @�x; @2x�/;

where S is the chemical source term and T the physical transport operator,i.e. convection and diffusion. Many methods for model reduction are applied tothe chemical source term of this system of reaction transport partial differentialequations (PDE), which describe the reactive flow. This means the model reductionmethod is only regarding the system of ordinary differential equations (ODE)modeling the kinetic source term

dt � WD d

dt� D S.�/:

Trajectories in the chemical composition space are relaxing to SIMs of lowdimension on their way towards equilibrium. In this sense, these SIMs represent theslow chemistry. The existence of a SIM is closely related to multiple time scales anda spectral gap in the eigenvalues of the Jacobian of the chemical source term. Themodel reduction method can be seen as a method for computation of such a manifoldpointwise. In general, some parameterization variables will be chosen, known asreaction progress variables and the method delivers the values of the other variablesin sense of species reconstruction. This can be used to simplify the computation ofthe full system of reaction transport PDE via applying it in situ or by evaluation ofa tabulated manifold [38] and hence saves a lot of computation time.

5.1.2 Common Model Reduction Methods

Early model reduction approaches like the quasi steady-state (QSSA) and partialequilibrium assumption (PEA) [47] have been performed “by hand”. Hence, specialexpert knowledge about the chemistry model has to be available. In contrast, modernnumerical model reduction methods compute automatically a reduced model. Manyof these modern techniques are based explicitly or implicitly on a time-scale analysisof the underlying ODE system with the purpose to identify a slow attractingmanifold in phase space where – after a short initial relaxation period – the systemdynamics evolve. For a detailed overview, see e.g. [24] and references therein.

An important method of this type is the intrinsic low dimensional manifoldmethod (ILDM). Maas and Pope introduced the ILDM method in 1992 [32].

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 163

It became very popular and widely used in the reactive flows community. A localtime-scale analysis is performed on the Jacobian JS of the system of ODEs modelingthe chemical kinetics S. An upper block triangular matrix is computed with theSchur decomposition.

The fast and slow submatrices can be fully decoupled with the so called Sylvesterequation [23]. Fast time scales are assumed to be fully relaxed. Together with theparameter equation the ILDM equations can be stated as

T �1

r; fastS.�/

P T� � c

!D 0;

where Tr,fast�1 is the projection into the fast eigenspace and the second line the

parameterization of the SIM. A drawback of this method is that a solution of theILDM equation does not have to exist. It has been shown that the ILDM methodidentifies the SIM to first order in the timescale separation [28, 40]. For recentdevelopments and extensions of the ILDM method, see e.g. [14] and referencestherein.

Another popular technique is the computational singular perturbation (CSP)method proposed by Lam in 1985 [30, 31]. The basic concept of this method is arepresentation of the dynamical system in a set of “ideal” basis vectors such that fastand slow modes are decoupled. This method improves the SIM approximation withone order per CSP iteration [48]. Also some iterative methods came into applicationthat are based on an evaluation of functional equations suitably describing thecentral characteristics of a slow attracting manifold, for example invariance andstability. Examples are Fraser’s algorithm [18, 21, 34] and the method of invariantgrids [15, 16, 24, 25]. Other methods are the invariant constrained equilibrium edgepreimage curve (ICE-PIC) method [37, 39], flamelet-generated manifolds [19] andmany more.

5.2 Models for Combustion Chemistry

We list formulae necessary for the formulation of a combustion model in thissection. For details we refer to text books as e.g. [29, 47].

A model comprises nspec chemical species composed by nelem chemical elements.The chemical source term S obeys the law of elemental mass conservation and anenergetic balance has to be fulfilled.

5.2.1 Mass Balances

In the following, the variables zi are given in terms of specific moles, which aredefined as the amount of species i (ni) divided by the total mass (m) of the system,

164 D. Lebiedz and J. Siehr

which is the same as the mass fractions (wi) of the species i divided by its molarmass (Mi):

zi D ni

mD wi

Mi

; i D 1; :::; nspec:

The elemental mass conservation of each element in the system is formulated as

Nzi DnspecXj D1

�ij zj ; i D 1; :::; nelem;

where ¦ij is the atomic composition coefficient – the number of element i in speciesj. There is also a restriction to the choice of the elemental specific moles, whichrequires that the mass fractions sum to one

nelemXiD1

NMi Nzi D 1;

with the molar mass of element i. This ensures the conservation of the total mass ofthe system.

5.2.2 Energy Balance

We consider systems within one of the four standard thermodynamic environmentswhich is either in an isothermal or adiabatic and either in an isochoric or isobaricstate. In the isothermal case only the temperature is fixed while in the adiabatic casesthe specific enthalpy h or the specific internal energy e respectively, are fixed.

5.2.3 ODE Models

The ODE model is considered as a constraint in the optimization problem for thecomputation of reduced models as discussed in a later section. Therefore, the ODEreaction system that is given by the source term (called S until now) is discussedin the following. This ODE system includes mass action kinetics and obeys theelemental mass conservation laws.

The mass balance in specific moles zi can be formulated as

dt zi D Sm WD !i

�; i D 1; :::; nspec:

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 165

The symbol ¡ refers to the overall mass density in the system in the right handside of this equation. It is given via the ideal gas law

� D m

VD p NM

RT:

The total mass and volume V are needed in the isochoric case; the total pressurep, the gas constant R, the temperature T, and the mean molar mass are necessary inthe isobaric case for computation of the right hand side. The molar net chemicalproduction rate is denoted by the vector ¨. It has to be computed based on aset of chemical elementary reactions and their kinetic parameters as described inSect. 5.2.4.

In our case, we formulate the energy balance via the right hand side of thetemperature equation dtT D Se. In the isothermal case Se D 0 holds. In the adiabaticcases energy or enthalpy conservation defines the concise form of Se.

5.2.4 Chemical Kinetics

We subsume the computation of ¨ in the remainder of this section. A chemicalcombustion mechanism generally is given as a set of nreac elementary reactionsinvolving nspec species (and eventually a third body M)

nspecXiD1

�0ij Xi $

nspecXiD1

�00ij Xi ; j D 1; :::; nreac

with the chemical species Xi and the forward and reverse stoichiometric coefficients�ij

0 and �ij00. The forward and reverse rate of reaction j is

rf;j D kf;j

nreacYiD1

c�0

ij

i

rr;j D kr;j

nreacYiD1

c�00

ij

i

with the concentrations ci of species i and the rate constants kf,j and kr,j. With the netstoichiometric coefficient

�ij D �00ij � �0

ij

the net rate of change ¨i of species i is given as

166 D. Lebiedz and J. Siehr

!i DnreacXj D1

�ij .rf;j � rr;j /:

In case a third body M takes part in reaction j, third body collision efficiencies ’i

for all species i D 1, : : : , nspec are to be given. The third body concentration

cM DnspecXiD1

˛i ci

is then multiplied to the products above.Formulas for the computation of the rate coefficients are stated in the following.

The elementary reactions in the mechanism considered here are given in Arrheniusformat and pressure dependent Troe format, resp. The three parameters A, b, andEa are given in the Arrhenius case for each reaction. The forward reaction ratecoefficient is computed via the extended Arrhenius formula

kf;j D ATb

1K e�EaRT :

A more complicated formula applies in case of pressure dependent reactions.Here, the forward rate constant is computed using Troe fall off curves, see e.g. theoriginal publications [22, 45].

The reverse reaction rate constant of a reaction is computed via evaluation of theequilibrium constant of the reaction. The equilibrium constant of reaction j in termsof concentrations is given as

Kc;j D�

pı

RT

��j

exp

��Sı

r;j

R� �H ı

r;j

RT

�;

with the standard pressure pı and the net change of the number of species

�j DnspecXiD1

�ij :

The change in entropy and enthalpy can be computed by using an evaluationof the NASA polynomials for their molar values of the species involved [13]. Thereverse rate of reaction j is

kr;j D kf;j

Kc;j

:

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 167

5.3 Optimization Problem

Lebiedz introduced a model reduction method based on the minimization of entropyproduction rate along trajectories in chemical composition space in [1]. The basicidea is that the SIM should be characterized by maximum relaxation of the systemdynamics under given constraints of fixed reaction progress variables. This approachhas been extended and refined in a number of following publications [2, 4, 6–8].

An optimization problem has to be solved for the approximation of points on themanifold. It can be written in specific moles and temperature as minimization of anobjective functional

minz;T

tfZt0

ˆ.z.t// dt

subject to

dt z.t/ D Sm.z.t/; T .t//

dt T .t/ D S e.z.t/; T .t//

0 D C.z.t�/; T .t�//

0 D zj .t�/ � zt�j ; j 2 Irpv

0 � z.t/; T .t/

and

t 2 Œt0; tf�

t� 2 Œt0; tf� .fixed/:

The dynamics of the model to be reduced are given via the ODE system in theconstraints. This guarantees to identify as solution of the optimization problem aspecial solution trajectory (piece).

All conservation laws, that the system obeys, are combined in the (nonlinear)function C. The dynamics contain differential forms of the balances of mass andenergy. The values of the conserved quantities have to be specified at some (fixed)point in time along a solution which we choose to be t*. They are also fulfilled alongthe whole time interval via the dynamics. This is the conservation of each chemicalelement and a specification of a fixed temperature, enthalpy or energy depending onthe assumed thermodynamic environment.

168 D. Lebiedz and J. Siehr

In order to approximate the SIM, the parameterization needs to be specified.The species (specific moles zi) which serve as reaction progress variables andtheir number have to be specified in advance. The number of progress variablesdetermines the chosen dimension of the SIM to be approximated. The indicesof the reaction progress variables are collected in the index set Irpv, a subset off1, : : : ,nspecg, and their values are fixed in the optimization problem. Hence, thesevalues are parameters of the optimization problem and change between differentcalls for the solution of the problem for deriving different points on the manifold.

Only positive values of specific moles and temperature have a physical meaning.This physical inequality constraint is included in the optimization problem. This isalso of technical importance in the optimization context with a realistic combustionmechanism as negative values of zi and T can result in undefined values (logarithmof a negative number) of the right hand sides Sm and Se for pressure dependentreactions. The positivity and the linear mass conservation relations define a polytopein the composition space, where the dynamics evolve.

The crucial point in time t* in [t0, tf] specifies the position where along theoptimal trajectory piece the fixation of the reaction progress variables (and theconstraint C) is applied. In first publications, e.g. [1, 3, 4], t* D t0 has been chosen.The inverse idea is the fixation at the end point t* D tf. The solution point andsolution trajectory piece of the optimization problem is supposed to be part of theSIM. Therefore, also in backward direction of time the trajectory is supposed to berelaxed.

To conclude, it can be stated that the essential degrees of freedom of theoptimization problem are the effective phase space dimension nspec � nelem minusthe number of reaction progress variables jIrpvj. The goal of solving the optimizationproblem is the determination (species reconstruction) of the “missing” values as amapping of the parameters into full composition space.

5.3.1 The Criterion

The choice of the criterion ˆ is crucial. It should measure the degree of chemicalforce relaxation along a trajectory, which should be minimal on the SIM. Severalcriteria have been tested, especially in [4, 5].

5.3.1.1 Entropy Based Criteria

In [1], the author introduced the concept of optimization based model reductionby identification of so called minimal entropy production trajectories (MEPT). Theconcept is based on choosing trajectories along which the entropy production isminimal. The entropy production rate of elementary reaction j (due to irreversibleprocesses) is defined as

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 169

diSj

dtD R.rf;j � rr;j / ln

�rf;j

rr;j

�:

The total entropy production rate is the sum over all elementary reactions. Hence,the criterion used in the optimization problem (the different suggestions for ˆ arenumbered in the following) would be

ˆ1.z.t// DnreacXj D1

diSj

dt:

In [2], this idea is extended and discussed with the definition of entropy in amathematical sense.

5.3.1.2 Curvature Based Criteria

If we regard the formula for entropy production rate in comparison to the formulafor the net chemical production rate, it is obvious that there is a connectionbetween minimizing the entropy production and the chemical production. The onlydifference is the mass density and a weighting induced by the multiplication with thenet stoichiometric coefficients instead of the logarithmic ratio of the rates. Curvaturebased criteria are an attempt in the same direction which starts from another pointof view.

The SIM to be approximated is considered to represent the slow dynamics. Thismeans the rate of change of the variable values is supposed to be small. The rate ofchange is related to the curvature of the trajectories as geometrical objects in phasespace, it is the second derivative

Rz D JSm.z.t/; T .t//Sm.z.t/; T .t//;

where we denote the Jacobian of a function S as JS. This can be interpreted as adirectional derivative of the chemical source term with respect to its own direction.In a notation that coincides with the general optimization problem, a choice for thecriterion ˆ would be

ˆ2.z.t// D jjJSm.z.t/; T .t//Sm.z.t/; T .t//jj22:

Interestingly, this can be seen as minimizing the length of a trajectory piece in asuitable Riemannian metric. The length of a continuously differentiable curve z ona Riemannian manifold is defined as the curve integral

L.z/ DZ

z

pgz.Pz.t/; Pz.t// dt

170 D. Lebiedz and J. Siehr

with a scalar product gz on the tangent space of the curve in each point. Thechoice of gz(S,S) D STJS

TJSS with the symmetric positive definite (if there areno conservations) matrix JS

TJS leads to ˆ2. Therefore, the solution curve can beinterpreted as a geodesic, i.e. a curve which minimizes the length of path betweentwo points along the manifold. This duality of minimal velocity change and minimallength arises naturally as a minimal curvature coincides with a minimal distance thatis covered and vice versa.

5.3.2 Theoretical Results

5.3.2.1 Existence of Solutions

In [6] and [8], it has been shown that there always exists a solution of theoptimization problem subject to a model which describes a combustion process asintroduced before if there exists a feasible solution at all.

The key is the proof of the compactness of the feasible domain of the opti-mization problem. In case of linear (mass conservation) constraints together withpositivity, the chemically realizable space is a closed, bounded polytope in thechemical composition space, which is compact by definition. In case of the nonlinearenthalpy or energy conservation, proper maps ensure compactness. Proper mapsare defined as maps where the preimage of a compact set is compact. In thisway, the feasible domain is the intersection of compact spaces and, therefore, alsocompact.

The objective function is a continuous function in the state (z,T) at t*. It takes itsminimum on a compact feasible set. Hence, a solution exists if the feasible set is notempty.

5.3.2.2 Quality of Solutions

In [6], it has been shown that the reverse mode of the method identifies the correctSIM for infinite time horizon in case of a linear test model and the Davis–Skodjetest model [18], which have an analytically defined one-dimensional SIM. Hence,we use the reverse mode for all numerical results presented here.

The proof exploits the fact that the general linear system as well as the Davis–Skodje model has an analytical solution. This solution is directly evaluated in theobjective function. The solution is identified via the standard sufficient conditionsfor optimality. A consideration of the limit t0 ! �1 finally shows that thedeviation between the given exact SIM and the solution of the optimization problemvanishes. The same result holds in case of an infinite spectral gap for a finite timehorizon.

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 171

5.4 Numerical Methods

The functions in the semi-infinite optimization problem have to be discretized.The resulting nonlinear programming (NLP) problem can be solved either by asequential quadratic programming (SQP) [36] or an Interior Point (IP) method, seee.g. the review [20].

5.4.1 Discretization

There are mainly two ways for discretization and solution of the semi-infiniteoptimization problem: the sequential and the simultaneous approach.

In the sequential approach, the solution of the ODE and the optimization are fullydecoupled. The initial values (z(t0),T(t0)) serve as optimization variables. Startingat t0 the system is integrated with a stiff ODE solver, e.g. via a BDF scheme [17].The integrand in the objective function is integrated itself, and the end point isevaluated in sense of a Mayer term objective functional. The optimization iteration isperformed after that based on the results of the integration and computed derivativeinformation. A single shooting is appropriate in the cases t* D t0 and t* D tf as wedeal with a stable ODE system, whereas a double shooting is needed in case of t* in(t0, tf) for the values at t*.

It might be beneficial in case of unstable or extremely stiff systems to use an all-at-once approach. The solution of the dynamic constraints and the optimization arecoupled in this so called simultaneous approach. The interval [t0, tf] is divided intoseveral sub-intervals. Polynomials are constructed via, e.g., a collocation method[10] on each sub-interval tangent to the vector field of the dynamics approximatingtheir solution, and the corresponding formulae are treated as constraints in theoptimization iteration. We use a Gauss–Radau formula in a collocation approachwith linear, quadratic, and cubic polynomials, respectively, because they have stiffdecay [10].

5.4.2 Solution of the Finite Dimensional Optimization Problem

In both cases, the result of the discretization is a finite-dimensional NLP problem.It can be solved using an SQP algorithm or an IP method. The SQP algorithm treatsthe inequality constraints with an active set strategy, see e.g. [35]. Newton’s methodis applied to the first order optimality conditions of a quadratic approximation of theobjective function of the NLP problem and a linear approximation of the constraints.Here, only equality and active inequality constraints are taken into account. Theactive set in the solution is iteratively identified by activating and deactivating con-straints. In contrast, the inequality constraints are coupled to the objective function

172 D. Lebiedz and J. Siehr

in an IP method via a barrier term forcing the iterates strictly into the interior ofthe feasible domain. The resulting equality constrained NLP problem is solvedwith a homotopy method. Newton’s method is applied to the first order optimalityconditions. The barrier parameter is driven to zero in progress of optimization. Inthis way a homotopy path to the solution of the NLP problem is followed.

5.4.3 Algorithms and Software

Application results of the model reduction method are presented in the followingsection. In general, for the solution of the problem we only use open source softwareand our own developments. We use the software package IPOPT [46] for numericaloptimization.

For the solution of linear equation systems within the optimization algorithm,HSL routines [27] and MUMPS [9], resp., are used. We use a collocation approachbased on a Gauss–Radau method [10] for discretization of the optimization problem.Alternatively, we use a shooting approach with a BDF integrator that has beendeveloped in [42, 43]. We use MATLAB for plotting as the only commercialsoftware.

Derivatives needed for the optimization are computed with the automatic differ-entiation package CppAD [11, 12]. The strategy of CppAD is based on operatoroverloading. The computer code which is supposed to be differentiated is savedwithin a CppAD tape. This tape can be “optimized” before evaluation. Furthermore,the criterion JSS can be seen as a directional derivative where the use of multiplelevels of AD (the evaluation of a tape within a tape) can be beneficial. The BDFintegrator used in the sequential optimization approach directly uses such a tape asinput variable. For generation of sensitivities, it incorporates several strategies ofsolving sensitivity differential equations and adjoint differential equations [42].

A user usually computes not only a single approximation point on the SIM but asequence of points needed e.g. in a CFD simulation. Hence, it is necessary to have aneffective warm start strategy that has been implemented in our code with a homotopymethod. A homotopy parameter drives the iterates from a previous solution to thesolution of new problem.

As most numerical codes which simulate combustion processes, the code alsoreads input files for thermodynamic and kinetic parameters. In our case, it is thestandard format for NASA polynomial coefficients [13] and the reaction mechanismwith the kinetic parameters for each reaction.

5.5 Results

Results for the application of the optimization based model reduction method areshown in this section.

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 173

Fig. 5.1 Results for the solution of the optimization problem in case of the Davis–Skodje modelwith a spectral gap ” D 3. The rings are the solution points for several values of the reactionprogress variable y1, for each of which the optimization is solved

5.5.1 Davis–Skodje Mechanism

The Davis–Skodje mechanism serves as first test case [18, 41]. It is given via atwo-dimensional ODE system

dy1

dtD �y1

dy2

dtD �y2 C . � 1/y1 C y2

1

.1 C 1/2

;

where ” > 1 is a measure for the spectral gap (time scale separation). The Davis–Skodje model is a standard test model for model reduction methods due to theadjustable time scale separation and an analytically given SIM as the graph of thefunction

y2 D y1

1 C y1

:

Results for the solution of the optimization problem are shown in Figs. 5.1and 5.2. In both cases, a small value for ” has been chosen, which poses a highchallenge for model reduction algorithms. The open rings denote solutions of theoptimization problem at the end point t* D tf. The optimization problem is solvedwith criterion ˆ2 in the objective function along a time horizon of tf – t0 D 5, wherehere yi play the role of zi (i D 1, 2) in the problem formulation. Here y1 is chosenas reaction progress variable and fixed to several different values. Additionally,trajectory pieces are shown which emanate from the solutions points (as solutions

174 D. Lebiedz and J. Siehr

Fig. 5.2 Results for the solution of the optimization problem in case of the Davis–Skodje modelas in Fig. 5.1 but with a spectral gap ” D 1.1

Table 5.1 Simple hydrogencombustion test mechanismfrom Gorban et al. [26]

kf kr

H2 $ 2H 2.0 216.0O2 $ 2O 1.0 337.5H2O $ H C OH 1.0 1,400.0H2 C O $ H C OH 1,000.0 10,800.0O2 C H $ O C OH 1,000.0 33,750.0H2 C O $ H2O 100.0 0.7714

Forward and backward rate coefficients do notdepend on temperature

of the initial value problem) and converge to equilibrium, shown as full dot. Thereverse part of the trajectory through a solution point is the optimal trajectory piececomputed in the optimization process. It can be seen that the results deliver aninvariant one-dimensional manifold. Only for the small value y1 D 0.01 the solutionis not invariant in Fig. 5.2. Invariance could be achieved with a larger time horizonas has been proved in [6].

5.5.2 Six Species Test Mechanism

Another small test mechanism, which has been introduced for model reductiontest purposes in [26] consists of six chemical species involved in six (in each caseforward and reverse) elementary reactions, cf. Table 5.1.

This test mechanism is regarded within an isochoric and isothermal thermody-namic environment. Hence, a choice of ¡ D 1 kg/m3 leads to the same numericvalues of concentrations and specific moles

zi D ci kg=m3 8i D 1; :::; nspec;

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 175

and we arrive at our standard formulation in specific moles. This mechanism is onlyan academic example. Units are meaningless here and are skipped in the following.

There are only the chemical elements hydrogen and oxygen involved. Thecorresponding mass conservation relations are chosen to be

2zH2 C 2zH2O C zH C zOH D 1

2zO2 C zH2O C zO C zOH D 2:

Hence, this mechanism yields a system with effectively four degrees of freedom.We want to compute a two-dimensional manifold. We chose the values of thefuel zH2 and the reaction product zH2O to serve as reaction progress variables.Their values are fixed at different values in the area where the value of zH2 ishigher than the equilibrium value and zH2O is lower than its equilibrium value.Combinations which violate the mass conservation and positivity are identified andignored automatically by the code.

The results are shown in Fig. 5.3, where again the open rings refer to differentsolutions of the optimization problem and trajectories emanating from these areconverging to equilibrium, shown as a full dot with a large value of H2O and nearlyall H2 is “burned.” An invariant two-dimensional manifold can be seen, wherethere is no relaxation phase of the trajectories from the solution points onto themanifold.

5.5.3 Ozone Decomposition

We consider an ozone decomposition mechanism, which includes atomic oxygen,dioxygen, and ozone, as a first realistic test problem. The mechanism is given inTable 5.2 with the coefficients as in [33] for the forward reactions.

The thermodynamic data is used in form of NASA polynomial coefficients.The model is formulated as in [8] within the framework described in Sect. 5.2.Reverse rate coefficients are derived from equilibrium thermodynamics. For massconservation the elemental specific mole has to be given. It is

NzO D .0:015 999 kg=mol/�1 D 62:5 mol=kg:

We consider a density of ¡ D 0.2 kg/m3 in the isochoric case and a pressure ofp D 105 Pa for isobaric conditions, respectively.

No positivity is demanded in the results presented in Figs. 5.4, 5.5, 5.6, and 5.7for the species, so negative values can be seen in the plots. The model leaves twodegrees of freedom; we choose to compute approximations of a one-dimensionalmanifold with zO as reaction progress variable. Again the results are shown asin the examples before. It can be seen that there is always a relaxation phase ofthe optimized trajectory piece onto that manifold, especially in the adiabatic casesshown in Figs. 5.6 and 5.7 far from equilibrium.

176 D. Lebiedz and J. Siehr

Fig. 5.3 Results (two-dimensional SIM) for the six species test mechanism. The value of zH2 andzH2O serve as reaction progress variables. The optimization problem is solved several times fordifferent combinations of these. We use the reverse mode (tf D t*) with an integration interval oftf �t0 D 10�5. The free zi are plotted versus the progress variables

Table 5.2 Ozone decomposition mechanism for the forward rates as in[33]

A/cm, mol, s b Ea/kJ mol

O C O C M $ O2 C M 2.9 � 1017 �1.0 0.0O3 C M $ O C O2 C M 9.5 � 1014 0.0 95.0O C O3 $ O2 C O2 5.2 � 1012 0.0 17.4

Collision efficiencies in reactions including M: ’O D 1.14, ’O2 D 0.40,’O3 D 0.92

5.5.4 Syngas Combustion Mechanism

We use a syngas combustion which is a part of the GRI 3.0 mechanism [44] asan example of a full detailed chemistry combustion mechanism. It consists of 33reactions of the GRI 3.0 mechanism which involve no other species than O, O2, H,

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 177

Fig. 5.4 Results of the approximation of a one-dimensional SIM for the ozone decompositionmechanism within an isothermal and isochoric environment. The value of zO serves as reactionprogress variable. The optimization problem is solved several times for different values ofzO between zero and the largest possible value allowed by elemental mass conservation. Theintegration interval is tf �t0 D 10�6 s. The free zi (in mol/kg) and temperature are plotted versus zO

Fig. 5.5 Results of the approximation of a one-dimensional SIM for the ozone decompositionmechanism modeled within an isothermal and isobaric environment. Again the integration intervalof tf �t0 D 10�6 s is used

OH, H2, HO2, H2O2, H2O, N2, CO, and CO2. Those 33 reactions are 31 Arrhenius-type and two pressure-dependent ones.

The overall syngas combustion reaction is

H2 C CO C O2 ! H2O C CO2

where N2 only serves as a collision partner. We assume a stoichiometric mixture ofsyngas with air in an adiabatic and isochoric environment. As fixed mass density weuse ¡ D 0.3 kg/m3.

178 D. Lebiedz and J. Siehr

Fig. 5.6 Results (one-dimensional SIM) for the ozone decomposition mechanism modeled withinan adiabatic and isochoric environment. As in the adiabatic case the reaction is faster, theintegration horizon is reduced to tf �t0 D 10�8 s

Fig. 5.7 Results (one-dimensional SIM) for the ozone decomposition mechanism modeled withinan adiabatic and isobaric environment. Again the integration horizon is tf � t0 D 10�8 s

We assume a ratio of nH2:nCO D 1:1, and a ratio of nO2:nN2 D 1:3.76 simi-lar to air. This leads to an unburned mixture of zCO D zH2 D 5.973 mol/kg andzN2 D 22.46 mol/kg. The specific internal energy of this mixture at a temperatureof T D 1,000 K is used as a fixed specific internal energy for SIM computation.

Results for the solution of the optimization problem are shown in Fig. 5.8. Themodel has eight degrees of freedom. We aim to compute an approximation of atwo-dimensional manifold near equilibrium. We chose the products of the overallreaction as parameterizing variables, namely zH2O and zCO2. The system has muchmore time scales involved in comparison to the examples before. The computationaleffort is much higher. In case of an application of the method within a CFDapplication, a higher dimensional parameterization would certainly be desirable.

5 Simplified Reaction Models for Combustion in Gas Turbine Combustion Chambers 179

Fig. 5.8 Plot of the results of an approximation of a two-dimensional SIM in the phase space ofthe syngas combustion model. The values of zH2O and zCO2 are used as reaction progress variables.An integration horizon of tf �t0 D 10�7 s is chosen

5.6 Conclusion

We have implemented a code for model reduction of combustion processes. Themodel reduction method is based on a suitable criterion for the minimizationof a relaxation of chemical forces along reaction trajectories. It can be shownthat there always exists a solution of the optimization problem which defines the

180 D. Lebiedz and J. Siehr

approximation of the slow manifold. Furthermore, it can be shown that the methodidentifies the exact slow manifold in case of a linear system and the Davis–Skodjetest model. The method has been successfully applied to many example problemsand models which arise in combustion chemistry.

Acknowledgements The authors acknowledge the financial support from the German ResearchCouncil (DFG) through the project B2 within SFB 568.The authors wish to thank the late Jurgen Warnatz (IWR, Heidelberg) for providing professionalmentoring for combustion research.

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