Flow Turbulence Combust (2015) 94:593–617 DOI 10.1007/s10494-015-9595-3
A Constrained Control Approach for the Automated Choice of an Optimal Progress Variable for Chemistry Tabulation ¨ Uwe Prufert · Sandra Hartl · Franziska Hunger · Danny Messig · Michael Eiermann · Christian Hasse
Received: 29 August 2014 / Accepted: 15 January 2015 / Published online: 6 February 2015 © Springer Science+Business Media Dordrecht 2015
Abstract Flame structure look-up table generating strategies, e.g. the flamelet-progress variable approach, are based on the definition of a suitable progress variable. This variable is usually obtained from a linear combination of the species’ mass fractions contained in the system. The most important issue is the unique mapping between the progress variable and the independent variable, e.g. space or time. This is ensured by claiming the monotonicity of the progress variable. For simple fuel-oxidiser compositions, this can be performed by analysing the flame structure. However, in the case of complex chemical systems, finding such a progress variable is a non-trivial task, since a lack of monotonicity in the main species mass fractions often exists. In this article it is investigated how a valid progress variable can be found by an automated procedure. Some recent investigations use the fact that finding a monotonous progress variable is equivalent to solving a constrained optimisation problem. The new approach is to construct an algorithm to find a valid progress variable, which is also optimal in the sense that the gradients of the tabulated quantities are minimized. This leads to smoother data and reduces the interpolation effort when reading from the table. The method solves a nonlinear optimal control problem, where the monotonicity of the progress variable is ensured as an inequality constraint by using a genetic algorithm. Finally the performance of the new algorithm is evaluated for homogeneous reactor calculations and laminar diffusion flamelet look-up tables. Keywords Constrained optimal control · Genetic algorithm · Flamelet-progress variable · FPV · Chemistry tabulation
U. Pr¨ufert · M. Eiermann TU Bergakademie Freiberg, Institute of Numerical Mathematics and Optimisation, Freiberg, Germany S. Hartl () · F. Hunger · D. Messig · C. Hasse TU Bergakademie Freiberg, Chair of Numerical Thermo-Fluid Dynamics, Freiberg, Germany e-mail:
[email protected]
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1 Introduction Simulating chemically reacting processes in turbulent flows requires suitable models to describe the turbulence-chemistry interaction (TCI). One widely used model is the steady laminar flamelet model [8]. The flamelet model is based on the assumption of fast but finite chemistry in thin flame sheets. This model was the starting point for a number of developments and some extensions have been made. One recent extension is the flamelet-progress variable (FPV) model [9] in order to capture different processes which are not described by the steady flamelet model as demonstrated for local extinction, NOx emission and radiation [3, 4]. Similarly, the flamelet generated manifolds (FGM) approach [14] and the flame prolongation of intrinsic low dimensional manifolds (FPI) [2] use the look-up over a progress variable as the parameter describing the progress in the combusting mixture. The latter are both based on premixed flames. Here, the chemical solution set Yk can be addressed as Yk (Z, C), where Z is the mixture fraction and C the normalised progress variable. In those TCI models, a suitable choice of the progress variable is crucial. Attention has been paid to this topic in [5–7]. Some requirements for the progress variable can be defined according to [5]: – – – –
The transport equation is conveniently solved in a combustion simulation. The parameter setup characterise uniquely each point in the thermo-chemical state space, hence it has to be strictly monotonically decreasing or increasing. Process controlling quantity and projection of the trend of the combustion process (flame region, slow conversion reactions). Appropriate mapping with respect to the species and temperature.
Identifying a suitable progress variable can become more complicated for complex flame structures and detailed chemistry. An optimisation procedure for finding a progress variable is therefore proposed in this investigation. This work can be seen as an extension of the previous work by Niu et al. [7] with the main difference that the chemical species gradients are directly minimized as a function of the independent variable instead of only constraining this quantity from above. The remaining constraint is the demand of monotonicity of the progress variable. Further it is not required that a valid initial guess of a progress variable must exist, which can become relevant in the application of this approach. Furthermore diffusion flames are analysed, in contrast to previous works, where mostly premixed flames are investigated. Considering diffusion flames gives a wider range of equivalence ratios since not only the reactive regime is considered and valid data sets over the whole equivalence ratio range can be obtained. Here the algorithm for finding a valid and optimal progress variable is applied to homogeneous reactor calculations as well as laminar diffusion flamelet tables. Both approaches are used for a well established methane-air flame [1] and an inverse diffusion oxyfuel flame [12] to verify the algorithm and the resulting progress variable within the flamelet-progress variable approach. The article is organised as follows: In Section 2 an introduction into the general progress variable formalism is given as well as a description of its computational procedure. Following in Section 3 the optimal control problem is defined. Its solution is a valid progress variable to a number of parameter dependent data sets. In Section 4 the result of the optimisation of the progress variable for the two test cases are presented. Finally, conclusions are given in Section 5.
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2 Definition of a Feasible Progress Variable In the following, we use the same notation as Niu et al. [7] for the progress variable definition. A chemical system including ns chemical species is considered. Let Yk (x) be the mass fractions of chemical species k with k = 1, ..., ns over a set X ⊆ R. In this article it is assumed that x ∈ X is ordered with respect to the relation ‘ 0, ∀δ > 0, ∀x ∈ X. normalised, if min(Yc (x)) = 0 and max(Yc (x)) = 1 holds. x∈X
x∈X
Note that throughout this article a normalised progress variable Yc is denoted by C but in the following, the optimisation method is developed in terms of a general progress variables Yc . It is worth mentioning that item (i) can never be fulfilled if all mass fractions Yk are constant on a non-empty subset of X. In this case, no progress variable exists. The requirement of strong monotonicity guarantees that the mapping f : x → Yc (x) is invertible. The strict monotonicity of the progress variable (i) is equivalent to c (x) (ia) dY > 0, ∀x ∈ X. dx Throughout this article, strong monotonously decreasing progress variables are considered as their negative and only strong monotonously increasing progress variables are treated. When using for example a global one-step algorithm, where no further species conversion takes place, different monotonous progress variables can be found, e.g. the product mass fraction or the negative mass fraction of fuel and oxidiser. When selecting species based on a simulation using a detailed chemical mechanism, the definition of a progress variable (if one exists) can become much more complicated. Based on the idea of using the progress variable in chemistry tabulation approaches it would be advantageous to have smooth species and temperature gradients for interpolation to avoid potential issues in the species look-up, e.g. small changes in Yc lead to large differences in Yk . For this reason, the following additional constraint is required: k (iii) dY dYc (x(Yc )) < ωk with ωk > 0. This means that the absolute value of the species derivative to the progress with respect variable should not exceed a threshold. The vector ω = ω1 , ..., ωns gives an individual measure for the derivative of each species with respect to the progress variable Yc .
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All Yc , which fulfill (i) and (iii), are denoted as feasible and the set of these Yc is denoted the feasible set. Note that every solution fulfilling these constraints is feasible. Depending on the choice of ωk the set of feasible solutions may be empty. Based on these formulations the optimisation problem used in this paper is derived in the following.
3 Finding a Progress Variable as an Optimisation Problem First the problem formulation used in this paper is presented in Section 3.1. In Section 3.2 a similar approach derived by Niu et al., considering a linear optimisation approach to minimise f (α) = c · α (see Eq. (24) in [7]) and using (iii) as a constraint, is reviewed. Further the choice of the coefficient vectors is discussed in Section 3.3 and the extension to a multi-parameter problem in Section 3.4. 3.1 Problem formulation The problem of finding a valid progress variable can be written in terms of optimisation: dYk (x(Yc (x; α))) minn f α∈R s dYc (x; α) subject to Yc (x + δ; α) − Yc (x; α) ≥ 0, ∀x ∈ X, ∀δ > 0 (1) and ns αk Yk (x), x ∈ X. Yc (x; α) := k=1
This means that a progress variable Yc is sought that minimises the species gradients and fulfills the criterion of monotonicity. It should be taken into account that the data Yk (x) is obtained by a numerical simulation and hence only exists on discrete points xn , n = 1, ..., nx . Thus, the discrete approximations of the derivatives, i.e. finite differences are used. The discrete counterparts of the problem read: (Yc ) n (α) := Yc (xn ; α) =
ns
αk Yk (xn ),
n = 1, ..., nx .
k=1
Y = {Yk,n }, k = 1, ..., ns , n = 1, ..., nx and Arranging thedata Yk,n := Yk (xn ) in a matrix setting Yαc = (Yc )1 (α), . . . , (Yc )nx (α) with α = [α1 , ..., αns ], it can be written as
Yαc = αY = Y α . Note that Yαc and α are vectors and Y is in general not square. First all constraints have to be discretised. Starting with the monotonicity criteria, (1), the discrete version can be written as: ns Yc (xj + δxj ; α) − Yc (xj ; α) = αk Yk (xj + δxj ) − Yk (xj ) > 0 k=1
for j = 1, ..., nx − 1, where δxj is given by δxj = xj +1 − xj . This can be written in terms of matrices as (2) Yαc := αY,
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where Y := [Y(:; 2, ..., nx ) − Y(:; 1, ..., (nx − 1))] ∈ Rns ,nx −1 . Second, the discrete formulation of the derivative of Yk with respect to the progress k variable Yαc is obtained by approximating the differential quotient dY dYc (x(Yc )). This leads to Yk (xj +1 (Yc )) − Yk (xj (Yc )) dYk (xj (Yc )) ≈ ns dYc αi (Yi (xj +1 (Yc )) − Yi (xj (Yc ))) i=1
(Y)k,j = Yαc j
(3)
for all k = 1, ..., ns and j = 1, ..., nx − 1. Defining the matrix (Y)kj (DY(α))kj := Yαc j we obtain DY(α) = Y · diag(1./αY), where ./ is the pointwise division with respect to the entries of the vector/matrix and diag is the diagonal operator diag(v) = V with Vii = vi and Vij = 0 for i = j . The problem now reads minn f (DY(α)) (4) α∈R
s
subject to Yαc = αY > 0. (5) n ,n s x Further, the function f : R → R can now be chosen to fit the requirements of the tabulation. Note that in the problem formulation a progress variable will be identified by its coefficient vector α. 3.2 Barrier formulation of the feasible set problem The main difference of the method shown here and the optimisation by Niu et al. [7] is the direct minimisation of the quantity of interest DY while the demand of the monotonicity of Yc is considered as a constraint. Before going into detail to the our approach for the optimisation problem, the approach by Niu et al. [7] is summarized to identify potential issues using the quantity DY as a constraint. In [7] an optimal control approach is used to identify a strictly monotonically progress variable by formulating a linear optimisation problem f (α) = c · α = ci αi (see eqn. (24) in [7]), with c ∈ Rns . By setting c = 0 the method in [7] only identifies a progress variable with a bounded gradient DY but does not minimise this quantity. The optimisation problem used by Niu et al. [7] is shown below: min (c · α)
α∈Rns
subject to dYc (x; α) > 0 , ∀x ∈ X dx (Y) dY k,j k = (x(Yc (x; α))) < ωk , Yαc j dYc Yαc =
and
(6)
(7)
where ωk defines an upper bound. They further assume that a valid progress variable, i.e. an α with Yαc > 0, exists.
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Based on the existence of a feasible progress variable α, Yαc > 0 yields and the eliminating of the non-differentiable absolute value in Eq. 7 was stated. From Eq. 7 the formulation (Y)k,j Yαc j ≥ , ∀k = 1, ..., ns , ∀j = 1, ..., nx − 1 (8) ωk is obtained. Since the left-hand side of Eq. 8 yields ∀k = 1, ..., ns the upper bound can be considering for every component k which results in: (Y)k,: α Yc ≥ max . (9) k=1,...,ns ωk Note that by ωk < ∞ the constraint (6) is also included in Eq. 9. Note further that for ωk → 0 the left hand side of Eq. 9 increases, i.e. the derivative Yαc is only bounded from k below and consequently they maximise Yαc when they minimise dY dYc . When a nonempty interior of the feasible region exists this optimisation problem can be solved with the barrier formulation of interior points methods [16]. The basic idea of barrier methods is the reformulation of the objective function with the inequality constraints into a logarithmic barrier term. For solving a nonlinear optimisation problems of the form (10)
min f (x)
x∈Rn
subject to xi ≥ 0, ∀j = 1, ..., m1 with m1 ∈ R, the reformulated into the equality constrained problem yields min
f (x) − μ n
x>0,x∈R
m2
ln(xj ),
(11)
(12)
j =1
where μ > 0 is the barrier parameter. This leads to the following barrier formulation of the optimisation problem: ⎛ ⎞ nx (Y)k,j α ⎠, ln Yc j − max min ⎝c · α − μ k=1,...,ns ωk
(13)
j =1
where no further constraints on the coefficients αk , k = 1, ..., ns , are present. This formulation generates mainly two potential problems: (i) (ii)
Depending on the choice of ω, the feasible set may be empty, hence an appropriate choice for the constraint ωk , k = 1, ..., ns is needed. Since no further constraints on the coefficients αk exists, the objective (13) is not convex. By for example introducing an additional constraint for the coefficients, e.g. |αk | ≤ a for a ∈ R convexity can be forces for the feasible set.
However, this approach is only valid if at least one feasible solution exists, i.e. a valid progress variable for all data sets has to be at hand. Unfortunately this is hard to achieve. Note that in Niu et al. [7] the feasibility of the progress variable is used for the elimination of the magnitude in Eq. 7. To prevent these difficulties, a direct minimization of the k quantity dY dYc instead of applying the constraint (9) is suggested. It is worth mentioning that the reformulation of Niu et al. treats the problem as a simple linear optimisation problem with a reasonable, user-defined bound on ω. This choice of ω is no longer needed here.
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3.3 The choice of coefficient vectors α In general, the method proposed has only the restriction that the set A has to be a bounded subset of Rn . Within this restriction, it is possible to use all pseudo-real computer numbers in the optimisation. However, this usually leads to a prohibitive increase in computational time for the optimisation problem especially for large chemical systems. Therefore, it is advantageous for the algorithm and the convergence time to set some restrictions to the coefficient vector. One thing to consider are the species with close to zero mass fraction since they do not contribute to the progress of the combustion. In the following species with Yi < , < 10e−6 , are set to αi = 0. Note that the choice of may be a little delicate since the remaining species cannot compensate the non-monotonicity of the major species when removing too many minor species. Basically this can be seen as a filter for numerical noise and is equivalent to the elimination of all species mass fraction with nearly zero values from the data set Y. It is important to note that these restrictions can potentially lead to small differences in the optimal value of the functional. But this does not necessarily impact the success of the overall optimisation procedure. However, in certain specific cases it might be feasible to remove these restrictions. In order to obtain a convex problem (see Section 3.2), α is chosen from a bounded and convex subset of Rns , e.g. α ∈ A = [−a, a]ns where a is a positive real number. A further restriction on α can be to consider coefficient vectors α ∈ {−a, −a +1, ..., 0, 1, ..., a −1, a} where a is a positive integer. This integer formulation of the problem simulates the widely used heuristic selection of species to form a progress variable, where a = 1 is chosen, cf. [5]. This integer valued coefficient space is of course a strong restriction but it is useful for the convergence time when applying the genetic algorithm. However an integer valued α is not necessary for applying a genetic algorithm (GA). Note that a scaling of α/a does not change the normalised progress variables and in general Yc (x, α/a) = 1/a Yc (x, α) holds. The following problem is considered: min f (DY(α))
(14)
subject to Yαc > 0 α ∈ A, A bounded, where Y is the data matrix and it is assumed that the data is ordered with respect to the variable x, i.e. it holds Y = {Yk (xj )} with x0 < x1 < ... < xnx . Since α can be integer valued, problem (14) cannot be solved by gradient based methods without a reformulation. However, by considering that in most applications the number of major species is not larger than hundred, genetic algorithms are a feasible choice. In this case, the objective function f does not need to be differentiable and can be chosen as non-continuous. This offers a more pragmatic choice of f . As mentioned before the monotonicity of the progress variable is the most important factor and it is especially for complex fuels not straightforward to find an initial guess where monotonicity holds. Therefore it is useful to include this information in the objective function. Based on this, information of the non-monotonicity was included and the objective function is changed to min fg (α) := f (DYα) + g(Yαc ) whereby the function g is a suitable penalty function for non-monotonicity. The structure of α will be directly included in the genetic algorithm.
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The GA for solving the optimisation problem (14) reads as follows: (i) (ii) (iii) (iv)
Set i = 0. Generate randomly the initial population A0 = {α1 , ..., αn0 }. Evaluate the population vl = f (αl ), l = 1, ..., ni . Let li = argminl=1,...,ni {vl }. If minl=1,...,ni {vl } < δ, return αli . of the best m individuals out of Ai . Let Bi be the set Set Ai+1 = {Bi GE(Bi )}, i = i + 1. Go to (ii).
In the nomenclature of genetic algorithms the set of vectors αk , k ∈ N forms a population while the members of the populations are individuals. A population at a fixed state is a generation. The function GE(B) covers the implementation of the genetics: Crossing and mutating the chromosomes α of the set B of individuals. 3.4 Extension on multi-parameter problems For the application of chemistry tabulation in turbulent combustion simulations, the data does not only depend on the spatial or temporal variable x, but also on some parameters such as boundary conditions, initial temperatures, mixture compositions, etc. These parameters are collected in a vector p, i.e. Yk (x; p) is given in the data set Yp , and result in a set of data depending on these parameters. In the following it is assumed that the data are given in the form Yp = (Y(p1 ); ...; Y(pn )) and Yp = (Y(p1 ); ...; Y(pn )). Further it has to be assumed that there is at least one k such that the kth row of Y(pl ) does not vanish for all l = 1, ..., n. This is a necessary condition for the existence of a progress variable. Now the algorithm can be applied to multiple parameter problems as well as to single parameter problems. In general, the choice of the functional and the penalty function is flexible and every userdefined function can be used. Since the choice of the functional should capture physical effects, the specific choice of the functional can be important and this effect should be considered for future investigations. Here, we test our method for all examples (see Section 4) with the following objective function fg : nd ns dYc max | (DY(α))j,k,i | + μ · min 0, − (x; α) dx, (15) fg (α) := j =1,...,nx dx X i=1 k=1
where nd is the number of data sets based on different flame simulations, ns the number of species, μ ∈ R+ the weighting of the non-monotonicity and nx the number of grid points. Furthermore the superscript indicates the dependency of the progress variable on the equivalence ratio and thus on the mixture fraction, respectively. The coefficients α are chosen from the ns -dimensional cube with a = 20 αk ∈ 1/20 · {−20, −19, ..., 0, 1, ..., 19, 20},
∀k = 1, ..., ns .
(16)
The second term in the objective is the penalty function, which integrates the monotonously decreasing part of the progress variable with respect to the independent variable. This ensures the validity of the progress variable and defines a measure for the non-monotonocity of a progress variable. When no monotonous initial solution exists, finding a manually defined progress variable becomes much more complicated and might result in a generally not entirely monotonous definition. Based on the definition for the penalty function it is possible to start the optimisation from a non-monotonous progress variable. For problems where it is known that at least one feasible solution exists it can easily be shown that the optimal solution will be reached when
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using this penalty function. However, when monotonicity does not exist in the complete mixture fraction range, also an optimal progress variable with respect to monotonicity and species gradients would be found.
4 Numerical Tests For validating the progress variable optimisation approach, the method is applied to two different diffusion flame configurations, a laminar diffusion methane-air flame [1] and an inverse diffusion oxyfuel flame according to [11]. Before analysing both flame configurations with the help of laminar diffusion flamelets (1D) formulated in mixture fraction space, homogeneous reactor calculations (0D) of chosen mixtures are investigated in respect to the complexity of the fuels and an optimized progress variable. The prior analysis of the diffusion flame compositions with homogeneous reactor simulations are performed isobaric and isothermal whereby T is chosen as the temperature at equilibrium of each given mixtures (T = Teq ). These 0D cases are investigated since the problem of finding a monotonous progress variable for capturing the progress in a chemically reacting system significantly reduces complexity. Further laminar diffusion flamelet calculations were performed. These laminar diffusion flamelet calculations build the basis for later 2D/3D flame calculations. For both diffusion flame configurations, flamelet look-up tables (FLUTs) were constructed which consist of flamelet solutions for different stoichiometric scalar dissipation rates (SDR) according to the steady laminar flamelet approach [8]. A preliminary study was performed to conclude that 50 different stoichiometric scalar dissipation rates χ ∈ [0, χq ] are sufficiently accurate for the considered cases. Based on these tables, the optimisation was performed for selected mixture fractions. These solutions were evaluated with the standard progress variable for methane-air flames, Yc = CO + CO2 , and an optimized progress variable computed with the genetic algorithm. The non-normalised progress variables and their monotonicity are analysed for all test cases and the main species profiles are evaluated in more detail. The chemistry was modeled by using the detailed GRI3.0 [10] mechanism with 53 species and 325 reactions. 4.1 Implementation The algorithm is implemented in Matlab [13]. The program provides functions to check the feasibility of the progress variable, i.e. it tests if Yc is monotonously increasing for all data and it detects monotonous species. User requirements for the progress variable definition are considered, e.g. selected species can explicitly be included or excluded and the sign of a selected species can be specified. To guarantee the monotonous decrease of the objective function in the evolution, every generation, except the first, contains the best individual from the former generation. It is further possible to add individuals to the first generation as an initial guess and the program creates a randomly generated first generation of individuals if not. The algorithm stops after a maximum number of iterations and the obtained progress variable is the individual of the last generation, which minimises the objective. Thus, the obtained solution does not necessarily have to be the optimum of the problem, but depending on the number of generations allowed, a sub-optimum is reached. However, if an optimal progress variable exists, it is found by the algorithm. But in general the existence of a valid progress variable
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cannot be guaranteed. Since the algorithm generates a monotonously decreasing sequence of its functional values and is zero-bounded, at least convergence can be stated. 4.2 Homogeneous reactor calculations The homogeneous reactors were calculated isothermal and isobaric, the initial associated temperature is T = Teq and the initial pressure 1 bar. In the following compositions based on two test cases are treated, a methane-air diffusion flame and a partial oxidation flame. Within the 0D homogeneous reactor calculations these configurations are investigated at rich, stoichiometric and lean conditions analogous to [15]. 4.2.1 Methane-air composition The first test case, based on the methane-air flame by [1], was chosen as a reference case since the flame structure of the non-premixed laminar methane-air flame is well understood and was analysed in much detail. The initial conditions for the methane-air calculations treated here can be found in Table 1: Using the genetic algorithm for the multi-parameter problem by choosing nd = 5 an appropriate progress variable to map the species data for these 5 equivalence ratios to one progress variable for all homogeneous reactors was obtained. After 500 generations, the progress variable coefficients α shown in Table 2 are obtained for the 0D case of the methane-air compositions. A comparison of the standard and the optimized non-normalised progress variables over time is shown in Fig. 1. Both progress variables are monotonous for all treated equivalence ratios and show a unique mapping over time. However the optimized one shows a higher gradient for dYc /dt, which results in a minimized gradient of the species with respect to the progress variable. The benefit can be seen in more detail in Fig. 2 where the species gradients dYH2 O /dYc , dYCO /dYc and dYCO2 /dYc are shown for = 2.5 against the normalised progress variable C. Even though both progress variables are monotonous, the optimized progress variable shows a reduced gradient compared to the standard choice. Thus the maximal gradient, which is the optimisation target, is smaller. It is important to note that the maximisation of the gradient dYc /dt has no negative influence on the time step during integration using the progress variable. An analysis carried out for the stoichiometric methane-air composition (not shown here) confirmed that the time step is not smaller when solving for the optimized progress variable, in fact it is mostly larger than for the full system. Thus, the solution of the full system with all species seems to exhibit more stiffness than the reduced one for this case. In general the choice of a suitable numerical algorithm for the solution of the
Table 1 Initial settings for the methane-air test case
Teq [K]
YCH4
YO2
YN2
0.5
1480.2
0.0284
0.2264
0.7452
1
2225.2
0.0552
0.2201
0.7247
1.5
1904.7
0.0806
0.2142
0.7052
2
1564.9
0.1046
0.2086
0.6868
2.5
1272.1
0.1274
0.2033
0.6693
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Table 2 Coefficients αk of the progress variables for the homogeneous reactor calculations computed by the genetic algorithm Nr.
species
1
H2
4
O2
7
α
Nr.
species
Nr.
α
species
α −1
2
H
−0.75
3
O
−1
5
OH
−0.45
6
H2 O
0.1
HO2
1
8
H 2 O2
0
9
C
0.25
0.95
10
CH
−0.55
11
CH2
−0.65
12
CH2 (S)
0.85
13
CH3
0.95
14
CG4
−0.95
15
CO
0.8
16
CO2
1
17
HCO
−0.8
18
CH2 O
0.95
19
CH2 OH
−0.7
20
CH3 O
−0.35
21
CH3 OH
1
22
C2 H
−1
23
C2 H2
−0.7
24
C2 H3
25
C2 H 4
−0.85
26
C 2 H5
−0.65
27
C 2 H6
0.85
28
HCCO
−0.95
29
CH2 CO
−1
30
HCCOH
0.5
31
N
−0.4
32
NH
−0.8
33
NH2
0.2
34
NH3
−0.95
35
NNH
−0.5
36
NO
1
−0.65
37
NO2
0.2
38
N2 O
−0.55
39
HNO
40
CN
0.15
41
HCN
−0.95
42
H2 CN
−0.95
0.75
43
HCNN
0.95
44
HCNO
0.2
45
HOCN
−0.6
46
HNCO
0
47
NCO
−0.35
48
N2
−1
49
AR
0
50
C 3 H7
−0.7
51
C3 H8
52
CH2 CHO
0.15
53
CH3 CHO
0.45
0.15
0.2 0
Φ=0.5 Φ=1 Φ=1.5 Φ=2 Φ=2.5
Yc
−0.2 −0.4 −0.6 −0.8
Standard Optimized
−1 −6
10
−4
10
−2
0
10 10 Time [s]
2
10
4
10
Fig. 1 Progress variables (standard and optimized) for the methane-air compositions based on the homogeneous reactor calculations
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3
2
2
1 0 −1 −2 0
Φ=2.5
3 dYCO2/dYC
3 dYCO/dYC
dYH2O/dYC
Standard Optimized
1 0 −1
0.5 C
1
−2 0
0.5 C
1
2 1 0 −1 −2 0
0.5 C
1
Fig. 2 Species gradient for the methane-air =2.5 composition based on the homogeneous reactor calculations
progress variable in physical space should be taken with care but it does not seem to be more restrictive (rather less) than in other reacting flow simulations. In general both progress variables allow a unique mapping for the species and are therefore valid but in terms of the interpolation of the tabulated results, the optimized progress variable is more feasible. 4.2.2 Oxyfuel composition The second case is based on a reference flame for partial oxidation according to [11] and was originally introduced by Stelzner et al. [12]. This flame is an inverse oxyfuel partial oxidation flame applying a molar mixture of 1:1 CH4 and CO2 as fuel and pure O2 as oxidizer. This setup is chosen because it provides a more complex chemical structure as analysed in [15]. In the fuel-rich region endothermic reactions occur at slow time scales whereas in the fuel-lean and the stoichiometric region very fast oxidation reactions take place due to very high temperatures in the oxyfuel environment. Similar to the methane-air composition rich, stoichiometric and lean conditions are investigated with Teq and the initial pressure of 1 bar. Note that this choice provides completely different regimes, namely the flame zone at =1 and a slow conversion reactions at = 2.5, within this oxyfuel composition. The initial conditions for the different equivalence ratios of the oxyfuel flame treated here can be found in Table 3. Table 3 Initial settings for the oxyfuel test case
Teq [K]
YCH4
YCO2
YO2
0.5
2652
0.0854
0.2342
0.6805
1
2783.2
0.1295
0.3546
0.5159
1.5
2591.5
0.1562
0.4285
0.4153
2
1966
0.1742
0.4781
0.3476
2.5
1432
0.1873
0.5139
0.2988
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605
When applying the multi-parameter genetic algorithm to the homogeneous reactor calculations all given data sets are considered, resulting in nd = 5. The optimized progress variable is a strictly feasible solution for all treated homogeneous reactor calculations. The values of α are presented in Table 4 and the standard progress variable for a methane-air flame (Yc = CO2 + CO) was again used for comparison. Figure 3 shows the non-normalised progress variables on the one hand over the whole progress variable range and on the other hand zoomed into the non-monotonic region of the standard progress variable. It can be seen that for the fuel-lean and stoichiometric cases both progress variables are monotonous and therefore suitable to map the species and temperature uniquely. However, for the fuel-rich cases, only the optimized progress variable can represent the progress and the standard choice is not valid. Having a closer look to the behavior of the progress variable in the fuel rich regions it is visible that the non-monotonicity even increases for higher equivalence ratios. Furthermore, a much higher gradient for dYc /dx can be seen for the optimized progress variable, which results in a lower gradient of the species over the progress variable and therefore smoother data for the interpolation. To highlight this, the main species H2 O, CO and CO2 are plotted over the normalised progress variables C for = 2.5 in Fig. 4. These results confirm the non-monotonicities or even inconsistencies of the standard progress variable which does not map the species uniquely. This would lead to interpolation errors when using this data. For getting a better idea of the non-monotonicities, Fig. 5 shows the species gradients over the progress variable. For clarity reasons this plot is scaled between -10 and 10 even though the maximal and minimal values for the standard progress variable are around |10e5|. The figure shows the gradient of three of the main species, namely dYH2 O /dYc , Table 4 Coefficients αk of the progress variables for the homogeneous reactor calculations computed by the genetic algorithm Nr.
species
α
Nr.
species
α
1
H2
−0.95
2
H
−0.4
4
O2
−1
5
OH
−0.15
7
HO2
8
H 2 O2
−0.7
9
C
−0.9
10
CH
−0.95
11
CH2
−0.9
12
CH2 (S)
−0.35
13
CH3
1
14
CH4
−1
15
CO
16
CO2
0.7
17
HCO
−0.75
18
CH2 O
1
19
CH2 OH
−0.3
20
CH3 O
−0.4
21
CH3 OH
0.6
22
C2 H
−0.7
23
C2 H2
0.6
24
C2 H3
25
C2 H 4
−0.95
26
C 2 H5
0.95
27
C 2 H6
0.95
Nr.
species
α
3
O
−0.7
6
H2 O
0.25
1
−1 0.85
28
HCCO
−0.85
29
CH2 CO
−0.6
30
HCCOH
31
N
0
32
NH
0
33
NH2
0
34
NH3
0
35
NNH
0
36
NO
0 0
−0.9
37
NO2
0
38
N2 O
0
39
HNO
40
CN
0
41
HCN
0
42
H2 CN
0
43
HCNN
0
44
HCNO
0
45
HOCN
0
46
HNCO
0
47
NCO
0
48
N2
0
49
AR
0
50
C 3 H7
0.85
51
C 3 H8
0.8
52
CH2 CHO
−0.5
53
CH3 CHO
0.7
606
Flow Turbulence Combust (2015) 94:593–617 1
0.8
0.6
0.75
0.2
0.7 Yc
0.4
Yc
Φ=0.5 Φ=1 Φ=1.5 Φ=2 Φ=2.5
0.85
0.8
0 −0.2
Standard Optimized
0.65 0.6
Φ=0.5 Φ=1 Φ=1.5 Φ=2 Φ=2.5
−0.4 −0.6 Standard Optimized
−0.8 −1
−6
10
−4
10
−2
0
10 10 Time [s]
2
0.55 0.5 0.45
4
10
−6
10
−4
10
10
−2
0
10 10 Time [s]
2
10
4
10
Fig. 3 Progress variables (standard and optimized) for the oxyfuel composition based on the homogeneous reactor calculations
0.8
0.3
0.6
0.2
0.7 0.6
0.4 0.2
0.1 0 0
Φ=2.5
YCO2
0.4
YCO
YH2O
Standard Optimized
0.5 C
0 0
1
0.5 0.4
0.5 C
1
0
0.5 C
1
Fig. 4 Main species over progress variables for the oxyfuel composition based on the homogeneous reactor calculations for = 2.5
10
5
5
0 −5 −10 0
0.5 C
1
Φ=2.5
10 dYCO2/dYC
10 dYCO/dYC
dYH2O/dYC
Standard Optimized
0 −5 −10 0
0.5 C
1
5 0 −5 −10 0
0.5 C
1
Fig. 5 Species gradient for the oxyfuel composition based on the homogeneous reactor calculations for = 2.5
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607
dYCO /dYc and dYCO2 /dYc , against the normalized progress variable. Compared to the standard progress variable the species gradient are significantly reduced and no extreme peaks can be seen which would lead to errors in further 2D/3D calculations. Even for the stoichiometric case (not shown here) the gradient of the species over progress variable is higher for the standard progress variable choice. We can conclude that in homogeneous reactor calculations the standard progress variable for oxyfuel compositions is only suitable for stoichiometric and lean equivalence ratios but fails for rich oxyfuel compositions. Only the optimized progress variables allow a unique mapping for the species as claimed by the algorithm. When using the approach of Niu et al. [7] it would be a principal task to find an initial monotonous progress variable. This is not necessary in the GA used here. 4.3 Diffusion flamelet calculations Laminar diffusion flamelet calculations are performed and analysed since they provide the possibility to investigate the whole mixture fraction range. Furthermore, flamelet look-up tables based on the mixture fraction and the progress variable can be built. As a first step the complete mixture fraction range is analysed in order to get an overview of the flame structure and the monotonicity of the species in this diffusion flamelet setup. Based on this the non-normalised progress variable as well as the main species will be compared for both progress variable definitions. 4.3.1 Methane-air flame For calculating the laminar diffusion flamelets an initial temperature of 298 K, initial pressure of 1 bar and 50 different stoichiometric scalar dissipation rates are used. Before going into the details of the progress variable optimisation, the flame structure of diffusion flamelets is analysed. First of all the main species over mixture fraction are shown for varying stoichiometric scalar dissipation rates in Fig. 6. Each line corresponds to one diffusion flamelet solution for one stoichiometric scalar dissipation rate. It can be seen that areas
0.1
0.02
χ
0.01 0 0
0.2
0.5 Mixture Fraction
0.05
0 0
1
χ
0.1
0.5 Mixture Fraction
1
χ
YCO
2
0.15 YCO
χ
2
YH O
YH
2
0.03
0.1
0.05
0.05 0 0
0.5 Mixture Fraction
1
0 0
0.5 Mixture Fraction
1
Fig. 6 Basic structure of species solutions when using laminar diffusion flamelets for the methane-air flame
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Flow Turbulence Combust (2015) 94:593–617
exist where the species span a wide range for the different χst (e.g. H2 ) and areas can be identified where overlapping of the lines appear (e.g. CO2 ). Both occurrences are important for a progress variable definition since overlapping is an evidence that species change their monotonicity for different mixture fraction values and are therefore not suitable as a progress variable. Further, a wide stretch of the species over different stoichiometric scalar dissipation rates does point out, that the progress variable has to span a wide range as well to capture the changes in the species mass fractions. Based on these species results fixed equivalence ratios at rich ( = 1.5, 2, 2.5), stoichiometric ( = 1) and lean ( = 0.5) conditions are chosen for the progress variable optimisation. These equivalence ratios are chosen to explicitly show that both the lean as well as the rich region can be successfully included in the optimisation. The optimisation tool is not limited to specified equivalence ratios. Depending on the application, the equivalence ratio range can be smaller or even larger. The location in mixture fraction space for these equivalence ratios are shown in Fig. 6 by constant lines. Concentrating on these specific mixture fractions the same species are analysed over the whole stoichiometric scalar dissipation rate range χst ∈ [0, . . . , χq ] in Fig. 7. The colors correspond to the monotonicity behavior of one species and one mixture fraction for all stoichiometric scalar dissipation rates. Blue corresponds to a monotonously increasing, green monotonously decreasing and beige to non-monotonous behavior. Figure 7 shows that all four main species change their monotonicity for changing mixture fractions but are mainly monotonous. For example H2 shows monotonously increasing, decreasing and non-monotonous processes. Based on these analysis the monotonic behavior for all species over all mixture fractions is shown in the monotonicity diagram in Fig. 8. The diagram illustrates the flammability limits, the stoichiometric mixture fractions, Zst = 0.0549, and the monotonic behavior of the species over all stoichimetric scalar dissipation rates. The progress variable should
Φ=0.5
Φ=1
0.02
Φ=1.5 0.16
H2O
H2
0.01 0.005
0.1
0.06 0
0.01 10 Scalar Dissipation Rate
0.01 10 Scalar Dissipation Rate
0.14 0.12 CO2
0.15 CO
0.12
0.08
0.2
0.1 0.05 0 0
Φ=2.5
0.14
0.015
0 0
Φ=2
0.1 0.08 0.06
0.01 10 Scalar Dissipation Rate
0.04 0
0.01 10 Scalar Dissipation Rate
Fig. 7 Species solutions against stoichiometric scalar dissipation rates when using laminar diffusion flamelets for the methane-air flame
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609
Fig. 8 Monotonicity diagram of the species in the methane-air diffusion flame, where blue illustrates monotonously increasing species, green monotonously decreasing species and beige illustrates non-monotonicity
be at least monotonous for a range of mixture fraction values near stoichiometry or in the flammability region respectively. Based on this diagram it can be concluded: (i)
No single species exists, which is monotonous for all mixture fractions when varying χst . (ii) Species exist, which are monotonous in the flammability region. (iii) The data can be split within the flammability area into sections where one or more monotonous species can be found. (iv) These sections are not unique. Within these sections, the species that potentially define a progress variable, are not unique. The multi-parameter genetic algorithm is applied with nd = 5 and yields a strictly feasible solution for all treated laminar flamelet calculations. The optimized progress variable coefficients α are shown in Table 5. Similar to the homogeneous reactor calculations, the non normalized progress variable profiles are compared for different equivalence ratios in Fig. 9. Since Ycstandard is monotonously decreasing in contrast to Ycoptimized , the negative optimized progress variable was used for comparison. Both progress variables are monotonous and show a unique mapping over the scalar dissipation rate. This results in a unique mapping of the species mass fractions. For the methane-air configurations both progress variables are therefore an appropriate choice. However in the optimized one the benefit of an optimized gradient (especially for the rich case) appears. This can be seen in Fig. 10 for the major species H2 O, CO and CO2 and different equivalence ratios. The species profiles show that both progress variables behave similarly in the lean and stoichiometric case but show large differences in the rich case. Even if the standard progress variable is monotonous over the scalar dissipation rates, sharp gradients can be seen for all of the three species at = 2.5. In contrast to this a much smaller species gradient appears for the optimized progress variable. Having a look at the species gradients in Fig. 11
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Table 5 Coefficients αk of the progress variables for the laminar diffusion flamelet calculations computed by the genetic algorithm Nr.
species
Nr.
α
species
Nr.
α
species
α −1
1
H2
1
2
H
−0.8
3
O
4
O2
−1
5
OH
−1
6
H2 O
7
HO2
−1
8
H 2 O2
9
C
11
CH2
14
CH4
10
CH
13
CH3
0.3
16
CO2
0.9
17
19
CH2 OH
0
20
22
C2 H
0.95
23
25
C2 H 4
−1
28
HCCO
−0.3
31
N
−0.7
34
NH3
−0.55
37
NO2
0.85
12
CH2 (S)
0.6
−1
15
CO
1
HCO
0
18
CH2 O
−0.75
CH3 O
0.85
21
CH3 OH
−0.9
C2 H2
−0.35
24
C 2 H3
−0.85
26
C2 H5
−1
27
C2 H6
−1
29
CH2 CO
−0.8
30
HCCOH
−1
32
NH
−0.8
33
NH2
−0.05
35
NNH
−0.3
36
NO
1
38
N2 O
−0.8
39
HNO
−0.55
−1
0.1
0.95 −0.8
0.05
40
CN
0.1
41
HCN
−0.85
42
H2 CN
−0.65
43
HCNN
−0.7
44
HCNO
−0.5
45
HOCN
0.2
46
HNCO
−0.85
47
NCO
0.25
48
N2
49
AR
−0.25
50
C 3 H7
0.5
51
C3 H8
52
CH2 CHO
−0.35
53
CH3 CHO
0.35
0.4 −0.4
this benefit appears much clearer. In the lean and stoichiometric case small gradients can be seen for both progress variables but when analyzing the rich case it is shown that the species gradients of the standard progress variable are much higher and show steep peaks
Φ=1
Φ=1.5
0.3
0.6
0.25
0.55
0.2
0.5
Yc,optimized
Yc,standard
Φ=0.5
0.15 0.1 0.05 0
Φ=2
Φ=2.5
0.45 0.4 0.35
−3
−1
1
10 10 10 Scalar Dissipation Rate
0.3
−3
−1
1
10 10 10 Scalar Dissipation Rate
Fig. 9 Progress variables (standard and optimized) for the methane-air flame based on the laminar diffusion flamelet calculations
Flow Turbulence Combust (2015) 94:593–617
H2O
0.2
0.5 C
Φ=1
0.2
0.1
0 0
1
Yi
Φ=0.5
0.05
0 0
Standard Optimized
CO CO2
Yi
Yi
0.1
611
0.5 C
0.1
0 0
1
Φ=2.5
0.5 C
1
Fig. 10 Main species over progress variables for the methane-air flame based on the laminar diffusion flamelet calculations
for small C. This effect is overcome when using the optimized progress variable. To ensure readability the plot is scaled between 10 and −10. When using the data for FPV 2D Flame simulations it can be an advantage for look-up strategies to use the optimized progress variable due to the smaller species gradient over progress variable. But in general both progress variables are monotonous. 4.3.2 Oxyfuel flame The initial temperature for the 1D oxyfuel flame is 300 K and the initial pressure 1 bar. Similar to the methane-air set up 50 stoichiometric scalar dissipation rates are used for generating the 1D flamelets. To get an idea of the different structure of this oxyfuel flame in comparison to the methane-air flame the laminar diffusion flamelets are investigated starting with an analysis of the major species and their monotonicity over the whole mixture fraction
H2O
10
Standard Optimized
CO CO2
Φ=0.5
10
Φ=1
10
0
dYi/Yc
5 dYi/Yc
dYi/Yc
5 0
−5 −10 0
Φ=2.5
0 −5
0.5 C
1
−10 0
0.5 C
1
−10 0
0.5 C
1
Fig. 11 Main species gradients over progress variables for the methane-air flame based on the laminar diffusion flamelet calculations
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Flow Turbulence Combust (2015) 94:593–617
range. Further discrete equivalence ratios, namely = 0.5 (lean), = 1 (stoichiometric) and = 1.5, 2, 2.5 (rich) are used for a detailed comparison. The species profile of H2 , H2 O, CO and CO2 are shown in Fig. 12. Each line corresponds to a laminar diffusion flamelet for a fixed stoichiometric scalar dissipation rate. In contrast to the methane-air flame the stoichiometric mixture fraction, Zst = 0.484, is much higher and the species do not span such a wide area. This makes it harder to find an appropriate progress variable with a smooth species gradient. Based on the diffusion flamelet profiles of the main species, 5 discrete mixture fraction values are chosen and analysed for varying stoichiometric scalar dissipation rates, see Fig. 13. Unfortunately there does not exist any monotonicity for these major species for the different equivalence ratios. This is on the one hand due to the small gradient for small scalar dissipation rates and on the other hand based on the steep gradient for high χst values. At least the species mass fractions of H2 and CO show high gradients in the main field. Similar to Fig. 8 and based on the monotonicity of the species for fixed equivalence ratios the monotonicity plot for each species and each mixture fraction/equivalence ratio is shown in Fig. 14. One point in this plot corresponds to a complete range of stoichiometric scalar dissipation rates. Further the flammability limits and the stoichiometric mixture fraction are shown. It is worth noting that the flammability range is much larger and so the monotonicity has to be fulfilled for many more data sets. From the monotonicity diagramm in Fig. 14 the following can be concluded: (i)
No single species exists, which is monotonous for all mixture fractions and all stoichiometric scalar dissipation rates. (ii) No single species exists, which is monotonous in the flammability area. (iii) The data can not be split within the flammability area into sections where we find one or more monotonous species. In contrast to the methane-air diffusion flame, where the standard progress variable showed monotonous results, no monotonous species or simple species combination exists in the flammability limits of the oxyfuel flame. Since Ycstandard is monotonously decreasing
0.03
0.01 0 0
χ 0.5 Mixture Fraction
2
YCO
YCO
0.6
0.3
χ 0.5 Mixture Fraction
0.4
0.5 Mixture Fraction
1
χ
0.2
0.1 0 0
0.1 0 0
1
0.4
0.2
χ
2
YH O
YH
2
0.2 0.02
1
0 0
0.5 Mixture Fraction
1
Fig. 12 Basic structure of species solutions when using laminar diffusion flamelets for the oxyfuel flame
Flow Turbulence Combust (2015) 94:593–617
−4
2
x 10
Φ=0.5
613 Φ=1
Φ=1.5 0.08
Φ=2
Φ=2.5
H2O
H2
0.06 1
0.04 0.02 0
0 0.01 10 1000 Scalar Dissipation Rate
0.04
0.2
0.03
0.15 CO2
CO
0
0.02 0.01 0
0 0.01 10 1000 Scalar Dissipation Rate
0.1 0.05
0 0.01 10 1000 Scalar Dissipation Rate
0
0 0.01 10 1000 Scalar Dissipation Rate
Fig. 13 Species solutions against stoichiometric scalar dissipation rates when using laminar diffusion flamelets for the oxyfuel flame
in contrast to Ycoptimized , the negative optimized progress variable was used for comparison in Fig. 15. Consequently and as a result of Fig. 15 the standard progress variable is nonmonotonous. However, when using the genetic algorithm an appropriat monotonously increasing progress variable can be found that uniquely maps the species data for all chosen flamelet solutions.
Fig. 14 Monotonicity diagram of the species in the oxyfuel flame, where blue illustrates monotonously increasing species, green monotonously decreasing species and beige illustrates non-monotonicity.
614
Flow Turbulence Combust (2015) 94:593–617 Φ=1
Φ=1.5
0.7
0.4
0.6
0.3
Yc,optimized
Yc,standard
Φ=0.5
0.5 0.4
0.1 0
0.2
−0.1 −3
−1
1
3
10 10 10 10 Scalar Dissipation Rate
Φ=2.5
0.2
0.3
0.1
Φ=2
−0.2
−3
−1
1
3
10 10 10 10 Scalar Dissipation Rate
Fig. 15 Progress variables (standard and optimized) for the oxyfuel flame based on the laminar diffusion flamelet calculations
The coefficients of the optimized progress variable α are shown in Table 6. Figure 16 provides a closer look to the main species mass fractions with respect to the standard and optimized progress variable. The non-normalised progress variable shows an overall small gradient for lean to stoichimetric cases. However, in rich cases the progress variable exhibits a non-monotonicity. It can be seen that the species profiles for the lean and stoichiometric cases are similar to the standard progress variable. However, in the rich case a significant improvement with respect to the steep gradients appears. In general and analogous to the homogeneous reactor calculation, the standard progress variable is valid for lean to stoichiometric cases but in the Table 6 Coefficients αk of the progress variables for the laminar diffusion flamelet calculations computed by the genetic algorithm Nr.
species
Nr.
α
species
Nr.
α
species
α −0.7
1
H2
0.55
2
H
−1
3
O
4
O2
−0.55
5
OH
−0.9
6
H2 O
−0.1
7
HO2
−1
8
H 2 O2
−0.95
9
C
−1
10
CH
−1
11
CH2
−1
12
CH2 (S)
−0.95
13
CH3
−1
14
CH4
−1
15
CO
16
CO2
17
HCO
−1
18
CH2 O
−1
0.45
0.6
19
CH2 OH
−0.95
20
CH3 O
−1
21
CH3 OH
−1
22
C2 H
−1
23
C2 H2
1
24
C2 H3
−1
25
C2 H 4
−1
26
C2 H5
−0.9
27
C2 H6
−1 −1
28
HCCO
−1
29
CH2 CO
−1
30
HCCOH
31
N
0
32
NH
0
33
NH2
0
34
NH3
0
35
NNH
0
36
NO
0 0
37
NO2
0
38
N2 O
0
39
HNO
40
CN
0
41
HCN
0
42
H2 CN
0
43
HCNN
0
44
HCNO
0
45
HOCN
0
46
HNCO
0
47
NCO
0
48
N2
49
AR
0
50
C 3 H7
−0.95
51
C 3 H8
52
CH2 CHO
−1
53
CH3 CHO
−0.95
0 −0.8
Flow Turbulence Combust (2015) 94:593–617
Yi
0.4
Φ=0.5
0.5
Φ=1
0.4
0.6
0.2
0.3
0.4
0.1
0.2
0.2
0.1 0 0.5 C
0 1 0
1
Φ=2.5
0.8
0.3
0 0 0.5 C
615
Φ=2.5− zoomed 0.5
Standard Optimized
H2O
0.4
CO CO2
0.3 0.2
0.5 C
0.478
1
0.48 C
Fig. 16 Main species over progress variables for the oxyfuel flame based on the laminar diffusion flamelet calculations. (left: = 0.5, 1, right: = 2.5 (original + zoomed))
rich case uniqueness is not achieved. This problem is overcome with the optimized progress variable and therefore only the optimized progress variable should be used for a proper flame simulation. Finally the gradients of the major species over the progress variable are shown in Fig. 17. Based on this plot the improvement in the species gradient when using the optimized progress variable can be clearly seen. The gradients for the lean, stoichiometric and rich cases are very high for the standard progress variable and show steep peaks. For clarity reason the plot was scaled between 10 and -10. In contrast to the methane-air flame even the gradients for the lean and stoichiometric cases are high and can lead to interpolation problems. Having a look at the species gradients for the optimized progress variable much lower values can be achieved and no more peaks appear which results in much smoother species data. Based on these results it can be concluded that both progress variables have very flat gradients over the scalar dissipation rates since the variation of the species for different χst is much lower than for the methane-air flame. Further non-monotonicities are shown when using the standard progress variable which can be overcome by the optimized progress variable.
5 Conclusions Turbulence-Chemistry-Interaction models are often based on chemical look-up table strategies, such as the laminar flamelet model. A recent approach is to use a dynamic (reactive) scalar, e.g. a progress variable, instead of a static parameter to describe the chemical state. H2O
Φ=0.5
10
10
5 dYi/Yc
5 dYi/Yc
Φ=1
0
0
0
−5
−5
−5
−10 0
−10 0
−10 0
0.5 C
1
Φ=2.5
5 dYi/Yc
10
Standard Optimized
CO CO2
0.5 C
1
0.5 C
1
Fig. 17 Main species gradients over progress variables for the oxyfuel flame based on the laminar diffusion flamelet calculations
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This progress variable serves to describe the dynamic behavior of the chemical system and is usually obtained from linear combinations of the species mass fractions. In order to define an invertible mapping between the independent variable and the progress variable, monotonicity of the progress variable is required. For this task some automated procedures are available and published in the literature but finding a valid and thus monotonous progress variable can become complicated for complex flame structures. In this investigation, an algorithm is constructed to find a valid progress variable, which is optimal in the sense that it minimises the derivative of the tabulated chemical species to k the progress variable dY dYc . A minimized derivative leads to smoother data and reduces the interpolation error when reading data from the flamelet look-up table. For this optimisation problem, a genetic algorithm was applied. Here, also in the case of an initial population consisting of unfeasible progress variables, the algorithm will provide a sub-optimal solution and will only fail to find a monotonous solution, if no valid progress variable exists. However, when introducing a penalty function that evaluates the measure of monotonicity, a progress variable can be found that might not be monotonous because it does not exist in the whole mixture fraction range, but is optimal with respect to monotonicity and species gradients. The procedure of finding such an optimized progress variable was applied to homogeneous reactor calculations as well as diffusion flamelet calculations for a laminar methane-air flame and a more complex oxyfuel flame. In order to evaluate the optimisation algorithm, a comparison was performed with a standard progress variable Yc = CO + CO2 usually used for methane-air flames. It could be shown, that a monotonous mapping was obtained for the methane-air flame using both the standard and the optimized progress variable. However the optimized progress variable gives smoother species gradients over progress variable, which is desired for interpolation issues. In the second example, an oxyfuel flame, the standard choice of a progress variable does not only show steeper gradients, it even fails to uniquely map the species for both the homogeneous reactor and the laminar diffusion flamelet calculations, because of its lack of monotonocity especially for rich mixtures. For both examples the automated algorithm finds a strongly monotonically progress variable that can be used in the flamelet-progress variable approach without analysing the flame structure in detail. Acknowledgments The authors kindly acknowledge the financial support by the Saxon Ministry of Science and Fine Arts and the European Union in the project “BioRedKat” (project number 100097882) and by the Federal Ministry of Education and Research of Germany in the framework of “Virtuhcon” (project number 03Z2FN11).
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