M U S C L E S. Modelling of UnSteady Combustion in Low Emission Systems. Project No. GRD Contract No. G4RD-CT

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1 M U S C L E S Modelling of UnSteady Combustion in Low Emission Systems Project No. GRD Work Package 2 Prediction of unsteady reacting flow and validation Task 2. Prediction of lean blow out in liquid fuelled combustion Deliverable Report 2.5 Integration of extended kinetic scheme in JPDF module - June Responsible Partner: Universität Karlsruhe Engler Bunte Institut Lehrstuhl und Bereich Verbrennungstechnik Engler Bunte Ring D 763 Karlsruhe Authors: Dipl.-Ing. Frank Wetzel Prof. Dr.-Ing. Nikolaos Zarzalis

2 MUSCLES Integration of extended kinetic scheme in JPDF module Contents Contents Contents 2 Introduction and Objectives 3 2 Theory and Formulation of the JPDF-Approach 4 2. Mathematical Bases Jpdf Approach for Closing the Mean Turbulent Reaction Rate Mixture Fraction Reaction Progress Variable Closure of the Mean Turbulent Reaction Rate 0 3 Code-Implementation / PrePDF 3 4 Validation of JPDF Reaction Model 5 5 Conclusion 8 Listings 9 L. Symbols 9 L.2 Literature 20 L.3 Figures 20 L.4 Tables 20 Project No. GRD

3 MUSCLES - Integration of extended kinetic scheme in JPDF module Introduction and Objectives Introduction and Objectives Within the framework of MUSCLES the EBI Karlsruhe proposed a turbulent reaction model, which is based on a presumed joint probability density function (JPDF) approach. To avoid problems occurring by the closure of the mean production term of species transport equations the 2-domain--step kinetic which was presented within Deliverable 2.4 has to be employed in order to model the instantaneous reaction rate. The concept of the 2-domain--step kinetic states, that on the one hand the reaction progress can be specified by a single reaction progress variable and that on the other hand the influence of the turbulent mixing process on the local stoichiometry is characterized by a mixture fraction variable. The present report documents the integration of the enhanced kinetic sub-module in an existing JPDF model, already used and verified within the scope of the CFD4C project. Results of the JPDF model are presented in Section 4 using the TASCflow CFD-solver. Project No. GRD

4 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach 2 Theory and Formulation of the JPDF-Approach 2. Mathematical Bases Probability Density Function, Mean and Variance Turbulent combustion can be seen as a stochastic process. Hence all variables can be assumed as stochastic variables φ, i.e. as a function of the continuous variable of space x & and of time t. A complete description of a variate φ provides the probability density function (PDF) P () ϕ, which is defined as a derivative of the distribution function D () ϕ [ 4 ]: φ φ ( ) d P φ() ϕ = Dφ() ϕ ( 2- ) dϕ The dependency of those two variables at one location x & is shown in Fig. 2-. Fig. 2-: Definition of the probability density function Thus the probability to find a variate φ inside a length-interval d ϕ of ϕ is given by: ( ϕ φ ϕ + dϕ) = Pφ ( ϕ) dϕ Pr ob. ( 2-2 ) This implicates the normalization condition of the probability density function: The mean value Q () φ of a variate ( x &,t) () ϕ d ϕ P = φ. ( 2-3 ) φ is: () φ = Q() ϕ P () ϕ φ dϕ Q. ( 2-4 ) Project No. GRD

5 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach To characterize the fluctuations of the flow the 2 nd central moment (variance) is used: 2 Q () φ = Q() ϕ Q() φ ( ) 2 P () ϕ φ dϕ. ( 2-5 ) Joint Probability Density Function For a function Q of several stochastic variables a multivariable probability density function, i.e. a joint probability density function (JPDF) can be assumed. It is given by: P ( ϕ,ψ) and ( ϕ,ψ) ( ) d P φ, χ( ϕ, ψ) = Dφ, χ( ϕ, ψ). ( 2-6 ) dϕdψ D are both defined in the 2-dimensional state-space ϕ ψ. Therefore the following equations for mean ( 2-7 ) and variance ( 2-8 ) can be derived: ( φ, χ) = Q( ϕ, ψ) Pφ χ( ϕ, ψ) Q dϕdψ, ( 2-7 ) 2 ( ) P ( ϕ, ψ) ( φ, χ) = Q( ϕ, ψ) Q( φ, χ) 2 Q φ, χ dϕdψ. ( 2-8 ) Stochastically Independent Variates, Marginal PDF The JPDF can be expressed as product of the two PDFs () ϕ χ ( x &,t) are stochastically independent: φ χ ( ϕ, ψ) = P ( ϕ) P ( ψ) φ χ P and P ( ψ) if the variates ( x &,t) φ χ φ and P,. ( 2-9 ) Every JPDF can be fractionised into a marginal PDF if taking the influence of only one variate into account: φ χ ( ϕ, ψ) dψ = P ( ϕ) P,. ( 2-0 ) φ Density as a Statistical Variable Due to density fluctuations in turbulent flow density has to be handled as a statistical variable. To avoid unclosed terms containing information about density fluctuation density-weighted averaging by FAVRE is used: & Q x, t Q & & =. ( 2- ) ( ) ( x) + Q ( x, t) Following BILGER s proposal [ ] and using ( 2-0 ) leads to a density-weighted PDF-formulation: Project No. GRD

6 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach P () ϕ = φ ρpρ φ( ρ, ϕ) dρ ρ 0, ( 2-2 ) with the time-averaged density determined from Favre-PDF: = ρ P φ d 0 ρ () ϕ ϕ. ( 2-3 ) The density-weighted mean-values and variances can be calculated as in the time-averaged case: () φ = Q() ϕ () ϕ P φ dϕ Q ( 2-4 ) 2 Q = Q P d, ( 2-5 ) 2 φ ϕ ( Q() ϕ () φ ) () ϕ while the Reynolds-averaged variables are given by: () φ () ϕ = ρ Q P () φ ϕ dϕ ρ Q. ( 2-6 ) Density-weighted PDFs in density-fluctuated flow can be modelled in the same way as timeaveraged PDFs in density-constant flow. 2.2 Jpdf Approach for Closing the Mean Turbulent Reaction Rate According to the MUSCLES contract, the 2-domain--step kinetic has to be applied to an existing JPDF model already used and verified within the scope of the CFD4C project (see CFD4C Deliverable Report.9 [ 2 ] ). The time averaged production rate of a species α at a location x & in terms of a multivariable PDF can be written as ω α = ρ ω α P & 0 N ρ ( ρ,t,y,,y ;x) dρ dt dy dyn. ( 2-7 ) The multivariable PDF P & ( ρ,t,y,,yn;x ) is obtained by its moments, which are determined as a function of x & by solving appropriate balance equations. The joint probability density function P & ( ρ,t,y,,yn;x ) can not only be used to determine the mean turbulent production rate but also to Project No. GRD

7 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach obtain all other thermo-chemical quantities, namely the time mean mass density and the Favre average species mass fractions: ρ = P & ( ) ρ,t,y 0,,YN;x dρ dt dy dyn ρ ( 2-8 ) Y P & α = ( ) 0,T,Y 0 0 Y 0 α ρ,,yn ;x dρ dt dy dyn ( 2-9 ) Correlations of the type determined directly as ω α Y β occurring in the transport equations of the species variances can be ω α Y β = ρ ( Y Y ) ω α β α 0 N ρ P & ( ρ,t,y,,y ;x) dρ dt dy dyn. ( 2-20 ) To overcome this closure problem, the concept of the 2-domain--step kinetic ( MUSCLES Deliverable 2.4 ) has been employed in order to model the instantaneous reaction rate. The model concept states, that the reaction progress can be specified by a single state variable, the reaction progress variable c, whereas the influence of the turbulent mixing process on the local stoichiometry is characterized by a mixture fraction variable f. Therefore this closure is based on the assumption that the temperature and the concentration fields are functions of c and f only Mixture Fraction Under the assumption of turbulent Schmidt numbers being equal the state of the mixture is readily defined by the mixture fraction variable. In this work the mixture fraction at any point in the flow field is defined by ( ZC + ZH) ( ZC + ZH) Ox ( ZC + ZH) F ( ZC + ZH) Ox f = ( 2-2 ) where Z C and Z H denote the element mass fractions of carbon (C) and hydrogen (H), respectively. The subscripts F and Ox denote the values of Z C and Z H in the fuel stream and in the air stream(s). The element mass fraction Z e of an atom e can be calculated from the known species mass fractions by Project No. GRD

8 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach W Z = ε e e ei Yi ( 2-22 ) Wi where the matrix ε ei specifies the number of atoms of type e in a molecule of type i. Since the mixture fraction is a conserved scalar, i.e. unaffected by chemical reactions, it satisfies the following transport equation using a standard eddy viscosity concept: ( ρ f ) & ( ) ν + ρ u f = ρ D + t f t Scf. ( 2-23 ) The statistical behaviour will be like that of other conserved scalars such as the mass fraction of an inert species. The influence of turbulent mixture fraction fluctuations on the scalar variables can be taken into account by solving a transport equation for g f 2 of the following form ( ρ g) ( u & g) ν D ν g C t f f 2 D f f t + ρ = ρ + t + g, ρ ρ. Sc ( 2-24 ) g Sc g The last term on the right hand side of equation ( 2-24 ) is the dissipation rate of mixture fraction fluctuations caused by mixing on the molecular level. A common approach to model the scalar dissipation rate χ f is to relate it to the turbulent time scale t t of the velocity fluctuations [ 5 ] g ε χ C f = 2ρD f f ρ = g,2 ρ g. ( 2-25 ) tt k Both modelling constants C g, and C g,2 are set to a value of 2.0. Following [ 3 ] a Gaussian PDF for the mixture fraction variable is presumed. The parts of the distribution in the ranges < f < 0 and < f < +, that are physically meaningless, are lumped into Dirac delta functions at f = 0 and f =, respectively. The two parameters c σ and c µ are determined implicitly by means of a look up table from the mean and the variance of the mixture fraction, respectively Reaction Progress Variable The reaction progress variable is defined as: Project No. GRD

9 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach ZO,bounded c =. ( 2-26 ) min( ZO,stoich,ZO,local) Here Z O, local denotes the local element mass fraction of oxygen Z = Z, ( 2-27 ) O,local O Z O,bounded the mass fraction of chemically bounded O atoms ZO,bounded =, ( 2-28 ) ZO Y O 2 and Z O, stoich the element mass fraction needed for complete combustion WO W Z O O,stoich 2 ZC + ZH WC 2 WH =. ( 2-29 ) Based on the conservation of elements he element mass fractions of carbon, hydrogen and oxygen can be calculated from the following relations ZO ( Z + Z ) f + ( Z + Z ) ( f ) ZC = CZ,C C H F c H Ox ( 2-30 ) ZH ZO,max ( ZC + ZH) f + ( Zc + ZH) ( f) ZC = ( 2-3 ) F Ox ( ( Z + Z ) f + ( Z + Z ) ( f) Z ) = ( 2-32 ) C H F c H Ox C where the constant C Z,C depends on the composition of the fuel F: CZ,C W ε C C,F WF =. ( 2-33 ) The maximum value of the O atom mass fraction is equal to the maximum oxygen mass fraction and therefore determined by the composition of the combustion air, i.e. Z O, max = Contrary to the mixture fraction variable, the reaction progress variable is not a conserved scalar and therefore a reaction source term has to be formulated and modelled ( ρ c) & ( u c) D t ν + ρ = ρ + c + ω c t Sc c. ( 2-34 ) Project No. GRD

10 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach Additionally the following transport equation for the fluctuation intensity d c 2 can be formulated : ( ρ d) ( u & d) ν ν + ρ = ρ D + d + C ρ t c c 2ρD c c + 2ω c t t c Sc d,. ( 2-35 ) d Scd The reaction source terms of the reaction progress variable c and the reaction progress variable variance d are closed by assuming a clipped Gaussian distribution which is described in the next subsection Closure of the Mean Turbulent Reaction Rate Based on equation ( 2-7 ) and assuming statistical independence of the two state variables c and f, the mean turbulent reaction rate appearing in equation ( 2-34 ) can be written as ( c,f ) ω c P & ( c;x) P & ω c = ρ ( f;x) 0 c f dc df 0. ( 2-36 ) ρ The unclosed source term ω c c of equation ( 2-35 ) can be calculated according to ( 2-20 ) as ( c,f)( c c ) ω c c P & ( c;x) P & ω c = ρ ( f;x) dc df 0 0 c f. ( 2-37 ) ρ With the help of equation ( 2-36 ) and ( 2-37 ) the transport equations for the mean progress variable c and its variance d is closed. Finally the mean species mass fraction Y α can be calculated according to Y Y P & P α = ρ 0 0 α c f & ( c;x) ( f;x) dc df. ( 2-38 ) To close the expressions above a clipped Gaussian distribution is assumed in the presented model. In the following the procedure how to obtain the values of the expectation values (see eq. ( 2-36 ) ( 2-37 ) ( 2-38 )) will be described. The clipped Gaussian probability distribution (Fig. 2-2) of a stochastic variable φ is given by with () φ = αδ() φ + βδ( φ) + [ H() φ H( φ ) ] γ g() φ P ( 2-39 ) Project No. GRD

11 MUSCLES - Extension of kinetic scheme to highly diluted Combustion 2 Theory and Formulation of the JPDF-Approach 2 2 ( φ c ) / 2c µ σ g() φ = e ( 2-40 ) cσ 2π where δ () φ and H () φ are the Dirac delta and Heavyside functions, respectively. The mean is denoted by c µ and the standard deviation by c σ. The delta functions at φ = 0 and φ = represent the tails of the distribution. Fig. 2-2: Schematic illustration of the Clipped-Gaussian pdf The distribution function P () φ satisfies the normalization condition () φ dφ P = ( 2-4 ) and the two parameters α and β are defined by and 0 µ α = g() φ dφ = erf 2 2σ ( 2-42 ) µ β = g() φ dφ = erf 2 2σ. ( 2-43 ) Therefore the normalization constant can be calculated from Project No. GRD

12 MUSCLES - Extension of kinetic scheme to highly diluted Combustion γ= ( erf µ 2 Theory and Formulation of the JPDF-Approach 2( α β ) 2σ + erf ( µ ) ) ( 2σ ). ( 2-44 ) The Clipped Gaussian distribution function has two parameters c µ and c σ. They are related to the expectation φ and the variance φ 2 of the random variable φ by (equations ( 2-4 ) ( 2-5 )) φ = φ Pφ (φ)dφ = β + 0 g(φ)dφ and ( ) ( ( 2-45 ) ) 2 φ 2 = φ φ Pφ (φ)dφ = β + 0 φ2 g(φ )dφ φ 2dφ. ( 2-46 ) As can be seen from equation ( 2-45 ) and ( 2-46 ) no explicit expression for the two parameters c µ and c σ can be obtained, therefore their values must be determined iteratively based on the local values of φ and φ 2 for which transport equations have to be solved. For computational convenience this is done prior to the cfd calculation and the result is stored in a look-up table. As no analytical function exists that maps the mean-values and variances of the statistical distributions, that are of the Clipped-Gaussian-type, onto their parameters, i.e. the median value c µ and standard-deviation c σ of the corresponding Gaussian distribution, this mapping is realized by means of a look-up table (see section 3). Via this table the parameters are calculated using bilinear interpolation of the tabulated values. The resolution of the table is chosen such way that the maximal interpolation error is less than 0-5. Fig. 2-3 illustrates the interrelationship of c, c 2, c µ and c σ. Fig. 2-3: Interrelationship of mean PDF c, variance c 2 and the parameters of the Gaussian- c µ and c σ. Project No. GRD

13 MUSCLES - Integration of extended kinetic scheme in JPDF module 3 Code-Implementation / PrePDF 3 Code-Implementation / PrePDF Reducing computational costs can be realised by solving parts of the computation prior to the cfd calculation, e.g. using a pre-processor. Within the pre-processing step calculations are carried out which would otherwise be reiterated many times during the cfd calculation. The result is stored in a look-up table (Fig. 3-) accessed by the cfd calculation (Fig. 3-2). The state space of all parameters has to be taken into account in order that the values needed by the cfd calculation can be determined from the look-up table using an appropriate interpolation. The efficiency of the pre-processor is directly associated with the algorithm used for tabulating. The tabulating algorithm should fulfil the following requirements: optimised memory requirements, optimised access time, flexible data structure (for using the tabulating method with different problem formulations) and portability of the program. To satisfy these Requirements the preprocessing software PrePDF is used. PrePDF has been presented by EBI within the scope of the CFD4C project [ 2 ]. PrePDF evaluates the PDF-integrals and stores the result in look-up tables. The complexity of reading values from the look-up table has been reduced using adaptive tabulation. Fig. 3-: Scheme of generating a PDF look-up table Project No. GRD

14 MUSCLES - Integration of extended kinetic scheme in JPDF module 3 Code-Implementation / PrePDF Fig. 3-2: Scheme of accessing a PDF look-up table Project No. GRD

15 MUSCLES - Integration of extended kinetic scheme in JPDF module 4 Validation 4 Validation of JPDF Reaction Model For testing the JPDF reaction model the matrix burner test case has been selected. The matrix burner has been developed for systematically analysing stationary highly turbulent premixed free jet flames. Fig. 4- shows the schematic setup of the matrix burner. The turbulence generator consists of a metal cylinder with 32 holes each including a helix swirl vane. The 32 holes are arranged in such way that the adjoining holes generate vorticity of opposite signs for providing a highly turbulent axial burner jet without integral swirl and a well defined flow pattern. Homogenisation of the premixed methan-air flow takes place in the burner nozzle. Flame stabilisation is realised by hydrogen pilot flames at the outer rim. Fig. 4-: Setup of the matrix burner system Project No. GRD

16 MUSCLES - Integration of extended kinetic scheme in JPDF module 4 Validation Flame Fuel Mixture φ Preheating T Reynolds D Tu l t - K - mm % mm R 408 methane R 403 methane Tab. 4- Definition of the matrix burner test cases Fig. 4-2: Comparison of JPDF-Calculation (Simulation) with experiment for Temperature, CH 4- and COconcentration of flame-configuration R408 (see Tab. 4-) Fig. 4-3: Comparison of JPDF-Calculation (Simulation) with experiment for Temperature CH 4- and COconcentration of flame-configuration R403 (see Tab. 4-) Project No. GRD

17 MUSCLES - Integration of extended kinetic scheme in JPDF module 4 Validation Fig. 4-2 to Fig. 2- show comparisons of JPDF-calculations with experimental data [ 6 ] for the flame-configurations specified in Tab. 4-. They all show, that the JPDF-model is capable of well reproducing the field measurements with respect to flame length and cone angle, especially for the flame with preheating temperatures of 673 K (R403). Temperature and CH 4 -concentration fields can be calculated by the JPDF-model with acceptable accuracy. Only the concentration of CO is overpredicted by the JPDF-model. Project No. GRD

18 MUSCLES - Integration of extended kinetic scheme in JPDF module 5 Conclusion 5 Conclusion The present report documents the integration of the 2-domain--step kinetic (presented in MUSCLES deliverable report D2.4) into an existing JPDF-model which was presented within the scope of the CFD4C project. A first test has shown that the JPDF-model applying the 2-domain-- step kinetic is capable of reproducing flow field measurements of the selected turbulent flameconfigurations with good accuracy. Project No. GRD

19 MUSCLES - Integration of extended kinetic scheme in JPDF module Annex Listings L. Symbols c Reaction progress variable c µ median value of the Gaussian function c σ standard-deviation of the Gaussian function D Distribution function D m 2 s Diffusivity Coefficient f Mixture fraction P Probability density function Sc Schmidt Number t s Time T K Temperature u & m s Velocity W k kg mol Molar weight of species k x & m Coordinates in space (x,y,z) Y kg kg Mass fraction of species k k Z k kg kg Element mass fraction of element k Greek ε Number of atoms of type e in molecule of type i ei ν t m 2 s ρ 3 kg m Mass density ω k kg m s Turbulent dynamic viscosity 3 Production rate of species k Subscripts F Ox stoich Fuel Oxidator stoichiometrical Operators () Reynolds fluctuation () Favre fluctuation () () Reynolds mean value Favre mean value Project No. GRD

20 MUSCLES - Integration of extended kinetic scheme in JPDF module Annex L.2 Literature [ ] BILGER, R. W. (975): A mote on favre averaging in veriable density flows. Combustion Science and Technology, Vol., P. 5-7 [ 2 ] BOCKHORN, H.; HOFFMANN, A.; GROßSCHMIDT, D. (2002): Final JPDF module for the model and validation report. Deliverable Report D.9. European Community, G4RD-CT [ 3 ] LOCKWOOD, F.C. and NAGUIB, A.S. (975): The prediction of the fluctuations in the properties of free, round jet, turbulent, diffusion flames. Combustion and Flame, Vol. 24, P [ 4 ] POPE, S. B. (985) : Pdf methods for turbulent reactive flows. Progress in Energy and Combustion Science, Vol., P.9-92 [ 5 ] TENNEKES, H. and LUMLEY, J.L. (972): A first Course in Turbulence. MIT Press, Cambrigde, Massachusetts. [ 6 ] ZAJADATZ, M. (2002): Matrixbrenner. PhD thesis, Engler-Bunte-Institut, Breich Verbrennungstechnik, Universität Karlsruhe (TH), Karlsruhe. L.3 Figures Fig. 2-: Definition of the probability density function 4 Fig. 2-2: Schematic illustration of the Clipped-Gaussian pdf Fig. 2-3: Interrelationship of mean 2 c, variance c and the parameters of the Gaussian-PDF c µ and c σ. 2 Fig. 3-: Scheme of generating a PDF look-up table 3 Fig. 3-2: Scheme of accessing a PDF look-up table 4 Fig. 4-: Setup of the matrix burner system 5 Fig. 4-2: Comparison of JPDF-Calculation (Simulation) with experiment for Temperature, CH 4 - and CO-concentration of flame-configuration R408 (see Tab. 4-) 6 Fig. 4-3: Comparison of JPDF-Calculation (Simulation) with experiment for Temperature CH 4 - and CO-concentration of flame-configuration R403 (see Tab. 4-) 6 L.4 Tables Tab. 4- Definition of the matrix burner test cases 6 Project No. GRD