Flamelet-Based Time-Scale Analysis of a High-Pressure Gasifier

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Flamelet-Based Time-Scale Analysis of a High-Pressure Gasifier S. N. P. Vegendla,*,† D. Messig,‡ S. Weise,‡ and C. Hasse† †

ZIK Virtuhcon, and ‡Institute of Energy Process Engineering and Chemical Engineering, Technische Universit€at (TU) Bergakademie Freiberg, Reiche Zeche, Fuchsmuehlenweg 9, 09599 Freiberg, Germany ABSTRACT: A computational fluid dynamics (CFD) multi-feed stream flamelet model is developed and coupled to a CFD solver to simulate single-phase gas turbulent reactive flow in a gasifier. The flow equations are solved using OpenFOAM, whereas the species mass fractions and the temperatures are taken from tabulated flamelet solutions. The look-up tables were generated for the three-feed stream system of fuel, oxidizer, and steam. The validity of the flamelet model for gasification conditions is investigated in detail using an analysis of the flow and the flamelet time scales. The flamelet time scale includes both mixing and chemical reactions. On the basis of the time scales, three zones can be identified in the gasifier, namely, the (i) flame zone, (ii) recirculation zone, and (iii) post-flame or reforming zone. Good agreement is found when comparing the CFD-flamelet simulation results to the experimental values at the outlet. The effects of different scalar dissipation rates on the species mass fractions and the outlet temperature are studied, which is shown to have a significant influence on the final results.

1. INTRODUCTION Gasification is a process where hydrocarbon feedstock is converted into syngas. The gasification process occurs under reducing conditions at high pressure [high-pressure partial oxidation (HP-POX)]. Computational fluid dynamics (CFD) has become an important tool to design the reactors and understand the physical processes. However, the reliability of the simulation results strongly depends upon the modeling approach. As discussed by Rehm et al.,1 the partial oxidation process can be achieved in three different ways in the HP-POX reactor: (i) noncatalytic partial oxidation of methane, (ii) catalytic partial oxidation of methane, and (iii) gasification of liquid fuels. In this paper, the noncatalytic partial oxidation of methane is investigated. The gasification reactor needs two sets of governing equations for reactions and flow properties, such as velocities, species mass fractions, etc. Industrial reactors operate in a turbulent regime. The Reynolds average (RA) approach is applied using phenomenological closures from the literature.2 The coupling to chemical reactions is calculated with a flamelet approach, and its further details are given below. Br€uggemann3 used different combinations of reactors (plug flow and perfectly stirred tank reactors) to model the species conversion. This study emphasized the strong effect of both mixing and chemical kinetics on the outlet results. In this study, the coupling of a CFD code and a flamelet model is used to describe the turbulence
chemistry interaction. Previous CFD studies of gasifiers/risers, e.g., using the eddy dissipation concept1 (EDC), transported probability density function (PDF),2 and presumed PDF,4 revealed that most of the computation time is spent solving the reaction source terms in the species balance equations when considering detailed chemical mechanisms. The flamelet model is used to reduce the computational time by generating a look-up table and to access the species mass fractions and temperature in diffusion flames as in industrial burners and gasifiers. Furthermore, the flamelet concept is well-established in turbulent combustion. However, the validity for gasification has not yet been established. r 2011 American Chemical Society

The advantage of the flamelet model over the simplified models, e.g., the EDC and the eddy break-up (EBU) model, is to predict correct temperature distribution near the burner rim.5 Also, Rehm et al.1 found deviations in comparison to the experimental values using an EDC concept in HP-POX simulations. In the flamelet model, the species mass fraction and the temperature equations are solved in mixture fraction space. In the two-feed stream, two additional equations are needed to be solved along with the flow model equations, e.g., mixture fraction and its variance, to access the flamelet look-up tables. The laminar flamelet model views the turbulent flame as an ensemble of locally one-dimensional laminar structures embedded within the turbulent flow field.6,7 These laminar structures are essentially one-dimensional; turbulence imposes strain on the structure. The above assumptions have been shown to be valid for many technical applications. However, it may fail for very slow reactions, such as the formation of nitric oxide (NO) and nitrous oxide (N2O). In the present work, the steadystate flamelet model is investigated in the gasification regime. In this paper, methane, steam, and oxygen are fed separately at the inlets of the gasifier. OpenFOAM is used to solve the flow and mean mixture fraction and its variance equations. The mass fractions of all species are obtained from the flamelet look-up tables.

2. MODELING 2.1. Flow. The time-averaged conservation equations for the turbulent single-phase gas flow are obtained from the basic governing equations of the flow after RA of the total mass, momentum, and scalar mixture fraction of fuel and steam and its variances. The gas-phase turbulence is modeled using a realizable k
ε with standard parameters.8 In these flow equations, velocity fluctuation terms are modeled using a Received: May 30, 2011 Revised: July 25, 2011 Published: August 01, 2011 3892

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gradient-diffusion hypothesis. The appropriate initial solution and boundary conditions of mass flux, temperature, and outlet pressure are specified according to Fox.9 Three streams are fed separately into the gasification chamber: the fuel, the oxidizer, and the steam. The oxidizer and steam are both regarded as the oxidizer. Thus, the model systems consists of three streams: one fuel stream and two oxidizer streams. A total of 11 partial differential equations (PDEs) need to be solved to obtain the flow variables of velocity (Ux, Uy, and Uz), pressure (p), turbulent kinetic energy (k) and its dissipation rate (ε), and mean mixture fraction of fuel (Zf) and steam (Zs) (eq 1), and its variances (eq 2). All of these equations are solved using a steady-state approach using a segregated solver based on the OpenFOAM solver (flameletFoam)
 ~Þ ∂ðFZ μt ~ ~ ∇Z ð1Þ þ ∇ðF~uZÞ ¼ ∇ σt ∂t
 ~ 00 2 Þ ∂ðFZ ~ 00 2 Þ ¼ ∇ μt ∇Z ~ 00 2 þ Cg μt ð∇Z ~ Þ2 þ ∇ðF~uZ ∂t σt ε ~ 00 2
Cd F Z k

Figure 1. Coupling scheme of the CFD code and the flamelet lookup table.

Table 1. HP-POX Feed Conditions (Case 1) operating conditions

ð2Þ

where μt and σt are the turbulent viscosity and the turbulent Schmidt number, respectively. In the flow equations, convection terms are discretized using Gauss upwind scheme, gradient terms are approximated using Gauss linear, convection terms in the mixture fraction and its variances are approximated using the Gauss limited linear, and Laplacian terms are approximated using Gauss linear uncorrected, in the OpenFOAM solver.

fuel

steam

oxidizer

feed ratios based on steam

6.8

1

9.738

temperature (K) fuel mixture fraction

657 1

506.8 0

506.8 0

steam mixture fraction

0

1

0.0677

steam/fuel mixture fraction variance

0

0

0

gas turbulent intensity (%)

10

10

outlet pressure (bar)

10 61

2.2. Turbulence
Chemistry Interaction. 2.2.1. Flamelet Model for Two-Feed Stream. The flamelet concept for non-premixed systems has been widely used in the modeling of turbulent combustion, and a recent overview is given by Peters.10 The Lewis numbers are set to unity for all species in this investigation. Assuming a one-dimensional behavior of the combustion phenomena in the normal direction to the flame front, flamelet equations for enthalpy and species mass fractions can be derived applying a coordinate transformation. The resulting equations for the temperature and the species mass fractions are given in eqs 3 and 4. In addition, the scalar dissipation rate [χ = 2D(rZ)2] represents the influence of the flow field on the flamelet structure.10 Enthalpy equation: F

χ ∂2 T
2 ∂Z2

Nc

∑ i Ωi ¼ 0 i ¼ 1 Cp h

ð3Þ

Species transport equation: F

χ ∂2 Yi þ Ωi ¼ 0 2 ∂Z2

ð4Þ

where Ωi is the reaction source term. In the present simulations, enthalpy defects are neglected. 2.2.2. Flamelet Look-up Table for Three-Feed Stream. The flamelet look-up table is built in a preprocessing step, using four independent variables: the mean fuel mixture fraction and its variance, the mean steam mixture fraction, and the scalar dissipation rate. The mean temperature and mean mass fractions for every species can be expressed as a function of these four independent parameters. First, the flamelet equations above are solved, which yield the species mass fractions and the temperature as a function of the fuel mixture fraction. The turbulent-averaged value is obtained by integration with the PDF; see eq 5 for Y~ i Z 1 Yi ðZÞPðZÞdZ ð5Þ Y~ i ¼ 0

Figure 2. Schematic diagram of HP-POX.1

where P(Z) is the PDF of the mixture fraction. In this study, we implemented the well-established β-PDF, which is constructed from the mean and the variance. To generate a look-up table, this procedure is carried out for various oxidizer (steam mass fraction varying from 0 to 1) compositions and various scalar dissipation rates. The generated look-up tables are accessed in each grid cell based on the current solution of the four parameters: the fuel mean mixture fraction and its variance, steam mean mixture fraction, and the scalar dissipation rate. Here, the steam mixture fraction variance is not included in the flamelet library. The two mean mixture fractions and its variances, one for the fuel and another one for the steam, are solved in the CFD solver.5 The access of the flamelet look-up tables are based on the mass fraction of steam in the oxidizer, which is obtained from eq 6. Mass fraction of steam in the oxidizer: Ys ¼

~s Z ~f 1
Z

ð6Þ

~ s and Z ~ f are obtained from the CFD code by solving the where Z transport equation for the mixture fraction of the steam and fuel, respectively. 3893

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Table 2. Time and Length Scales integral scale zone (L/Lmax)

time (s)

length (m)


3

6  10


3

8  10


3

9  10


3

3  10

outlet (0) 0.347

to 0.01

0.02 to 0.1 5  10

near inlet (0.937)

Kolmogorov scale


4

to 0.12

time (s)
5

3  10

to 0.01


4

3  10

to 0.04


7

8  10

to 0.03

flamelet time scale (s)

length (m)

to 1  10


4

to 7  10


4

to 1  10


3


6

5  10


5

2  10


6

1  10

to 1  10


5

5  10
6

to 3  10


5

4  10
6

to 4  10


5

1  10
8 to 1  10
6

2.2.3. CFD-Flamelet Algorithm. In a CFD-flamelet coupling code, in each grid cell, the generated flamelets for different mass fractions of steam in the oxidizer are accessed using the steam mass fraction from eq 6. After the correct look-up table entry is located, the rest of the procedure is the same as in a two-feed stream model to assign the species concentrations and temperature at a specified mixture fraction of the fuel and its variance.11 As seen in Figure 1, the CFD-flamelet coupling is used to retrieve the species mass fractions and temperature from the flamelet look-up tables, without solving the species continuity equations in CFD directly. The CFD code solves the equations for the velocity field, pressure, the turbulent kinetic energy and its dissipation, and mixture fraction and its variance. The obtained information for the mixture fraction and its variance is used for the retrieval, as explained above. The scalar dissipation and its stoichiometric rates are calculated using the fuel mixture fraction variance from the CFD, as described in eqs 7
11. The scalar dissipation rate in each computational grid cell is obtained using the following equation (eq 7): ε ~ 00 2 χ~ ¼ Cχ Z k χst ¼ χ~Z 0

ð7Þ ferfc
1 ðZst Þ

1

Figure 3. Two-dimensional temperature field along the length of the geometry.

ð8Þ

~, Z ~ 00 2 ÞdZ ~ ferfc
1 ðZÞPðZ

ferfc
1 ðZÞ ¼ expf
2ðerfc
1 ð2ZÞÞ2 g

ð9Þ

The complementary inverse error function at stoichiometric mixture fraction is ferfc
1 ðZst Þ ¼ constant value ¼ expf
2ðerfc
1 ð2Zst ÞÞ2 g

ð10Þ

The calculated β-PDF value for the complementary inverse error function (eq 11) is stored in the flamelet look-up table, which is directly accessed in the CFD code. Z 1 ~, Z ~ 00 2 ÞdZ ~ ferfc
1 ðZÞPðZ ð11Þ P value ¼ 0

~, Z ~ 00 Þ : β-PDF PðZ 2

The scalar dissipation rate at stoichiometric value in each grid cell is calculated using eq 8. The complementary inverse error function values are interpolated from the flamelet look-up tables at a specified mean mixture fraction and its variance.

3. HP-POX FEED SYSTEM The HP-POX consists of three-feed system: (i) fuel, (ii) pure steam, and (iii) combination of O2 and steam. Two kinds of independent mixture fractions are fed in a three-feed system, i.e., fuel and steam, as explained above. The initial and boundary conditions are given in Table 1, to solve all of the transport

Figure 4. Flamelet time scales for different scalar dissipation rates, at a given pressure of 61 bar (Zst = 0.177) and using a constant HP-POX outlet oxidizer composition.

equations for hydrodynamic properties and flamelet equations. The schematic diagram of the HP-POX can be seen in Figure 2.1 In this work, two case simulations were performed in a twodimensional (2D) axi-symmetric grid of 14 108 computational cells with gradual refinement toward the inlet. A detailed chemical mechanism of GRIMECH 3.0 with 53 species is reduced to 28 species to generate the flamelet look-up table by excluding molecules representing “N” element and C3 and higher hydrocarbons, because the latter has no impact, as shown by Rehm et al.1 The applicability of GRI-Mech 3.0 for high-pressure conditions is shown by Rehm et al.1 and Tranter et al.12 3894

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4. RESULTS AND DISCUSSION 4.1. Time and Length Scales (Case 1). The flow and flamelet time scales are analyzed for the HP-POX reactor for given operating conditions, as shown in Table 1.

4.1.1. Flow Length and Time Scales. The integral and Kolmogorov time and length scales are obtained with the following definitions:

timescales

Table 3. Outlet Species Mass Fractions of Synthesis Gas χ

χ

wet basis

mass fraction experimental equilibrium (10
5, s
1) (10
4, s
1) simulation CH4

0.005

8.9  10
4

0.00172

0.0058

9  10
4

CO

0.59

0.614

0.6125

0.6051

0.6125

CO2

0.116

0.0842

0.0843

0.08459

0.08398

H2

0.081

0.0824

0.0821

0.08056

0.08183

H2O

0.203

0.2133

0.2142

0.2188

0.2156

N2

0.005

0.00503

0.00504

0.00514

0.005

O2

0.0

0.0

0.0

0.0

0.0

T (K)

1702

1677

1680

1700

1692

lengthscales

integral k τI ¼ ε LI ¼

k3=2 ε

Kolmogorov rffiffiffi υ τη ¼ ε !1=4 υ3 Lη ¼ ε

where υ is the kinematic viscosity of the gas. 4.1.2. Flamelet Time Scale. The species mass fraction equation in the mixture fraction space is rewritten as follows: F∂Yi χ ∂2 Y i ¼F þ Ωi ¼ β i 2 ∂Z2 ∂τ

ð12Þ

Figure 5. Axial profiles of the (i) temperature, (ii) species mass fraction, (iii) integral and Kolmogorov time and length scales and flamelet time scales, along the center of the reactor (where L = 0 is the outlet). 3895

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Figure 6. Flow length and time scales and species time scales in the (i) flame zone and (ii) post-flame and recirculation zone.

The times scales of the reaction species are reciprocal of the Eigen values of the Jacobian matrix (see eq 13). In this paper, a single time scale of the reaction for all species is presented as from Rao and Rutland.13 The assumption is made that each species has the same flamelet time scale. This time scale is chosen as a derivative of O2 in the lean-fuel conditions and as a derivative of CH4 in the rich-fuel conditions. 3 2 dβi ¼ 1 dβi ¼ 1 6 7 6 dYj ¼ 1 3 3 3 dYj ¼ m 7 7 6 333 333 7 6333 7 ð13Þ Jacobian ¼ 6 6333 333 333 7 7 6 dβ dβi ¼ m 5 4 i¼m 3 3 3 dYj ¼ 1 dYj ¼ m In Table 2, the length and time scales of the flow and flamelet time scales are presented at three different positions of the HPPOX reactor: the reforming/post-flame zone, recirculation zone, and flame zone. As seen in Figure 3, these three zones are illustrated in the temperature contour plot. It is important to note that the very high flame temperature is caused using pure oxygen and neglecting radiation. As seen in Table 2, the HP-POX flow and flamelet time scales vary significantly from zone to zone. In this work, the time-scale analyses are shown at different heights of the reactor. In Figure 4, the results of a numerical experiment are shown on the basis of the oxidizer composition at the outlet. The flamelet time scale is plotted for different scalar dissipation rates. The flamelet time scale varies strongly over the fuel mixture fraction. The flamelet time scales are lowest in the vicinity of stoichiometric mixture. In the lean- and rich-fuel conditions, the time scale increases when reaching the mixture fraction limit of 0 and 1, respectively. As seen in Figure 4, the time scale on the lean side is less sensitive for all of the scalar dissipation rates. On the other hand, it strongly depends upon the scalar dissipation rate on the richfuel side, which is applicable for gasification. The mean outlet fuel mixture fraction is also shown in Figure 4. As seen in Figure 4, no significant differences are found when comparing the time scales at high pressures for different scalar dissipation rates for a given stoichiometric mixture fraction. On the other hand, it is interesting to note that significant deviations are found at low pressures (not shown here). In Table 3, the composition for the major species and the temperature at the outlet are shown. The species mass fractions show reasonable agreement with the experimental values. In the simulations, overpredicted CO could be due to the higher

Figure 7. Radial profiles of time and length scales (integral and Kolmogorov) at different heights of the reactor (where L = 0 is the outlet) (Symbols: solid line, integral length; dotted line, integral time; dashed-dot-dot line, Kolmogorov length; and dashed-dot line, Kolmogorov time).

conversion of CO2 than in the experimental results. Hence, an underprediction of CO2 is observed with the presently developed model. The outlet species mass fraction values are very sensitive to the scalar dissipation rate, as seen in Figure 4 (reactive time scales vary significantly in the rich-fuel condition). This is especially important because the scalar dissipation rates are generally very small, and this shows the importance to model this quantity. As seen in Table 3, the overpredicted simulation temperatures are mainly due to the non-equilibrium effect and predicted low mixture fraction of the fuel than in the feed (error ∼ 0.386%). In the simulated temperature, the former contributes a value of 3 K and later contributes a value of 12 K. The low prediction of the mixture fraction is due to the numerical error by solving the partial differential equations. At the outlet, the simulated stoichiometric scalar dissipation rate is around 10
5 s
1. When the stoichiometric scalar dissipation rate is increased to 10
4 s
1, the predicted values are much closer to the experimental observations. This should be interpreted as a measure of the uncertainty in the simulation and experimental results. As seen in Figure 5, the flame zone is found near the inlet. As seen in iii of Figure 5, the lower order of the magnitude of length and time scales is found near the inlet, as compared to the rest of the reactor (Table 2). In the present profiles, the observed peaks at dimensionless lengths of 0.1, 0.6, and 0.95 are mainly due to the change of the physical dimensions (radius) of the geometry (see Figure 2). As seen in iii of Figure 5, the obtained flamelet time scales are the combination of the mixing and reaction time scales from the flamelet. Near the flame zone, the time scales are a unit order of magnitude lower when compared to the rest of the reactor. 3896

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Figure 8. Radial profiles of the (i) temperature and (ii) species mass fraction at a reactor height of L/Lmax = 0.97 from the outlet.

As seen in Table 2, near the burner, the flamelet time scales are much smaller compared to the integral/Kolmogorov time scale. Also, in the reforming and recirculation zones, the flamelet time scales are found still lower than the flow time scales. As seen in iii of Figure 5, the maximum reaction time scales are calculated from the hydrogen species from the flamelet code. As seen in Figure 6, the species reaction time scales of CH4, CO, CO2, and H2 are much smaller than the flow time scales (Kolmogorov and integral) in the flame zone. In the post-flame zone, the reaction time scales of all of the species are smaller than the flow time scales, except H2 species (slower formation reaction rates of H2). This important finding shows that the flamelet approach is a suitable model under these gasification conditions, except for H2 species. As seen in Figure 7, the integral and Kolmogorov length and time scales vary in radial direction. Three zones can be identified with the present profiles of the length and time scales. The flame zone is near the inlet (L/Lmax of 0.97), where the length and time scales are found to be very low with respect to the other zones. The recirculation and reforming zones are located beside and further downstream of the flame zone (see Figure 3). From these profiles, it can be concluded that the Kolomogrov time scales drop around 2 orders of magnitude, when comparing the HP-POX inlet and outlet sections of the reactor. As seen in Figures 8 and 9, the temperatures are high near the inlet, where the species mass fractions are rapidly changing because of combustion of the reactants (methane). In the recirculation zone, the species mass fractions and temperature are found to be almost constant across the radial direction because of the high residence time of the fluid elements. 4.2. Case 2 Simulations. To further evaluate the applicability of the flamelet approach for gasification, a second operating point

ARTICLE

Figure 9. Radial profiles of the (i) temperature and (ii) species mass fraction at a reactor height of L/Lmax = 0.937 from the outlet.

Table 4. HP-POX Feed Conditions (Case 2) operating conditions

fuel

steam

oxidizer

feed ratios based on steam

6.05

1

8.59

temperature (K) fuel mixture fraction

641 1

522 0

522 0

steam mixture fraction

0

1

0.05814

steam/fuel mixture fraction variance

0

0

0

gas turbulent intensity (%)

10

10

outlet pressure (bar)

10

61

Table 5. Outlet Species Mass Fractions of Synthesis Gas χ

wet basis

χ

mass fraction experimental equilibrium (10
5, s
1) (10
4, s
1) simulation CH4

8.8  10
4

0.0014

0.00177

0.0057

1  10
3

CO

0.603

0.611

0.6094

0.6023

0.6099

CO2

0.101

0.08499

0.08504

0.0853

0.085

H2

0.07933

0.08192

0.08159

0.0801

0.08145

H2O

0.21

0.216

0.217

0.2214

0.217

N2

0.00497

0.005027

0.00502

0.0051

0.005

O2

0.0

0.0

0.0

0.0

0.0

T (K)

1689

1681

1685

1705

1694

was simulated, and the feed compositions are shown in Table 4. The different feed conditions are considered, as compared to Table 1. As seen in Table 5, the outlet species mass fractions and temperature in the simulations are shown in reasonably good 3897

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agreement with the experimental values. An underpredicted outlet CO2 mass fraction is found when compared to experimental results. The outlet species mass fraction values again are very sensitive to the scalar dissipation rate (see 10
5
10
4 s
1), as explained above (see case 1). When the stoichiometric scalar dissipation rate is increased (10
4 s
1), the simulated species mass fractions are closer to the experimental values. When the simulation results are compared to the experimental values in cases 1 and 2, the developed CFD-flamelet model yields good agreement. The flamelet approach originally developed for combustion has been shown to be valid for gasification, and the results are very promising (see Tables 3 and 5).

h = enthalpy (kJ mol
1) k = total kinetic energy (m2 s
2) L = length (m) P = β-PDF r and R = radius (m) T = temperature (K) t = time (s) u = velocity (m s
1) Y = mass fraction Z = mean mixture fraction Cg = constant value of 2.86 in the mixture fraction variance equation Cd = constant value of 2 in the mixture fraction variance equation

5. CONCLUSION A high-pressure gasifier using methane as feedstock was modeled using a CFD-flamelet approach. A three-feed system flamelet model was formulated, and the flamelet solutions were stored in a look-up table. In the gasifier, the flow length and time scales, i.e., Kolomogorov and integral, and the flamelet time scales were analyzed. On the basis of the time scales, the gasifier was classified into three different zones: flame zone, recirculation zone, and reforming zone. In the gasifier, the flamelet time scales were much smaller than the Kolomogorov time scale, except for the H2 species flamelet time scale at the outlet. This is also valid for the rich-fuel conditions of gasification. The flamelet model, which is widely used in modeling turbulent combustion, was investigated concerning its applicability for gasification. The obtained results at the reactor outlet with the present developed CFD-flamelet model show good agreement with the experimental observations. In the simulations, the overpredicted CO was due to higher conversion of CO2 than in the experimental results. Also, the found deviations in the simulations were due to underpredicted simulated values of stoichiometric scalar dissipation rates than in the experiments. The sensitivity of the stoichiometric scalar dissipation rates were studied on species mass fractions and temperature. The simulated results were much closer to the experiments when increasing the stoichiometric scalar dissipation rate by 1 order of magnitude at the outlet.

Greek Letters

’ AUTHOR INFORMATION Corresponding Author

*Telephone: +49(0)3731394824. Fax: +49(0)3731394555. E-mail: [email protected].

’ ACKNOWLEDGMENT The authors gratefully acknowledge financial support through the Federal Ministry of Education and Research and Federal Ministry of Economics and Technology of Germany in the framework of Virtuhcon and COORVED under project numbers 040201030 and 040201035. We thank the German Federal Ministry of Economics and Technology (BMWiNO, 0327113B), the Saxon Ministry of Science and the Fine Arts (SMWK), and the Lurghi GmbH/Air Liquide for supporting our work. ’ NOMENCLATURE Cp = specific heat at constant pressure (kJ mol
1 K
1) D = diffusion coefficient (m2 s
1)

χ = scalar dissipation rate (s
1) β = RHS of eq 12 (s
1) μ = viscosity (Pa s) υ = kinematic viscosity (m2 s
1) F = density (kg m
3) ε = turbulent kinetic energy dissipation rate (m2 s
3) τ = time (s) σ = Schmidt number value of 0.7 Ω = reaction source term (kg m
3 s
1) Cχ = constant value of 2 Subscripts

f = fuel i and j = species indices s = steam st = stoichiometric t = turbulent η = Kolmogorov I = integral Superscripts


= mean value ∼ = Favre average 00 2 = variance

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