Eddy-Dissipation Modeling of a Turbulent Diffusion

IMAT 2010 – 3rd International Meeting of Advances in Thermofluids, Singapore 30th November 2010

Eddy-Dissipation Modeling of a Turbulent Diffusion Flame with a Reaction Rate Based on the Kolmogorov Time Scale

Khalid M. Saqr1* , Hassan I. Kassem1, Hossam. S. Aly2, Mohsin M. Sies1 and Mazlan Abdul Wahid1 1

High-Speed Reacting Flow Laboratory, Faculty of Mechanical Engineering Universiti Teknologi Malaysia, 81310 Skudai, Johor - MALAYSIA 2 Department of Aeronautical Engineering, Faculty of Mechanical Engineering Universiti Teknologi Malaysia, 81310 Skudai, Johor - MALAYSIA * Corresponding author: [email protected] ABSTRACT The chemical reaction rate is highly dependent on the turbulence time scale in the eddydissipation model of diffusion flames. In the present work, two turbulence time scales and their effect on the flame predictions are investigated. The first time scale is the large eddy time scale based on the ratio between turbulence dissipation rate and kinetic energy. The second time scale is the Kolmogorov time scale, which is the time scale of the smallest scale of turbulence. Results were compared based on several comparisons of temperature profiles with measurements of a methane-air turbulent diffusion co-flowing jet flame. 1 INTRODUCTION The numerical simulation of turbulent non-premixed flames encounters a major difficulty when it comes to calculating the reaction rate. This difficulty comes, in principle, from the interaction between turbulence and reaction regions [1]. In non-premixed flames, the combustion occurs on the interface between fuel and oxidizer. For this reason, the mixing rate, which radically depends on turbulence, controls the chemical reaction [2]. In the work of Spalding [3], the reaction rate is controlled by the large-eddy mixing time. This time is proportional to the ratio between the local turbulence kinetic energy ( k ) and its dissipation rate ( ε ). Magnussen et. al. proposed several mechanisms to couple the effect of molecular mixing due to turbulence and IMAT2010-012

chemical reaction [1, 4, 5]. Their most known and established model is the Eddy Dissipation Model (EDM). The EDM is, in its essence, a modification of the Spalding eddy-break-up model [3]. The foundation of such work is to pass over the influence of chemical kinetics by replacing the chemical time scale of a certain one-step reaction by a turbulent time scale, proportional to ε / k . This is based on the paradigm that the mixing of reactants and products is the factor that determines the chemical reaction rate, in a turbulent flows [6]. Thus, the most important factor in determining the accuracy of the EDM in simulating turbulent nonpremixed flames is the method by which the turbulence time scale is calculated. Therefore, in the present paper we investigate the effect of two different time scales on the performance of EDM in predicting a Page 116


respectively. The diffusive term in the equation is subjected to N

∑Y ψ

k

=0

(3) The chemical reaction rate is governed by 

N

∑ω

k

=0

k =1

(4)

77.5

1000.00

Air Methane

Figure 1. Flame Dimensions in mm. [7]

The configuration of the considered flame is illustrated in figure 1. Methane is injected through a 4 mm diameter jet nozzle, concentric with a 1m long annular burner. The flame configuration is illustrated in figure 1. The mass flow rate of methane and air were fixed at 1.72×10-4 kg/s and 118×10-4 kg/s for all cases, respectively. The governing equations of the reacting flow field are: Total mass conservation: ∇ ⋅ (ρV ) = 0 (1)

∂x

(2)

∂x

∂y

k =1

(5) and in Y direction:

N ∂p ∂τ xy ∂τ yy + ρ ∑ Yk f ky + + ∂y ∂x ∂y k =1

(6) The following version of the energy equation was derived based on [19], [20] and [6]:

( )

N   ∇ ⋅ (V (ρE + p )) = ∇ ⋅  k e ∇T − ∑ hkψ k + τ ⋅ V  k =1   

hk  +∑ ωk k =1 M k N

(7) E = h−

where

p

ρ

+

V2 2

(8) N

T2

k =1

T1

h = ∑ Yk ∫ Cp k dT

and mass conservation for species k 

Configuration.

The conservation of momentum for a reacting flow in X direction: N ∂τ yx ∂p ∂τ ∇ ⋅ (ρuV ) = − + xx + + ρ ∑ Yk f kx

∇ ⋅ (ρvV ) = −

2 PROBLEM FORMULATION

∇ ⋅ (ρ (V +ψ k )Yk ) = ω k

k

k =1

2.00

nonpremixed coflowing turbulent jet flame. The physical reason behind the use of the Kolmogorov time scale is the assumption that molecular mixing occurs within time scale corresponding to the dissipation level of turbulent scales rather than the time scale of the large eddy. Since molecular mixing is the governing parameter of the reaction rate in the eddybreak up theory of combustion, the reaction rate becomes function of the Kolmogorov time scale. The authors have recently reported several investigations of nonpremixed flames using the EDM with a time scale corresponding to the large eddy time scale [7-10]. Other reports of using the EDM with the large eddy time scale, which is the original formulation of the model, can be found in numerous publications such as [11-18]. However, to the best of the authors’ knowledge, there is published literature that reports the use of EDM with Kolmogorov turbulence time scale.

(9)



where ψ k and ω k are the mass diffusion velocity and reaction rate for species k , IMAT2010-012

Page 117




2

S=

(

1 ∇ ⋅V + ∇ ⋅V T 2

and

)

(10)

The standard k − ε turbulence model can be given by the following equations: Turbulence kinetic energy ( k ) equation:  µ   ∇ ⋅ (ρVk ) = ∇ ⋅  µ + T ∇k  σk    + µ T S 2 − ρε (11) Dissipation rate equation:  µ   ∇ ⋅ (ρVε ) = ∇ ⋅  µ + T ∇ε  σε   

ε

+ Cε 1

µ T S 2 − Cε 2 ρ

ε2

k k (12) Eddy viscosity: k2 (13) µ T = ρC µ ε where Cε 1 =1.44, Cε 2 =1.92, C µ =0.09, σ k =1.0, σ ε =1.3 The large eddy time scale (LTS) is given

k and the Kolmogorov time scale ε

by

1

ν  2 (KTS) is given by   ε  The eddy dissipation model computes the chemical reaction rate by taking the minimum value of the following equations for fuel, oxidizer and products, respectively: 

ω = ρAY

1

Fuel

Fuel



ω

= ρA

Oxygen 

ω

Pr oducts

=ρ

The computational domain was spatially discretized to 80×103 variable density quadrilateral grid cells. The direction of cell size growth tends to locate the smallest cells in the region where the fuel interfaces with the air, hence, provide further accuracy for the solution of the flow field in the reaction zone. The reacting flow field variables in each cell were assumed to have an average value at the cell centre [7]. This first order upwind scheme was used in order to calculate the pressure, velocity, and temperature through the governing equations [21]. A momentum-weighted averaging method was implemented in order to interpolate the cell-center velocity values to the face values [22]. Solution convergence was assumed to be achieved when the residuals of all governing equations reached 1×10-4. The numerical method and grid has been recently reported by the authors in details in [7-10]. 4 RESULTS AND DISCUSSION The predicted axial temperature is plotted against experimental measurements for the same flame from Ref. [23] in figure 2. The EDM yielded over-predicted axial flame temperature when the reaction rate was governed by the KTS. 2500

(14)

τT

YOxygen 1

λ

3 NUMERICAL MODEL AND DETAILS

τT

A⋅ B 1 YPr oduct τT 1+ λ

where τ T is the turbulence time scale.

(15) (16)

2000

Temperature (K)



τ = µ 2S − δ∇ ⋅ V  3  

1500

1000

Measurements LTS KTS

500

0 0

100

200

300

400

Distance (mm)

IMAT2010-012

Page 118


Figure 2. Computed and measured axial flame temperature

Figure 3. Predicted and measured radial temperature profile at an axial location of 150 mm 2500

Temperature (K)

2000

1500

1000

500


0 0

10

20

30

40

50

Distance (mm)


2000

Temperature (K)

The predicted radial temperature profiles reveal very interesting behavior of the EDM/KTS. As shown in figures 3-6, the EDM/ KTS predicted radial temperature profiles that are in better qualitative agreement with measurements than such predicted by the EDM/LTS. The trend of the temperature profiles predicted by EDM/KTS is very consistent with the trend of measurements in comparison with the profiles predicted by EDM/LTS. However, the qualitative agreement was compromised. The EDM/KTS over predicted the flame radial temperature profiles at different axial locations, as shown in figures 3-5. In figure 6, interestingly, the EDM/KTS showed better qualitative and quantitative agreement with measurements than the EDM/LTS at an axial location of 425 mm. It is can also be noted that the quantitative agreement of the EDM/KTS predictions with measurements far from the flame centerline is much better than such agreement near to the centerline. This is evident from figures 4-6. The quantitative agreement of the EDM/KTS predictions after the radial location of 20 mm from the flame centerline is better than such agreement in the radial location of 0-20 mm.

1500

1000

500


0 0

10

20

30

40

50

60

Distance (mm)


2500

1800

Temperature (K)

1600

Temperature (K)

2000

1500

1000

1400 1200 1000 800 600

500



400 200 0

0 0

10

20

Distance (mm)

IMAT2010-012

30

40

10

20

30

40

50

60

Distance (mm)

Figure 6. Predicted and measured radial temperature profile at an axial location of 425 mm Page 119


5 CONCLUSION A new physical assumption concerning the time scale of the chemical reaction rate of turbulent nonpremixed flames was presented and examined in this paper. The assumption states that the mixing, which controls the reaction rate in the Spalding’s eddy-breakup theory, is governed by the Kolmogorov time scale instead of the large eddy time scale as the original EDM formulation postulated. The new assumption was examined using a coflowing nonpremixed jet flame case study. It was found that the EDM with the Kolmogorov time scale yields results that are in better qualitative agreement with measurements than the original formulation which uses the large eddy time scale. However, the EDM with the Kolmogorov time scale over predicted both the axial and radial flame temperature profiles. REFERENCES [1.] B. Magnussen, On the Structure of Turbulence and a Generalized Eddy Dissipation Concept for Chemical Reaction in Turbulent Flow, in: 19th AIAA Aerospace Science Meeting, St Louis, Missouri, USA, 1981. [2.] J. Warnatz, M. Maas, R. Dibble, Combustion: physical and chemical fundamentals, modeling and simulation, experiments, pollutant formation, Springer, 2006. [3.] D.B. Spalding, Mixing and chemical reaction in steady confined turbulent flames, in: 13th Symposium (Int.) on Combustion, The Combustion Institute, 1970. [4.] B.F. Magnussen, B.H. Hjertager, On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion, Symposium (International) on Combustion, 16(1) (1977) 719-729. IMAT2010-012

[5.] B.F. Magnussen, B.H. Hjertager, J.G. Olsen, D. Bhaduri, Effects of turbulent structure and local concentrations on soot formation and combustion in C2H2 diffusion flames, Symposium (International) on Combustion, 17(1) (1979) 13831393. [6.] N. Peters, Turbulent Combustion, Cambridge University Press, UK, 2000. [7.] K.M. Saqr, H.S. Aly, M.M. Sies, M.A. Wahid, Effect of Free Stream Turbulence on NOx and Soot Formation in Turbulent Diffusion CH4-Air Flames, International Communications on Heat and Mass Transfer, 37 (6) (2010) 611-617. [8.] K.M. Saqr, M.M. Sies, M.A. Wahid, Analyzing the effect of free stream turbulence on gaseous non-premixed flames in: Proceedings Of the 10th Asian International Conference on Fluid Machinery, American Institute of Physics, KL, Malaysia, 2009, pp. 891-900. [9.] K.M. Saqr, M.M. Sies, M.A. Wahid, Numerical investigation of the turbulence combustion interaction in non-premixed CH4/Air Flames, International Journal of Applied Mathematics and Mechanics, 5(8) (2009) 69-79. [10.] K.M. Saqr, M.M. Sies, H.M. Ujir, M.A. Wahid, Whirling Flames for Fuel Economy and Low NOX combustion in: Proceedings Of the 10th Asian International Conference on Fluid Machinery, American Institute of Physics, KL, Malaysia, 2009, pp. 225-235. [11.] R. Holder, C.X. Lin, Numerical simulation of reactive turbulent flows over bluff body flame holders: A Parametric study, in: 2008 Proceedings of the ASME Summer Heat Transfer Conference, HT 2008, Jacksonville, FL, 2009, pp. 129-143. Page 120


[12.] D. Achim, J. Naser, Y.S. Morsi, S. Pascoe, Numerical investigation of full scale coal combustion model of tangentially fired boiler with the effect of mill ducting, Heat and Mass Transfer/Waermeund Stoffuebertragung, (2009) 1-13. [13.] F. Lopez-Parra, A. Turan, Computational study on the effects of non-periodic flow perturbations on the emissions of soot and NOx in a confined turbulent methane/air diffusion flame, Combustion Science and Technology, 179(7) (2007) 1361-1384. [14.] J.S. Baik, Y.J. Kim, Effect of nozzle shape on the performance of high velocity oxygen-fuel thermal spray system, Surface and Coatings Technology, 202(22-23) (2008) 5457-5462. [15.] L.Y. Jiang, I. Campbell, K. Su, Combustion modeling in a model combustor, Hangkong Dongli Xuebao/Journal of Aerospace Power, 22(5) (2007) 694-703. [16.] A. Habibi, B. Merci, G.J. Heynderickx, Impact of radiation models in CFD simulations of steam cracking furnaces, Computers and Chemical Engineering, 31(11) (2007) 1389-1406. [17.] K. Majidi, CFD modeling of nonpremixed combustion in a gas turbine combustor, in: U.S. Rohatgi, D. Mewes, J. Betaille, I. Rhodes (Eds.) American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FED, Montreal, Que., 2002, pp. 971977. [18.] S.S. Kumar, V. Ganesan, Flow investigations in an aero gas turbine engine afterburner, in: Transportation 2005, Orlando, FL, 2006, pp. 15-23. [19.] [19] K. Kuo, Principles of Combustion, John Wily and Sons. New York, USA, 1986. IMAT2010-012

[20.] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, Edwards Publishing. PA, USA, 2001. [21.] K.C. Karki, S.V. Patankar, PressureBased Calculation Procedure for Viscous Flows at All Speeds in Arbitrary Configurations, AIAA Journal, 27 (1989) 1167-1174. [22.] C.M. Rhie, W.L. Chow, Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation., AIAA Journal, 21 (1983) 1525-1532. [23.] S.J. Brookes, J.B. Moss, Measurements of Soot and Thermal Radiation From Confined Turbulent Jet Diffusion Flames of Methane, Combustion and Flame, 116 (1999) 49-61.

Page 121