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NUMERICAL INVESTIGATIONS OF TIME DEPENDENT TURBULENT SWIRLING FLAME CHARACTERISTICS IN 3-D FURNACES

Dr.Ramiz A. Kameel*, Prof.Dr.Essam E. Khalil Faculty of Engineering, Department of Mechanical Power Engineering

* Presently with CEB, 3 Rashdan Str., Dokki, Cairo Cairo, Egypt

ABSTRACT

This paper reports a numerical method designed to predict the transient flow characteristics in three-dimensional furnace configurations at high Reynolds numbers. The paper introduces a developed Computational Fluid Dynamics (CFD) program (3DTCOMB) to predict the airflow and combustion characteristics in the furnaces. The transient nature of the flame propagation is described and explained in the present work through a “squashing and stretching” process. The agreement between the results of numerical techniques based on the Arrhenius reaction model and the experimental results is qualitative in nature. The predicted transient results would aid the future development of the combustion models on a large-scale furnace. Attention and emphasis should be directed to the micro-scale modeling.

INTRODUCTION Combustion chamber designers always endeavour to achieve the optimum operating conditions that give maximum combustion efficiency together with minimum pollutant formation rate. 1 This can be attained if details of the combustion process are well known. Since the Sixties, after the advance of computers, numerical procedures became the most adequate tool, which aids the design procedure at sufficient accuracy and proper cost. The mathematical and numerical simulation of the combustion processes involve some of the following physicochemical phenomena: aerodynamic flow pattern, heat and mass transfer, turbulence, two-phase flow, chemical kinetics, and radiation. The exact treatment of any of these individual processes is very complex and requires excessively large computing facilities. To aid the designer and to ease his task, mathematical models that simplify the exact treatment of these processes are proposed, and embodied in the numerical procedure to yield the conservation equations solvable. 2 Furnace design would be greatly facilitated by the use of the mathematical and numerical procedures. These procedures can simulate different operating conditions for various designs to obtain the optimum design conditions. Different combinations of inlet velocities, combustor geometry, jet temperatures and equivalence ratios may produce very different flame and flow patterns, which in turn, may lead to undesired thermo-acoustically induced combustion instabilities. 3

Numerical simulation of reacting flows – developing spatially and transiently – is ideally designed to properly address most of the designer requirements. In the earlier investigations, the calculations of furnace flow properties were carried out using steady and two-dimensional procedures 4,5. These calculations are verified with experimental data and indicated good agreements with the experimental data. In the recent investigations, the numerical calculations were carried out using the three-dimensional models 6 and using the time-dependent models 3. The main purposes of this paper are, firstly, the documentation and testing of the present numerical tool, developed to predict the transient reacting flows. Second aim is to investigate the transient nature of reacting flows in a furnace, documenting the flame progress to enhance the future modelling developments. Third purpose is to mark up several observations to enhance the physical modeling of the reacting flows and flame propagation.

CALCULATION PROCEDURE CONSERVATION EQUATIONS The present program 3DTCOMB solves the elliptic form of the governing differential equations in three-dimensional cylindrical configurations under transient conditions. The different governing partial differential equations are typically expressed in a general form as:

)1(Srr

1

rr

rr

1

xx

Wr

1Vr

rr

1U

xt

Φ+���

����

�

θ∂Φ∂Γ

θ∂∂+��

�

����

�

∂Φ∂Γ

∂∂+��

�

����

�

∂Φ∂Γ

∂∂=

Φρθ∂

∂+Φρ

∂∂

+Φρ∂∂

+Φρ∂∂

The corresponding terms expressed in the general elliptic form are indicated in Table 1. Various assumptions are necessarily added to yield the differential equations soluble. The use of the present turbulence model will be justified and explained briefly in the following subsection. The representation of the thermodynamic properties is obtained from references as discussed later. TURBULENCE MODEL The turbulence characteristics are modeled using the two-equation k-ε turbulence model. The effective diffusion coefficients and source terms for the two differential equations are listed in Table 1. The values

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AIAA 2004-2383

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of the kinetic energy and dissipation rate are corrected near the wall using the wall functions. 7 THERMODYNAMIC PROPERTIES The density of mixtures of air, the combusting gas and the combustion products can be adequately represented for present purposes, by the equation of a perfect gas

)2(TR

PM=ρ

With M and P determined with the aid of the appropriate mass fractions and Dalton’s law of partial pressure. The specific heat of gases was calculated from the expressions

)4(CmC

)3(TbaC

Pii

iPmix

OiOiPi �=

+=

The definition of the mixture stagnation enthalpy is as follows:

[ ] )5(2WVUTCmHmh 222Pi

iifufu ++ρ+�+=

And includes the heat of reaction, Hfu, which should be specified from the composition of the fuel. The constants in equation (3) and (5) are given in Table 2. COMBUSTION MODELS The results obtained with two combustion models are introduced in this paper; these are described in the following sub-sections. When attempting to simulate the turbulent combustion process, flow and combustion chemistry simplifications are carried. Model 1: this model assumes a physically controlled, one step reaction, as shown in Equation 6, with supposing the fuel and oxidant cannot exist at the same place at any time and the reaction is infinitely fast and equilibrium is attained. 8 The only species equation to be solved is that for the mixture fraction f, and this equation has no source as represented in Table 1. The other terms are calculated based on the Equations 7-9. The chemistry reaction is

)6(OH2COO2CH 2224 +→+ Model configurations can be summarized as follow 8-10: In regions of flame where oxidant and products exist

0 < f < fst mfu = 0 (7) mox = (1 – f / fst) In regions of flame where fuel and products exist 1 > f > fst mfu = (f - fst) / (1 – fst) (8) mox = 0 And mpr = 1 – mox – mfu (9) The temperature is calculated from the equation (5) Model 2: In contrast to first model, a finite reaction rate is introduced in this model. In particular, both mfu and mox can coexist at the same location at the same time. In solving the PDE for mfu, its source term (Rfu) (rate of the fuel consumption) is represented by an Arrhenius type source term or by an eddy-break-up term. The smaller of these two expressions at a given point in the flow field dominates the reaction and is used; the eddy dissipation rate often slowing chemical reaction. 8-10 Rfu = A ρ2 mfu mox exp(-E/RT) (10) Or Rfu = CR g

2 (ρε/k) (11) These constants are represented in Table 2. In terms of the number of differential equations considered, model 2 requires the solution of the f, g, mfu equations. Further modifications were added in several researches, but these modifications were not fully justified. 3

SOLUTION PROCEDURE COMPUTATIONAL GRID GENERATION The present program utilizes the modified hyperbolic formula 11 for grid node generation. The present formula generates non-uniform orthogonal grid. The present program is designed to simulate the flow domain by 144,000 cells (x, r, θ) ≡ (80, 60, 30) to attain the enough accurate predictions with efficient cost. NUMERICAL SCHEME The present program utilizes the power-law scheme that was proposed and recommended 12 to avoid the simple first-order upwind difference scheme.

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Table 1: Terms of the Partial Differential Equations (PDE) Conservation of Φ Γ SΦ Mass 1 0 0 Axial momentum U µeff -∂P/∂x + SU Radial momentum V µeff -∂P/∂r + SV Tangential momentum W µeff -(1/r)(∂P/∂θ) – ggr ρ (1-β∆t) + SW Stagnation enthalpy h µeff/σh 0 Mixture fraction f µeff/σf 0 Concentration fluctuation g µeff/σg Cg1Gg1 – Cg2 ρ g ε/k Mass fraction of fuel mfu µeff/σfu Rfu Kinetic energy k µeff/σk G - ρ ε Dissipation rate ε µeff/σε C1 ε G/k – C2 ρ ε2/k µeff = µlam + µ t µ t = ρ Cµ k

2 / ε G = µt [2{( ∂U/∂x)2 +(∂V/∂r)2 +(∂W/r∂θ + V/r)2}+(∂U/∂r + ∂V/∂x)2 +(∂V/r∂θ + ∂W/∂r – W/r)2 +(∂U/∂θ + ∂W/∂x)2] SU =∂/∂x(µeff ∂U/∂x)+(1/r)∂/∂r(rµeff ∂V/∂x)+(1/r)∂/∂θ(µeff ∂W/∂x) SV =∂/∂x(µeff ∂U/∂r)+(1/r)(rµeff ∂V/∂r)+(1/r)∂/∂θ(µeff r ∂(W/r)/∂r)-2(µeff /r)( ∂W/r∂θ+V/r)+ρW2/r SW =∂/∂x(µeff ∂U/r∂θ)+(1/r)∂/∂r(rµeff((1/r)∂V/∂θ-W/r))+ (µeff /r)(r∂(W/r)/∂r+(1/r)∂V/∂θ)+(1/r)∂/∂θ(µeff(∂W/r∂θ+2V/r))- ρVW/r Cg1 = µeff ((∂mfu/∂x)2+(∂mfu/∂r)2+(∂mfu/∂ r∂θ)2) C1 = 1.44, C2 = 1.92, Cµ = 0.09, Cg1 = 2.8, Cg2 = 2.0, CR = 1.0 σh = 0.9, σfu = 0.9, σf = 0.9, σg = 0.9, σk = 0.9, σε = 1.225

The power-law scheme is not particularly expensive to compute and provides an extremely good representation of the exponential behaviour 12. Although the power-law scheme is only first-order approximation on the basis of truncation error, the power-law difference scheme is a conservative formulation and does not demonstrate numerical oscillations. TIME-DEPENDENT PROCEDURE This program has the capability to switch between the steady state and unsteady state bases. The time interval value is calculated according to the stability condition of the partial differential equation 13. The program is designed to provide storage for the values of variable Φ at time t and at time t + ∆t to employ the iterations within the time step to get the convergent solution within the time step. The fully implicit scheme is utilized in this model. It is found that the proper value of ∆t is 0.1 or less. INITIAL AND BOUNDARY CONDITIONS The initial conditions for the solution are taken to represent the actual value of each variable Φ at the time t=0. The boundary conditions are obtained from measurements. Near the wall, the log-law of the wall function is applied to correct and find the values of turbulence and dissipation. The velocity of the airflow tends to be zero at the wall surface. Gradients of mfu, f, and g were assumed to zero at all walls. At the inlet, the dissipation rate was determined from the equation.

)12(y03.0

kC

a

23

µ=ε

At the exit, all gradients were assumed to be zero; this assumption has a negligible influence on the upstream flow predictions. CONVERGENCE AND STABILITY The convergence should be attained over the time step. The simultaneous and non-linear character of the finite difference equations necessitates that special measures are employed to procure numerical stability (convergence): these include under relaxation of the solution of the momentum and turbulence equations by under relaxation factors which relate the old and the new values of Φ as follows:

( ) )13(1 oldnew Φγ−+Φγ=Φ Where γ is the under-relaxation factor, which was varied between 0.2 and 0.3 for the three velocity components as the number of iteration increases. For the turbulence quantities, γ was taken between 0.2 and 0.4 and for other variables between 0.2 and 0.6. The required iterations for convergence in the time interval are based on the nature of the problem and the numerical conditions (grid nodes, under-relaxation factor, initial guess, etc.). So the time (on the computer processor) required for obtaining the results is based on many factors. NUMERICAL COST The required time for solution, using model 2, per equation per cell is 1.41x10-5 in an 800 MHz PC. Using this configuration, it is required about 15 hours to attain the steady state solution. Model 1 takes about 75% of the required time of the second model.

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Table 2: Combustion Model Constants and Thermodynamic Properties [8]

Gas Composition Molecular weight ao x 10-3 (kJ/kg K) bo x 10-3 (kJ/kg K) Methane CH4 16 1000 2.055 Oxygen O2 32 888.1 0.0977 Carbon dioxide CO2 44 1740.2 0.3072 Nitrogen N2 28 823.8 0.1983 Water vapor H2O 18 1002.3 0.0865 Combustion model constants Hfu = 4.07 x 104 kJ/kg, A = 1010 m3/kg s, E/R = 1.84 x 104 K

VALIDATION PROCEDURE

This section will present the program validation test results performed on a carefully selected application that represent the basic furnace flows; i.e. flow over backward facing step and the comments on the assessment and validation process. FLOW OVER A BACKWARD-FACING STEP Properties of a turbulent flow downstream of a plane sudden expansion were reported in previous literatures 14 with aid of smoke and, in the vicinity of the reattachment position, with tufts. In a subsequent numerical investigation 15 the flow properties of plane and axisymmetric flow were predicted using two-dimensional turbulent flow model. The results tended to be in good agreement with the experimental data. In the present work, the flow properties of a three-dimensional axisymmetric configuration are predicted using transient prediction model. The repeatability of the determination of reattachment is remarkably good, as showing in Figure 1. It is observed that the recirculation zones were not identical on both sides of the symmetry line. An interaction pairing occurred in the recirculation zone with periodical expansion and constriction of wall recirculation zones. CHEMICALLY REACTING FLOW The predictions of gas temperature and axial velocity distributions along the centreline in the swirling reacting flows of furnace 16 are carried out in the present work to verify the present combustion models. The predicted results were obtained using model 2, as shown in Figure 2. Furnace dimensions are summarized in Table 3. The boundary conditions were obtained from the measured values. 16 . Table 3: Dimensions of Furnace of [16]

Furnace diameter Df 300 mm Furnace length Lf 900 mm Fuel inlet diameter D1 12 mm Inlet diameter of the air annulus D2 27 mm Outlet diameter of the air annulus D3 55 mm Diameter of exit Dout 90 mm Swirl number S 0.52

Figure 1: Measured and Predicted Variation of Recirculation

Zone Length with Expansion Ratio, Re = 5 x 104

Figure 2: Predicted temperature and axial velocity Results

The observed velocity decay was due to the expansion and the sudden increase of velocities was then due to reaction and density reduction during combustion. Good agreements were also observed in the flame region at the entrance region of the furnace (flame brush). The discrepancy near furnace exit may be regarded to the outflow floating boundary conditions in the presence of the flame.

TRANSIENT SIMULATION This section describes time dependent flame propagation and behaviour obtained with the present numerical model. This was carried out on the furnace described in the Table 3. 16 For combustion model 1 no special treatment should be followed to start the flame simulation. In model 2 the start of ignition simulation is needed. Simulation starts by diffusion of the fuel and combustion air to be mixed; this needs a catalyst. This catalyst is simply a spark as assumed earlier. 18

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Definitely, reviewing the Arrhenius reaction model, it can be deduced that the flame does not start without high temperature mixture. So, at the present work, a simulated spark at the burner downstream is carried out. The temperature is raised artificially in this region to at small time interval (from start 0 second to 0.3 second).

RESULTS AND DISCUSSIONS The predicted mixture fraction and fuel mass fraction distributions are presented here together with corresponding temperature patterns. The model 1 is so simple and depends on simplified assumptions, that the chemical reaction is influenced by the reactants concentration decay. The fast chemical reaction may represent the real process in the well-mixed fuel and oxidant mixture. The fast chemical reaction, also, need a very fast flame propagation with unconfined diffusion. Model 1 successfully predicts the flame nature qualitatively; the longitudinal and lateral wavy nature of the flame was so clear in the transient simulation using the first model. The prediction procedure using this model doesn’t require large computing time but, it can not be adequately used for design. In the present swirling reacting flow, the Arrhenius model is a fair representative model. Such model is so sensitive to the boundary conditions especially near or at the flame regions. In high temperature regions, the switching between the Arrhenius model and the eddy-break-up model was frequently observed, which consequently influences the transient prediction and the steady sate results. When Model 2 is incorporated, the results were recorded for 100 seconds. The steady state results were attained at about 70 seconds approximately after ignitions. All special sequences of events in the numerical simulation are represented in Table 4. The spatial location of the maximum temperature was recorded with time in terms of axial and radial distances. The axial and radial loci of the maximum temperatures are represented in Figure 3. One can observe that the flame had expanded longitudinally and laterally after 11 seconds of the ignition. Table 4: Simulation Special Events

Time, s Event 0 Start of numerical simulation 0 Start of the ignition spark simulation 0.3 End of the ignition spark simulation 10 Products exiting the furnace 11 Longitudinal flame propagation 20 - 25 Rich fuel volume depletion 35 - 40 Rich fuel volume depletion 40 Maximum heat release 41 Flame filling furnace 45 Lateral flame expansion

Figure 3: Axial and Radial Locus of Maximum Temperature.

Figure 4: Temporal Rich Fuel Volume development

The flame reaches the end of furnace after 41 seconds from start of ignition. This observed phenomenon has a great influence on the outflow condition at the exit of the furnace. High disturbances in the flow and the turbulence were increasingly observed.

Figure 5: Temporal Heat Release of the Flame

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The flame impinged on rear wall and back flow to the half of the furnace is observed after 3-4 seconds from impinging the furnace end wall. At this instance, the hot gases were reverted backward toward the burner and expanded laterally toward the furnace side walls. The volume of the rich fuel was recorded with the time as shown in Figure 4. The volume changed periodically with the time. The volume of rich fuel was computed for two conditions (mfu > 0.1 and mfu > 0.15). The rich fuel volume is between 13.33 % and 22.5 % of furnace volume. As indicated in the figure the steady state is achieved at 110 seconds from the ignition and the steady state results were presented for centerline previously in Figure 2. The flame heat release is displayed in the both Figures 5 and 6 in the form of heat release and volumetric heat release rate variation with the time. The flame is generally characterized by the zone, which has local temperature greater than 1000 oK. The heat release from the flame was tracked with the time. The flame heat release rate at the instant of ignition was shown as maximum here. The flame heat release rate decreased after the ignition to the minimum level. The heat release rate increased to reach its maximum point at 40 seconds from the ignition. The rate of the heat release decreases after 40 seconds due to the depletion of the flame structure. The temperature temporal distribution in the furnace is represented in Figure 7. The propagation scenario of the flame can be extracted from the Figure. The transient nature of the flame propagation is described and explained in the present work through a “squashing and stretching” process. The most important events are displayed here in the Figure 7. The flame envelope can be identified based on the flame temperature definition.

Figure 6: Temporal Volumetric Flame Heat Release

CONCLUSIONS This paper introduces a developed Computational Fluid Dynamics (CFD) program (3DTCOMB) to predict the airflow and flame characteristics in the furnaces, based on two combustion models. It was found that the Arrhenius rate-based model is proper for furnace flow predictions. In the non-swirl confined flame simulation, the outflow conditions should be considered with great care. The temporal nature of the flame propagation is described and explained in the present work. One can define the flame propagation as a “squashing and stretching” process. It is observed that the numerical results based on the Model 1 have insufficient agreement with the previous experimental and numerical results. However, this first model successfully predicted a qualitative preliminary indication to the flame performance. The simulation process based on the first model is quicker than the simulation process based on the model 2. The agreement between the numerical results based on the Arrhenius reaction rate model and the experimental results is qualitative and further refinements and developments are needed. The time dependent flame behaviour would aid the future development of the combustion models on a large-scale furnace. Attention and emphasis should be directed to the micro-scale modeling. The present program (3DTCOMB) will be subjected to further development in future to extend its capabilities to more adequately describe the combustion-time processes and turbulence-chemistry interactions .The final goal is to support the engineers and designers of the furnaces and combustors.

REFERENCES [1] Khalil, E. E., 1981, Numerical computation of combustion generated pollution, La Rivista dei Combustibili, Vol. XXXV, fasc. 4, p. 193, 1981. [2] Khalil, E. E., 1979, Acta Astronautica 6, 449. [3] Grinstein, F. F., and Fureby, C., 2003, LES studies of the flow in a swirl gas combustor, AIAA 2003-0484, 41st AIAA Aerospace Sciences Meeting and Exhibit, Reno, 2003. [4] Khalil, E. E., and Whitelaw, J. H., 1974, The calculation of local-flow properties in two-dimensional furnaces, I.J.H.M.T. Vol. 18, p. 775, 1975. [5] Hutchinson, P., Khalil, E. E., Whitelaw, J. H., and Wigley, G., 1976, The calculation of furnace-flow properties and their experimental verification, Journal of Heat Transfer, ASME, 276, May 1976. [6] Khalil, E. E., 2001, Computer aided design of flow regimes and heat transfer in combustion chambers,

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Time 5 6 7.5 10 Sec.

Time 11 12.5 15 20 Sec.

Figure 7 – a: Temperature Distribution in a Confined non-swirling Flame

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Time 22.5 25 30 40 Sec.

Figure 7 – b: Temperature Distribution in a Confined non-swirling Flame

AIAA 2001-0950, 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 2001.

[7] Launder, B. E., and Spalding, D. B., 1974, The numerical computations of turbulent flows, Computer Methods in Applied Mechanics and Engineering 3 (1974) 269-289. [8] Khalil, E. E., 1983, Modeling of furnaces and combustors, Abacus Press, 1st Edition, U.K., 1983. [9] Beér, J. M., and Chigier, N. A., 1972, Combustion Aerodynamics, Halsted-Wiley Int., New York, 1972. [10] Gupta, A. K., and Lilley, D. G., 1985, Flow field modeling and diagnostics, Abacus Press, 1985. [11] Kameel, R., and Khalil, E. E., 2002, Generation of the grid node distribution using modified hyperbolic equations, 40th Aerospace Sciences Meeting & Exhibit, Reno, Nevada, AIAA-2002-656, 12-15 January 2002. [12] Patankar, S. V. 1980, Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, WDC, 1980. [13] Anderson, D. A., Tannehill, J. C., and Pletdher, R. H., 1980, Computational fluid mechanics and heat transfer, Hemisphere 1980. [14] Abbott, D. E., and Kline, S. J., 1962, Experimental investigation of subsonic turbulent flow over single and

double backward facing steps, J. Basic Eng. 84, 317, 1962. [15] Gosman, A. D., Khalil, E. E., and Whitelaw, J. H., 1979, The calculation of two-dimensional turbulent recirculating flows, Turbulent Shear Flows I, Springer-Verlag Berlin Heidelberg, 1979. [16] Baker, R. J., Hutchinson, P., Khalil, E. E., and Whitelaw, J. H., 1974, Measurements of three orthogonal velocity components in confined co-axial jet flows with and without swirl and combustion, Proc. 15th Symposium on Combustion. See also Imperial College, Mech. Eng. Dept. Report HTS/74/29. [17] Khalil, E. E., Spalding, D. B., and Whitelaw, J. H., 1975, The calculation of local flow properties in two-dimensional furnaces, Int. J. Heat Mass Transfer, Vol. 18, p. 775-791, 1975. [18] Spalding, D. B., 1976, Mathematical models of turbulent flames: a review, Combustion Science and Technology, Vol. 13, p. 3-25, 1976. [19] Kameel, R., and Khalil, E. E., 2004,”Heat Transfer Characteristics In Furnaces: Effect Of Combustion And Heat Transfer Modelling”, 42nd Aerospace Sciences Meeting & Exhibit, Reno, Nevada, AIAA-2004-803, 5-8 January 2004.

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NOMENCLATURE

A Pre-exponential coefficient ao, bo Constants in the specific heat Cp Specific heat at constant pressure C1, C2 Constants in turbulence model Cg1, Cg2 Constants in combustion model Cµ Turbulent viscosity constant CR Eddy-break-up constant D Diameter E Activation Energy f Mixture fraction = (ϕ - ϕA) / ( ϕF - ϕA) g Square of the fluctuation of concentration ggr Acceleration gravity = 9.81 h Stagnation enthalpy Hfu Heat of reaction of fuel i Stoichiometric mass of oxygen per unit

mass of fuel k Kinetic energy of turbulence

= ½ (u2 + v2 + w2) M Molecular weight m Mass fraction P Pressure R Universal gas constant r Radial distance from center-line Rfu Rate of chemical reaction Sφ Source or sink term of any variable S T

Swirl number Absolute temperature

U Flow mean velocity in the axial direction u Fluctuating component of axial velocity V Radial mean velocity v Fluctuating component of radial velocity W Tangential mean velocity w Fluctuating component of tangential

velocity x Axial direction X Axial distance from burner exit Y Radial distance from center line ya Width of burner annular

Greek symbols

Γ Effective diffusion coefficient. µ Viscosity ρ Density σφ Prandtl and Schmidt number for any

variable ε Dissipation of energy ϕ The dependent variable (mfu – (mox / i)) φ General dependent variable

Subscripts

A Air stream eff Effective F Fuel stream f Furnace fu Fuel i Species ox Oxidant pr Product st Stoichiometric mixture value t Turbulent