AIAA 93-0245 Structure and Properties of the Heat Release Locations in a Diffusion Flame Kuldeep Prasad University of Illinois, Urbana, IL

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January 11 -1 4, 1993 / Reno, NV For permission to copy or repubilsh, contact the American institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

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AIAA-93-0245 STRUCTURE AND PROPERTIES OF THE HEAT RELEASE LOCATIONS IN A DIFFUSION FLAME

Kuldeep Prasad L

University of Illinois Urbana. IL ,i

Abstract

This study deals with a fundamental numerical invtm- tigation of chemically reacting fluid flows through 2-D burnem. We make use of a detailed set of finite chemical kinetic rate equations to numerically simulate a lami- nar diffusion flame. The code has been constructed to consider the viscous effects in a mixing layer, heat con- duction, the multi-component diffusion and convection of important species, the iinite rate reactions of these species, and the resulting interactions between the fluid mechanics and the chemistry. The numerical model has been used to obtain a detailed description of the lead- ing edge of laminar diffusion flames o b t a e d above 2-D methane/air burners. It is shown that the leading edge flame, a flame holding point for the rest of the diffu- sion flame, is dominated by the kinetic aspects of the fueI/oxidizer species and is mainly responsible for heat transfer to any upstream boundary surface.

Introduction

Most practical combustion systems involve flames in re gions of mixing oxidizer and fuel vapors as in gas tur- bin-, rocket engines and commercial furnaces. The overall combustion is governed by a complex interplay of chemical reactions, transport and gas dynamic pro- cesses that arestmnglydependent on physical boundary conditions and type of chemical system. The ability to predict the coupled effects of various complex transport processes and chemical kinetics in these system is crit- ical in predicting flammability limits, stability criteria and extinction limits. The studies described here were originally motivated by conditions pertinent to combus tion of heterogeneous solid rocket propellants, but are descriptive of the flames in burners commonly referred to as “wolfhard” burners, which are often used for lab- oratory scale studies of diffusion flames, combustion ki- netics and pollutant formation.

A numerical model was constructed for the purpose of understanding the interplay of diffusion, convection and chemical reactions, and the resulting flame com- plex when a laminar mixing flow undergoes exotbemic chemical reactions. Fig. 1 provides a schematic d i k gram of the burner geometry and the computationd domain in which the solution a desired. The elk-

description of a diffusion controlled flame in such a configur&on pictures a thin flame in the midng region

that extends all the way to the burner surface. In the present study there was particular concern with that region of the flame near the burner surface, where heat loss from the reaction region to surrounding flow lim- its the temperature rise and thus limits reaction rates. In this region close to the initial point of fuel-oldd&er contaet the assnmptjon of diffusion limited flames are inapplicable, and detailed consideration of coupled ther- mal and species diffusion and chemical reaction rates is required for a realistic description of the flame complex.

Near the burner surface, the concentration gradients of primary reactants are high, and midng rat& cor- respondingly high. Chemical reaction rates are low be- cause of the relatively low temperatures, so that a small region of partially premixed reactants develops, which is increasingly heated as it moves outwards towards a flame that stabilizes in the mixing region. The leading edge of this flame (referred to here as the ‘leading edge flame” or “LEF” ) differs from the trailing diffusion lim- ited flame in that it stands in a region of premixed g a s g that are continuously premixed in the approach flow and lead to very concentrated reaction because of the pre- mixing. The flame cannot move upstream all the way to the burner surface, and must stabilize at some location

vide heat release rate consistent with multi-dimensional heat outflow. The details of this complex flame region ate not well known because it is usually very small, and difficult to observe experimentally. It is equally diffi- cult to describe analytically because of the necessity to consider-coupled convective flow, species diffusion, heat transfer and reaction processes in at least two space di- mensions.

The numerical model examine the leading edge por- tion of the diffusion flame. It is shown that the leading edge portion of the diffusion flame has the maximum rate of heat release per unit volume and is dominated by the kinetic aspects of the fuel-oxidizer species. This complicated type of flame arrangement has been o b served experimentally by Phillips[fj, Liebman et al. [2] and Ishikawa [3). Wichman [4] has constructed a math- ematical model of the leading edge portion of a diffu- sion flame formed over two initiallyseparated co-flowing streams of fuel and oxidizer. The temperature field was analyzed in the pure diffusion flame region, in the pre- mixed flame region, in the pure quenching region, and in the Utriple point region” where all of the separate zone meet. Dold [5] have developed a low heat-release model for triple flame strucutre and have shown that

where the temperature dependent reaction rate can pro- L

Copyright @ 1992 by the America1 Institute of Aeronautics and Astronautics, Inc. All rights reserved

triple-flame propagation speed depend on the transverse mixture fraction gradient and is bounded above by the maximum adiabatic laminar flame speed of the system.

Most of the computationally oriented combustion studies that have appeared in the literature that use detailed chemistry, have focused on steady, one- dimensional, i.e. freely propagating or burner stabilized pre-mixed flames [6] and counter flow pre-mixed [7] or diffusion flames. A comprehensive survey of the numer- ical techniques currently employed in detailed combus- tion modeling have been provided by Oran et al. [8].

McMurtry et al. [9] studied the effect of chemical heat release on a subsonic, temporally developing mix- ing layer. They solved both the compressible form of the governing equations as well as a more computation- ally efficient form of the equations valid for low Mach numbers. Reactions were modeled with a binary, single step irreversible reaction. Drummond et al. [lo] used a detailed hydrogen kinetic scheme to develop an implicit procedure for studying a reacting mixing layer. Bussing et al. [Ill developed a finite volume method for calcu- lation of compressible chemically reacting flows. The techniques include the implicit treatment of the chem- ical source term, point implicit multiple grid accelera- tor and a constant CFL condition. Eklund et al. [12] used both Runge Kutta and Adam predictor corrector method for computations involving reaction rate terms.

Smooke et al. [13] used a detailed chemistry t r ans port combustion model for studying axisymmetric lam- inar diffusion flames in which a cylindrical fuel stream is surrounded by a co-flowing oxidizer jet. They used vorticity and stream function to eliminate pressure as one of the dependent variables. The diffusion velocities were obtained using a modified Fick's Law. They also utilized a thin infinitely fast, global reaction model as a starting point for the determination of good initial solu- tion estimates for their finite rate axisymmetric model.

In this research a numerical model has been con- structed for the study of 2-D multi-component chemi- cally reacting fluid flows with detailed kinetics, variable transport and thermodynamic properties. The model is used to obtain a detailed description of the leading edge portion of laminar diffusion flames.

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Mathematical Formulation

Modeling gas phase reactive flows is based on a gen- erally accepted set of time dependent, coupled partial differential equations maintaining conservation of total density, momentum, total energy and individual species density. These equations describe the convective mo- tion of the fluid, the chemical reactions among the con- stituent species, and the diffusive transport processes &'

such as thermal conduction and molecular diffusion

Governing Equations

A strong conservation form of the two dimensional, un- steady, compressible Navier Stokes equations, used to describe gas phase reactive flows for a system contain- ing N species undergoing M elementary chemical reac- tions, [14], can be written as follows

atf+ a 2 + ayP = a,R' + a, S + I?

pu2 + P

(pet + P ) u

(1) - - - - where f, E , F , R, S and I? are vectors defined as

PU P V

9" (pet + P ) v

P t v

r 0 1 0 1

r n 1

L Here p and p t are the total m a s density and the indi- vidual species density; u, v are the bulk fluid velocity components in the x and y direction respectively and et is the total energy per unit mass. f,,t and f,,t are the body forces per unit m a s acting on the kth species in x and y directions respectively. The viscous stress terms rZr, r,, and r,, appearing in the conservation equations are given by

The terms qr and q, appearing in the total energy equation are the net rate of heat flux in the two co- ordinate directions and can be expressed as

The thermodynamic pressure is defined as that the diffusion velocities introduce no net momentum to the fluid flow i.e. L

(5)

where w k is the molecular weight of the k th species, and Re is the universal gas constant. The caloric equation of state is used to define the enthalpy of the individual species, which in turn is used to define the total energy as follows

pet = p e + T (u +u2)

eaction Kinetic Model

The types of reactions of importance in combustion in- elude unimolecular decomposition reactions, bimolecu- lar exchange and dissociation reactions, and three-body recombination reactions. For a system containing N

= g p k ( /L: + L: Cp,kdT) - P + ( u z + u') species undergoing a set of M elementary chemical re- action, the general F h reaction can be expressed as

P z (6)

k = l

The specific heat for each species is obtained by using N N

a sixth order polynomial in temperature. c V ; , ; s k ~ ~ c V ~ , i s k i = 1 ,..., M (10) k = l k=1

7

c p , k = C ~ I , ~ T ' - ' (7)

The various specific heat coefficients and the heat of

JANNAF thermochemical tables. The various diffusion coefficients such as thermal conductivity, binary diffu- sion coefficients, viscosity and thermal diffusion coeffi- cient are obtained from a rigorous treatment of kinetic

model the inter-molecular potential function based on which the collision integrals are evaluated. The various diffusion coefficients are then obtained using detailed ki- netic theory and are functions of the temperature, pres- sure and the various species properties.

The rate of production of the kth species due to the i th I = 1 reaction is given by

fi ( c j ) v : ]

j=1

(11)

L

formationsfor the various species are obtained from the w ; , k = (v& - &)

and the total rate of production of the k f h species is obtained by using

theory[l5]. A Lennard Jones potential is constructed to M

i J k = x i ) ; , k k = 1,. . ., N (1'4 ;=I

The k ~ , i and kb,; are the forward and reverse reaction rate constants. Each kj,; is a function of temperature usually given by the modified Arrhenius expression

kj,; = A ; P + exp (13) Diffusion Velocity Model

The diffusion velocities in a multi-component reacting flow mixdue may arise because of concentrations gradi- The reverse reaction rate constant kb,; is calculated ents, pressure gradients, differential body forces and due from k,,i and the equilibrium constant(in concentration to the Soret effect. The diffusion velocities for each of units) h;,; hy the laws of microscopic reversibility. the N species in both the x and y direction are obtained by solving the exact diffusion equation given by I ; * , ;=% i = l , ..., M (14)

k c , ;

(yt - x k ) a p The equilibrium constant K P , i for the ith general re- ?d*) action is defined by using the standard change in Gibb's

N

3X j = 1 free energy as follows :

(& - x k ) ap N _- (15)

AGP . - XkX.

RlnIip,; -- t = 1, ..., M T ax' -E ( e ( % - v k ) ) -I p ay 'Y j=1

Here x k and yk are the mole and m a s fraction Of the kth species respectively. Since only N-1 equations of the above N equations are independent of each other, the above equations are solved subject to the constraint,

The equilibrium constant Kc,; in terms of concentra- tions is then obtained from

( R ? ' ) ~ L = ~ ( ~ ~ . ~ - ' ' : ~ ' ) i = 1 , . . . ,M (16) N i

K c , i =

d equations, along with unstable reaction rate equations. The presence of stiffness and unstable behavior is em-

-./

heddkd in the reaction rate equations. This usually dic- tates intolerably small time stips for time dependent Governing Equations in the 'Eansformed Plane

The physical domain(x,y) is highly compressed in both the x,y directions in the region where fuel and oxidizer mix. The grid is required t o be uniform in the computa- tional domain to maintain a required order of accuracy. The governing equations (l), written in the physical do- main(x,y), must therefore be transformed to an appro priate computational domain(f, q) for solution. Let the general transformation be given hy

7 = t;E = E ( 2 , Y . t ) ; v = 7 ( Z > Y , t )

arq + aeE + a p = a,R + a,s + K

(17)

(18)

then the governing equations (1) can be written as

where

K = I?/J

Here J is the transformation Jacobian:

solutions. The approach used in the present computations to

couple the fluid dynamic equations with the reac- tion rate t e rm is known as the time step splitting approach[l6]. In this approach the individual processes are solved independently and the changes resulting from the separate partial calculations are coupled (added) to- gether. The qualitative criteria for its validity is that the values of the physical variables must not change too quickly over a time step from any of the individual proceues. Since the reaction rate terms are chiefly re- sponsible for the stiffness and unstable behavior, they are decoupled from the fluid dynamics over the smallest fluid dynamic time marching step. The reaction rate equations at each grid point are then subcycled over each fluid dynamic time marching step.

The numerical scheme used in the present work for integrating the fluid dynamic equations is a predictor corrector explicit time marching procedure [lu, [B]. The model is second order accurate in both time and space. The complete set of Navier Stokes equations (18) are considered in the transformed plane.

Numerical Formulation

The governing equations for a chemically reacting vis- cous fluid Aow are parabolic in time. Assuming that the flow quantities are known in the flow field at time level t", the purpose of the numerical procedure is to advance the solution to a new level t"+' using a large enough time step, At.

The major problem in solving equation (18) stems from the presence of widely separated fast and slow ki- netic rates, which results in a system of stiff governing

/'

where subscripts I and K are the indices of the grid point location in the z and y direction respectively and superscript n represents the nth time step. The above explicit scheme is second-order accurate in both space and time. The forward and backward differences are alternated between the predictor and corrector steps as well as between the two spatial derivatives in a sequen- tial fashion.

The derivatives appearing in the viscous t e r m R, and S are differenced so as to maintain second order accu- racy. This is accomplished by differencing the deriva- tive term in R in the opposite direction to that used for 8E/af . The coefficient of the derivative term in R was averaged over the grid locations over which the

differencing was done. The r) derivative t,erm in R was approximated with a central difference. The coefficients "9 direction. A symmetric boundary condition was L of the cross-derivative term were evaluated at the grid location where the central differencing wzs performed.

Stokes equations exhibit numerical oscillations because of inadequate mesh refinement in regions of large gradi- ents. A set of fourth order dissipation t e r m are explic- itly added in the manner suggested by Jameson [19]. These artificial dissipation t e r n are formally of the same order to or smaller than the truncation error in- volved in the spatial and time difference formulas used to represent the derivatives. These artificial dissipa- tion terms therefore do not affect the formal accuracy of the present formulation. Exact evaluation of the dif- fusion velocities is extremely important while comput- ing flames that are limited by the diffusion of fuel into

where IMAX is the total number of grid points in the

employed at the center line of the computational domain by use of the antisymmetric reflection of tangential ve-

Computations involving the compressible Navier locity IJ, and symmetric reflection of all other variables. A slip wall boundary condition is employed at the lateral boundary by assuming that the flux of all transported properties across the wall is zero.

For a diffusion flame computation, the initial condi- tions at each point in the domian were the same as inlet conditions. Single step finite rate chemistry cornput+ tions were used to raise the temperature of the gases to a temperature close to the adiabatic flame temper- ature. The output of these computations was used as initial conditions for the complete 2-D Navier Stokes solver with realistic kinetics.

- the oxidizer and vice versa. The diffusion velocities are mainlv resDonsible for the observed flame shape and Integration of the Reaction Rate Equations . ' the height of the stoichiometric tip obtained above the burner surface. Calculating the diffusion velocities re- quires solution of equations (8) subject to the constraint

Lagrange multipliers was used to incorporate the con-

The kinetic integrator used for integration of the reac- tion rate equations

(23) dpt . - = W k dt

equation (9). In the present approach the method of ~~

consists of an exponentially fitted trapezoidal rule [ZO] , rnr, rnn,

straint equation. A least squares approach was then used to minimize the residual and the resulting matrix .

1"I~lL'J system of equations are solved by Gaussian elimination to obtain the diffusion velocities. Since the matrices formed at the various grid locations were independent of each other, the procedure of constructing the matri- where p; is the approfirnation to the ces and inverting them was vectorized over the various grid points.

pn+i = p; + h [u n+l + (1 - ut) 4 1 k = 1,. . . , N (24)

solution to equation 24 at the current time, p, h is the tirne steplength (= t"+' - t " ) , CJ; = CJk(py,T") is the net rate of production of species evaluated at time level t", T', the temperature at time t" and Ub is a degree Initial and Boundary Conditions of implicitness or 'tuning" factor.

A specific solution of the reactive flow equations is deter- The kinetic integrator attempts to identify three d i s mined by the initial conditions and the boundary con- tinct regimes referred to as the induction, heat release ditions that describe the geometry of the system and and equilibration regimes. These three regimes are not =change of m=, momentum and ene ra occuring be- only physically different but also exhibit distinct math- tween the system and the rest of the physical world. ematical Characteristics [23], (241. During the induc- The total density, and v momentum and the various tion regime the concentration of intermediate species in- species densities are prescribed at both the inner and crease by many orders of magnitude from very small ini- outer portion of the inflow boundary. The pressure gr& tal concentrations to valuessufficient to initiate exother- dients normal to the burner surface are assumed to be mic chemical reactions. In this regime the coupling zero. At the outflow boundary, the normal gradients with the energy equation is weak and reactions are e+ of total density, momentum and of all species densities sentially isothermal. The full chemical mechanism is was assumed to be zero. The Pressure at large distances active during the heat release regime and is exhibited from the burner surface is assumed to be equal to the by sharply defined changes in temperature and molar ambient pressure. The pressure at the outflow bound- concetrations. The equilibration regime is character- ary was then interpolated using ized by monotonic approach of all species and temper-

ature towards their chemical equilibrium values. Dur-

tions are dominated by positive time constants, and the temperature also exhibits a positive time constant.

L k k W k

P ( r M A x , K ) = P ( r M A x - l , K ) + - p ( r M A x - l . K ) ) ing induction and early heat-release, the species equ+

\ ( X ( I M A X , K ) - X ( r M A x - - I , K ) )

X ( 1 M A X . K ) *

The associated ODE’S are unstable and are mathemat- chiometric tip) is approximately 6 mm. The flow being ically non-stiff. Since very small steps are required considered is a fuel lean flow and there is a significant for integrating unstable equations, a simple predictor- outflow of excess oxygen through the computational do- corrector scheme with Jacobi-Newton point iteration main. The concentration of major product species such is used. However, during late heat release and equi- as COz, H20 and GO seem to show a high concentra- libration when the temperature and species equations tion in the leading edge flame and the diffusion flame. exhibit negative time constants, large stepsizes can be I t is believed that the conversion of CO to COz is re- used, so Newton-Raphson iteration with calculation of sponsible for much of the heat released in a diffusion the full Jacobian matrix is the optimal convergence flame. The heat release rate profiles seem to support method. The governing ordinary differential equations that result. Concentration profiles for major products are stable during late heat release and equilibration, but have been shown in Figs. 4 5 and 6. In all surface plots are characterized by widely differing time constants re- the “x” axis is a direction perpendicular to the burner sulting in a system of mathematically stiff equations. surface (distance from the burner surface), the “y” axis

is parallel to the burner surface and the “z” axis con- tains the concentrations of the various species and other dependent varaibles. All dimensions are in SI units.

Fig. 7 provides surface plots for heat release rates (Joules/meter3) per unit volume in the computational

The to ob- domain above a 2-D burner. The results show a lead- tain relatively rigorous solutions to 2D laminar diffuion ing edge Portion of the diffusion flame with a define- flame problem when the reaction kinetics are known. able standoff distance from the burner surface. This The solutions provide the concentration fields of reac- Portion of the flame consumes the reactants that have tants and the spatial distribution ofthermodynamic and mixed in the stand-off space, yielding a very high VOl- Enematic variables. from the solution one can trace umetric heat release rate typical of kinetically limited the progress of reactions along streamlines, identify the premixed flames. The rapid heat release in the leading sites ofmaximum heat release rate, and characterize the edge flame region increases the temperature ofthe gases flow of heat upstream that heats, p ~ 0 1 y ~ ~ and i & t s downstream of the LEF, to a point where reaction rates

are at least as large as the rate at which diffusion occurs. A schematic diagram ofthe 2 - ~ diffusion flame burner T h e flame after the LEF is therefore again limited by

is shown in ~ i ~ . 1. using the numerical model and the fi- the diffusion process between the fuel and the oxidizer. nite difference scheme given in the previous sections, an The LEF therefore serves as a flame holding site for unsteady 2 - ~ flame propagation model was formulated the rest of the diffusion flame. The upstream heat flow for predicting the Characteristics of methane/air diffu- is dominated bY this leading edge flame, which, how- sion flame. The inner and outer duct halfthickness were ever, stands isolated in a relatively cold reactant flow 0.9 mm and 6.0 - respectively, The of the field and does not reach adiabatic stoichiometric flame gases flowing through both ducts was kept at 10 cmlsec, temperatures. and the temperature of the gases flowing into the com- The LEF unlike the rest of the diffusion flame is in a putational domain was taken as 298OK. The inner duct region of high heat losses and lower temperature. As a contains a mixture of 60% Methane and 40% Nitrogen. result, its characteristics are substantially dependent on The outer duct contains 80% Oxygen and 20% Nitro- kinetic rates. The development of an appreciable pre- gen. The divider between the fuel and oxidizer ducts is mixed volume, followed by very large reaction rates in assumed to be infinitesimally thin. The adiabatic flame the LEF region results in relatively high concentrations temperature obtained using the NASA thermochemical ofintermediate species in the LEF. Figs. 8 and 9 provide equilibrium package was approximately 280OoK. The contour and surface plots of intermediate species CHO computational domain consisted of a stretched 64*32 in the region above the 2-D burner. Similar behavior grid, concentrated near the contact point of the fuel was observed for all the intermediate species used in and the oxidizer. Validation of the various numerical this simulation. The diffusion flame on the other hand procedures was done for the case of pre-mixed flames showed a relatively small concentration of most inter- ~ 5 1 . mediate species.

The concentration (moles/meter3) profiles for major The surface plots for pressure(Newton/meter2) and product species CHq and Oz have been shown in Figs. magnitude of total temperature gradient(K/meter) are 2 and 3 respectively. I t is observed that methane is shown in Figs. 10 and 11 respectively. The temperature consumed almost completely within 3 mm. above the profiles indicate that the diffusion flame is located in burner surface. The flame height (height of the stoi- a region where the temperatures are extremely high.

J

Results and Discussion

of this study show that it is

d’ reactant mixtures.

-’

I csnve 11"s Zero gradient bovndarv condition

Computational Domain

However the temperature gradient plots indicate that the region of the diffusion flame is also nearly adiabatic. The heat release rate profiles indicate that as compared to the leading edge flame, the heat release rate in the rest of the diffusion flame is relatively low, as seen from Fig. 7. The low heat release rate in the diffusion flame can be explained by the small conductive and convective heat losses in the diffusion flame.

The LEF flame region, on the other hand, is located in a region where large conductive heat losses are taking place(Fig. 11). The LEF is mainlyresponsible for heae ing the incoming gases to a temperature large enough for near infinite rate reactions to occur. As indicated by the temperature profiles, the LEF is located in a re- gion where temperatures in a given mass element are changing rapidly. Under steady state conditions a great quantity of heat must be released in this region to bal- ance the conductive and convective heat losses. Due to the large temperature gradients observed in the LEF re- gion, the LEF is mainly responsible for providing heat transfer to any upstream boundary surface.

The large heat release rates in the leading edge flame region results in a rapid increase in temperature. In order to satisfy the equation of state the densities in this region must decrease. The flow under considera tion is an incompressible flow, since the Mach numbers are essentially zero. The pressures must therefore es sentially stay constant. However it was observed that the leading edge flame region showed a slight increase in pressure. Surface plots for pressure (Newton/meter') have been shown in Fig. 10. Although the pressure g r a dients are small, the momentum of the gases flowing out of the burner is also small. The prevure gradients are therefore responsible for slowing the incoming gases and turning these gases around the leading edge flame region as shown by the pattern of the particle path lines and by the velocity plots. This indirectly helps in expand- ing the flow and in reducing the density of the gases in the leading edge flame region. The velocity plots (me- ter/sec) for a 2-D diffusion flame calculation have been presented in Fig. 12. The particle path lines have been show in Fig. 13. The decrease in flow velocity into the LEF has the effect of an apparent increase in LEF flame speed, insofar as that flame speed determines the stable position of the flame (flame stand off distance). The stabilization of the LEF is attributed to the balance be- tween flow divergence effects and the heat loss effects, such that the flame will position itself in the mixing flow at a location where its effective flame speed matches the flow speed. This is important in typical combus- tion situations where progress of a flame depends on the upstream heat flow, and explains the high propagation speeds reported for flames propagating along the sto- ichiometric surfaces of stratified oxidizer-fuel systems

:/Condition

\

L I Symmetry ! Boundan

t t t t l t t t t

Figure 1: Schematic diagram of burner geometry and computational domain

[2], and the high apparent flame speeds in a diffusion flame burner [25]. The particle path lines, Fig. 13 in- dicate two large recirculation cells that are established between the hot surfaces of the flame and the cooler shield. The presence of these recirculation cells reduces the total area available for the flow of the combustion gases and hence the velocities are increased due to the combined effects of natural convection and a reduced flow area. The leading edge flame region gives rise to large velocity gradients which results in viscous stresses and generation of vorticity. The expansion of the gases and the large increase in the velocity of the gases tends to increase the velocities of the gases further away from the leading edge flame due to the viscous stresses.

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Parametric Studies

Systematic studies of varying flow velocity, ambient pressure and concentration of the gas flowing through the inner tube have been carried out to determine the structure and properties of the leading edge flame as a function of the various flow parameters.

Effect of Gas Velocity on the LEF Systematic studies of varying flow velocities with the LEF stabilizing on the burner have been performed. The effect of gas velocity is investigated by comparing three cases of different velocities. sL

Figure 2: Surface plot of major reactants CH,

I

I ‘In.. I

Figure 4: Surface plot of major products GO2

Figure 3: Surface plot of major reactants O2 Figure 5: Surface plot of major products HzO

Figure 6: Contour plot of major products CO

.91. z

Figure 8: Contour plot of intermediate species C X O

Figure 7: Surface plot of heat release rate per uni t vol- ume Figure 9: Surface plot of intermediate species CXO

Figure 10: Surface plot of Pressure .91. Z

Figure 12: Velocity vector plots above 2-D burner

Figure 11: Surface plot of Temperature gradient

4.

,D1. z

Figure 13: Streamline profiles above 2-D burner

Casel: Velocity = 10 cm/sec

Case2: Velocity = io cm/sec

Cases: Velocity = 40 cm/sec

assumption that lateral diffusion as the primary effect responsible for the development of the mixing layer in the LEF region has been found to he reasonable.

The amount of heat released in the leading edge flame is dependent on the state of “premixedness” of the reac-

ea& case the composition in the fuel flow is set to tants and on kinetics of the reaction. When the temper- 40% C H ~ and 60% N ~ , The oxidizer flow is atlire is below the ignition temperature which is about of 80% 0 2 and 20% Nz 1250 Kelvin, no appreciable reaction will take place and

is found that at higher flame velocities the flame rapid accumulation of the partially premixed reactants stand off distance (FSOD) increases as seen in figure 15, due to high Concentration gradients will occur. As the the temperature gradient in the flame increases and the velocity ofthe gases increasS the flow of Premixed equilibrium flame temperatures rea& a higher value. reactants flowing through the leading edge flame would The values of the maximum temperature gradient are increase, and as a result the amount of heat released at higher for flames at higher flow velocities, consistent the leading edge flame i n c r e w . The maximum with the requirement that the flame heats up a higher perature gradient decreases as the velocity of the gases

flux of reactants to remain stationary in the higher flow increase because the flame (LEF) has stabilized at a velocity. larger distance from the burner surface.

Figure 14 indicates that the maximum heat release In response to the increase of gas velocity, the LEF in the leading edge flame region reach a higher value at moves Out to asite corresponding to larger midng time. large velocities. The temperature gradient at the burner The longer mixing time prepares a broader mixing re- surface as seen in figure 16 shows that the gion, hence the LEF is more nearly one dimensional temperature gradient decreases as the flow velocity is with higher flame temperature, possibly due to lower increased, This is because the flame as a larger lateral heat loss. An increase in temperature with in- distance from the burner surface when the flow velocity creasing velocity results through flame stabilization at a is increased. location of larger mixing time, so that a larger partially

field is one that premixed region and more net heat are availaible. develops as the gas convects downstream away from the

Effect of Dilution on LEF velocities leave the burner surface, the concentrations gradients at the interface of the two flows are initially Systematic studies of diluting the fuel flow with the LEF very steep, and mixing proceeds very rapidly by molec- stabilizing on the burner have been performed. The ef- ular diffusion as the flow away from the surface. fect of dilution is investigated by comparing three cases Under the condition of laminar flow, it may be assumed of different fuel flow concentrations. fro simplicity that near the burner surface diffusive mix- ing will proceed primarily in the lateral direction, at a rate that is independent of mean convective veloc- ity in the vertical direction. The concentration profile will smooth out with time or distance. The details will depend on the relative diffusion constants, molecular weights of the gases involved and the effects of heat-up In each case the gas velocity is set to 10 cm/sec. The from the flame. These effects will he neglected in this oxidizer flow have the compositions of 80% 0 2 and 20% phenomenological analysis. Nz

When the gas velocity is increased, a specific con- The effect of reducing the concentration of methane centration profile which corresponds to a given flow- in the inner flow shows a decrease in the maximum heat diffusion time will occur at a distance out from the release rate at the leading edge flame as seen in figure 17. surface that is proportional to the flow velocity. In The flame stand off distance (FSOD) increases as the other words the location of these concentration profiles inner flow is diluted as seen in figure 18. Figure 19 show is stretched out in the “y” direction when the convec- the effect of dilution on the temperature gradient at tion velocity is increased. The numerical results indicate the burner surface. Results indicate that the maximum approximately linear stretching of the constant concen- temperature gradient at the burner surface reduces as tration lines in the axial direction upon increasing flow the flame moves away from the burner surface and as velocity. The results suggest that the axial diffusion the concentration of fuel flow is reduced. which will further complicate the stretching of the mix- The Leading Edge Flame is a distinctly multidimen- .

L ing region is small compared to lateral diffusion and the sional, nonadiabatic flame that positions itself in the

the LEF region, the

burner surface. As the two gas flows with equal upward L

* C-1: 40% CH4 60% Nz

Case2: 30% CH4 70% N2

c-3: 20% c~~ 80% N~

mixing fan at the point where its upstream propagation flame stand off distance (FSOD) and the gas velocity. speed matches the initial flow velocity. The stabiliza- As the ambient pressures are reduced the mixing times tion of the leading edge flame is influenced by both the increase. The characteristic time of chemical reactions lateral heat loss effect and flow divergence effect. increase as ambient pressures are reduced because of

The primary effect imposed by the dilution of the reduction in species density and decrease in the rate of inner flow is to decrease the heat of reaction of the mix- chemical reactions. ture. A second effect is a reduction in diffusion rate The diffusion coefficients are inversely proportional of fuel into oxidizer due to lower concentration gradi- to pressures. However the flux associated with the dif- ents. A third effect is a reduced oxidizer-fuel collision fusion velocities is independent of pressure. The con- frequency because of ineffective collisions involving dilu- vective mass flux through the burner surface reduces as ent molecules. As the dilution level increases, a rapid the ambient pressures are reduced because of reduction increase of mixing time is needed for obtaining a station- in density. The velocity of the gases flowing through ary flame. The LEF retreats from the burner surface to the burner surface was kept constant. A third effect of locations of reduced lateral heat loss that will permit reducing ambient pressure is reduction in chemical re- the same flame temperature as before dilution. If the action rate and smaller adiabatic flame speeds. Thus convective velocity is kept constant while increasing the as ambient pressures are reduced the mixing time as dilution the wider LEF region means less lateral tem- well as the characteristic times of chemical reaction are perature gradient, hence the heat loss is reduced. As larger. the inner flow is diluted, the temperature at the leading The FSOD increases as ambient pressures are reduced edge flame reduced. This is inspite of the fact that the because of larger mixing times and characteristic time flame positions itself far enough out in the flow so that of chemical reactions. The LEF stabilizes at a larger the enthalpy deficit due to dilution would be compen- distance from the burner surface as ambient pressure is sated for by reduced lateral heat losses, leading to only reduced. It is because of the larger characteristic time a small effect of dilution on temperature of stationary of chemical reactions that we observe diffusion flames at flames. very low ambient pressures to be extremely diffused in

nature. Also because of the smaller rate of chemical re-

-/

d Effect of Pressure on the LEF

Systematic studies of varying ambient pressure with the LEF stabilizing on the burner have been performed. The effect of pressure is investigated by comparing three cases of different ambient pressures.

Casel: Pressure=l.O atm

Case2: Pressure=0.5 atm

Case3: Pressure=O.l atm

In each case the composition in the fuel flow are set to 40% CH, and 60% N z . The oxidizer flow have the compositions of 80% 02 and 20% Nz

The maximum rate of heat release in the leading edge flame zone decreases as the ambient messure is reduced

action the net heat release rate is smaller at smaller am- bient pressures. This is because the lateral heat losses are smaller at larger flame stand off distances.

References [l] Phillips, H., “Flame in a Buoyant Methane

Layer,” Tenth Symposium (International) on Combustion, Cambridge, 1965, pp.1277-1283.

[2] Liebman, I., Corry. J., and Perlee, H.E., “Flame Propagation in Layered Methane Air System,” Combust. Sci. Technol. 1957 (1970). (1970).

[3] Ishikawa, N., “Flame Structure and Propagation Through an Interface of Layered Gases,” Com- bust. Sci. Technol. 31:185 (1983).

as seen in figure 20. The flame stand off distance in- creases as the pressure is reduced as seen in figure 21. [4] Wichman, I. s., “On the Quenching of a Diffusion

Flame Near a Cold Wall,” Combust. Sci. Technol. Figure 22 show temperature gradient at the burner sur- Any n.n ,<Ann\

0’I:ZYD-JIJ ( l Y 5 Y J . face as a function of changing ambient pressure. It is observed that maximum temperature gradient on the burner surface decreases as flame moves away from the burner surface and as the pressure is reduced.

The stabilization of the LEF as the ambient pres- sures are reduced is influenced by both the mixing times

j and the characteristic time of chemical reactions. The mixing time is estimated here as the ratio between the

[5] Dold, J . W., “Flame Propagation in a Nonuni- form hlixture: Analysis of a Slowly Varying Triple Flame,” Combust. Flame 76:71-88 (1989).

[6] Margolis, S.B., “Time Dependent Solution of a Premixed Flame in a Stagnation Flow,” J. Comp. Phys. 27:410 (1978).

[7] Sato, J. and Tsuji, H., “ Extinction of a Premixed flame in a Stagnation Flow Considering Gerneral Lewis Number,” Comb. Sci. and Technol. 33:193

[8] Oran, E.S. and Boris, J.P., “Numerical Simula- tion of Reactive Flow,” Elsevier Science Puhlish- ing Co., 1987.

[9] McMurtry, P.A., Jou, W.H., Riley, J.J. and Met- calfe, R.W., “Direct Numerical Simulation of a Reacting Mixing Layer with Chemical Heat Re- lease,” AIAA 85-0143, (1985).

[lo] Drummond, J.P. and Hussaini, M.Y., “A Detailed Model of a Supersonic Reacting Mixing Layer,” AIAA 861427, (1987).

[ll] Bussing, T.R.A. and Murman, E.M., “A Finite Volume Method for the Calculation of Compress ible Chemically Reacting Flow,” AIAA 85-0331, (1985).

[12] Eklund, D.R., Drummond, J.P. and Hassan, H.A., “Efficient Calculation of Chemically React- ing Flow,” AIAA 25:6 (1987).

[13] Smooke, M.D., Turnbull, A.A., Mitchell, R.E. and Keyes, D.E., in “Mathematical modeling in combustion and Related Topics,” Brauner, C.M., and Schmidt-Laine, C., “Solution of Two Dimen- sional Adsymmetric Laminar Diffusion Flames by Adaptive Boundary Value Methods,” 261-300, (1988).

[14] Williams, F.A., “Combustion Theory The Funda- mental Theory of Chemically Reacting Flow Sys- tems,” 1985.

[15] Chapman, S. and Cowling, T.G., “The Math- ematical Theory of Nonuniform Gases,” Cam- bridge University Press, Cambridge, England.

[16] Yanenko, N.N., “The Method of Fractional Steps,” SpringerVerlag, New York.

[17] MacCormack, R.W. and Paullay, A.J., Com- putational Efficiency Achieved by Time Splitting of Finite Difference Operators,” AIAA 72-154, (1972).

[18] MacCormack, R.W. and Baldwin, B.S., “A Nu- merical Method for Solving the Navier-Stokes Equations with Application to Shock-Boundary Layer Interaction,” AIAA 75-1, (1975).

[19] Jamgon, A,, Schmit, W. and Turkel, E., “Nu- merical Solution of the Euler Equations by Finite Volume Methods Using Runge Kutta Time Split- ting Schemes,” AIAA 81-1259, (1981).

Figure 14: Effect of velocity on heat release rate above burner surface

[20] Miranker, W.L., “Numerical Methods for Stiff Equations and Singular Perturbation Problems,” D. Reidel Publishing Co., 1981.

[21] Shampine, L.F., “Type-Insensitive ODE Codes Based on Implicit A-stable Formulas,” Math. of L Comput. 39:159 (1982).

[22] Babcock, P.D., Stutzman, L.F. and Brandon, D.M., “Improvements in a Single-Step Integration Algorithm,” Simulation, 333 (1979).

[23] Radhakrishnan, K. and Pratt, D.T., “Fast Algc- rithm for Calculating Kinetics in Turbulent Re- ,acting Flow,” Combust. Sci. Technol. 58:155-176 (1988).

[24] Radhakrishnan, K., ‘‘ New Integration Techniques for Chemical Kinetic Rate Equations 1. Efficiency Comparison,” NASA-TP-2372, (1984).

(251 Prasad, K., “A Numerical Study of Reacting Fluid Flows through Two-dimensional Burners,” Ph. D. thesis, Georgia Institute of Technology, June 1991.

2

Figure 15: Effect of velocity on flame stand-off distance Figure 17: Effect of dilution on heat release rate above burner surface

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Figure 16: Effect of velocity on temperature gradient above burner surface Figure 18: Effect of dilution on flame stand off distance

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Figure 20: Effect of pressure on heat release rate

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Figure 21: Effect of pressure on flame stand off distance

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Figure 22: Effect of pressure on temperature gradient

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